Annals of Communications in Mathematics: Volume 5, Number 1 (2022)

ISSN: 2582-0818 Volume 5, Number 1 (2022) Annals of Communications in Mathematics An International Journal http://www.technoskypub.com/journal/acm/ Annals of Communications in Mathematics Volume 5, Number 1 (2022) Table of Contents Articles 1. Some separation axioms in soft ideal topological spaces F. H. Khedr, O. R. Sayed, and S. R. Mohamed Annals of Communications in Mathematics, Vol. 5 (1) (2022), 1-17 2. Some types of interior filters in quasi-ordered Gamma -semigroups Daniel A. Romano Annals of Communications in Mathematics, Vol. 5 (1) (2022), 18-31 3. Fuzzy Minimal and Maximal e-Open Sets M. Sankari, S. Durai raj and C. Murugesan Annals of Communications in Mathematics, Vol. 5 (1) (2022), 32-37 4. Existence Results for Fuzzy Differential Equations with ψ-Hilfer Fractional Derivative K. Kanagarajan, R. Vivek, D. Vivek, E. M. Elsayed Annals of Communications in Mathematics, Vol. 5 (1) (2022), 38-54 5. On Ⓢ-closed sets and semi Ⓢ-closed in nano topological spaces I. Rajasekaran, N. Sekar and A. Pandi Annals of Communications in Mathematics, Vol. 5 (1) (2022), 55-62 6. L-fuzzifying proximity, L-fuzzifying uniform space and L-fuzzifying strong uniform space Mohammed M. Khalaf Annals of Communications in Mathematics, Vol. 5 (1) (2022), 63-73 ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 5, Number 1 (2022), 1-17 ISSN: 2582-0818 © http://www.technoskypub.com SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES F. H. KHEDR, O. R. SAYED∗ AND S. R. MOHAMED A BSTRACT. For dealing with uncertainties researchers introduced the concept of soft sets. In this paper, a new class of soft sets called soft delta pre ideal open sets in soft ideal topological space related to the notion of soft pre ideal regular pre ideal open sets is introduced. Also, some new soft separation axioms based on the soft delta pre ideal open sets are investigated. 1. I NTRODUCTION AND P RELIMINARIES In 1999, Molodtsov [22] initiated the theory of soft sets as a new mathematical tool for dealing with uncertainties. Also, he applied this theory to several directions (see, for example, [23-26]). The soft set theory has been applied to many different fields (see, for example, [1-2], [4-5], [7-9], [16-21], [27], [31-32], [34]). Later, few researches (see, for example, [3], [6], [10-11], [15], [22], [28-30], [33]) introduced and studied the notion of soft topological spaces. In [13-14], the authors initiated the notion of soft ideal. They also introduced the concept of soft local function. These concepts were discussed with a view to find new soft topologies from the original one, called soft topological spaces with soft e EL-Sheikh [8] introduced the notions of I-open e e open ideal (X, τe, E, I). soft sets, pre-Ie e e soft sets, α-I-open soft sets, semi-I-open soft sets and β-I-open soft sets to soft topological spaces. He studied the relations between these different types of subsets of soft e topological spaces with soft ideal. Also, he introduced the concepts of I-continuous soft, e e e e pre-I-continuous soft, α-I-continuous soft, semi-I- continuous soft and β-I-continuous soft functions and discussed their properties. Jafari [11] introduced the concept of pre regular preopen set in general topological space. This paper extends this set to soft topological e e spaces with soft ideal, soft pre-I-regular pre-I-open set, and study some of its properties. e Also, the concept of a soft delta pre-I-open set is given and some of its properties are e investigated. Finally, a soft delta pre-I-separation axioms are given. 2010 Mathematics Subject Classification. 54A05, 54A20, 54D10. ˜ Key words and phrases. Soft ideal topological space; soft delta pre-I-open. Received: December 28, 2021. Accepted: March 20, 2022. Published: June 30, 2022. *Corresponding author. 1 2 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED Definition 1.1. [23]. Let X be an initial universe set, P (X) the power set of X, that is, the set of all subsets of X, and A a set of parameters. A pair (F, A), where F is a map from A to P (X), is called a soft set over X. In what follows by SS(X, A) we denote the family of all soft sets (F, A) over X. Definition 1.2. [23] Let (F, A), (G, A) ∈ SS(X, A). We say that the pair (F, A) is a soft subset of (G, A) if F (p) ⊆ G(p), for every p ∈ A. Symbolically, we write (F, A) ⊑ (G, A). Also, we say that the pairs (F, A) and (G, A) are soft equal if (F, A) ⊑ (G, A) and (G, A) ⊑ (F, A). Symbolically, we write (F, A) = (G, A). Definition 1.3. [23]. Let Λ be an arbitrary index set and {(Fi , A) : i ∈ Λ} ⊆ SS(X, A). Then (1) The soft union of these soft sets is the soft set S (F, A) ∈ SS(X, A), where the map F : A → P (X) is defined as follows: F (p) = {Fi (p) : i ∈ Λ}, for every p ∈ A. Symbolically, we write (F, A) = ⊔{(Fi , A) : i ∈ Λ}. (2) The soft intersection of these soft sets is the soft Tset (F, A) ∈ SS(X, A), where the map F : A → P (X) is defined as follows: F (p) = {Fi (p) : i ∈ Λ}, for every p ∈ A. Symbolically, we write (F, A) = ⊓{(Fi , A) : i ∈ Λ}. Definition 1.4. [33]. Let (F, A) ∈ SS(X, A). The soft complement of (F, A) is the soft set (H, A) ∈ SS(X, A), where the map H : A → P (X) defined as follows: c H(p) = X\F (p), for every p ∈ A. Symbolically, we write (H, A) = (F, A) . Obvic c ously, (F, A) = (F , A) [10]. For two given subsets (M, A), (N, A) ∈ SS(X, A) [30], we have (i) ((M, A) ⊔ (N, A))c = (M, A)c ⊓ (N, A)c ; (ii) ((M, A) ⊓ (N, A))c = (M, A)c ⊔ (N, A)c . Definition 1.5. [23]. The soft set (F, A) ∈ SS(X, A), where F (p) = φ, for every p ∈ A is called the A-null soft set of SS(X, A) and denoted by 0A . The soft set (F, A) ∈ SS(X, A), where F (p) = X, for every p ∈ A is called the A-absolute soft set of SS(X, A) and denoted by 1A . Definition 1.6. [33]. The soft set (F, A) ∈ SS(X, A) is called a soft point in X, denoted by xe , if for the element e ∈ A, F (e) = {x} and F (e′ ) = φ for all e′ ∈ A\{e}. The set of all soft points of X is denoted by SP (X, E). The soft point xe is said to be in the soft set (G, A), denoted by xee ∈ (G, A), if for the element e ∈ A and x ∈ F (e). Definition 1.7. [33]. Let SS(X, A) and SS(Y, B) be families of soft sets. Let u : X → Y and p : A → B be mappings. Then the mapping fpu : SS(X, A) → SS(Y, B) is defined as: (1) The image of (F, A) ∈ SS(X, A) under fpu is the soft set fpu (F, A) = (fpu (F ), B) in SS(Y, B) such that S ( u(F (x)), p−1 (y) ̸= φ −1 x∈p (y) fpu (F )(y) = φ, otherwise for all y ∈ B. −1 (2) The inverse image of (G, B) ∈ SS(Y, B) under fpu is the soft set fpu (G, B) = −1 −1 −1 (fpu (G), A) in SS(X, A) such that fpu (G)(x) = u (G(p(x))) for all x ∈ A. Proposition 1.1. [10]. Let (F, A), (F1 , A) ∈ SS(X, A) and (G, B), (G1 , B) ∈ SS(Y, B). The following statements are true: SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES 3 (1) If (F, A) ⊑ (F1 , A), then fpu (F, A) ⊑ fpu (F1 , A). −1 −1 (2) If (G, B) ⊑ (G1 , B), then fpu (G, B) ⊑ fpu (G1 , B). −1 (3) (F, A) ⊑ fpu (fpu (F, A)). −1 (4) fpu (fpu (G, B)) ⊑ (G, B). c c −1 −1 (5) fpu ((G, B) ) = (fpu (G, B)) . (6) fpu ((F, A) ⊔ (F1 , A)) = fpu (F, A) ⊔ fpu (F1 , A). (7) fpu ((F, A) ⊓ (F1 , A)) ⊑ fpu (F, A) ⊓ fpu (F1 , A). −1 −1 −1 (8) fpu ((G, B) ⊔ (G1 , B)) = fpu (G, B) ⊔ fpu (G1 , B). −1 −1 −1 (9) fpu ((G, B) ⊓ (G1 , B)) = fpu (G, B) ⊓ fpu (G1 , B). Definition 1.8. [33]. Let X be an initial universe set, A a set of parameters, and τe ⊆ SS(X, A). We say that the family τe defines a soft topology on X if the following axioms are true: (1) 0A , 1A ∈ τe. (2) If (G, A), (H, A) ∈ τe, then (G, A) ⊓ (H, A) ∈ τe. (3) If (Gi , A) ∈ τe for every i ∈ I, then ⊔{(Gi , A) : i ∈ I} ∈ τe . The triplet (X, τe, A) is called a soft topological space. The members of τe are called soft c open sets in X. Also, a soft set (F, A) is called soft closed if the complement (F, A) c belongs to τe . The family of all soft closed sets is denoted by τe . Definition 1.9. Let (X, τe, A) be a soft topological space and (F, A) ∈ SS(X, A). (1) The soft closure of (F, A) [30] is the soft set c seCl(F, A) = ⊓{(S, A) : (S, A) ∈ τe , (F, A) ⊑ (S, A)}. (2) The soft interior of (F, A) [33] is the soft set seInt(F, A) = ⊔{(S, A) : (S, A) ∈ τe, (S, A) ⊑ (F, A)}. Definition 1.10. [13]. (1) Let Ie be a non-null collection of soft sets over a universe X with a fixed set of e parameters E. Then I⊆SS(X, E) is called a soft ideal on X with the same set E if e then (F, E) ⊔ (G, E) ∈ I; e (a) (F, E), (G, E) ∈ I, e (b) (F, E) ∈ Ie and (G, E) ⊑ (F, E), then (G, E) ∈ I, (2) Let (X, τe, E) be a soft topological space and Ie be a soft ideal over X. Then e is called a soft ideal topological space. Let (F, E) ∈ e SS(X, E), the (X, τe, E, I) soft operator ∗ : SS(X, E) → SS(X, E), defined by ∗ e ∗ / Ie ∀ Oxe ∈ (F, E) (I, τe) or (F, E) = ⊔{xe ∈ SP (X, E) : Oxe ⊓ (F, E) ∈ τe} is called the soft local function of (F, E) with respect to Ie and τe, where Oxe is a τe-soft open set containing xe . Theorem 1.2. [13]. Let (X, τe, E, Ie ) be a soft ideal topological space and (F, E), (U, E) ∈ SS(X, E). Then we have (1) The soft closure operator seCl∗ : SS(X, E) → SS(X, E), defined by seCl∗ (F, E) = ∗ (F, E) ⊔ (F, E) , satisfies Kuratwski’s axioms. ∗ ∗ (2) If (U, E) ∈ τe, then (U, E) ⊓ (F, E) ⊑ [(U, E) ⊓ (F, E)] . e is Definition 1.11. [14]. A soft subset (F, E) of a soft ideal topological space (X, τe, E, I) ∗ ∗ said to be τ -soft dense if seCl (F, E) = 1E . 4 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED e be a soft ideal topological space and (F, E) ∈ SS(X, E) Definition 1.12. [8]. Let (X, τe, E, I) and xe ∈ SP (X, E). Then: ∗ e (1) (F, E) is said to be a soft-I-open set if (F, E) ⊑ seInt(F, E) . The complement e e e of soft-I-open set is called soft-I-closed and we denote the set of all soft-I-open e e e e e (resp. soft-I- closed) sets by S IO(X, E)(resp. S IC(X, E)). e (2) (F, E) is said to be a soft pre-I-open set if (F, E) ⊑ seInt(e sCl∗ (F, E)). The e e complement of a soft pre-I-open set is called soft pre-I-closed and the family e e e of all soft pre-I-open (resp. soft pre-I-closed) sets in (X, τe, E, I) is denoted by e IO(X, e e IC(X, e SP E)(resp. SP E)). e (3) (F, E) is said to be a soft α-I-open set if (F, E) ⊑ seInt(e sCl∗ (e sInt(F, E))). e e The complement of a soft α-I-open set is called soft α-I-closed and the fame e e ily of all soft α-I-open(resp. soft α-I-closed) sets in (X, τe, E, I) is denoted by e IO(X, e e IC(X, e Sα E)(resp. Sα E)). e e IO(X, e (4) xe is called a soft pre-I-Interior point of (F, E) if there exists (G, E) ∈ SP E) e e such that xe ∈ (G, E) ⊑ (F, E), the set of all soft pre-I-interior points of e e (F, E) is called the soft pre-I-interior of (F, E) and is denoted by sepIInt(F, E). e e IO(X, e Consequently, sepIInt(F, E) = ⊔{(G, E) : (G, E) ∈ SP E), (G, E) ⊑ (F, E)} e (5) xe is called a soft pre-I-closure point of (F, E) if (F, E) ⊓ (H, E) ̸= 0E e e e e (H, E). The set of all soft pre-Ifor every (H, E) ∈ SP IO(X, E) and xe ∈ e closure points of (F, E) is called the soft pre-I-closure of (F, E) and is denoted by e e e sepICl(F, E). Consequently, sepICl(F, E) = ⊓{(H, E) : (H, E) ∈ sepIC(X, E), (F, E) ⊑ (H, E)}. e e e Theorem 1.3. [8]. Every soft α-I-open (resp. soft α-I-closed) set is soft pre-I-open (resp. e soft pre-I-closed). e be a soft ideal topological space, and (F, E), (G, E) ∈ SS(X, E). Theorem 1.4. [8]. Let (X, τe, E, I) Then the following hold: e e (1) sepICl((F, E)c ) = (e spIInt(F, E))c . c e e (2) sepIInt ((F, E) ) = (e spICl(F, E))c . e e e (3) sepIInt[(F, E) ⊓ (G, E)] ⊑ sepIInt(F, E) ⊓ sepIInt(G, E). e e e (4) (F, E) ∈ SP IC(X) ⇐⇒ (F, E) = sepICl(F, E). e IO(X) e e (5) (F, E) ∈ SP ⇐⇒ (F, E) = sepIInt(F, E). e e e (6) sepIInt(e spIInt(F, E)) = sepIInt(F, E). e e (7) If (F, E) ⊑ (G, E), then sepIInt(F, E) ⊑ sepIInt(G, E). e e (8) If (F, E) ⊑ (G, E), then sepICl(F, E) ⊑ sepICl(G, E). 2. S OFT PRE -Ie - REGULAR PRE -Ie - OPEN SETS e e In this section, we introduce the concept of soft pre-I-regular pre-I-open set in which e the notion of soft pre-I-open set is involved and study some of its properties. Also, we e e present other notions called extremely soft pre-I-disconnected and soft pre-I-regular sets. e Definition 2.1. Let (X, τe, E, Ie ) be a soft ideal topological space. A soft pre-I-open set e e e e (F, E) is said to be soft pre-I-regular pre-I-open if (F, E) = sepIInt(e spICl(F, E)). The SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES 5 e e e e complement of a soft pre- I-regular pre-I-open set is said to be soft pre-I-regular pre-Iclosed. Example 2.2. Suppose that (X, τe, E, Ie ) is a soft ideal topological space, where X = {h1 , h2 , h3 }, E = {e}, τe = {1E , 0E , {(e, {h1 })}, {(e, {h2 })}, {(e, {h1 , h2 })} and Ie = e IO(X, e {0E , {(e, {h1 })}. Then one can deduce that SP E) = {1E , 0E , {(e, {h1 })}, {(e, {h2 })}, {(e, {h1 , h2 })}}. e e We have {1E , 0E , {(e, {h1 })}, {(e, {h2 })}} are soft pre-I-regular pre-I-open sets. e be a soft ideal topological space and (F, E) ∈ SS(X, E).(F, E) Definition 2.3. Let (X, τe, E, I) e e e is said to be soft pre-I-regular if it is soft pre-I-open and soft pre-I-closed. e be a soft ideal topological space, where X = {h1 , h2 , h3 }, Example 2.4. Let (X, τe, E, I) E = {e}, τe = {1E , 0E , {(e, {h2 , h3 })}} and Ie = {0E , {(e, {h1 })}}. Then one can dee IO(X, e duce that SP E) = {1E , 0E , {(e, {h2 , h3 })} , {(e, {h2 })} , {(e, {h3 })}, {(e, {h1 , h2 })}, {(e, {h1 , h3 })}, we have {{(e, {h2 })}, {(e, {h3 })}, {(e, {h1 , h2 })}, {(e, {h1 , h3 })}, 1E , 0E } are soft pree e e I-open and soft pre- I-closed sets. So, they are soft pre-I-regular. e e e Remark. (1) Soft pre-I-regular set is soft pre-I-regular pre-I-open; e e e (2) Soft pre-I-regular pre-I-open set is soft pre-I-open; e e e (3) Soft-I-open set and soft pre-I-regular pre-I-open set are independent. e be the soft ideal topological space as in Example 2.2. We Example 2.5. Let (X, τe, E, I) e e e have (F, E) = {(e, {h1 })} is soft pre-I-regular pre-I-open but not soft pre-I-closed. e be the soft ideal topological space as in Example 2.4. We Example 2.6. Let (X, τe, E, I) e e e get (F, E) = {(e, {h2 , h3 })} is a soft-I-open set but it is not soft pre- I-regular pre-Ie e open and we have (G, E) = {(e, {h2 , h3 })} is soft pre-I-open set but it is not soft pre-Ie regular pre-I-open. e be a soft ideal topological space, where X = {h1 , h2 , h3 }, E = Example 2.7. Let (X, τe, E, I) {e}, τe = {1E , 0E , {(e, {h1 })}, {(e, {h3 })}, {(e, {h1 , h3 })}, {(e, {h2 , h3 })} and Ie = e IO(X, e {0E , {(e, {h1 })}. Then SP E) = {1E , 0E , {(e, {h1 })}, {(e, {h3 })}, {(e, {h1 , h3 })}, {(e, {h2 , h3 })}}.Therefore e e e {(e, {h1 })} is soft pre-I-regular pre-I-open set but it is not soft-I-open. e e e Remark. The soft intersection of two soft pre-I-regular pre-I-open sets is not soft pre-Ie regular pre-I-open, in general, as can be shown by the following example. e be the soft ideal topological space as in Example 2.4. Then, Example 2.8. Let (X, τe, E, I) e e the soft sets {(e, {h1 , h2 })} and {(e, {h1 , h3 })} are both soft pre-I-regular pre-I-open e e sets, but their intersection {(e, {h1 })} is not soft pre-I-regular pre-I-open. e be a soft ideal topological space and (F, E), (G, E)∈SS(X, E). Theorem 2.1. Let (X, τe, E, I) Then the following statement are true. e e e spICl(G, e (1) If (F, E) ⊑ (G, E), then sepIInt(e spICl(F, E)) ⊑ sepIInte E)) e IO(X, e e e (2) If (F, E)∈SP E), then (F, E) ⊑ sepIInt(e spICl(F, E)). e IO(X, e e e spIInt(e e e (3) For every (F, E)∈SP E), we have sepIInt(e spICl(e spICl(F, E)))) = e e sepIInt(e spICl(F, E)). e e e (4) If (F, E) and (G, E) are disjoint soft pre-I-open sets, then sepIInt(e spICl(F, E)) e e and sepIInt(e spICl(G, E)) are disjoint. e e e (5) If (F, E) is a soft pre-I-regular pre-I-open set, then sepICl((F, E)c ) is a soft pree e I-regular pre-I-closed set. 6 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED e e e e (6) If (F, E) is a soft pre-I-regular pre-I-open set, then sepIInt(F, E) is soft pre-Ie regular pre-I-open. Proof. (1) Follows from Theorem 1.4 (7) and (8). e IO(X, e e e e e SP (2) (F, E) ∈ E) =⇒ (F, E) = sepIInt(F, E) ⊑ sepIInt(e spICl(F, E)). e e e IO(X, e e SP (3) It is obvious that sepIInt(e spICl(F, E))∈ E). So, by (2), we have e e e e spIInt(e e e sepIInt(e spICl(F, E)) ⊑ sepIInt(e spICl(e spICl(F, E)))). On the other e e e e spIInt(e e e hand, sepIInt(e spICl(F, E)) ⊑ sepICl (F, E), which implies that sepICl(e spICl(F, E))) ⊑ e e e e e e e sepICl(e spICl(F, E)) = sepICl(F, E). Hence sepIInt(e spICl(e spIInt(e spICl(F, E)))) ⊑ e e e e spIInt(e e e sepIInt(e spICl(F, E)). Therefore, we obtain sepIInt(e spICl(e spICl(F, E)))) e e = sepIInt(e spICl(F, E)). e (4) Since (F, E) and (G, E) are disjoint soft pre-I-open sets, (F, E) ⊓ (G, E) = 0E e e e which implies that (F, E)⊓e spICl(G, E) = 0E and so (F, E)⊓ sepIInt(e spICl(G, E)) = e e e e e e 0E . Since sepIInt(e spICl(G, E) is soft pre-I-open, sepICl(F, E) ⊓ sepIInt(e spICl(G, E) = e e e e 0E . Hence sepIInt(e spICl(F, E)) ⊓ sepIInt(e spICl(G, E)) = 0E . e e e e (F, E)) (5) Given that (F, E) is soft pre-I-regular pre-I-open set. So spICl  c (F, E)= sepIInt(e c e e e e spIInt((F, E)c )). which implies that (F, E) = sepIInt(e spICl(F, E)) = sepICl(e e e spIInt(e e e e Therefore, sepIC((F, E)c ) = sepICl(e spICl((F, E)c ))). Hence sepICl((F, E)c ) e e is soft pre-I-regular pre-I-closed set. c e e e (6) By (5), if (F, E) is soft pre-I-regular  pre-I-open, then c sepICl((F, E) ) is soft e e e e pre-I-regular pre-I-closed. Hence sepICl((F, E)c ) is soft pre-I-regular pree e e e I-open which implies that sepIInt(F, E) is soft pre-I-regular pre-I-open set. □ e is said to be soft-I-submaximal e Definition 2.9. A soft ideal topological space (X, τe, E, I) if each τ ∗ -soft dense subset is soft open. e be a soft ideal topological space, the following are equivaLemma 2.2. Let (X, τe, E, I) lent: e (i) Every soft pre-I-open set is soft open. e is soft-I-submaximal. e (ii) (X, τe, E, I) Proof. (i)=⇒(ii). Suppose that (F, E) is a τ ∗ -soft dense set, then seCl∗ (F, E) = 1E which implies that seInt(e sCl∗ (F, E)) = 1E . Hence (F, E) ⊑ seInt(e sCl∗ (F, E)) = 1E . Therefore, by (i), we have (F, E) is soft open. e (ii) =⇒(i). Let (G, E) be a soft pre-I-open subset of X. Then (G, E) ⊑ seInt(e sCl∗ (G, E)) = ∗ ∗ ∗ c (U, E), say. Then seCl (G, E) = seCl (U, E), so that seCl [(U, E) ⊔ (G, E)] = seCl∗ ((U, E)c ) ⊔ seCl∗ (G, E) = (U, E)c ⊔ seCl∗ (G, E) = 1E and thus (U, E)c ⊔ (G, E) is τ ∗ -soft dense set in X. Thus (U, E)c ⊔ (G, E) is soft open. Now, we have (G, E) = ((U, E)c ⊔ (G, E)) ⊓ (U, E), is the intersection of two soft open sets, so that (G, E) is soft open. □ e Theorem 2.3. In a soft-I-submaximal soft ideal topological space, the intersection of any e e finite number of soft pre-I-open sets is soft pre-I-open. Proof. It’s clear from Lemma 2.2. □ e Theorem 2.4. In a soft-I-submaximal soft ideal topological space, the intersection of any e e e e finite number of soft pre-I-regular pre-I-open set is soft pre-I-regular pre-I-open. SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES 7 e e Proof. Let {(Fi , E) : i = 1, 2, ..., n} be a finite family of soft pre-I-regular pre-Ie is soft-I-submaximal, e open sets. Since the space (X, τe, E, I) then, by Theorem 2.2, e ⊓{(Fi , E) : i = 1, 2, ..., n} is soft pre-I-open. Therefore, ⊓{(Fi , E) : i = 1, 2, ..., n} ⊑ e e sepIInt(e spICl(⊓(F i , E)). Also, for each i = 1, 2, ..., n, ⊓(Fi , E) ⊑ (Fi , E) which e e e e implies that sepIInt(e spICl(⊓(F epIInt(e spICl(F i , E) )) ⊑ s i , E)). Also, each (Fi , E) e e e e is soft pre-I-regular pre-I-open implies that (Fi , E) = sepIInt(e spICl(F i , E)) which ime e e e plies that sepIInt(e spICl(⊓(Fi , E))) ⊑ ⊓(Fi , E) and so ⊓(Fi , E) = sepIInt(e spICl(⊓(F i , E)). e e Hence ⊓(Fi , E) is soft pre-I-regular pre-I-open. □ e e Remark. It should be noted that an arbitrary union of soft pre-I-regular pre-I-open sets is e e e e soft pre-I-regular pre-I-open. But the Intersection of two soft pre-I-regular pre-I-closed e e sets fails to be soft pre-I-regular pre-I-closed as can be shown by the following example. e as in Example 2.2. Example 2.10. Consider the soft ideal topological space (X, τe, E, I) We have the two soft sets (F, E) = {(e, {h1 , h3 })} and (G, E) = {(e, {h2 , h3 })} are e e soft pre-I-regular pre-I-closed sets but their intersection (F, E) ⊓ (G, E) = {(e, {h3 })} e e is not soft pre-I-regular pre-I-closed set. e be a soft ideal topological space and let (F, E), (G, E) ∈ Theorem 2.5. Let (X, τe, E, I) SS(X, E), then the following hold. e e e e (1) If (F, E) is soft pre-I-closed, then sepIInt(F, E) is soft pre-I-regular pre-I-open. e e e e (2) If (F, E) is soft pre-I-open, then sepICl(F, E) is soft pre-I-regular pre-I-closed. e e (3) If (F, E) and (G, E) are soft pre-I-regular pre-I-closed sets, then (F, E) ⊑ e e (G, E) if and only if sepIInt(F, E) ⊑ sepIInt(G, E). e e (4) If (F, E) and (G, E) are soft pre-I-regular pre-I-open sets, then (F, E) ⊑ (G, E) e e if and only if sepICl(F, E) ⊑ sepICl(G, E). e e (1) Since (F, E) is soft pre-I-closed, we have (F, E) = sepICl(F, E). Now, we obtain e e spIInt(F, e e e spIInt(e e e sepIInt(e spICl(e E)) = sepIInt(e spICl(e spICl(F, E)))) = e e e e e sepIInt(e spICl(F, E)) = sepIInt(F, E). Hence sepIInt(F, E) is soft pre-I-regular e pre-I-open. e spIInt(e e e e spIInt(e e e spIInt(F, e (2) Now, we have sepICl(e spICl(F, E))) = sepICl(e spICl(e E)) = e spIInt(F, e e e e sepICl(e E) = sepICl(F, E). Hence sepICl(F, E) is soft pre-I-regular e pre-I-closed. e e (3) Given (F, E) and (G, E) are soft pre-I-regular pre-I-closed sets. Therefore, we e e e e have (F, E) = sepICl(e spIInt(F, E)) and (G, E) = sepICl(e spIInt((F, E)). e e Clearly, we have sepIInt(F, E) ⊑ sepIInt(G, E) whenever (F, E) ⊑ (G, E). e e Conversely, suppose that sepIInt(F, E) ⊑ sepIInt(G, E). Now, we obtain (F, E) = e spIInt(F, e sepICl(e E)) ⊑ e spIInt(G, e sepICl(e E))) = (G, E). Hence (F, E) ⊑ (G, E). e e (4) Given (F, E) and (G, E) are soft pre-I-regular pre-I-open sets. Therefore, we e e e e obtain (F, E) = sepIInt(e spICl(F, E)) and (G, E) = sepIInt(e spICl(G, E)). e e e e Suppose (F, E) ⊑ (G, E), sepICl(F, E) = sepICl(e spIInt(e spICl(F, E))) ⊑ e spIInt(e e e e e e sepICl(e spICl(G, E))) = sepICl(G, E). Therefore, sepICl(F, E) ⊑ sepICl(G, E). Proof. 8 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED e e e e Conversely, sepICl(F, E) ⊑ sepICl(G, E). Now, we obtain that (F, E) = sepIInt(e spICl(F, E)) ⊑ e e sepIInt(e spICl(G, E)) = (G, E). Therefore, (F, E) ⊑ (G, E). □ e be a soft ideal topological space and (F, E)∈SS(X, E). Definition 2.11. Let (X, τe, E, I) e A soft subset (F, E) is said to be soft-I-rare if seInt∗ (F, E) = 0E . e be a soft ideal topological space where X = {h1 , h2 , h3 }, Example 2.12. Let (X, τe, E, I) E = {e}, τe = {1E , 0E , {(e, {h2 })}} and Ie = {0E , {(e, {h2 })}}, then τ ∗ = {1E , 0E , {(e, {h2 })}, {(e, {h1 , h3 })}}. Take (F, E) = {(e, {h1 })}, so seInt∗ (F, E) = 0E . Hence we get (F, E) = {(e, {h1 })} is e a soft-I-rare set. e is said to be Definition 2.13. A subset (F, E) of soft ideal topological space (X, τe, E, I) soft nowhere dense set if seInt(e sCl(F, E)) = 0E . e be the soft ideal topological space as in Example 2.12. Example 2.14. Let (X, τe, E, I) Take (F, E) = {(e, {h1 })}, seInt(e sCl(F, E)) = 0E , so (F, E) is soft nowhere dense. e be a soft ideal topological space and (F, E), (U, E)∈ SS(X, E). Lemma 2.6. Let (X, τe, E, I) If (U, E) is soft open set, then (U, E) ⊓ seCl∗ (F, E) ⊑ seCl∗ ((U, E) ⊓ (F, E)). Proof. Since (U, E)∈e τ , by Theorem 1.2 we obtain (U, E) ⊓ seCl∗ (F, E) = (U, E) ⊓ ∗ ∗ [(F, E) ⊔ (F, E) ] = [(U, E) ⊓ (F, E)] ⊔ [(U, E) ⊓ (F, E) ] ⊑ [(U, E) ⊓ (F, E)] ⊔ ∗ ∗ [(U, E) ⊓ (F, E)] (Theorem 2.1) = seCl ((U, E)⊓(F, E)). Hence (U, E)⊓e sCl∗ (F, E) ⊑ ∗ seCl ((U, E) ⊓ (F, E)). □ e be a soft ideal topological space and (F, E)∈SS(X, E). Lemma 2.7. Let (X, τe, E, I) Then e (1) sepIInt(F, E) = (F, E) ⊓ seInt(e sCl∗ (F, E)). e (2) sepICl(F, E) = (F, E) ⊔ seCl(e sInt∗ (F, E)). (1) Since (F, E)⊓e sInt(e sCl∗ (F, E)) ⊑ seInt(e sCl∗ (F, E)) = seInt(e sInt(e sCl∗ (F, E))) = ∗ ∗ ∗ ∗ seInt(e sCl (F, E)⊓e sInt(e sCl (F, E))) ⊑ seInt(e sCl ((F, E)⊓e sInt(e sCl (F, E)))), e then we have (F, E)⊓e sInt(e sCl∗ (F, E)) is soft pre-I-open set contained in (F, E) e e and so (F, E)⊓e sInt(e sCl∗ (F, E)) ⊑ sepIInt(F, E). On other hand, sepIInt(F, E) ∗ e e e is soft pre-I- open, sepIInt(F, E) ⊑ seInt(e sCl (e spIInt(F, E))) ⊑ seInt(e sCl∗ (F, E)) e e and so sepIInt(F, E) ⊑ (F, E) ⊓ seInt(e sCl∗ (F, E)). Hence sepIInt(F, E) = ∗ (F, E) ⊓ seInt(e sCl (F, E)). e e (2) By (1), sepIInt(F, E) = (F, E) ⊓ seInt(e sCl∗ (F, E)). So, (e spIInt(F, E))c = ∗ c c c e [(F, E)⊓e sInt(e sCl (F, E))] , sepICl((F, E) ) = (F, E) ⊔[e sInt(e sCl∗(F, E))]c c c ∗ c e and sepICl((F, E) ) = (F, E) ⊔ seCl(e sInt ((F, E) )). Assume, (F, E)c = e (G, E). Hence sepICl(G, E) = (G, E) ⊔ seCl(e sInt∗ (G, E)). □ Proof. e be a soft ideal topological space and (F, E)∈SS(X)E . Theorem 2.8. Let (X, τe, E, I) Then the following hold. e (1) The empty set is the only soft subset which is nowhere dense and soft pre-I-regular e pre-I-open; e e e (2) If (F, E) is soft pre-I-regular pre-I-closed, then every soft-I-rare set is soft pree I-open. SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES 9 e e (1) Suppose that (F, E) is soft nowhere dense and soft pre-I-regular pre-Ie e open. Then, by Lemma 2.7(1) we have (F, E) = sepIInt(e spICl(F, E)) = ∗ e e e sepICl(F, E) ⊓ seInt(e sCl (e spICl((F, E))). Therefore (F, E) ⊑ sepICl(F, E) ⊓ e e seInt(e sCl∗ (e sCl(F, E))) ⊑ sepICl(F, E) ⊓ seInt(e sCl(F, E)) = sepICl(F, E) ⊓ 0E = 0E . e e e spIInt(F, e (2) Suppose that (F, E) is soft pre-I-regular pre-I-closed. Then (F, E) = sepICl(e E)) = ∗ ∗ e e e sepIInt(F, E)⊔e sCl(e sInt (e spIInt(F, E))) ⊑ sepIInt(F, E)⊔e sCl(e sInt (F, E)) = e e e sepIInt(F, E) ⊔ 0E = sepIInt(F, E). Therefore, (F, E) = sepIInt(F, E). Hence e (F, E) is soft pre- I-open. □ Proof. e is called soft extremely pre-Ie Definition 2.15. A soft ideal topological space (X, τe, E, I) e e e disconnected if the soft pre- I-closure of every soft pre-I-open set is soft pre-I-open. e be a soft ideal topological space where X = {h1 , h2 , h3 }, Example 2.16. Let (X, τe, E, I) E = {e},e τ = {1E , 0E , {(e, { h1 })}, {(e, {h2 , h3 })}} and Ie = {0E , {(e, { h1 })}, {(e, { h3 })}, {(e, {h1 , h3 })}}. e e Then SP IO(X, E) = {1E , 0E , {(e, { h1 })}, {(e, {h2 , h3 })}, {(e, { h2 })}, {(e, {h1 , h2 })}}. e is extremely soft pre-I-disconnected. e So, (X, τe, E, I) e be a soft ideal topological space. Then the following are Theorem 2.9. Let (X, τe, E, I) equivalent: e is extremely soft pre-I-disconnected; e (1) (X, τe, E, I) e e e (2) Every soft pre-I-regular pre-I-open set is soft pre-I-regular. e is extremely soft pre-I-disconnected e Proof. (1) ⇒ (2): Assume that (X, τe, E, I) and e e e e (F, E) is soft pre-I-regular pre-I-open. Then (F, E) is soft pre-I-open and so, sepICl(F, E) e e e e is a soft pre-I-open set. Hence (F, E) = sepIInt(e spICl(F, E)) = sepICl(F, E). Hence e e (F, E) is soft pre-I-closed. Therefore (F, E) is soft pre-I-regular. e e (2) ⇒ (1): Suppose that (F, E) is soft pre-I- open. Then, by Theorem 2.5(2), sepICl(F, E) c e e e e is soft pre-I-regular pre-I-closed which implies that (e spICl(F, E)) is soft pre-I-regular e open. Hence (e e e e pre-IspICl(F, E))c is soft pre-I-regular. Therefore, (e spICl(F, E))c is e e e e soft pre-I-closed and so sepICl(F, E) is soft pre-I-open. Hence (X, τe, E, I) is soft exe tremely pre-I-disconnected. □ e is a soft extremely pre-I-disconnected e Theorem 2.10. Let (X, τe, E, I) space and (F, E)∈SS(X, E). Then the following are equivalent: e (1) (F, E) is soft pre-I-regular. e e (2) (F, E) = sepICl(e spIInt(F, E)); c e e (3) (F, E) is soft pre-I-regular pre-I-open; e e (4) (F, E) is soft pre-I-regular pre-I-open. e e Proof. (1) ⇒ (2): Suppose that (F, E) is soft pre-I-regular. Then (F, E) is soft pre-Ie e e open and soft pre-I-closed and so (F, E) = sepIInt(F, E) and (F, E) = sepICl(F, E). e spIInt(F, e Hence (F, E) = sepICl(e E)). e spIInt(F, e e spIInt(F, e (2) ⇒ (3): Suppose that (F, E) = sepICl(e E)). Then (F, E)c = (e spICl(e E)))c = c c e e e e sepIInt(e spICl((F, E) )). So (F, E) is soft pre-I-regular pre-I-open. c e e e (3) ⇒ (4): Since (F, E) is soft pre-I-regular pre-I-open, then (F, E)c is soft pre- Ie e regular (Theorem 2.9). So, (F, E) is soft pre-I-open and soft pre-I-closed, thus (F, E) = 10 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED e e e e e sepIInt(F, E) = sepIInt(e spICl(F, E)). Hence (F, E) is soft pre-I-regular pre-I-open. (4) ⇒ (1): The proof follows from Theorem 2.9. □ e is said to be sepIR-door e Definition 2.17. A soft ideal topological space (X, τe, E, I) space e e e e if every soft subset of τe is either soft pre-I-regular pre-I-open or soft pre-I-regular pre-Iclosed. e Example 2.18. The soft ideal topological space of Example 2.1 is a sepIR-door space. e be a sepIR-door e e Theorem 2.11. Let (X, τe, E, I) space. Then every soft pre-I-open set in e e the space is soft pre-I-regular pre-I-open. e e is a sepIR-door e Proof. Let (F, E) be a soft pre-I-open subset of X. Since (X, τe, E, I) e e e e space, then (F, E) is either soft pre-I-regular pre-I- open or soft pre- I-regular pre-Ie e closed. If (F, E) is soft pre-I-regular pre-I-open, then the proof is complete. If (F, E) is e e e e soft pre-I-regular pre-I-closed, so (F, E) = sepIInt(e spICl(F, E)). □ e be a soft ideal topological space and (F, E)∈SS(X)E . Theorem 2.12. Let (X, τe, E, I) e e e If (F, E) is both soft α − I-open and soft α − I-closed, then (F, E) is soft pre-I-regular e open set. pre-I- e e Proof. Suppose (F, E) is a soft α − I-open and soft α − I-closed set. Then (F, E) is soft e e e pre-I-open and soft pre-I-closed set (Theorem 1.3). Hence (F, E) is soft pre-I-regular e pre-I-open set. □ e be a soft ideal topological space and (F, E)∈SS(X)E . If Theorem 2.13. Let (X, τe, E, I) e e e (F, E) is soft α−I- open and soft pre-I-regular pre-I-open, then (F, E) = seInt(e sCl∗ (e sInt(F, E))). e Proof. Suppose that (F, E) is a soft α−I-open set. Then (F, E) ⊑ seInt(e sCl∗ (e sInt(F, E))) e e e e and (F, E) is soft pre-I-regular pre-I-open. Hence, we have (F, E) = sepIInt(e spICl(F, E)) = e e spIInt(e e e e e sepIInt(e spICl(e spICl(F, E)))) ⊒ sepIInt(e sCl∗ (e sInt(e spICl(F, E)))) ⊒ ∗ ∗ e sepIInt(e sCl (e sInt(F, E))) ⊒ seInt(e sCl (e sInt(F, E))). Therefore (F, E) = seInt(e sCl∗ (e sInt(F, E))). □ 3. e sδpIe - OPEN SETS e e In this section, we define the soft delta pre-I-open set by using the notion of soft pre-Ie regular pre-I-open sets and study some of their properties. e Definition 3.1. A soft point xe ∈ SP (X, E) is called a seδpI-cluster point of (F, E) if e e (F, E) ⊓ (U, E) ̸= 0E for every soft pre-I-regular pre-I-open set (U, E) containing xe . e e The set of all seδpI-cluster points of (F, E) is called the seδpI-closure of (F, E) and is dee e e noted by seδpICl(F, E). The complement of an seδpI-closed set is called an seδpI-open set. e e e IO(X, e We denote the collection of all seδpI-open set (resp. seδpI-closed) sets by SδP E) e IC(X, e (resp. SδP E)). e be as in Example 2.4. Then, we have 1E , 0E , {(e, {h1 , h2 })}, Example 3.2. Let (X, τe, E, I) e e {(e, {h1 , h3 )}, {(e, {h2 })}, {(e, {h3 })} are soft pre-I-regular pre-I-open sets. Therefore e e SδP IC(X, E) = {1E , 0E , {(e, {h1 , h2 })}, {(e, {h1 , h3 })}, {(e, {h2 })}, {(e, {h3 })}, {(e, {h1 })}} SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES 11 and e IO(X, e SδP E) = {1E , 0E , {(e, {h1 , h2 })}, {(e, {h1 , h3 })}, {(e, {h2 })}, {(e, {h3 })}, {(e, {h2 , h3 })}}. e is called Definition 3.3. A soft set (F, E) in a soft ideal topological space (X, τe, E, I) e e seδpI-neighborhood of a soft point xe if there exists an seδpI-open set (U, E) such that e (U, E) ⊑ (F, E). xe ∈ e be the soft ideal topological space as in Example 2.4. Then Example 3.4. Let (X, τe, E, I) e we have {(e, {h2 , h3 })} is an seδpI-neighborhood of a soft point xe = {(e, {h2 })}. Ine IO(X, e e {(e, {h2 , h3 })} ⊑ {(e, {h2 , h3 })} and {(e, {h2 , h3 })} ∈ SδP deed, {(e, {h2 })} ∈ E). e be a soft ideal topological space and (F, E), (G, E) ∈ SS(X, E) Lemma 3.1. Let (X, τe, E, I) and {(Ui , E) : i ∈ Λ} ⊆ SS(X, E). Then the following hold. e (1) (F, E) ⊑ seδpICl(F, E); e e (2) If (F, E) ⊑ (G, E), then seδpICl(F, E) ⊑ seδpICl(G, E); e e (3) seδpICl{⊓{(Ui , E) : i ∈ Λ}} ⊑ ⊓ {e sδpICl(Ui , E) : i ∈ Λ}; e e (4) ⊔ {e sδpICl(U eδpICl{⊔(U i , E) : i ∈ Λ} ⊑ s i , E) : i ∈ Λ}; e e e (5) seδpICl{(F, E) ⊔ (G, E)} = seδpICl(F, E) ⊔ seδpICl(G, E). Proof. (2) (3) (4) (5) e e e (F, E) and (G, E) be a soft pre-I-regular (1) Let xe ∈ pre-I-open set containe e seδpICl(F, ing xe . Therefore (F, E) ⊓ (G, E) ̸= 0E implies xe ∈ E) which e implies that (F, E) ⊑ seδpICl(F, E); e e (H, E) ̸= e e seδpICl(F, E). Then (F, E) ∩ ∅ Suppose (F, E) ⊑ (G, E) and xe ∈ e e for every soft pre-I-regular pre-I-open set (H, E) containing xe . Since (F, E) ⊑ e e seδpICl(G, (G, E), then (G, E) ⊓ (H, E) ̸= 0E . Therefore xe ∈ E). So, e e seδpICl(F, E) ⊑ seδpICl(G, E); e Since ⊓(Ui , E) ⊑ (Ui , E) for each i ∈ Λ, by (2) seδpICl{⊓{(U i , E) : i∈ Λ}} ⊑ e e e seδpICl(Ui , E) for each i ∈ Λ. So, seδpICl{⊓{(Ui , E) : i ∈ Λ}} ⊑ ⊓ {e sδpICl(U i , E) : e e i ∈ Λ}. Therefore seδpICl{⊓{(Ui , E) : i ∈ Λ}} ⊑ ⊓{e sδpICl(Ui , E) : i ∈ Λ}; e e eδpICl{⊔{(U Since (Ui◦ , E) ⊑ ⊔i∈Λ (Ui , E) for each i◦ ∈ Λ, seδpICl(U i , E) : i◦ , E) ⊑ s e e i ∈Λ}}. Hence ⊔{e sδpICl(U , E) : i ∈Λ} ⊑ s e δp ICl{⊔{(U , E) : i ∈ Λ}}; i i e e seδpICl{(F, E) ⊔ (G, E)}. Then ((F, E) ⊔ (G, E)) ⊓ (H, E) ̸= Let xe ∈ e e 0E , for every soft pre-I-regular pre-I-open set (H, E) containing xe . Hence (F, E) ⊓ (H, E) ̸= 0E or (G, E) ⊓ (H, E) ̸= 0E which implies that e e e e seδpICl(F, E) ⊔ seδpICl(G, E). Therefore seδpICl{(F, E) ⊔ (G, E)} ⊑ xe ∈ e e e e seδpICl(F, E) ⊔ seδpICl(G, E). Also, seδpICl{(F, E) ⊔ (G, E)} ⊒ seδpICl(F, E) ⊔ e e e e seδpICl(G, E). So, seδpICl{(F, E) ⊔ (G, E)} = seδpICl(F, E) ⊔ seδpICl(G, E). □ e Lemma 3.2. Arbitrary soft intersection of seδpI-closed sets in a soft ideal topological e e space (X, τe, E, I) is seδpI-closed. e IC(X), e Proof. Suppose that (F, E) = ⊓{(Fi , E) : (Fi , E) ∈ SδP i ∈ Λ}. Then e e we obtain seδpICl(F , E) = (F , E), i ∈ Λ. Thus s e δp ICl[⊓{(F , E) : i ∈ Λ}] ⊑ i i i e e ⊓ {(Fi , E), i ∈ Λ}. Therefore, seδpICl(F, E) ⊑ (F, E).Therefore (F, E) = seδpICl(F, E) e and (F, E) is seδpI-closed. □ e be a soft ideal topological space and (F, E) ∈ SS(X, E). Lemma 3.3. Let (X, τe, E, I) Then 12 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED e e IC(X), e (1) seδpICl(F, E) = ⊓{(Gi , E) : (Gi , E) ∈ SδP (F, E) ⊑ (Gi , E), i ∈ Λ}, e e IRC(X), e (2) seδpICl(F, E) = ⊓{(Gi , E) : (Gi , E) ∈ SP (F, E) ⊑ (Gi , E), i ∈ Λ}. e/ e e (1) Let xe ∈⊓{(G i , E) : (Gi , E) ∈ SδP IC(X), (F, E) ⊑ (Gi , E), , i ∈ Λ}. e/ (Gi , E) and xe ∈ e/ (F, E) e e Then there exists (Gi◦ , E) ∈ SδP IC(X) such that xe ∈ ◦ c e e e (Gi◦ , E)c , then as (F, E) ⊑ (Gi◦ , E). Since (Gi◦ , E) ∈ SδP IO(X) and xe ∈ c e/ seδpICl(F, e E). Conwe have (Gi◦ , E) ⊓ (F, E) = 0E . Therefore xe ∈ e e e IO(X) e / seδpICl(F, E). Then there exists (U, E) ∈ SδP versely, suppose xe ∈ e/ (U, E)c and e (U, E) and (U, E) ⊓ (F, E) = 0E . Thus xe ∈ such that xe ∈ c c e (U, E) ∈ SδP IC(X). We can replace (U, E) with (Gi , E) for some i ∈ Λ and e e IRC(X), e obtain (F, E) ⊑ (Gi , E). So xe ∈ / ⊓{(Gi , E) : (Gi , E) ∈ SP (F, E) ⊑ (Gi , E), i ∈ Λ}. (2) The proof is similar to the proof of (1). □ Proof. e be a soft ideal topological space, (F, E) ∈ SS(X, E) and Lemma 3.4. Let (X, τe, E, I) e e seδpICl(F, xe ∈ SP (X, E). Then xe ∈ E) if and only if (U, E) ⊓ (F, E) ̸= 0E for e e e every seδpI-open (soft pre-I-regular pre-I-open) set (U, E) containing xe . e/ seδpICl(F, e e e Proof. Suppose that xe ∈ E). Then, {e sδpICl(F, E)}c is an seδpI-open set containing xe that doesn’t intersect (F, E). That is, (U, E) ⊓ (F, E) = 0E , where e (U, E) = {e sδpICl(F, E)}c . The converse is obvious. □ e e e for any (F, E) ∈ SS(X, E); Corollary 3.5. (1) seδpICl(F, E) is seδpI-closed in (X, τe, E, I) e e open) if and only if it is the (2) (F, E) ∈ SS(X, E) is seδpI-closed (resp. seδpIe e e soft intersection (resp. soft union) of soft pre-I-regular pre-I-closed (soft pre-Ie regular pre-I-open) sets. e be a soft ideal topological space and (F, E) ∈ SS(X, E). Lemma 3.6. Let (X, τe, E, I) e e e containing (F, E). Then seδpICl(F, E) is the smallest seδpI-closed set in (X, τe, E, I) e e Proof. Let {(Fi , E) : i ∈ Λ} be the collection of all soft seδpI-closed subsets of (X, τe, E, I) e e containing (F, E). So by Lemma 3.2, seδpICl(F, E) = ⊓{(Fi , E) : i ∈ Λ} is seδpI-closed. Since (F, E) ⊑ (Fi , E) for each i ∈ Λ, we have (F, E) ⊑ ⊓{(Fi , E) : i ∈ Λ} = e e e seδpICl(F, E). Thus seδpICl(F, E) is a soft seδpI-closed set containing (F, E). Also, e e since seδpICl(F, E) = ⊓{(Fi , E) : i ∈Λ}, then seδpICl(F, E) ⊑ (Fi , E) for each i ∈ Λ. e e e containing Consequently, seδpICl(F, E) is the smallest seδpI-closed set in (X, τe, E, I) (F, E). □ e e e Remark. It is clear that by Corollary 3.1, every soft pre-I-regular pre-I-open set is seδpIopen. However, the converse is not true as shown by the following example. e be the soft ideal topological space as in Example 2.4, then Example 3.5. Let(X, τe, E, I) e e we have {(e, {h1 , h2 })}, {(e, { h1 , h3 })} are soft pre-I-regular pre-I-closed sets. Thus e by Corollary 3.5, {(e, {h1 , h2 })} ⊓ {(e, { h1 , h3 })} = {(e, {h1 })} is seδpI-closed, and e e e so {(e, { h2 , h3 })} is seδpI-open. But sepIInt(e spICl{(e, { h2 , h3 })}) = 1E , and so e e open. {(e, { h2 , h3 })} is not soft pre-I-regular pre-I- SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES 13 e e Remark. The soft union of even two seδpI-closed sets need not be a seδpI-closed set as shown by the following example. e be the soft ideal topological space as in Example 3.4. {(e, {h2 })}, Example 3.6. Let(X, τe, E, I) e e {(e, {h3 })} are soft pre-I-regular pre-I-closed sets. Thus, by Corollary 3.5, {(e, {h2 })}, e e {(e, {h3 })} are seδpI-closed. However, {(e, { h2 , h3 })} is not seδpI-closed. 4. S OFT DELTA PRE -Ie - SEPARATION AXIOMS In this section, we introduce the concept of soft separation axioms using soft delta pree I-open sets. We define a seδpIe − T0 space, a seδpIe − T1 space and a seδpIe − T2 space and study some of their properties. e is said to be a seδpIe − T0 Definition 4.1. A soft ideal topological space (X, τe, E, I) space if for every pair of soft points xe , ye ∈SP (X, E) such that xe ̸= ye , there exe IO(X, e ists (F, E) ∈ SδP E), containing one of them but not the other. e be a soft ideal topological space, where X = {h1 , h2 }, E = Example 4.2. Let (X, τe, E, I) {e1 , e2 }, τe = {1E , 0E , {(e1 , {h1 })}, {(e1 , {h2 }), (e2 , {h1 })}, {(e1 , {h1 }), (e2 , {h2 })}, {(e1 , X), (e2 , {h1 })}} e is a seδpIe − T0 space. and Ie = {0E , {(e1 , {h1 })}}. The space (X, τe, E, I) e is said to be a seδpIe − T1 Definition 4.3. A soft ideal topological space (X, τe, E, I) space if for every pair of soft points xe , ye ∈ SP (X, E) such that xe ̸= ye , there exe/ (F, E) and ye ∈ e IO(X), e e (F, E), ye ∈ e (G, E), ist (F, E), (G, E) ∈SδP such that xe ∈ e xe ∈ / (G, E). e in Example 4.2 is a seδpIe − T1 Example 4.4. The soft ideal topological space (X, τe, E, I) space. e is said to be a seδpIe − T2 Definition 4.5. A soft ideal topological space (X, τe, E, I) space if for every a pair soft points xe , ye ∈ SP (X, E) such that xe ̸= ye , there exe IO(X, e e (G, E) and (F, E) ⊓ e (F, E), ye ∈ ist (F, E), (G, E) ∈ SδP E), such that xe ∈ (G, E) = 0E . Example 4.6. The soft ideal topological space (X, τe, E, Ie ) in Example 4.2 is a seδpIe− T2 space. Remark. From Corollary 3.5(2) we have: e is seδpIe − T0 if and only if for every (1) A soft ideal topological space (X, τe, E, I) e pair of distinct soft points xe , ye of SP (X, E), there exists a soft pre-I-regular e pre-I-open set containing one of the soft point but not the other. e is seδpIe−T1 if and only if for every pair (2) A soft ideal topological space (X, τe, E, I) e of distinct soft points xe , ye of SP (X, E), there exists a soft pre-I-regular pree e I-open set (U, E) in SS(X, E) containing xe but not ye and a soft pre-I-regular e pre-I-open set (V, E) in SS(X, E) containing ye but not xe . e is seδpIe − T2 if and only if for every (3) A soft ideal topological space (X, τe, E, I) e pair of distinct soft points xe , ye of SP (X, E), there exists a soft pre-I-regular e pre- I-open set (U, E) and (V, E) in SS(X, E) containing xe and ye , respectively, such that (U, E) ⊓ (V, E) = 0E . e is an seδpIe − Ti space, then it is seδpIe − Ti−1 , i = 1, 2. The Remark. If (X, τe, E, I) converse need not be true as shown in the following example. 14 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED e is the soft ideal topological space in Example 2.4. Example 4.7. Suppose that (X, τe, E, I) e regular pre-I-open e The soft pre-Isets are {1E , 0E , {(e, {h1 })}, {(e, {h2 })}}. It is clear e from Remark 4.1, that (X, τe, E, I) is seδpIe − T0 space but it is not seδpIe − T1 . e Remark. It is easy to see from Remark 4.1 and the fact that every soft pre-I-regular e e e e pre-I-open set is soft pre-I-open, that if a space (X, τe, E, I) is seδpI − Ti , then it is soft pre-Ie− Ti , i = 0, 1, 2. The converse need not be true as seen from the following example. e be a soft ideal topological space where X = {h1 , h2 , h3 , h4 }, Example 4.8. Let (X, τe, E, I) E = {e}, τe = {1E , 0E , (e, {h1 }), (e, {h3 }), (e, {h1 , h3 })} and Ie = {0E , (e, {h1 })}. e IO(X, e Then we have that SP E) = {1E , 0E , {(e, {h1 })}, {(e, {h1 , h3 })}, {(e, {h3 })}, {(e, {h1 , h2 , h3 })}, {(e, h1 , h3 , h4 e e and the soft pre-I-regular pre-I-open sets are {1E , 0E , {(e, {h1 })}, {(e, {h3 })}. Then we e e have (X, τe, E, I) is soft pre-I − T0 but not seδpIe − T0 . e be a soft ideal topological space. A space (X, τe, E, I) e is Theorem 4.1. Let (X, τe, E, I) e T2 . seδpIe − T2 if and only if it is soft pre-I- e is a soft pre-Ie − T2 space. Let xe , ye ∈ SP (X, E) such Proof. Assume that (X, τe, E, I) e that xe ̸= ye , then by the assumption, there exist disjoint soft pre-I-open sets (U, E) and (V, E) containing xe and ye , respectively. Since (U, E) ⊓ (V, E) = 0E and (V, E) is e e e e soft pre-I-open, then sepICl(U, E) ⊓ (V, E) = 0E and thus, sepIInt(e spICl(U, E)) ⊓ e e e e e (V, E) = 0E . Similarly, since sepIInt(e spICl(U, E)) is soft pre-I-open, then sepIInt(e spICl(U, E)) ⊓ e e e e e sepICl(V, E) = 0E which implies that sepIInt(e spICl(U, E)) ⊓ sepIInt(e spICl(V, E)) = e e e e 0E . Now, (U, E) ⊑ sepIInt(e spICl(U, E)) and (V, E) ⊑ sepIInt(e spICl(V, E)) as e e e e e (U, E) and (V, E) are soft pre-I-open sets. Thus, sepIInt(e spICl(U, E)) and sepIInt(e spICl(V, E)) e e are disjoint soft pre-I-regular pre-I- open sets containing xe and ye , respectively. Hence e is seδpIe − T2 . The converse of proof follows from Remark by Remark 4.1 (3), (X, τe, E, I) 4.3. □ e is a seδpIe − T0 space if and Theorem 4.2. A soft ideal topological space (X, τe, E, I) e only if for each pair of distinct soft points xe and ye of SP (X, E), seδpICl({x e }) ̸= e seδpICl({ye }). e be a seδpIe− T0 space and xe , ye ∈ SP (X, E), such that xe ̸= ye . Proof. Let (X, τe, E, I) c e Then, there exists a seδpI-open set (G, E) containing xe but not ye and therefore (G, E) is e e a seδpI-closed set which contains ye but not xe . By Lemma 3.6, we have seδpICl({y e }) ⊑ c e e e e (G, E) and xe ∈e / sδpICl({y }). Hence s e δp ICl({x }) = ̸ s e δp ICl({y }). Conversely, e e e e suppose that xe , ye ∈ SP (X, E), xe ̸= ye . Then by the assumption, seδpICl({x e }) ̸= e seδpICl({ye }). Hence there exists at least one soft point ze ∈ SP (X, E) such that e/ seδpICl({y e/ sδpICl({y e e e e seδpICl({x ze ∈ e }) and ze ∈ e }), say. We claim that xe ∈e e }). If e e e e xe ∈seδpICl({ye }), by Lemma 3.1(2), seδpICl({xe }) ⊑ seδpICl({ye }) which is a contrac e/ seδpICl({y e e e {e diction to the fact that ze ∈ sδpICl({y e }). Thus xe ∈ e })} . But by Lemma c e e 3.1(1) and Corollary 3.5(1) we have {e sδpICl({y eδpI-open set that doesn’t cone })} is a s e e tain ye . Therefore (X, τe, E, I) is a seδpI − T0 space. □ e is a seδpIe − T1 space if and only Theorem 4.3. A soft ideal topological space (X, τe, E, I) e if the soft singleton sets of SS(X, E) are seδpI-closed. SOME SEPARATION AXIOMS IN SOFT IDEAL TOPOLOGICAL SPACES 15 e is a seδpI−T e 1 space and xe ∈ SP (X, E). Let ye ∈ SP (X, E)⧹{xe }. Proof. Suppose that (X, τe, E, I) e e (U, E) but Then xe ̸= ye and so there exists a seδpI-open set (U, E) such that ye ∈ c c e e e e e SδP IO(X)}. xe ∈ / (U, E). Consequently, ye ∈ (U, E) ⊑ xe . Now, xe = ⊔{(U, E) : ye ∈ xce ∈ e Therefore, xe is seδpI-closed. Conversely, Let xe , ye ∈ SP (X, E) such that xe ̸= ye e e and xe , ye are seδpI-closed sets. Then by the assumption, xce is a seδpI-open set containc e ing ye but not xe . Similarly, ye is a seδpI-open set containing xe but not ye . Therefore e is a seδpIe − T1 space. (X, τe, E, I) □ e Definition 4.9. A soft function fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) is called seδpI-continuous −1 e IO(X) e e IO(Y e if fpu (G, E) ∈ SδP for every (G, E) ∈ SδP ). Theorem 4.4. If (Y, σ e, E, Ie2 ) is seδpIe − T0 space and fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) e e e is seδpI-continuous and soft injective, then (X, τe, E, I1 ) is seδpI − T0 space. Proof. Let xe , ye ∈ SP (X, E) such that xe ̸= ye . Since fpu is soft injective and e/ (G, E) e 0 , there exists (G, E) ∈ SδP e IO(Y e e (G, E), fpu (ye ) ∈ (Y, σ e, E, Ie2 ) is seδpI−T ) containing xe and fpu (xe ) ∈ −1 e e IO(X) e with fpu (xe ) ̸= fpu (ye ). Since fpu is seδpI-continuous, then we have fpu (G, E) ∈ SδP −1 −1 e/ f (G, E). Therefore (X, τe, E, Ie1 ) is seδpIe − T0 e fpu (G, E) and ye ∈ such that xe ∈ pu space. □ Definition 4.10. A soft function fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) is said to be soft point e e e δpI-closure one-to-one if xe , ye ∈ SP (X, E) such that seδpICl({x eδpICl({y e }) ̸= s e }), e e implies seδpICl({fpu (xe )}) ̸= seδpICl({fpu (ye )}). e Theorem 4.5. If fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) is soft point δpI-closure one-to-one and (X, τe, E, Ie1 ) is seδpIe − T0 space, then fpu is one-to-one. Proof. Let xe , ye ∈ SP (X, E) such that xe ̸= ye . Since (X, τe, E, Ie1 ) is seδpIe − T0 , e e then by Theorem 4.2 we have seδpICl({x eδpICl({y e }) ̸= s e }). But fpu is soft point e e e δpI-closure one-to-one implies that seδpICl({fpu (xe )}) ̸= seδpICl({f pu (ye )}). Hence fpu (xe ) ̸= fpu (ye ). Therefore fpu is one-to-one. □ e Definition 4.11. A soft function fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) is said to be seδpIe IC(Y e e IC(X). e closed if fpu (G, E) ∈ SδP ) for every (G, E) ∈ SδP Theorem 4.6. Let (X, τe, E, Ie1 ) be seδpIe − T1 and fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) is e e e seδpI-closed surjective function. Then (Y, σ e, E, I2 ) is seδpI − T1 . Proof. Suppose that ye ∈ SP (Y ). Since fpu is soft surjective, then there exists xe ∈ SP (X, E) such that fpu (xe ) = ye . Since (X, τe, E, Ie1 ) is seδpIe − T1 , then from Theorem 4.3, we obe IC(X). e e IC(Y e tain xe ∈ SδP Again by the hypothesis, we have fpu (xe ) = ye ∈ SδP ). e e Hence, (Y, σ e, E, I2 ) is seδpI − T1 . □ e Theorem 4.7. Let fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) be a soft injective and seδpIe e e e continuous function. If (Y, σ e, E, I2 ) is seδpI − T1 , then (X, τe, E, I1 ) is seδpI − T1 . Proof. It’s similar to the proof of Theorem 4.4. □ e is seδpIe − T2 space, then for all xe , ye ∈ SP (X, E) with Theorem 4.8. If (X, τe, E, I) e/ seδpICl(F, e IO(X) e e e (F, E) and ye ∈ xe ̸= ye there exists (F, E) ∈ SδP such that xe ∈ E). 16 F. H. KHEDR, O. R. SAYED AND S. R. MOHAMED e is seδpIe − T2 , Proof. Let xe , ye ∈ SP (X, E) such that xe ̸= ye . Since (X, τe, E, I) e IO(X) e e (F, E) then there exist two disjoint soft sets (F, E), (G, E) ∈ SδP such that xe ∈ c e e e e and ye ∈ (G, E). Clearly we have, (G, E) ∈ SδP IC(X), seδpICl(F, E) ⊑ (G, E)c e/ seδpICl(F, e and therefore, ye ∈ E). □ e Theorem 4.9. Let fpu : (X, τe, E, Ie1 ) → (Y, σ e, E, Ie2 ) be a soft injective and seδpIe e e e continuous function. If (Y, σ e, E, I2 ) is seδpI − T2 , then (X, τe, E, I1 ) is seδpI − T2 . Proof. Since fpu is soft injective, so fpu (xe ) ̸= fpu (ye ) for each xe , ye ∈ SP (X, E) and e IO(Y e xe ̸= ye . Now, (Y, σ e, E, Ie2 ) being seδpIe − T2 , there exists (F, E), (G, E) ∈ SδP ) e e such that fpu (xe ) ∈ (F, E), fpu (ye ) ∈ (G, E) and (F, E) ⊓ (G, E) = 0E . Sup−1 −1 e pose that (U, E) = fpu (F, E) and (V, E) = fpu (G, E). Then by seδpI-continuity, −1 −1 e e e fpu (F, E) = (U, E), ye ∈ e fpu (G, E) = (V, E) (U, E), (V, E) ∈ SδP IO(X). Also, xe ∈ −1 −1 and (U, E) ⊓ (V, E) = fpu (F, E) ⊓ fpu (G, E) = 0E . Hence (X, τe, E, Ie1 ) is seδpIe − T2 . □ 5. C ONCLUSION The present work is devoted to define and study new classes of soft sets, namely soft e e e delta pre-I-open sets and soft pre-I-regular pre-I-open sets in soft ideal topological space. Also, a new class of soft separation axioms, namely soft delta pre ideal- Ti spaces, i = 0, 1, 2 is introduced. We believe that it would be interesting to extend this approach to other structures such as Fuzzy soft topology, fuzzifying soft topology etc. We intend to investigate all these issues in future research works. 6. ACKNOWLEDGEMENTS The authors are thankful to the Editor-in-Chief, Prof. G. Muhiuddin for the technical comments and to the anonymous referee(s) for a careful checking of the details. R EFERENCES [1] H. Aktas and N. Caǧman, Soft sets and soft groups, Inf. Sci., 177 (2007), 2726-2735. [2] M. I. Ali, F. Feng, X. Liu, W. K. Min and M. 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Kovkov, Soft sets technique and its application, Nech. Siste. Myagkie Vychisleniya, 1 (2006), 8-39. [27] D. Pei and D. Miao, From soft sets to information systems, In: X. Hu, Q. Liu, A. Skowron, T. Y. Lin, R. R. Yager and B. Zhang, eds., Proceedings of Granular Computing, IEEE, 2(2005), 617-621. [28] E. Peyghan, B. Samadi and A. Tayebi, About soft topological spaces, J. New Results in Sci., 2 (2013), 60-75. [29] O. R. Sayed and A. M Khalil, Some Properties of Soft θ-topology, Hacettepe Journal of Mathematics and Statistics, 44 (5) (2015), 1133–1145. [30] M. Shabir and M. Naz, On soft topological spaces, Comput. Math. Appl., 61 (2011), 1786-1799. [31] Y. C. Shao and K. Qin, The lattice structure of the soft groups, Procedia Engineering, 15 (2011), 3621-3625. [32] P. Zhu and Qiaoyanwen, Operations on soft sets revisited, J. Applied Math., 2013 (2013), 7 pages. http://dx.doi.org/10.1155/2013/105752. [33] I. Zorlutuna, M. Akdag, W. K. Min and S. Atmaca, Remarks on soft topological spaces, Ann. Fuzzy Math. Inform., 3 (2012), 171-185. [34] Y. Zou and Z. Xiao, Data analysis approaches of soft sets under incomplete information, Knowl. Base. Syst., 21 (2008), 941-945. F. H. K HEDR D EPARTMENT OF M ATHEMATICS , FACULTY OF S CIENCE , A SSIUT U NIVERSITY, A SSIUT 71516, EGYPT Email address: [email protected] O. R. S AYED D EPARTMENT OF M ATHEMATICS , FACULTY OF S CIENCE , A SSIUT U NIVERSITY, A SSIUT 71516, EGYPT Email address: o [email protected] S.R. M OHAMED D EPARTMENT OF M ATHEMATICS , FACULTY OF S CIENCE , M INIA UNIVERSITY, M INIA , EGYPT. Email address: [email protected] ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 5, Number 1 (2022), 18-31 ISSN: 2582-0818 © http://www.technoskypub.com SOME TYPES OF INTERIOR FILTERS IN QUASI-ORDERED Γ-SEMIGROUPS DANIEL A. ROMANO∗ A BSTRACT. In this article, the concepts of interior, weak-interior and quasi-interior filters in a quasi-ordered Γ-semigroup are introduced and recognize some of their fundamental properties. In addition to the above, the relationships between these three classes of filters in quasi-ordered Γ-semigroups are considered. One of the specifics in this analysis, among others, is that the requirement that a filter in a quasi-ordered Γ-semigroup S has to be a sub-semigroup in S is omitted. Instead of this requirement in the determination of these three classes of filters the consistency requirement is incorporated. 1. I NTRODUCTION The concept of interior ideals of a semigroup S has been introduced by S. Lajos in [13] as a subsemigroup J of S such that SJS ⊆ J. The interior ideals of semigroups have been also studied by G. Szász in [23, 24]. In [6, 7] N. Kehayopulu and M. Tsingelis introduced the concepts of interior ideals in ordered semigroups. W. Jantanan, O. Johdee and N. Praththong in [3] and M. M. Krishna Rao in [9] also wrote about interior ideals in ordered semigroup. Classes of weak-interior ideals and quasi-interior ideals in semigroups were introduced in articles [9, 10] by M. M. Krishna Rao. S. Tarsuslu and G. Çuvalcıoğlu in [25] also dealt with quasi-interior ideals in semigroups. These types of ideals in quasiordered semigroups were analyzed in [18] by D. A. Romano. The notion of a Γ-semigroup was introduced in 1984 in [20] by M. K. Sen. Then many authors took part in the development of this concept as well as many of its properties and substructures (see, for example [1, 2, 16, 21, 22]). Classes of weak-interior ideals and quasi-interior ideals in Γ-semigroup were introduced in articles [9, 11] by M. M. Krishna Rao. Thus Y. B. Jun and S. Lajos ([5]), Y. I. Kwon ([12]), S. K. Lee and S. S. Lee ([14]), S. K. Lee and Y. I. Kwon ([15]), K. Hila ([1]), A. Iampan ([2]), N. Kehayopulu and M. Tsingelis ([8]) and Jyothi V. et al. ([4]) analyzed the filters in quasi-ordered Γ-semigroups. 2010 Mathematics Subject Classification. 06F05. Key words and phrases. quasi-ordered Γ-semigroup, filter, interior filter, weak-interior filter, quasi-interior filter in quasi-ordered Γ-semigroup. Received: February 20, 2022. Accepted: May 10, 2022. Published: June 30, 2022. *Corresponding author. 18 INTERIOR FILTERS IN Γ-SEMIGROUPS 19 As we know, although many results on semigroups (about ordered semigroups) can be transferred into Γ-semigroups (res. into po-Γ-semigroups) just putting a symbol Γ in the appropriate place, while for some other results for the transfer needs subsequent technical changes ([8], page 97), we decide to look at these phenomena by considering some special filter substructures in quasi-ordered Γ-semigroups. The paper [19] discusses the design of interior, weak-interior and quasi-interior filters in quasi-ordered semigroups. In this article, the concepts of interior (Subsection 3.1), weak-interior (Subsection 3.2) and quasi-interior (Subsection 3.3) filters in a quasi-ordered Γ-semigroup are introduced and analyze of their fundamental properties are discussed. For the purposes of this report, the requirement that the mentioned filters in a quasi-ordered Γ-semigroup S be subsemigroups of S has been omitted. Instead, the consistency requirement is incorporated into the determination of these filters. The chosen orientation allows that the families of these filters form complete lattices. In addition to the above, the relationships between these three classes of filters in quasi-ordered Γ-semigroups are considered. Thus, for example, some satisfactory conditions were found that filters and interior filters coincide in a quasi-ordered Γ-semigroup. Also, if a quasi-ordered Γ-semigroup S satisfies one additional condition, then filters, interior filters and weak interior filters of S are coincide. 2. P RELIMINARIES Let S be a set. A relation ⪯ ⊆ S × S is a quasi-order on S if holds (1) (∀x ∈ S)(x ⪯ x), (2) (∀x, y, z ∈ S)((x ⪯ y ∧ y ⪯ z) =⇒ x ⪯ z). If the quasi-order relation is present on the set S, then we say that the set S is quasi-ordered. A quasi-order relation ⩽ on a set S is a partial order on S if the following holds (3) (∀x, y ∈ S)((x ⩽ y ∧ y ⩽ x) =⇒ x = y). In this case, for the set S is said to be an ordered set, or, in short, to be a po-set. A quasiorder in a set S does not have to be an order in S, in the general case. If S is a semigroup with respect to the internal binary operation ’·’, then the quasi-order ⪯ must be compatible with the operation in the following sense (4) (∀x, y, u ∈ S)(x ⪯ y =⇒ (ux ⪯ uy ∧ xy ⪯ yu)). Let S and Γ be two non-empty sets. S is called a Γ-semigroup if there exist mapping from S × Γ × S to S, written as (x, a, y) 7−→ xay satisfying the identity (5) (∀x, y, z ∈ S)(∀a, b ∈ Γ)((xay)bz = xa(ybc)). It is known that the concept of Γ-semigroups is a generalization of the notion of semigroups. A Γ-semigroup S is called a quasi-ordered Γ-semigroup if holds (6) (∀x, y, u ∈ S)(∀a ∈ Γ)(x ⪯ y =⇒ (xau ⪯ yau ∧ uax ⪯ uay)). A sufficient number of examples of ordered Γ-semigroups can be found in the literature (see, for example, [1]). Example 2.1. Let S =: M2×2 be a semigroup of real matrices of type 2 × 2 over the field R of real numbers and Γ = M2×2 . The ternary operation in S over R is the standard multiplication of matrices. Then M2×2 is an ordered Γ-semigroup under the relation ⩽ defined by (∀A, B ∈ M2×2 )(A ⩽ B ⇐⇒ (∀ ij ∈ {1, 2})(Aij ⩽ Bij )). 20 D. A. ROMANO Let S be a quasi-ordered Γ-semigroup. By a sub-semigroup of S we mean a non-empty subset A of S such that (7) (∀x, y ∈ S)(∀a ∈ Γ)((x ∈ A ∧ y ∈ A) =⇒ xay ∈ A). A non-empty subset J of a quasi-ordered Γ-semigroup S is called a right ideal of S if JΓS ⊆ J and the following holds (8) (∀u, v ∈ S)((v ∈ J ∧ u ⪯ v) =⇒ u ∈ J). A non-empty subset J of a Γ-semigroup S is called a left ideal of S if (8) and SΓJ ⊆ J holds. A subset J is called an ideal of S if it is both a left and a right ideal of S. It is obvious that a (left, right) ideal in a quasi-ordered Γ-semigroup S is a sub-semigroup in S. A subset F of a quasi-ordered Γ-semigroup S is said to be a right filter of S if the following holds (9) (∀x, y ∈ S)(∀α ∈ Γ)(xαy ∈ F =⇒ y ∈ F ) and (10) (∀x, y ∈ S)((x ∈ F ∧ x ⪯ y) =⇒ y ∈ F ). A subset F of a quasi-ordered Γ-semigroup S is said to be a left filter of S if (10) holds and the following implication is valid (11) (∀x, y ∈ S)(∀α ∈ Γ)(xαy ∈ F =⇒ x ∈ F ). A subset F of a quasi-ordered Γ-semigroup S is said to be a (two side) filter of S if it is both a left filter and a right filter of S. Remark. Our determination of (left, right) filters in a quasi-ordered Γ-semigroup S here differs from the determination of the concept of filters in the texts [1, 8, 15, 17]: We omit the requirement that (left, right) filter should be a sub-semigroup of S since there is no such condition in the determination of ideals in such semigroups. In addition, it is also not required that a filter be a non-empty subset of Γ-semigroup S. In each individual case (F is an interior filter, a weak-interior filter or a quasi-interior filter, which will be discussed in the next section), we will comment on the determination of the filter with additional condition (A) (∀x, y ∈ S)(∀a ∈ Γ)((x ∈ F ∧ y ∈ F ) =⇒ xay ∈ F ) and what it produces in these special cases. Of course, it could be said that the substructure in a quasi-ordered Γ-semigroup determined on this way is a generalized (left, right) filter, and then, for simplicity of writing, omit the adjective ’generalized’. 3. T HE MAIN RESULTS The material presented in this section is a central part of this paper. In this section, we introduce the concepts of interior filters, weak-interior filters and quasi-interior filters in a quasi-ordered Γ-semigroup and analyze their basic properties. In addition, some sufficient conditions have been found that connect these three classes of filters in such Γ-semigroups. 3.1. Interior filters. The concept of interior ideal of a semigroup S has been introduced by S. Lajos in [13] as a sub-semigroup J of a semigroup S such that SJS ⊆ J. The interior ideals of semigroups have been also studied by G. Szász in [23, 24]. Let S be a quasi-ordered Γ-semigroup. A non-empty sub-semigroup J of S is called an interior ideal of S if (8) and the following SΓJΓS ⊆ J holds. So, a subset J of a quasi-ordered Γ-semigroup S is an interior ideal of S if the following are valid: (a) J ̸= ∅, (7) (∀x, y ∈ S)(∀a ∈ Γ)((x ∈ J ∧ y ∈ J) =⇒ xay ∈ J), INTERIOR FILTERS IN Γ-SEMIGROUPS 21 (12) (∀x, u, v ∈ S)(∀a, b ∈ Γ)(x ∈ J =⇒ uaxbv ∈ J) and (8) (∀u, v ∈ S)((v ∈ J ∧ u ⪯ v) =⇒ u ∈ J). In the following definition we create the concept of interior filters of a quasi-ordered Γsemigroup as a dual of the concept of ordered interior ideals in such a semigroup. Definition 3.1. A subset F of a quasi-ordered Γ-semigroup S is an interior filter of S if (10) is valid and the following holds (13) (∀x, y ∈ S)(∀a ∈ Γ)(xay ∈ F =⇒ (x ∈ F ∨ y ∈ F )), (14) (∀u, v, x ∈ S)(∀a, b ∈ Γ)(uaxbv ∈ F =⇒ x ∈ F ). Remark. It should be noted that an interior filter F in a quasi-ordered Γ-semigroup S does not have to be either a non-empty subset of S or a sub-semigroup of S. Instead, the consistency requirement (13) is incorporated into the determination of this filter. Thus, the ∅ and S are trivial filters in a quasi-ordered Γ-semigroup S However, in the analysis that follows, we will not omit to analyze this substructure when it additionally satisfies condition (A). Without much difficulty, by direct verification the following proposition can be shown to be valid: Proposition 3.1. Let J be an interior ideal of a quasi-ordered Γ-semigroup S. Then, the set S \ J is an interior filter of S. If the ideal J of S satisfies the condition (P) (∀x, y ∈ S)(∀a ∈ Γ)(xay ∈ J =⇒ (x ∈ J ∨ y ∈ J)), then the filter S\J satisfies the condition (A). Proof. Let J be an interior ideal of a quasi-ordered Γ-semigroup S. This means that J satisfies the conditions (a), (7), (12) and (8). Put F =: S\J. Since the contraposition of (7) gives (13) and the contraposition of (12) gives (14), it remains to prove (10). Let x, y ∈ S be such that x ∈ S\J and x ⪯ y. If it were y ∈ J, we would have x ∈ J by (8), which is contrary to the assumption x ∈ / J. So it has to be y ∈ S\J. If the ideal J satisfies condition (P), then the statement (A) holds for the filter F since (A) is a contraposition of (P). □ Example 3.2. Let S = {0, 1, 2, 3, 4} and ’·’ defined on S as follows: · 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 3 2 3 0 3 3 3 3 4 0 4 2 3 4 Put Γ = S. A mapping S × Γ × S −→ S is defined as xay = usual product of x, y, a ∈ S. The quasi-order relation on S is given by ⪯ = {(0, 0), (0, 1), (0, 2), (0, 3),(0, 4), (1, 1), (2, 1), (2, 2), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 4)}. Then S forms a quasi-ordered Γ-semigroup. By direct verification one can establish that the sets {1, 2, 3, 4}, {1, 2, 4} and {1} are interior filters in Γ-semigroup S. The following theorem connects the concept of filters and the concept of interior filters in a quasi-ordered Γ-semigroup. Theorem 3.2. Any filter in a quasi-ordered Γ-semigroup S is an interior filter in S. 22 D. A. ROMANO Proof. Let F be a filter in a quasi-ordered Γ-semigroup S. This means that F satisfies the conditions (9), (10) and (11). Let us prove the condition (13) and (14). Let x, y ∈ S and a ∈ Γ be such that xay ∈ F . Then x ∈ F and y ∈ F by (9) and (11) because F is a right and left filter in S. Let u, v, x ∈ S and a, b ∈ Γ be such that uaxbv ∈ F . Then uax ∈ F by (11) since F is a right filter in S. Also, from uax ∈ F it follows x ∈ F by (9) since F is a left filter in S. This proves that F satisfies the condition (14). So, F is an interior filter in S. □ The inverse of the previous theorem is not valid as the following example shows: Example 3.3. Let S = {a, b, c, d} and operations ’·’ defined on S as follows: · a b a a c b c d c b d d d d c d d d d d d d d d Put Γ = S. A mapping S × Γ × S −→ S is defined as xay = usual product of x, y, a ∈ S and ⪯=: {(a, a), (a, b), (a, c), (b, b), (c, c), (d, d)}. Then S forms a quasi-ordered Γsemigroup. By direct verification one can establish that the sets {a, b} and {a, b, c} are interior filters in Γ-semigroup S but they are neither left filters nor right filters in S. The inverse of the Theorem 3.2 can be proved in one special case: A quasi-ordered Γ-semigroup S is called intra-regular ([8]) if holds (IR) (∀x ∈ S)(∀a ∈ Γ)(∃u, v ∈ S)(∃b, c ∈ Γ)(x ⪯ ubxaxcv). It is called left (right) regular if holds (LR) (∀x ∈ S)(∀a ∈ Γ)(∃b ∈ Γ)(∃u ∈ S)(x ⪯ ubxax) ((RR) (∀x ∈ S)(∀a ∈ Γ)(∃b ∈ Γ)(∃v ∈ S)(x ⪯ xaxbv), res.) Theorem 3.3. Let S be an intra-regular quasi-ordered Γ-semigroup. Then any interior filter in S is a filter in S. Proof. Let F be an interior filter in S. This means that F satisfies conditions (10), (13) and (14). (9) and (11) need to be proved. Let x, y ∈ S and a ∈ Γ be such that xay ∈ F . Then for any b ∈ Γ there exist elements u, v ∈ S and c, d ∈ Γ such that xay ⪯ uc(xay)b(xay)dv by (IR). Thus uc(xay)b(xay)dv ∈ F by (10). On the one hand, we have that (ucx)ayb(xaydv) ∈ F gives y ∈ F by (14), while, on the other hand, we have that (ucxay)bxa(ydv) ∈ F gives x ∈ F by (14). Therefore, F is a filter in S. □ We can, also, show that if S is a right (left) regular quasi-order Γ-semigroup, then any interior filter in S is a left (right) filter in S. Theorem 3.4. Let a quasi-ordered Γ-semigroup S is a right regular. Then any interior filter in S is a left filter in S. Proof. Let F be an interior filter in a quasi-ordered right regular Γ-semigroup S. The validity of formula (11) needs to be proven. Let x, y ∈ S and a ∈ Γ be such that xay ∈ F . Then for any b ∈ Γ there exist the elements u ∈ S and c ∈ Γ such that xay ⪯ (xay)b(xay)cv according (RR). Thus (xay)bxa(ycv) ∈ F by (10). Hence x ∈ F according to (14). □ INTERIOR FILTERS IN Γ-SEMIGROUPS 23 However, since in [8], Proposition 6, it is shown that if an ordered Γ-semigroup S is a left (right) regular, then S is intra-regular, the previous theorem immediately follows from Theorem 3.3 if ⪯ is an order relation. If we replace the condition (RR) in a quasi-ordered semigroup S with the condition (B) (∀x, v ∈ S)(∀a ∈ Γ)(x ⪯ xav) the validity of Theorem 3.4 will be preserved. Theorem 3.5. Let S be a quasi-ordered Γ-semigroup that satisfies the condition (B). Then any interior filter in S is a left filter in S. Proof. If a quasi-ordered Γ-semigroup S satisfies condition (B), then it is a right regular Γ-semigroup. Indeed, from (∀x, z ∈ S)(∀a ∈ Γ)(x ⪯ xaz) follows (∀x, v ∈ S)(∀a, b ∈ Γ)(x ⪯ xa(xbv) = (xax)bv) which means that S is a right regular Γ-semigroup. Hence, F is a left filter in S according Theorem 3.4. □ The condition (B) can be weakened. Theorem 3.6. Let S be a quasi-ordered Γ-semigroup that satisfies the condition (C) (∀x ∈ S)(∀a ∈ Γ)(x ⪯ xax). Then any interior filter in S is a left filter in S. Proof. Let S be a quasi-ordered Γ-semigroup that satisfies the condition (C) and let F be an interior filter in S. If x, y ∈ S and a, b ∈ Γ are elements such that xby ∈ F , then xaxby ∈ F by (10) because xby ⪯ (xax)by follows from x ⪯ xax by (6). Thus x ∈ F by (14). □ Our next theorem connects the terms ’interior ideal’ and ’interior filter’ in a quasiordered Γ-semigroup S. Theorem 3.7. If F (̸= S) is an interior filter of a quasi-ordered Γ-semigroup S, then the set F c is an interior ideal of S. In addition to the previous one, if F satisfies the condition (A), then the ideal F c satisfies the condition (P). Proof. It should be shown that the set F c satisfies the following conditions (a), (7), (12) and (8). The condition F ̸= S ensures that the set F c is inhabited. Let x, y ∈ S and a ∈ Γ be arbitrary elements such that x ∈ / F and y ∈ / F . Then xay ∈ / F or xay ∈ F . The second option xay ∈ F would give x ∈ F ∨ y ∈ F by (13), which contradicts the assumptions. Therefore, it must be xay ∈ / F. Let x, u, v ∈ S and a, b ∈ Γ be arbitrary elements such that x ∈ / F . Then uaxbv ∈ / or uaxbv ∈ F . The second option uaxbv ∈ F would give x ∈ F by (14), which contradicts the assumption. Therefore, it must be uaxbv ∈ / F. Let u, v ∈ S be such that u ⪯ v and v ∈ F c . If u ∈ F , we would have v ∈ F according to (10), which is contrary to the assumption v ∈ / F . So, it has to be u ∈ F c . □ The family Intf(S) of all internal filters of a quasi-ordered Γ-semigroup S is not empty because S ∈ Intf(S) and ∅ ∈ Intf(S). Additionally, the following applies: Theorem 3.8. The family Intf(S) of all interior filters of a quasi-ordered Γ-semigroup S forms a complete lattice. Proof. Let {Fi }i∈I be a non-empty family of interior filters of a quasi-ordered Γ-semigroup S. 24 D. A. ROMANO S (a) Let x, y ∈ S and a ∈ Γ be such that S xay ∈ i∈I Fi . ThenSthere exists an index k ∈ I such that xay S ∈ Fk . Thus x ∈ Fk ⊆ i∈I Fi or y ∈ Fk ⊆ i∈I Fi by (13). This means that the set i∈I Fi satisfies the condition (13).S Let u, v, x ∈ S and a ∈ Γ be such that uaxbvS ∈ i∈I Fi . Then there exists an index k ∈ S I such that uaxbv ∈ Fk . Thus x ∈ Fk ⊆ i∈I Fi by (14). This means that the set i∈I Fi satisfies the condition (14). S Let u, v ∈ S be such that u ∈ Si∈I Fi and u ⪯ v. Then there exists an index k ∈ I such that u ∈ Fk . Thus v ∈ S Fk ⊆ i∈I Fi . We conclude that the set i∈I Fi is an interior filter of S. (b) LetT X be the family of all interior filters of the quasi-ordered Γ-semigroupTS contained in i∈I Fi . Then ∪X is the maximal interior filter of S contained in i∈I fi , according to (a). S (c) If we put ⊔i∈I fi = i∈I Fi and ⊓i∈I Fi = ∪X, then (Intf(S), ⊔, ⊓) a is a complete lattice. □ The previous theorem supports our commitment expressed in the determination of the concept of filters in quasi-ordered Γ-semigroups: The union of sub-semigroups {Fi }i∈I of S does not have to be a sub-semigroup of S, in the general case. Corollary 3.9. For any subset X of a quasi-ordered Γ-semigroup S there is the maximal interior filter contained in X. Proof. The proof of this Corollary is obtained directly from part (b) of the evidence in the previous theorem. □ Corollary 3.10. For any element x ∈ S there is the maximal interior filter Fx in a quasiordered Γ-semigroup S such that x ∈ / Fx . Proof. One should take X = {u ∈ S : u ̸= x} and apply the previous corollary. □ 3.2. Weak-interior filters. This subsection is devoted to designing the concept of weakinterior filters of a quasi-ordered Γ-semigroup and recognizing its fundamental properties. The concept of weak interior ideals in Γ-semigroup was introduced in [11] as follows: - A non-empty subset J of a Γ-semigroup S is said to be left weak interior ideal of S if J is a Γ-sub-semigroup of S and SΓJΓJ ⊆ J. The last inclusion should be understood in the following sense (15) (∀u, x, y ∈ S)(∀a, b ∈ Γ)((x ∈ J ∧ y ∈ J) =⇒ uaxby ∈ J). -A non-empty subset J of a Γ-semigroup S is said to be right weak-interior ideal of S if J is a Γ-sub-semigroup of S and JΓJΓS ⊆ J. The last inclusion should be understood in the following sense (16) (∀x, y, v ∈ S)(∀a, b ∈ Γ)((x ∈ J ∧ y ∈ J) =⇒ xaybv ∈ J). - A non-empty subset J of a Γ-semigroup S is said to be weak interior ideal of S if J is a left weak interior ideal and a right weak interior ideal of S. A weak interior ideal of a Γ-semigroup S need not be an interior ideal of Γ-semigroup S (see, for example, [11], Remark 3.1 and Example 3.1). Dual of concept of (left, right) weak interior ideals in a quasi-ordered Γ-semigroup is created in the following way: Definition 3.4. Let F be a subset of a quasi-ordered Γ-semigroup S. - F is a left weak interior filter of S if (10) and (13) are valid and the following holds (17) (∀x, y, u ∈ S)(∀a, b ∈ Γ)(uaxby ∈ F =⇒ (x ∈ F ∨ y ∈ F )); INTERIOR FILTERS IN Γ-SEMIGROUPS 25 - F is a right weak interior filter of S if (10) and (13) are valid and the following holds (18) (∀x, y, v ∈ S)(∀a, b ∈ Γ)(xaybv ∈ K =⇒ (x ∈ K ∨ y ∈ K)); - F is a weak interior filter of S if F is a left weak interior filter and a right weak interior filter of S. Remark. The requirement for a weak-interior filter to be a sub-semigroup of a quasiordered Γ-semigroup is omitted similarly as in the case of interior filter determination. Proposition 3.11. Let J be a left weak interior ideal of a quasi-ordered Γ-semigroup S. Then, the set S \ J is a left weak interior filter of S. If the left weak interior ideal J of S satisfies the condition (P), then the weak interior filter S\J satisfies the condition (A). Proof. Let J be a left weak interior ideal of a quasi-ordered Γ-semigroup S. This means that J satisfies the conditions (a), (7), (8) and (15). (10), (13) and (17) need to be proved. (13) is the contraposition of (7) so it is a valid formula. (17) is the contraposition of (15). Thus, (17) is a valid formula. (10) is obtained from (8) as shown in Proposition 3.1. □   0 b Example 3.5. Let Q be a field of rational numbers, S := { | b, d ∈ Q} be 0 d the semigroup of matrices over the filed Q and Γ = S. The ternary operation in S over Q is the standard multiplication of matrices. Then S is an ordered Γ-semigroup. Then 0 0 F =: { | d ∈ Q ∧ d ̸= 0} is a left weak interior filter of the Γ-semigroup S and 0 d F is neither a left filter nor a right filter, not a weak interior filter and not an interior filter of the Γ-semigroup S. The following theorem proves that the concept of left weak interior filter in a quasiordered Γ-semigroup is well determined. Theorem 3.12. Let F (̸= S) be a left weak interior filter of a Γ-semigroup with apartness S. The the set F c is a left weak interior ideal in S. In addition to the previous one, if F satisfies the condition (A), then the ideal F c satisfies the condition (P). Proof. Let F be a left weak interior filter in a quasi-ordered Γ-semigroup. This means that F satisfies conditions (10), (13) and (17). It should be shown that the set F c satisfies the following conditions (a), (7), (15) and (8). The condition F ̸= S ensures that the set F c is inhabited. So, F c ̸= ∅. Let a ∈ Γ and x, y ∈ S be arbitrary elements such that x ∈ / F and y ∈ / F . Then xay ∈ F or xay ∈ / F . The first option would give x ∈ F or y ∈ F by (13) which is contrary to the assumptions x ∈ / F and y ∈ / F . Therefore, it must be xay ∈ / F . This means that F satisfies the condition (7). Let a, b ∈ Γ and u, x, y ∈ S be such that x ∈ / F and y ∈ / F . Then uaxby ∈ F or uaxby ∈ / F . The first option would give x ∈ F or y ∈ F by (17) which is contrary to the assumptions x ∈ / F and y ∈ / F . So, it has to be uaxby ∈ F c . □ The connection between the concept of right filters and the concept of left weak interior filters in a quasi-ordered Γ-semigroup is described in the following theorem. Theorem 3.13. Every right filter of a quasi-ordered Γ-semigroup S is a left weak interior filter of S. Proof. Let F be a right filter of a quasi-ordered Γ-semigroup S. This means that (9) and (10) are valid formulas. It should be proved that (10), (13) and (17) are valid formulas. Since (9) or (11) implies (13), we only need to prove (17). 26 D. A. ROMANO Let u, x, y ∈ S and a, b ∈ Γ such that (uxa)by = uaxby ∈ F . Then y ∈ F because F is a right filter of S. So, F is a left weak interior filter of S. □ Analogous to the previous one, it can be proved: Theorem 3.14. Any left filter of a quasi-ordered Γ-semigroup is a right weak interior filter in S. Therefore: Theorem 3.15. Any filter in a quasi-ordered Γ-semigroup S is a weak interior filter in S. The following theorem can be considered as the inverse of the Theorem 3.13. Theorem 3.16. Let a quasi-ordered Γ-semigroup S satisfy the condition (C). Then any left weak interior filter in S is a right filter in S. Proof. Let F be a left weak interior filter in S. This means that (10), (13) and (17) are valid. (10) and (11) need to be proved. Let x, y ∈ S and a ∈ Γ be arbitrary element such that xay ∈ F . Since S satisfies the condition (C), we have y ⪯ yby for any b ∈ Γ. Then xay ⪯ xa(yby) by (6). Thus xayby ∈ F by (10). Hence y ∈ F by (17). This means that F is a right filter in S. □ Analogous to the previous one, it can be proved: Theorem 3.17. Any right weak interior filter in a quasi-ordered Γ-semigroup S is a left filter in S, if S satisfies the condition (C). Therefore: Theorem 3.18. If a quasi-ordered Γ-semigroup S satisfies the condition (C), then any weak interior filter in S is a filter in S. The relationship between interior filter and weak interior filter in a quasi-ordered Γsemigroup is described by the following theorem. Theorem 3.19. Every interior filter of a quasi-ordered Γ-semigroup S is a left weak interior filter of S. Proof. Let F be an interior filter of a quasi-ordered Γ-semigroup S. This means that F satisfies (10), (13) and (14). It only needs to be proven (17). Let u, x, y ∈ S and a, b ∈ Γ be arbitrary elements such that uaxby ∈ F . Then x ∈ F because F is an interior filter of S. Thus x ∈ F ∨ y ∈ F which means that F is a left weak interior filter in S. □ Of course, analogous to the previous one, it can be shown: Theorem 3.20. Any interior filter in a quasi-ordered Γ-semigroup S is a right weakinterior filter in S. Therefore: Theorem 3.21. Any interior filter in a quasi-ordered Γ-semigroup S is a weak interior filter in S. Combining Theorem 3.18 and Theorem 3.15, we obtain the inverse of Theorem 3.21 Theorem 3.22. Let a quasi-ordered Γ-semigroup S satisfy the condition (C). Then any weak interior filter in S is an interior filter in S. INTERIOR FILTERS IN Γ-SEMIGROUPS 27 The family Wl Intf(S) of all left weak interior filters of a quasi-ordered Γ-semigroup S is not empty because S ∈ Wl Intf(S) and ∅ ∈ Wl Intf(S). Actually: Theorem 3.23. The family Wl Intf(S) of all left weak interior filters of a Γ-semigroup S forms a complete lattice. Proof. Let {Fi }i∈I be a family of left weak interior filters of a quasi-ordered Γ-semigroup S. S (a) Let x, y ∈ S and a ∈ Γ be such thatS xay ∈ i∈I Fi . ThenSthere exists an index k ∈ I such that xay S ∈ Fk . Thus x ∈ Fk ⊆ i∈I Ki or y ∈ Fk ⊆ i∈I Fi by (13). This means that the set i∈I Fi satisfies the condition (13).S Let u, x, y ∈ S and a, b ∈ Γ be such that uaxby Sexists an index S ∈ i∈I Fi . Then there k ∈ I such that uaxby S ∈ Fk . Thus x ∈ Fk ⊆ i∈I Fi or y ∈ Fk ⊆ i∈I Fi by (17). (17). This shows that the set i∈I Fi satisfies the condition S F . Then there exists Let x, y ∈ S be such that x ⪯ y and x ∈ i i∈I S an index k ∈ I S such that x ∈ Fk . Thus y ∈ Fk ⊆ i∈I Fi by (10). So, the set i∈I Fi satisfies the condition (10). S Hence, we conclude that the set i∈I Fi is a left weak interior filter of S. (b) Let X be the family of all left weak interior filters of Γ-semigroup S contained T T in i∈I Fi . Then ∪X is the maximal left weak interior filter of S contained in i∈I Fi , according to (a). S (c) If we put ⊔i∈I Fi = i∈I Fi and ⊓i∈I Fi = ∪X, then (Wl Intf(S), ⊔, ⊓) a is a complete lattice. □ Corollary 3.24. For any subset X of a quasi-ordered Γ-semigroup S there is the maximal left weak interior filter contained in X. Proof. The proof of this Corollary is obtained directly from part (b) of the evidence in the previous theorem. □ Corollary 3.25. For any element x ∈ S there is the maximal left weak interior filter Fx in a quasi-ordered Γ-semigroup S such that x ∈ / Fx . Proof. One should take X = {u ∈ S : u ̸= x} and apply the previous corollary. □ Without major difficulties, the previous claims concerning the left weak interior filters can be transformed into the claims concerning the right weak interior filters. 3.3. Quasi-interior filters. In this subsection, firstly, we will recall the determination of the notions of left, right, and two-sided quasi-interior ideals in a quasi-ordered Γsemigroups. - A non-empty subset J of a quasi-ordered Γ-semigroup S is said to be left quasi-interior ideal of S, if J is a Γ-sub-semigroup of S and (8) and SΓJΓSΓJ ⊆ J are hold. Thus, the following formulas (a) J ̸= ∅, (7) (∀x, y ∈ S)(∀a ∈ Γ)((x ∈ J ∧ y ∈ J) =⇒ xay ∈ J), (8) (∀x, yS)((x ⪯ y ∧ y ∈ J) =⇒ x ∈ J), and (19) (∀u, v, x, y ∈ S)(∀a, b, c ∈ Γ)((y ∈ J ∧ u ∈ J) =⇒ uaxbvcy ∈ J) are valid formulas in a Γ-semigroup S. - A non-empty subset J of a quasi-ordered Γ-semigroup S is said to be right quasi-interior 28 D. A. ROMANO ideal of S, if J is a Γ-sub-semigroup of S and (8) and JΓSΓJΓS ⊆ J are hold. Thus, the following formulas (a) J ̸= ∅, (7) (∀x, y ∈ S)(∀a ∈ Γ)((x ∈ J ∧ y ∈ J) =⇒ xay ∈ J), (8) (∀x, yS)((x ⪯ y ∧ y ∈ J) =⇒ x ∈ J), and (20) (∀x, y, u, v ∈ S)(∀a, b, c ∈ Γ)((x ∈ J ∧ y ∈ J) =⇒ xaubycv ∈ J) are valid formulas in S. - A non-empty subset J of a quasi-ordered Γ-semigroup S is said to be quasi-interior ideal of S, if J is a left quasi-interior ideal and a right quasi-interior ideal of S. In this subsection, we introduce the notion of left (right) quasi-interior filters as a generalization of ordered interior filters of a quasi-ordered Γ-semigroup and study its properties. Definition 3.6. Let S be a quasi-ordered Γ-semigroup. - A subset F of S is said to be left quasi-interior filter of S if (10) and (13) are valid and the following holds (21) (∀u, v, x, y ∈ S)(∀a, b, c ∈ Γ)(uaxbvcy ∈ K =⇒ (x ∈ F ∨ y ∈ F )), - A subset F of S is said to be right quasi-interior filter of S if (10) and (13) are valid and the following holds (22) (∀x, y, u, vS)(∀a, b, c ∈ Γ)(xaubycv ∈ K =⇒ (x ∈ F ∨ y ∈ F )), - A subset F of S is said to be quasi-interior filter of S if it is both a left quasi-interior filter and a right quasi-interior filter of S. Remark. As in the previous two cases of interior filters and weak interior filters in quasiordered Γ-semigroups here as well, in determining the concept of quasi-interior filters of a quasi-ordered Γ-semigroup S, we omit the requirement that this filter will be a subsemigroup of S. However, we will not avoid considering this class of filters if they meet this additional condition.   a 0 Example 3.7. Let Q be a field of rational numbers, S := { | a, b ∈ Q} be b 0 the semigroup of matrices over the filed Q and Γ = S. The ternary operation in S over Q is the standard multiplication of matrices. Then S is an ordered Γ-semigroup. Then 0 0 F =: { | d ∈ Q ∧ d ̸= 0} is a right quasi-interior filter of S. d 0 Theorem 3.26. Let S be a quasi-ordered Γ-semigroup. If F (̸= S) is a left quasi-interior filter of S, then the set F c is a left quasi-interior ideal of S. In addition to the previous one, if F satisfies the condition (A), then the ideal F c satisfies the condition (P). Proof. That the set F c is non-empty follows from the condition F ̸= S. Let x, y ∈ S and a ∈ Γ be arbitrary element such that x ∈ / F and y ∈ / F . Then xay ∈ F or xay ∈ / F . The first option would give x ∈ F or y ∈ F by (13) which is contrary to assumptions. Therefore, it must be xay ∈ / F . This means xay ∈ F c . Let x, y ∈ S be such that x ⪯ y and y ∈ F c . Suppose x ∈ F . Then it would be y ∈ F by (10) which is contrary to the hypothesis y ∈ / F . So it has to be x ∈ / F . This proves the validity of formula (8). Let x, y, u, v ∈ S and a, b, c ∈ Γ be arbitrary elements such that x ∈ / F and y ∈ / F. Then uaxbycv ∈ F or xaybzcu ∈ / F . The first option uaxbycv ∈ F would give x ∈ F or y ∈ F by (21) because F is a left quasi-interior gilter of S, which contradicts the hypotheses ∈ / F and yF . So, it must be xaybzcu ∈ / F . These prove that (19) holds for F c. □ INTERIOR FILTERS IN Γ-SEMIGROUPS 29 Analogous to the previous, it can be proved: Theorem 3.27. Let S be a quasi-ordered Γ-semigroup. If F (̸= S) is a right quasi-interior filter of S, then the set F c is a right quasi-interior ideal of S. The notion of quasi-interior filters is a generalization of the notion of filters in quasiordered Γ-semigroups as shown by the following theorem. Theorem 3.28. Every interior filter of a quasi-ordered Γ-semigroup S is a left quasiinterior filter of S. Proof. Let F ne an interior filter of a S. This means that F satisfies the conditions (10), (13) and (14). That F is a left quasi-interior filter is sufficient to prove (21). Let x, y, u, v ∈ S and a, b, c ∈ Γ be such that uaxb(vcy) = uaxbvcy ∈ F . Then x ∈ F by (14) because vcy ∈ S and F is an interior filter of S. Thus x ∈ F ∨ y ∈ F . So, the set F is a left quasi interior filter of S. □ The reverse of the previous theorem can be proved if the quasi-ordered Γ-semigroup S satisfies one additional condition. Theorem 3.29. Suppose that a quasi-ordered Γ-semigroup S satisfies one additional condition: (B) For every elements x, v ∈ S and a ∈ Γ the following holds x ⪯ xav. Then the interior filters and the left quasi-interior filters in S coincide. Proof. Suppose that a quasi-ordered Γ-semigroup S satisfies the condition (C) and let F be a left quasi-interior filter in S. This means that F satisfies the conditions (10), (13) and (21). It needs to be proven (14). Let x, u, v ∈ S and a, b, c ∈ Γ be arbitrary elements such that uaxbv ∈ F . On the other hand, we have uaxbv ⪯ uaxbvcx according to (B). Hence uaxbvcx inF by (10). Thus x ∈ F according to (21). So, the set F is an interior filter in S. □ Since any filter in a quasi-ordered Γ-semigroup S is an interior filter in S, according to Theorem 3.2, immediately from the Theorem 3.28 it follows: Corollary 3.30. Every filter of a quasi-ordered Γ-semigroup S is a left quasi-interior filter of S. Theorem 3.31. The family Ql intf(S) of all left quasi-interior filters of a quasi-ordered Γ-semigroup S forms a complete lattice. Proof. Let {Fi }i∈I be a family of left quasi-interior filters of a quasi-ordered Γ-semigroup S. S (a) Let x, y ∈ S and a ∈ Γ be arbitrary elements such that S there S xay ∈ i∈I Fi . Then F or y ∈ F ⊆ exists an index k ∈ I such that xay ∈ F . Thus x ∈ F ⊆ i k k k i∈I Fi i∈I S the condition (13). by (13). This means that the set i∈I Fi satisfies S Let x, y ∈ S be such that x ⪯ y and S an index k ∈ I S x ∈ i∈I Fi . Then there exists such that x ∈ Fk . Thus y ∈ Fk ⊆ i∈I Fi by (10). So, the set i∈I Fi satisfies the condition (10). S Let x, y, u, v ∈ S and a, b, c ∈ Γ be arbitrary elements such that uaxbvcy ∈ S i∈I Fi . Then there S exists an index l ∈ I such that uaxbvcy ∈ Fk . Thus x ∈ Fk ⊆ i∈I Fi or y ∈ Fk ⊆ S i∈I fi because Fk is a left quasi-interior filter of S. Hence, i∈I Fi is a left quasi-interior filter of S. 30 D. A. ROMANO T (b) Let X be the family of all left quasi-interior filters contained in i∈I Fi . Then ∪X is the maximal left quasi-interior S filter contained in X, according to (a) in this proof. (c) If we put ⊔i∈I Fi = i∈I Fi and ⊓i∈I Fi = ∪X, then (Ql intf(S), ⊔, ⊓) is a complete lattice. □ Analogous to the previous, it can be proved: Theorem 3.32. The family Qr intf(S) of all right quasi-interior filters of a quasi-ordered Γ-semigroup S forms a complete lattice. 4. C ONCLUSIONS Various types of filters in ordered Gamma -semigroups are the subject of studies by several authors in the second decade of this century. One of the main specifics of determining the concept of filter F in an ordered Γ-semigroup S is that the filter F must be Γ-subsemigroup of S. In this text, the concepts of interior Γ-filters, weak interior Γ-filters and quasi-interior Γ-filters in a quasi-ordered Γ-semigroup are introduced and analyzed. 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Algebra and its Applications (pp. 225-–239), Springer Proceedings in Mathematics and Statistics, vol 174. Springer, Singapore. 2016 [23] G. Szasz, Interior ideals in semigroups, In: Notes on Semigroups, Karl Marx Univ. Econ., Dept. Math. Budapest, No 5(1977), 1–7. [24] G. Szasz, Remark on interior ideals of semigroups, Studia Scient. Math. Hung., 16(1981), 61–63. [25] S. Tarsuslu (Yılmaz) and G. Çuvalcıoğlu, Intuitionistic fuzzy quasi-interior ideals of semigroups, Notes on Intuitionistic Fuzzy Sets, 27(4)(2021), 36-–43. D. A. ROMANO I NTERNATIONAL M ATHEMATICAL V IRTUAL I NSTITUTE , KORDUNA ŠKA S TREET 6, 78000 BANJA L UKA , B OSNIA AND H ERZEGOVINA Email address: [email protected] ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 5, Number 1 (2022), 32-37 ISSN: 2582-0818 © http://www.technoskypub.com FUZZY MINIMAL AND MAXIMAL e-OPEN SETS M. SANKARI∗ , S. DURAI RAJ AND C. MURUGESAN A BSTRACT. The aim of this article is to introduce fuzzy minimal e-open and fuzzy maximal e-open sets in fuzzy topological space. Further, we investigate some properties with these new sets. 1. I NTRODUCTION AND P RELIMINARIES Zadeh[7] established fuzzy set in 1965 and Chang[2] introduced fuzzy topology in 1968. Ittanagi and Wali[3] instigated the notions of fuzzy maximal and minimal open sets. The notion of fuzzy e-open set introduced by Seenivasan and Kamala[4]. In this paper , we introduce fuzzy minimal e-open and fuzzy maximal e-open sets. Further some of their related results investigated. The following terminologies “fuzzy e-open,fuzzy minimal e-open and fuzzy maximal e-open respectively abbreviated as Fe-O, FMIe-O and FMAe-O. ” Definition 1.1. A fuzzy subset ξ of a space X is called fuzzy regular open [1] (resp. fuzzy regular closed) if ξ =Int(Cl(ξ)) (resp.ξ =Cl(Int(ξ))). The fuzzy δ-interior of a fuzzy subset ξ of X is the union of all fuzzy regular open sets contained in ξ. A fuzzy subset ξ is called fuzzy δ-open [5] if ξ = Intδ (ξ). The complement of fuzzy δ-open set is called fuzzy δ-closed (i.e, ξ = Clδ (ξ)). Definition 1.2. A proper nonempty fuzzy open set E of Xis said to be a FMIO[3] set if (i)E and 0X are only fuzzy open sets contained in E. (ii) 1X and E are only fuzzy open sets containing E. Definition 1.3. A fuzzy subset ζ of a fts X is called Fe-O [4] if ζ ≤cl(intδ ζ)∪int(clδ ζ) and fuzzy e-closed set if ζ ≥cl(intδ ζ)∩int(clδ ζ). Let U be a fuzzy subset of a fts X. Then the fuzzy e-closure and e-interior [4] of U T µ : µ ≥ U, µ is fuzzy e-closed in X and eInt(U) = are defined as follows: eCl(U) = S {λ ≤ U, λ is Fe-O in X}. 2010 Mathematics Subject Classification. 54A40, 03E72. Key words and phrases. Fuzzy regular open set; Fuzzy δ-open set; Fuzzy e-open set; Fuzzy minimal e-open set;Fuzzy maximal e-open set. Received: March 18, 2022. Accepted: May 25, 2022. Published: June 30, 2022. *[email protected]. 32 FUZZY MINIMAL AND MAXIMAL e-OPEN SETS 33 2. FUZZY MINIMAL e-OPEN SETS Definition 2.1. A proper nonzero Fe-O set E in fts (X, τ) is said to be FMIe-O iff Fe-O set contained in E is 0X or E. Lemma 2.1. Let (X, τ) be a fts. (i) If E1 is FMIe-O and E2 is Fe-O in X, then E1 ∩ E2 = 0X or E1 ⊂ E2 . (ii) If E1 and E2 are FMIe-O , then E1 ∩ E2 = 0X or E1 = E2 . Proof. (i) Let us assume that E2 is Fe-O in X such that E1 ∩ E2 , 0X . Since E1 is FMIe-O , and E1 ∩ E2 ⊂ E1 , then E1 ∩ E2 = E1 implies that E1 ⊂ E2 . (ii) Suppose that E1 ∩ E2 , 0X , then clearly from(ii), E1 ⊂ E2 and E2 ⊂ E1 as E1 and E2 are FMIe-O . Hence E1 = E2 . □ S Ei , then E = E j for Theorem 2.2. Let E and Ei are FMIe-O sets for any i ∈ M. If E ⊆ i∈M any j ∈ M. S Proof. Suppose E ⊆ i∈M Ei , then E = E ∩ S i∈M ! Ei = S (E ∩ Ei ). By deploying lemma i∈M 2.1(ii), E ∩ Ei = 0X or E = Ei as E and Ei are FMIe-O sets. If E ∩ Ei = 0X , then E = 0X which contradicts that E is a FMIe-O set. Hence if E ∩ Ei , 0X then E = E j for any j ∈ M. □ ! S E i = 0X Theorem 2.3. If E and Ei are FMIe-O sets for any i ∈ M and E , Ei , then E∩ i∈M for any i ∈ M. ! S Ei , 0X , then E ∩ Ei , 0X for any i ∈ M. By deploying lemma 2.1(ii), ! S □ E i = 0X . E = Ei contradictory to E , Ei . Hence E ∩ Proof. Let E ∩ i∈M i∈M Theorem 2.4. If Ei is !a FMIe-O for any i ∈ M (|M| ≥ 2) and Ei , E j for any distinct S i, j ∈ M, then Ei ∩ E j = 0X for any j ∈ M. i∈M\{ j} ! S   Ei ∩ E j , 0X ⇒ (Ei ∩ E j ) , 0X . By lemma i∈M\{ j} i∈M\{ j} ! S 2.1(ii), Ei = E j , a contradiction. Hence □ Ei ∩ E j = 0X for any j ∈ M. Proof. Let S Ei ∩ E j , 0X . Then i∈M\{ j} Theorem 2.5. If Ei is a FMIe-O for any i ∈ M, (|M|!≥ 2) and !Ei , E j for any distinct S T S i, j ∈ M. If K is a proper fuzzy set of M, then h s = 0X . Ei s∈K i∈M\K Proof. Let S i∈M\K Ei ! T S s∈K ! E s , 0X . It implies that S (Ei ∩ E s ) , 0X for i ∈ M\K and s ∈ K implies that Ei ∩ E s , 0X for some ! s ∈ K. By lemma2.1(ii), Ei = h s , ! i ∈ M and T S S □ E s = 0X . which is a contradiction. Hence Ei i∈M\K s∈K Theorem 2.6. If Ei is a FMIe-O for any i ∈ "M such that # "Ei , E# j for any distinct i, j ∈ M. S S E k = 0X . If S is a proper nonzero fuzzy set of M, then Ei ∩ i∈M\k k∈S 34 M. SANKARI, S. DURAI RAJ AND C. MURUGESAN Proof. Assume that ∪ [Ei ∩ Ek ] , 0X for i ∈ M\k,k ∈ S . Clearly, for some i ∈ M,k ∈ S we □ have [Ei ∩ Ek ] , 0X . By deploying lemma 2.1(ii) Ei = Ek , a contradiction. Theorem 2.7. If Ei and Ek are FMIe-O " sets#for"any i #∈ M and k ∈ S and if ∃ an n ∈ S S S Ei . En 1 such that Ei , En for any i ∈ M, then n∈K i∈M Proof. Assume that ∃ an n ∈ S such that Ei , En for any i ∈ M, then # " S Ei for some n ∈ K. ⇒ En ⊂ " S n∈K # # " S Ei . En ⊂ i∈M i∈M ⇒ Ei , En for any i ∈ M, by theorem 2.2, which is a contradiction. Hence # " S Ei . i∈M " S n∈K # En 1 □ Theorem 2.8. # If" Ei is #a FMIe-O for any i ∈ M such that Ei , E j for any distinct i, j ∈ M, " S S Ei for any proper nonzero subset K of M. Ek ⫋ then k∈K i∈M Proof. Let m ∈ M\K, then " {Em#|m ∈ M\K} of FMIe-O # Em is a FMIe-O set of the family " S S S S [Em ∩ Ei ] = Em . [Em ∩ Ek ] = 0X . Also Em ∩ Ei = Ek = sets. Clearly Em ∩ i∈M i∈M k∈K # " k∈K # " S S Ei , then Em = 0X which is a contradiction that Em is a FMIe-O set. If Ek = i∈M k∈K # # " " S S □ Ei . Ek ⫋ Hence k∈K i∈M Theorem 2.9. If Ei is a FMIe-O set for any i ∈ M such that Ei , E j for any distinct i, j ∈ M,"then #c S (i) E j ⊂ Ei for some j ∈ M. S i∈M\{ j} (ii) Ei , 1X for any j ∈ M. i∈M\{ j} Proof. (i) By hypothesis, E# i , E j for any distinct i, j ∈ M. " S Ei ∩ E j = 0X which is true for any j ∈ M. By theorem 2.3, i∈M i Sh E i ∩ E j = 0X ⇒ i∈M ⇒ Ei ∩ E j = 0X (By Lemma 2.1(ii)) ⇒ Ei ⊂ E j c S ⇒ Ei ⊂ E j c . Hence proved. i∈M\{ j} S (ii) Let j ∈ M such that E i = 1X i∈M\{ j} ⇒ E i = 0X S ⇒ Ei is not a FMIe-O set, a contradiction. Hence Ei , 1X for any j ∈ M. i∈M\{ j} □ Corollary 2.10. If Ei is a FMIe-O set for any i ∈ M such that Ei , E j for any distinct i, j ∈ M, then Ei ∪ E j , 1X for any distinct i, j ∈ M. Proof. Similar to that of “Theorem 2.9(ii).” □ FUZZY MINIMAL AND MAXIMAL e-OPEN SETS 35 Theorem 2.11. If E" i is a #FMIe-O sets# for any i ∈ M such that Ei , E j for any distinct " c S S Ei for any j ∈ M. Ei ∩ i, j ∈ M, then E j = i∈M i∈M\{ j} #c # " #c " # " S S S T S S Ei Ei E j ∩ Ei = Ei Proof. For any j ∈ M ⇒ i∈M\{ j} i∈M\{ j} " ! !i∈M " !c # i∈M\{ j} c# S S S S = Ej ∩ Ei ∩ Ei Ei " i∈M\{ j} i∈M\{ j} i∈M\{ j} = 0X ∪ E j = E j for any j ∈ M. □ Proposition 2.12. Let G be a FMIe-O set. If xα ∈ G, then G ⊂ G1 for any fuzzy open neighbourhood G1 of xα . Proof. Let G1 be an Fe-O neighbourhood of xα such that G 1 G1 . Clearly G ∩ G1 is an Fe-O such that G ∩ G1 ⊊ G and G ∩ G1 , 0X . This implies that G is a FMIe-O set which a contradiction. □ Proposition 2.13. Let G be a FMIe-O set in a fuzzy topological space (X, τ). Then G = T {G1 : G1 fuzzy e-open neighbourhood of xα for any xα ∈ G} Proof. By deploying proposition 2.12 and as G is an Fe-O neighbourhood of xα , we have T G⊂ G1 : G1 fuzzy e-open neighbourhood of xα ⊂ G . This completes the proof. □ Theorem 2.14. Let G be a FMIe-O set. Then the following conditions are equivalent. (i) G is FMIe-O set. (ii)G ⊂ eCl(K) for any nonzero fuzzy subset K of G. (iii)eCl(G) = eCl(K) for any nonzero fuzzy subset K of G. Proof. (i) ⇒ (ii): By deploying “proposition 2.12”for any xα ∈ G and Fe-O neighbourhood M of xα , we have K = (G ∩ K) ⊂ (M ∩ K) for any proper nonzero fuzzy subset K ⊂ G. Therefore, we have (M ∩ K) , 0X and xα ∈ eCl(K). It follows that G ⊂ eCl(K). (ii) ⇒ (iii): For any proper fuzzy subset K of G, eCl(G) ⊂ eCl(K). Also by(ii) eCl(G) ⊂ eCl (eCl(K)) = eCl(K). Hence proved. (iii) ⇒ (i): Let us assume that G is not FMIe-O . Then there exists a proper Fe-O D such that D ⊂ G.Then ∃ yα ∈ G such that yα < D. Then eCl({yα }) ∈ Dc implies that eCl({yα }) , eCl(G), a contradiction. This completes our proof. □ 3. FUZZY MAXIMAL e-OPEN SETS AND ITS PROPERTIES Definition 3.1. A proper nonzero Fe-O set F of a fts (X, τ) is said to FMAe-O if any Fe-O set which contains F is either F or 1X . 0.5 Example 3.2. [6] Let X = {a, b, c, d}. Then fuzzy sets γ1 = a1 + b0 + 0c + d0 ; γ2 = 0.5 a + b + 0.5 0.5 0.4 0 0 0 1 0.5 0.5 0.5 0.5 0 0 0 c + d ; γ3 = a + b + c + d ; γ4 = a + b + c + d and γ5 = a + b + c + d ;are defined as follows: Consider the fuzzy topology τ = {0X , γ1 , γ2 , γ3 , γ4 , γ5 , 1X }. Here γ3 is FMIe-O and γ4 FMAe-O set. Lemma 3.1. Let (X, τ) be a fts. Then (i) If F1 is a FMAe-O and F2 is Fe-O in X, then F1 ∪ F2 = 1X or F2 ⊂ F1 . (ii) If F1 and F3 are FMAe-O sets, then either F1 ∪ F3 = 1X or F1 = F3 . 36 M. SANKARI, S. DURAI RAJ AND C. MURUGESAN Proof. (i) Assume that F2 1 F1 . Clearly, F1 ⊂ (F1 ∪ F2 ) a contrary to F1 is a FMAe-O set if F1 ∪ F2 , 1X . Hence, F1 ∪ F2 = 1X . (ii) Let F1 and F3 are FMAe-O sets. Then from(i) F3 ⊂ F1 and F1 ⊂ F3 implies that F1 = F3 . □ Theorem 3.2. If F1 , F2 and F3 are FMAe-O sets such that F1 , F2 and (F1 ∩ F2 ) ⊂ F3 , then either F1 = F3 or F2 = F3 . Proof. Suppose that F1 , F2 and F3 are FMAe-O sets with F1 , F2 , (F1 ∩ F2 ) ⊂ F3 and if F1 , F3 , then (F2 ∩ F3 ) = F2 ∩ (F3 ∩ 1X ) = F2 ∩ [F3 ∩ (F1 ∪ F2 )], by lemma 3.1(ii) = F2 ∩ [(F3 ∩ F1 ) ∪ (F3 ∩ F2 )] = [F2 ∩ F3 ∩ F1 ] ∪ [F2 ∩ F3 ∩ F2 ] = [F2 ∩ F1 ] ∪ [F2 ∩ F3 ] = F2 ∩ [F1 ∪ F3 ] = F 2 ∩ 1X = F2 (F2 ∩ F3 ) = F2 ⇒ F2 ⊂ F3 . As F2 and F3 are FMAe-O sets, F3 ⊂ F2 . Hence F2 = F3 . □ Theorem 3.3. For any distinct FMAe-O sets F1 , F2 , F3 [F1 ∩ F2 ] 1 [F1 ∩ F3 ]. Proof. Consider [F1 ∩ F2 ] ⊂ [F1 ∩ F3 ] for any distinct FMAe-O sets F1 , F2 and F3 . Then [F1 ∩ F2 ] ∪ [F2 ∩ F3 ] ⊂ [F1 ∩ F3 ] ∪ [F2 ∩ F3 ] = [F1 ∪ F3 ] ∩ F2 ⊂ [F1 ∪ F2 ] ∩ F3 = 1X ∩ F 2 ⊂ 1X ∩ F 3 = F2 is contained in F3 a contradiction to F1 , F2 and F3 are distinct . Hence [F1 ∩ F2 ] 1 [F1 ∩ F3 ]. □ Remark. Proofs of “Theorem 3.4, Corollary 3.5, Theorem 3.6 and Theorem 3.7” are similar to proofs of “Theorem 2.9, Corollary 2.10, Theorem 2.11 and Theorem 2.8” respectively. Hence the proofs are omitted. Theorem 3.4. If Fi is a FMAe-O sets for any i ∈ M, M is a finite set and Fi , F j for any distinct i, j ∈# M, then " c T (i) Fi ⊂ F j for any j ∈ M i∈M\{ T j} (ii) Fi , 0X for any j ∈ M. i∈M\{ j} Corollary 3.5. If Fi is a FMAe-O sets for any i ∈ M, M is a finite set and Fi , F j for any distinct i, j ∈ M then,Fi ∩ F j , 0X for any distinct i, j ∈ M. Theorem 3.6. If Fi is a FMAe-O #sets"for any i#c∈ M, M is a finite set and Fi , F j for any " T T Fi ∪ distinct i, j ∈ M, then F j = Fi for any j ∈ M. i∈M i∈M\{ j} Theorem 3.7. If Fi is a FMAe-O sets for any i ∈ M, M is a finite set and Fi , F j for any T T Fk . Fi ⫋ distinct i, j ∈ M and if K is a proper nonzero fuzzy subset of M, then i∈M k∈K Theorem 3.8. If Fi is a FMAe-O sets for any i ∈ M, M is a finite set and Fi , F j for any T Fi is a fuzzy subset, then F j is a fuzzy subset for any j ∈ M. distinct i, j ∈ M and if i∈M FUZZY MINIMAL AND MAXIMAL e-OPEN SETS 37 #c # " " T T Fi for any j ∈ M. Fi ∪ Proof. By “Theorem 3.6”, we have F j = i∈M i∈M\{ j} # " " # S T Fi ∪ Fj = Fic . i∈M i∈M\{ j} S Since M is finite, Fic is fuzzy e-closed. Hence F j is fuzzy e-closed for any j ∈ i∈M\{ j} M. □ Theorem 3.9. If Fi is a FMAe-O set for any i ∈ M, M is a finite set and Fi , F j for any T Fi = 0X ,then {Fi /i ∈ M} is the set of all FMAe-O sets of fts X. distinct i, j ∈ M. If i∈M Proof. Suppose that ∃ another FMAe-O Fk of a fts X such that Fk , Fi , ∀i ∈ M. Clearly, T T Fi = Fi , 0X , by Theorem 3.4(ii), a contradiction. 0X = i∈M i∈(M∪k)\{k} Hence {Fi /i ∈ M} is the family of all FMAe-O sets of fts X. □ 4. ACKNOWLEDGEMENTS The authors are thankful to the referees for their suggestions and commands to develop this manuscript. R EFERENCES [1] K. K. Azad, On fuzzy semi-continuity, fuzzy almost continuity and fuzzy weakly continuity, J. Math. Anal. Appl., 82(1981), 14-32. [2] C. L. Chang Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190. [3] B. M. Ittanagi and R. S. Wali On fuzzy minimal open and fuzzy maximal open sets in fuzzy topological spaces, International J. of Mathematical Sciences and Applications,1(2),2011. [4] V. Seenivasan and K. Kamala, Fuzzy e-continuity and fuzzy e-open sets, Annals of Fuzzy Mathematics and Informatics, 8(1)(2014),141- 148.. [5] Supriti Saha, Fuzzy δ-continuous mappings, J. Math. Anal. Appl. 126 (1987) 130-142.. [6] A. Swaminathan and M. Sankari, Some remarks on fuzzy mean open, closed and clopen sets, J. Appl. Math. and Informatics Vol. 39(2021), No. 5-6, pp. 743-749 Some remarks on fuzzy mean open, closed and clopen sets, J. Appl. Math. and Informatics Vol. 39(2021), No. 5-6, pp. 743-749. [7] L. A. Zadeh Fuzzy sets, Information and control 8 (1965), 338-353. M. S ANKARI D EPARTMENT OF M ATHEMATICS , L EKSHMIPURAM C OLLEGE OF A RTS AND S CIENCE , N EYYOOR , K ANYAKU MARI , TAMIL NADU -629 802, I NDIA . Email address: [email protected] S. D URAI RAJ D EPARTMENT OF M ATHEMATICS , P IONNEER K UMARASWAMI C OLLEGE OF A RTS AND S CIENCE , NAGER COIL , TAMIL NADU -629 003, I NDIA . Email address: [email protected] C. M URUGESAN R ESEARCH S CHOLAR , P IONNEER K UMARASWAMI C OLLEGE OF A RTS AND S CIENCE ,V ETTURINIMADAM , K ANYAKUMARI , TAMIL NADU -629 003, I NDIA .(A FFILIATED TO M ANONMANIAM S UNDARANAR U NIVER SITY, T IRUNELVELLI ) Email address: [email protected] ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 5, Number 1 (2022), 38-54 ISSN: 2582-0818 © http://www.technoskypub.com EXISTENCE RESULTS FOR FUZZY DIFFERENTIAL EQUATION WITH ψ-HILFER FRACTIONAL DERIVATIVE K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED∗ A BSTRACT. This manuscript concerns the fuzzy differential equation involving ψ-Hilfer type fractional derivative with nonlocal condition. By using successive approximation, we obtain the existence, uniqueness results of solution for ψ-Hilfer fuzzy differential equation. Further, nonlocal conditions are extended to the existence results. Furthermore, an application is shown to demonstrate the theoretical conclusions utility. 1. I NTRODUCTION Consider the ψ-Hilfer fuzzy fractional differential equation of the kind ( α,β,ψ Da+ x(t) = f (t, x(t)), for all t ∈ [a, b], Pm 1−γ,ψ Ia+ x(a) = x0 = i=1 Ci x(ti ), γ = α + β(1 − α), (1.1) where x ∈ R, 0 < α < 1, β ∈ [0, 1], f : [a, b] × E → E is a fuzzy function. Moreover, α,β,ψ 1−γ,ψ are the ψ-Hilfer fractional integral and derivative, which will be given in Ia+ , Da+ the next section. ti (i = 1, 2, . . . , m) satifies a < t1 ≤ t2 ≤ . . . < b and Ci is a real number, x0 ∈ R. Here nonlocal conditions are more effective than the initial conditions 1−γ Ia+ x(0) = x0 in terms of physical problems. x is said to be a solution of (1.1). The fundamental concept of theory of differential equation is a rich and beautiful field of pure and applied mathematics which deals with many disciplines including engineering, physics, economics, biology. There are many branches of theory of differential equation is fuzzy fractional differential equations that in recent year. The theoretical development of fractional differential equation is the Riemann-Liouville’s or Caputo sense have been excellently given in [1, 2, 3, 4, 5, 6], it has gathered significant not only in mathematical research but also in other applied sciences. In this way of fractional derivative concept that we should considered depends on the experimental data that best fits in the theoretical model. Hilfer has suggest a new generalized form of the fractional derivative, the so-called Hilfer fractional derivative(HFD) that merge the wide number of definition of fractional differential operators. For many definition on HFD and interesting applications, one can 2010 Mathematics Subject Classification. 34A08, 34B15, 34A12. Key words and phrases. Fuzzy differential equations; ψ-Hilfer fractional derivative; Existence; Uniqueness; Lipschitz condition. Received: March 19, 2022. Accepted: May 30, 2022. Published: June 30, 2022. *Corresponding author. 38 FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE 39 refer to [7, 8, 21]. Inspired by the definitions of the HFD and the concepts of fractional derivative of a function with respect to the another ψ kernal function suggest a new idea of fractional derivative,the so-called ψ-HFD. Recently, the topics of existence and uniqueness for the solution to the linear and nonlinear fuzzy differential equations with ψ-HFD has been further investigated and discuss by many researches in various aspects. In [9] the existence and uniqueness of RiemannLiouville fuzzy fractional differential equation has been demonstrate by Arshad and the concept of fuzzy type Riemann-Liouville differentiabilty based on Hukuhara differentiability in [15] by using the Hausdroff measure of noncompactness. Furthermore, the existence and uniqueness for fuzzy fractional differential equation with ψ-Hilfer under Liouville-Caputo generalized Hukuhara differentiability has been investigated in [10], and further see [11, 12, 13, 14, 19, 22]. In [16], the existence results for extremal solutions of interval fractional function integro-differential equation by using the monotone iteration approaches associated with the method of upper and lower solution was investigated. This paper is organized as follows: In Section 2, we give some preliminary facts that we need in what follows. In section 3, we present our main results on the existence results of solution by using successive approximation method. An illustrative example is given to show the practical usefulness of the analytical results. Conclusion is given in section 4. 2. P RELIMINARIES Now in this section we give some definitions and lemmas useful in our subsequent discussion. We denote by E the sapce of all fuzzy numbers on R. For c ∈ R, p ∈ [1, ∞], Let Xcp (a, b) denote the space of all complex-valued Lebesgue measurable functions f on a finite interval [a, b] for which ∥f ∥Xcp < ∞ with the norm ∥f ∥Xcp =  Rb a f (t) p dt t 1/p < ∞. Definition 2.1. [20] A fuzzy number is a fuzzy set x : R → [0, 1] which satisfies the following conditions: (i) x is normal, that is, there exists t0 ∈ R such that x(t0 ) = 1; (ii) x is fuzzy convex in R, that is, for λ ∈ [0, 1],
 x λt1 + (1 − λ)t2 ≥ min x(t1 ), x(t2 ) , for any t1 , t2 ∈ R; (iii) x is a upper semicontinuous on R; (iv) [x]0 = cl{z ∈ R | x(z) > 0} is compact. Denote by C([a, b], E) the set of all continuous fuzzy function and by AC([a, b], E) the set of all absolutely continuous fuzzy functions on the intervals [a, b] with values in E. Let γ ∈ (0, 1), by Cγ,ψ [a, b] we denote the space of continuous functions defined by Cγ,ψ [a, b] = {f : (a, b] → E : (ψ(t) − ψ(a))1−γ f (t) ∈ C[a, b]}. Let L([a, b], E) be the set of all fuzzy functions x : [a, b] → E such that the functions t 7→ D0 [x(t), b 0] belongs to L1 [a, b]. r If x is a fuzzy numbers on R, we define [x] = {z ∈ R | x(z) ≥ r} the r-level of x, with r ∈ (0, 1]. From condition (i) and (iv), it follows that the r-level set of x ∈ E, [x]r , is a nonempty compact interval for any r ∈ [0, 1]. We denote by [x(r), x(r)] the r-level of a fuzzy number x. For x1 , x2 ∈ E, and λ ∈ R, the sum x1 + x2 and the product λ · x1 40 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED are defined by [x1 + x2 ]r = [x1 ]r + [x2 ]r , [λ · x1 ]r = λ[x1 ]r , for all r ∈ [0, 1], where [x1 ]r + [x2 ]r means the usual addition of two intervals of R and λ[x1 ]r means the usual scalar product between λ and an real interval. For x ∈ E, we define the diameter of the r-level set of x as diam[u]r = u(r) − u(r). Definition 2.2. [19] Let x1 , x2 ∈ E. If there exists x3 ∈ E such that x1 = x2 + x3 , then x3 is called the Hukuhara difference of x1 and x2 and it is denoted by x1 ⊖ x2 . We note that x1 ⊖ x2 ̸= x1 + (−)x2 . Definition 2.3. [19] The distance D0 [x1 , x2 ] between two fuzzy numbers is defined as
D0 [x1 , x2 ] = supr∈[0,1] H [x1 ]r , [x2 ]r , for all x1 , x2 ∈ E, where H([x1 ]r , [x2 ]r ) = max{|u1 (r) − u1 (r)|, |u1 (r) − u1 (r)|} is a Hausdorff distance between [x1 ]r and [x2 ]r . Triangular fuzzy numbers are defined as a fuzzy set in E that is specifed by an ordered triple x = (a, b, c) ∈ R3 with c ∈ [a, b] such that [x]r = [x(r), x(r)] are the end points of rlevel sets for all r ∈ [0, 1], where x(r) = a + (b − a)r and x(r) = c − (c − b)r. In general, the parametric form of a fuzzy number x is a pair [x]r = [x(r), x(r)] of function x(r), x(r), r ∈ [0, 1], which satisfy the following conditions: u(r) is a monotonically increasing left-continuous function, u(r) is a monotonically decreasing left-continuous function, and u(r) ≤ u(r), r ∈ [0, 1]. Definition 2.4. [17] The generalized Hukuhara difference of two fuzzy numbers x, y ∈ E (gH-difference for short) is defined as follows: x ⊖gH y = ω ⇔ x = y + ω, or y = x + (−1)ω. A function x : [a, b] → E is called d-increasing (d − decreasing) on [a, b] if for every r ∈ [0, 1] the function t 7→ diam[x(t)]r is nondecreasing (nonincreasing) on [a, b]. If x is a d-increasing or d- decreasing on [a, b], then we say that x is d-monotone on [a, b]. Definition 2.5. [7, 8] The left-sided ψ-fractional integral of order α > 0, x ∈ Xcp (a, b) for -∞ < a < t < ∞ is defined by Z t ′ 1 α,ψ
ψ (τ )(ψ(t) − ψ(τ ))α−1 x(τ )dτ. (2.1) Ia+ x (t) = Γ(α) a Definition 2.6. [7, 8] The ψ-fractional derivative associated with the generalized fractional integrals (2) are defined, for 0 ≤ a < t < ∞, n = [α] + 1, by n  1 d n−α,ψ
α,ψ
Ia+ x (t) Da+ x (t) = ′ ψ (t) dt  n Z t ′ 1 1 d = ψ (τ )(ψ(t) − ψ(τ ))n−α−1 x(τ )dτ. (2.2) Γ(n − α) ψ ′ (t) dt a Let x ∈ L([a, b], E), then the ψ-Hilfer fractional integral of order α of the fuzzy function x is defined as follows: Z t ′ 1 α,ψ
ψ (τ )(ψ(t) − ψ(τ ))α−1 x(τ )dτ, t ≥ a. xα,ψ (t) = Ia+ x (t) = Γ(α) a Since [x(t)]r = [x(r, t), x(r, t)] and 0 < α < 1, we can considered the fuzzy ψfractional integral of the fuzzy function x based on lower and upper functions, that is, α,ψ
α,ψ
α,ψ
[ Ia+ x (t)]r = [ Ia+ x (r, t), Ia+ x (r, t)], t ≥ a, FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE 41 where α,ψ
Ia+ x (r, t) = 1 Γ(α) and α,ψ
Ia+ x (r, t) = 1 Γ(α) Rt a Rt a ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 x(r, τ )(τ )dτ , ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 x(r, τ )(τ )dτ . In addition, it follows that the opeartor xα,ψ (t) is linear and bounded from C([a, b], E) to C([a, b], E). Indeed, we have c ≤ ∥x∥0 Z 1 Γ(α) t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 dτ = a where ∥z∥0 = supt∈[a,b] D0 [z(t), b 0].
α ∥x∥0 ψ(t) − ψ(a) , Γ(α + 1) Definition 2.7. [7, 8] Let order α and type β satisfy n − 1 < α ≤ n and 0 ≤ β ≤ 1, with n ∈ N . The fuzzy ψ-Hilfer generalized Hukuhara fractional derivative(or ψ-Hilfer gH-fractional derivative) (left-sided/right-sided), with respect to t, with a function t ∈ C1−γ,ψ [a, b], is defined by α,β,ψ
Da+ x (t) = β(1−α),ψ
Ia+  1 d ψ ′ (t) dt  (1−β)(1−α),ψ Ia+ β(1−α),ψ ψ (1−β)(1−α),ψ
f Ia+ x (t), = Ia+ ′
x (t) if the gH-derivative x(1−α),ψ (t) exists for t ∈ [a, b], where (1−α),ψ x(1−α),ψ (t) := Ia+ Lemma 2.1. [7, 8] Let α,ψ Ia+
x (t) = 1 Γ(1 − α) Z t ′ ψ (τ )[ψ(t) − ψ(τ )]−α x(τ )dτ. a according to Eqs(2.1). Then α,ψ Ia+ (ψ(t) − ψ(a))β−1 (t) =
α+β−1 Γ(β) ψ(t) − ψ(a) , Γ(α + β) α ≥ 0, β > 0. 1−α,ψ 1 Lemma 2.2. [7, 8] Let α > 0, 0 ≤ γ < 1. If x ∈ Cγ,ψ [a, b] and Ia+ x ∈ Cγ,ψ [a, b], then
α−1 (I 1−α,ψ x)(a) α,ψ α,ψ
ψ(t) − ψ(a) . Ia+ Da+ x (t) = x(t) − a+ Γ(α) β(1−α),ψ Lemma 2.3. [7, 8] Let x ∈ L1 (a, b). If Da+ β(1−α),ψ α,β,ψ α,ψ Da+ Ia+ x = Ia+ β(1−α),ψ Da+ x, x exists on L1 (a, b), then for all t ∈ (a, b]. Lemma 2.4. [20] If x ∈ AC([a, b], E) is a d-monotone fuzzy function, where [x(t)]r = [x(r, t), x(r, t)] for 0 ≤ r ≤ 1, a ≤ t ≤ b, then for 0 < a < 1, we have that α,β,ψ α,β,ψ α,β,ψ (i) [(Da+ x)(t)]r = [Da+ x(r, t), Da+ x(r, t)] for t ∈ [a, b], if x is d-increasing α,β,ψ α,β,ψ α,β,ψ r (ii) [(Da+ x)(t)] = [Da+ x(r, t), Da+ x(r, t)] for t ∈ [a, b], if x is d-decreasing 42 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED Proof. Let x ∈ AC([a, b], E) be a d-monotone fuzzy function, then [x(t)]r = [x(r, t), x(r, t)]. If x is d-monotone then either x is d-increasing or d-decreasing, for any r ∈ [0, 1] To prove(i): Assume that x is d-increasing, ′ d d x(r, t)], x(r, t), dt [x (t)] = [ dt by definition of fuzzy ψ-Hilfer gH-fractional derivative   α,β,ψ
r  β(1−α),ψ 1,ψ (1−β)(1−α),ψ
β(1−α),ψ 1,ψ (1−β)(1−α),ψ
x (r, t) Da+ x (t) = Ia+ fψ Ia+ x (r, t), Ia+ fψ Ia+  1,ψ  β(1−α),ψ 1 d (1−β)(1−α),ψ
= Ia+ Ia+ x (r, t), ′ ψ (t) dt 1,ψ   1 d (1−β)(1−α),ψ
β(1−α),ψ x (r, t) Ia+ Ia+ ψ ′ (t) dt   α,β,ψ = Daα,β,ψ x(r, t), Da+ x(r, t) . + To prove(ii): Assume that x is d-decreasing, ′ d d x(r, t), dt x(r, t)], [x (t)] = [ dt by definition of fuzzy ψ-Hilfer gH-fractional derivative  α,β,ψ
r  β(1−α)′ ψ 1,ψ (1−β)(1−α),ψ
 β(1−α),ψ 1,ψ (1−β)(1−α),ψ
Da+ x (t) = Ia+ fψ Ia+ x (r, t), Ia+ fψ Ia+ x (r, t) 1,ψ   β(1−α) 1 d (1−β)(1−α),ψ
Ia+ x (r, t), = Ia+ ′ ψ (t) dt  1,ψ  1 d β(1−α),ψ (1−β)(1−α),ψ
Ia+ Ia+ x (r, t) ′ ψ (t) dt   α,β,ψ α,β,ψ = Da+ x(r, t), Da+ x(r, t) . This completes the proof. □ Lemma 2.5. If x ∈ AC([a, b], E) is a d-monotone fuzzy function t ∈ (a, b] and α ∈ (0, 1), α,ψ we set z(t) := Ia+ and z(1−α),ψ (t) is d-increasing on (a, b] then and α,ψ α,β,ψ
Ia+ Da+ x (t) = x(t) ⊖ Pm
1−γ Ci x(ti ) [ψ(t) − ψ(a)] Γ(γ) i=1 α,β,ψ α,ψ
Da+ Ia+ x (t) = x(t). Proof. Let x ∈ AC([a, b], E) be a d-monotone fuzzy function then by using ψ-HFD, we have, α,β,ψ
Da+ x (t) = β(1−α),ψ
Ia+  d 1 ψ ′ (t) dt  (1−β)(1−α),ψ
Ia+ x(t). FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE 43 α,ψ By applying Ia+ on the both sides, we get   1 d (1−β)(1−α),ψ
α,ψ α,β,ψ
α,ψ β(1−α),ψ
Ia+ Da+ x(t) = Ia+ Ia+ Ia+ x(t) ψ ′ (t) dt   1 d (1−β)(1−α),ψ
α+β(1−α),ψ
Ia+ x(t) = Ia+ ψ ′ (t) dt   1 d (1−γ),ψ
γ,ψ
= Ia+ Ia+ x(t) ψ ′ (t) dt
γ,ψ γ,ψ = Ia+ Da+ x(t),   1−γ,ψ γ,ψ d Ia+ x(t), where Da+ x(t) = ψ′1(t) dt α,ψ α,ψ
and we get Ia+ Da+ x (t) = x(t) ⊖ I (1−γ),ψ a+ Γ(α) x(a)[ψ(t) − ψ(a)]α−1 (1−γ),ψ I α,ψ α,β,ψ
α−1 Ia+ Da+ x (t) = x(t) ⊖ a+ . Γ(α) x(a)[ψ(t) − ψ(a)] Applying initial condition, we get m X (1−γ),ψ Ia+ x(a) = x0 = Ci x(ti ). (2.3) i=1 Pm C x(t ) α,ψ α,β,ψ
That is, Ia+ Da+ x (t) = x(t) ⊖gH i=1Γ(γ)i i [ψ(t) − ψ(a)]γ−1 , if z(t) is d-increasing on [a, b] or z(t) is d- decreasing on [a, b] and z(1−α),ψ (t) is dincreasing on (a, b]. Pm C x(t ) α,ψ α,β,ψ
In similar, Ia+ Da+ x (t) = x(t) + (−1) i=1Γ(γ)i i [ψ(t) − ψ(a)]γ−1 . α,β,ψ α,ψ Next we have, to prove that Da+ Ia+ x(t) = x(t). Let x ∈ L1 (a, b), β(1−α),ψ α,β,ψ α,ψ Da+ Ia+ x(t) = Ia+ = (1−β)(1−α),ψ DIa+ x(t) β(1−α),ψ 1−β(1−α),ψ Ia+ DIa+ x(t) β(1−α),ψ = Ia+ β(1−α),ψ Da+ x(t) 1−β(1−α),ψ α,β,ψ α,ψ Da+ Ia+ x(t) = x(t) ⊖ Ia+ x(a)[ψ(t) − ψ(a)]β(1−α)−1 = x(t). Γ(β(1 − α)) On the other hand, since x ∈ AC([a, b], E),there exists a constant K such that K = supt∈[a,b] D0 [x(t), b 0]. Then Z t ′ 1 α,ψ b ψ (τ )(ψ(t) − ψ(τ ))α−1 x(τ )ds D0 [Ia+ x(t), 0] = Γ(α) a Z t ′ 1 α,ψ b ψ (τ )(ψ(t) − ψ(τ ))α−1 |x(τ )|dτ sup D0 [Ia+ x(t), 0] ≤ Γ(α) t∈[a,b] a Z t ′ K α,ψ D0 [Ia+ x(t), b 0] ≤ ψ (τ )(ψ(t) − ψ(τ ))α−1 dτ Γ(α) a K α,ψ D0 [Ia+ x(t), b 0] ≤ (ψ(t) − ψ(a))α , Γ(α + 1) α,ψ and Ia+ x(t) = 0 at t = a. This completes the proof. □ 44 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED Lemma 2.6. Let χ : [a, b] → R+ be a continuous function on the interval [a, b] and satisfy Daα,β,ψ χ(t) ≤ g(t, χ(t)), t ≤ a, where g ∈ C([a, b] × R+ , R+ ). Assume that + m(t) = m(t, a, ξ0 ) is the maximal solution of the initial value problem α,β,ψ 1−γ,ψ
Da+ ξ(t) = g(t, ξ), Ia+ ξ (a) = ξ0 ≥ 0, t ∈ [a, b]. (2.4) Then, if χ(a) ≤ ξ0 , we have χ(t) ≤ m(t), t ∈ [a, b]. Lemma 2.7. Consider the initial value problem as follows: α,β,ψ 1−γ
= g(t, χ(t)), Ia+ χ (a) = χ0 = 0, for all t ∈ [a, b] Da+ (2.5) Let η > 0 be a given constant and B(χ0 , η) = {χ ∈ R : |χ − χ0 | ≤ η}. Assume that the real-valued function g : [a, b] × [0, η] → R+ satisfies the following conditions: (i) g ∈ C([a, b] × [0, η], R+ ), g(t, 0) = 0, 0 ≤ g(t, χ) ≤ Mg , for all (t, x) ∈ [a, b] × [0, η]; (ii) g(t, χ) is nondecreasing in χ for every t ∈ [a, b]. Then problem (6) has at least one solution defined on [a, b] and χ(t) ∈ B(χ0 , η). Proof. The problem (2.5) is equivalent to the following fractional integral equation: χ(t) = χ0 + 1 Γ(α) Given: χ(a) = χ0 = 0 χ(t) = 0 + ∗ ∗ 1 Γ(α) Choose t > a such that t ≤  Rt a Rt a ′ ψ (s)(ψ(t) − ψ(s))α−1 g(s, χ(s))ds. ′ ψ (s)(ψ(t) − ψ(s))α−1 g(s, χ(s))ds. ηΓ(1+α)
1/α Mg  + a , and put b∗ = min{t∗ , b}. Let us de- fine a sequence {x}∞ n=0 of successive approximation of problem (2.5) on [a,b] as follows: Z t ′ Mg 1 ψ (s)(ψ(t) − ψ(s))α−1 g(s, χn (s))ds (ψ(t) − ψ(a))α , χn+1 (t) = + Γ(α + 1) Γ(α) a Then, for n=0, we have Z t ′ 1 ψ (s)(ψ(t) − ψ(s))α−1 g(s, χ0 (s))ds χ1 (t) = Γ(α) a Mg (ψ(t) − ψ(a))α χ1 (t) ≤ Γ(α + 1) χ0 (t) = χ1 (t) ≤χ0 (t) ≤ η, t ∈ [a, b]. Hence g(t, η) is nondecreasing in χ for every t ∈ ([a, b∗ ]) and proceeding recursively, we find that, 0 ≤ χn+1 (t) ≤ χn (t) ≤ ..... ≤ χ0 (t) ≤ η, n = 0, 2, 3, . . ., it follows that, the sequence {χn }∞ n=0 is uniformly bounded for all n ≥ 0. Moreover., n Daα,β,ψ χ(t) = g(t, χ (t)) ≤ M , we get the equicontinuity of the sequence {χn }. Indeed, + for a ≤ t1 ≤ t2 ≤ b∗ and by using Mean-Value Theorem, we have 2M2 2M2 (t2 − t1 )α ≤ (t2 − t1 )α τ α,ψ , ∀τ ∈ [t1 , t2 ] ⊆ [a, b∗ ]. |χn (t2 ) − χn (t1 )| ≤ Γ(α + 1) Γ(α + 1) FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE 45
1/α ϵ . Γ(1 + α)τ α Thus, if |t2 − t1 | ≤ δ, we have |χn (t2 ) − χn (t1 )| ≤ ϵ, where δ = 2M g n Hence by using Arzela-Ascoli Theorem and the monotonicity of the sequence {χ }. Therefore limn→∞ χn (t) = χ(t) is uniformly on [a, b∗ ]. Thus, χ ∈ C([a, b∗ ], [0, η]) and χ(t) is a solution of the problem (2.5). This completes the proof. □ 3. M AIN RESUILTS In this section, we discuss the existence and uniqueness of solution of problem (1.1) to initial value problem by using successive approximation method under generalized lipschitz condition of the right-hand side. Lemma 3.1. Let γ = α + β(1 − α), where 0 < α < 1, 0 ≤ β ≤ 1, let f : (a, b] × E → E be a fuzzy function such that t 7−→ f (t, x) belongs to Cγ,ψ ([a, b], E) for any x ∈ E. Then a d-monotone fuzzy function x ∈ C([a, b], E) is a solution of problem (1.1) if and only if x satisfies the integral equation Pm
γ−1 i=1 Ci x(ti ) x(t) ⊖gH ψ(t) − ψ(a) Γ(γ) Z t ′ 1 ψ (τ )(ψ(t) − ψ(τ ))α−1 f (τ, x(τ ))dτ, t ∈ [a, b] (3.1) = Γ(α) a 1−γ,ψ and the fuzzy function t 7−→ Ia+ f (t, x) is d-increasing on (a, b]. Proof. First, we have to prove the necessary condition. Let x ∈ C([a, b], E) be a d-monotone solution of problem (1.1), and 1−γ,ψ let z(t) := x(t)⊖gH (Ia+ x(a)), t ∈ [a, b]. Because x is d-monotone on [a, b], it follows that t 7−→ z(t) is d-increasing on [a, b]. From (1.1) and Lemma 2.12 we have that Pm
1−γ α,ψ α,β,ψ
i=1 Ci x(ti ) [ψ(t) − ψ(a)] t ∈ [a, b]. (3.2) Ia+ Da+ x (t) = x(t) ⊖gH Γ(γ) Since f (t, x) ∈ Cγ,ψ ([a, b], E) for any x ∈ E, and from (1.1), it follows that α,ψ α,β,ψ
α,ψ Ia+ Da+ x (t) = Ia+ f (t, x(t)) Z t ′ 1 ψ (τ )(ψ(t) − ψ(τ ))α−1 f (s, x(τ ))dτ, f or = Γ(α) a t ∈ [a, b]. (3.3) In addition, since z(t) is d-increasing on (a, b], it follows that t 7−→ fα,ψ (t, x) is also d-increasing on (a, b]. Consequently, combining (3.2) and (3.3) proves the necessity condition. Next, we prove that the sufficiency. Let x ∈ C([a, b], E) be a d-monotone fuzzy function x satisfies the integral equation and such that t 7−→ fα,ψ (t, x) is d-increasing on (a, b]. Because of the continuity of the fuzzy function f , the fuzzy function t 7−→ fα,ψ (t, x) is continuous on (a, b] and fα,ψ (a, x(a)) = limt→a+ fα,ψ (t, x) = 0. Then Pm
1−γ Ci x(ti ) α,ψ ψ(t) − ψ(a) + Ia+ f (t, x(t))(t), x(t) = i=1 Γ(γ) m X 1−β(1−α),ψ 1−γ,ψ x(t) = Ci x(ti ) + Ia+ f (t, x(t))(t), Ia+ i=1 and 46 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED 1−γ,ψ Ia+ x(0) = Pm i=1 Ci x(ti ). α,β In addition, since t 7−→ fα,ψ (t, x) is d-increasing on (a, b], by applying Da+ on both sides, we obtain that Pm  
γ−1 α,β,ψ i=1 Ci x(ti ) [ψ(t) − ψ(a)] x(t) ⊖gH Da+ Γ(γ)   Z t ′ 1 α,β,ψ α−1 = Da+ ψ (τ )(ψ(t) − ψ(τ )) f (τ, x(τ ))dτ Γ(α) a α,β,ψ α,ψ = Da+ Ia+ f (t, x(t)). Thus,  α,β,ψ  α,ψ α,β,ψ α,β,ψ α,ψ Da+ Ia+ Da+ x(t) = Da+ Ia+ f (t, x(t)) α,β,ψ α,ψ α,β,ψ α,β,ψ α,ψ Da+ Ia+ Da+ x(t) = Da+ Ia+ f (t, x(t)) α,β,ψ Da+ x(t) = f (t, x(t)). This completes the proof. □ Theorem 3.2. Let f ∈ C([a, b] × B(x0 , h), E) and assume that the following conditions hold: (i) There exists a positive constant Mf such that D0 [f (t, z), b 0] ≤ Mf , for all (t, z) ∈ [a, b] × B(x0 , h); (ii) For every t ∈ [a, b] and every z, ω ∈ B(x0 , h), D0 [f (t, z), f (t, ω)] ≤ g(t, D0 [z, ω]), where g(t, ·) ∈ C([a, b] × [0, ψ], R+ ) satisfies the condition in Lemma 2.14 provided that the problem (2.5) has only the solution χ(t) = 0 on [a, b]. Then, the following successive approximations given by x0 (t) = x0 and for n = 1, 2, . . . , Pm
γ−1 n i=1 Ci x(ti ) x (t) ⊖gH ψ(t) − ψ(a) Γ(γ) Z t ′ 1 ψ (τ )(ψ(t) − ψ(τ ))α−1 f (τ, xn−1 (τ ))dτ (3.4) = Γ(α) a converge uniformly to a unique solution of problem (1.1) on some intervals [a, T ] for some n T ∈ (a, b] provided that the function t 7−→ Iaα,ψ + f (t, x (t)) is d-increasing on [a, T ]. 1/α   hΓ(1+α) ∗ ∗ , where M = max{Mg , Mf }, and Proof. Choose t > a such that t ≤ M setting T = min{t∗ , b}. Let S = {x : ω(a) = x0 and ω(t) ∈ B(x0 , h), for all t ∈ [a, T ]}, clearly S is a set of continuous fuzzy functions x. Next, we consider the sequence of continuous fuzzy function {xn }∞ n=0 given by: 0 x (t) = x0 for all t ∈ [a, T ], and for n = 1, 2, . . . xn (t) ⊖gH = 1 Γ(α) Pm
γ−1 Ci x(ti ) ψ(t) − ψ(a) Γ(γ) i=1 Z t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 f (τ, xn−1 (τ ))dτ [a, T ]. (3.5) a Step 1: First of all, we prove that xn (t) ∈ C([a, T ], B(x0 , h)). For n ≥ 1 and for any t1 , t2 ∈ [a, T ] with t1 < t2 , we have FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE  Pm n D0 x (t1 ) ⊖gH ≤ 1 Γ(α) + Z t1 a 1 Γ(α) Z
γ−1 n Ci x(ti ) ψ(t) − ψ(a) , x (t2 ) ⊖gH Γ(γ) i=1 ′ 47 Pm
γ−1 Ci x(ti ) ψ(t) − ψ(a) Γ(γ) i=1  ψ (τ )[(ψ(t1 ) − ψ(τ ))α−1 − (ψ(t2 ) − ψ(τ ))α−1 ]D0 [f (τ, xn (τ ), b 0]dτ t2 t1 ′ ψ (τ )[(ψ(t2 ) − ψ(τ ))α−1 ]D0 [f (τ, xn (τ ), b 0]dτ. 1 The second integral on right-hand side of the last inequality has the value Γ(α+1) (ψ(t2 ) − 1 α α ψ(t1 )) . For the first integral, it has the value Γ(α+1) [(ψ(t1 )−ψ(a)) −(ψ(t2 )−ψ(a))α ]. Hence, we get Mf D0 [xn (t1 ), xn (t2 )] ≤ [(ψ(t2 ) − ψ(t1 ))α + (ψ(t2 ) − ψ(t1 ))α − (ψ(t2 ) − ψ(a))α ] Γ(α + 1) 2Mf (ψ(t2 ) − ψ(t1 ))α , ≤ Γ(α + 1) and it follows that the last expression converges to 0 as t1 → t2 , which proves that xn is a continuous function on [a, T ] for all n ≥ 0. In addition, it follows that xn (t) ∈ B(x0 , h) for all t ∈ [a, T ] if and only if Pm
γ−1 n i=1 Ci x(ti ) ψ(t) − ψ(a) ∈ B(0, h), f orall t ∈ [a, T ]. x (t) ⊖gH Γ(γ) Indeed, if we suppose that xn−1 (t) ∈ S, for all t ∈ [a, T ] and for n ≥ 2, then from Pm  
γ−1 i=1 Ci x(ti ) D0 xn (t) ⊖gH ψ(t) − ψ(a) ,b 0 Γ(γ) Z t ′ 1 ψ (τ )(ψ(t) − ψ(τ ))α−1 D0 [f (τ, xn−1 (τ )), b 0]dτ ≤ Γ(α) a Mf [ψ(t) − ψ(a)]α ≤ h, ≤ Γ(α + 1) it follows that xn (t) ∈ S, forall t ∈ [a, T ]. Hence by mathematical induction, xn (t) ∈ S for all t ∈ [a, T ] and for n ≥ 1. Next, we have to prove that the sequence xn (t) converges uniformly to a continuous function x ∈ C([a, T ], B(x0 , h)). By assumption (ii) and mathematical induction, we have for t ∈ [a, T ] Pm Pm  
γ−1 n
γ−1 i=1 Ci x(ti ) i=1 Ci x(ti ) ψ(t) − ψ(a) , x (t) ⊖gH ψ(t) − ψ(a) D0 xn+1 (t) ⊖gH Γ(γ) Γ(γ) Z t ′ 1 ≤ ψ (τ )[ψ(t) − ψ(τ )]α−1 g(τ, ψ n−1 (τ ))dτ Γ(α) a ≤ ψ n (t), n = 0, 1, 2, . . . , (3.6) where ψ n (t) is defined as follows: ψ n (t) = and ψ 0 (t) = 0, 1, 2, . . ., 1 Γ(α) Rt M Γ(α+1) [ψ(t) a ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 g(τ, ψ n−1 (τ ))dτ − ψ(a)]α . Thus, we have, for t ∈ [a, T ] and for n = 48 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED α,β,ψ n+1 α,β,ψ n D0 [Da+ x (t), Da+ x (t)] ≤ D0 [f (t, xn (t)), f (t, xn−1 (t))] ≤ g(t, D0 [xn (t), xn−1 (t)]) ≤ g(t, χn−1 (t)). Let m ≥ n and t ∈ [a, T ], then we can obtain α,β,ψ α,β,ψ n α,β,ψ m Da+ D0 [xn (t), xm (t)] ≤ D0 [Da+ x (t), Da+ x (t)] α,β,ψ n α,β,ψ n+1 ≤ D0 [Da+ x (t), Da+ x (t)] α,β,ψ n+1 α,β,ψ m+1 + D0 [Da+ x (t), Da+ x (t)] α,β,ψ m+1 α,β,ψ m + D0 [Da+ x (t), Da+ x (t)] ≤ g(t, χn−1 (t)) + g(t, χn−1 (t)) + g(t, D0 [xn , xm (t)]) ≤ 2g(t, χn−1 (t)) + g(t, D0 [xn , xm (t)]). From (ii), because we have that the solution χ(t) = 0 is a unique solution of problem (2.5) α,β,ψ 1−γ,ψ Da+ χ(t) = g(t, χ(t)), Ia+ χ(a) = χ0 = 0 for all t ∈ [a, b]. That is, g(·, χn−1 (.)) : [a, T ] → [0, Mg ] uniformly converges to 0, for every ϵ > 0, there exists a natural numbers n0 such that α,β,ψ Da+ D0 [xn (t), xm (t)] ≤ g(t, D0 [xn (t), xm (t)]) + ϵ, for m ≥ n ≥ n0 . Now, we consider D0 [xn (a), xm (a)] = 0 < ϵ, it follows that, we have for t ∈ [a, T ], D0 [xn (t), xm (t)] ≤ λϵ (t), m ≥ n ≥ n0 , (3.7) where λϵ (t) is the maximal solution to the following problem α,β,ψ Da+ λϵ (t) = g(t, λϵ (t)) + ϵ, 1−γ,ψ (Ia+ )λϵ (a) = ϵ. It follows that, {χϵ (·, ω)} converges uniformly to the maximal solution χ(t) = 0 of problem (2.5) on [a, T ] as ϵ → 0. From (3.7), we can find n0 ∈ N large enough such that, for n, m > n0 , Pm 
γ−1 n i=1 Ci x(ti ) [ψ(t) − ψ(a)] , sup D0 x (t) ⊖gH Γ(γ) t∈[a,T ] Pm 
γ−1 i=1 Ci x(ti ) xm (t) ⊖gH [ψ(t) − ψ(a)] ≤ ϵ. (3.8) Γ(γ) Since (E, D0 ) is a complete metric space and (3.8) holds, it follows that {xn (t)} converges uniformly to x ∈ C([a, b], B(x0 , h)). Hence, we obtain Pm Pm  
γ−1

γ−1 i=1 Ci x(ti ) i=1 Ci x(ti ) [ψ(t) − ψ(a)] = lim xn (t) ⊖gH [ψ(t) − ψ(a)] x(t) ⊖gH n→∞ Γ(γ) Γ(γ) Z t ′ 1 = ψ (τ )[ψ(t) − ψ(τ )]f (τ, x(τ ))dτ, for allτ ∈ [a, T ] Γ(α) a Due to Lemma 3.1 the function x(t) is a solution to (1.1) on [a, T ]. Step 2: To show that the solution x is uniqueness, assume that y : [a, T ] → E is another solution of problem (1.1) on [a, T ]. Denote k(t) = D0 [x(t), y(t)]. Then k(a) = 0 and FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE 49 for every t ∈ [a, T ] we have Daα,β,ψ k(t) ≤ D0 [f (t, x(t)), f (t, y(t))] + Daα,β,ψ k(t) ≤ g(t, D0 (x(t), y(t))) + Daα,β,ψ k(t) ≤ g(t, k(t)) + It follows that, we obtain k(t) ≤ m(t), if k(a) ≤ ξ0 ∀t ∈ [a, T ], where m is a maximal solution of the problem Daα,β,ψ k(t) ≤ g(t, k(t)) + Daα,β,ψ m(t) ≤ g(t, m(t)), + Ia1−γ,ψ k(a) = 0 + Ia1−γ.ψ m(a) = 0. + Clearly, m(t) = 0. Therefore x(t) = y(t), for all t ∈ [a, T ]. Hence x is a solution of uniqueness. This completes the proof. □ Corollary 3.3. Let f ∈ C([a, b], E). Assume that there exists positive constants L, Mf such that, for every z, ω ∈ E,   D0 f (t, z), f (t, ω) ≤ LD0 [z, ω], D0 [f (t, z), b 0] ≤ Mf . Then the following successive approximations given by x0 (t) = x0 and for n = 1, 2, 3, . . . n x (t) ⊖gH Pm
γ−1 Ci x(ti ) 1 ψ(t) − ψ(a) = Γ(γ) Γ(α) i=1 Z t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 f (τ, xn−1 (τ ))dτ a converge uniformly to a unique solution of problem (1.1) on some intervals [a, T ] for some n T ∈ (a, b] provided that the function t 7−→ Iaα,ψ + f (t, x(t )) is d-increasing on [a, T ]. Example 3.1. Let γ = α + β(1 − α), where 0 < α < 1, 0 ≤ β ≤ 1, and λ ∈ R. We consider the linear fuzzy fractional differential equation under ψ-HFD and assume that the following conditions hold: ( α,β,ψ Da+ x(t) = λx(t) + p(t), Pm 1−γ,ψ Ia+ x(a) = x0 = i=1 Ci x(ti ), t ∈ (a, b] γ = α + β(1 − α). (3.9) Then x satiefies the integral equations x(t) ⊖gH λ = Γ(α) Z t Pm
γ−1 Ci x(ti ) ψ(t) − ψ(a) Γ(γ) i=1 ′ ψ (τ )(ψ(t) − ψ(τ )) a α,ψ α,ψ =λIa+ x(t) + Ia+ p(t), α−1 1 x(τ )dτ + Γ(α) Z t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 p(τ )dτ a where p ∈ C([a, b], E) and we also assume that the right-hand side of the above integral equation of diameter is increasing . We see that f (t, x) = λx + p satisfies the assumption of Corollary 3.3. To find the explicit solution of (3.9), we apply the method of successive 50 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED approximations. Setting u0 (t) = u0 and xn (t) ⊖gH Pm Ci x(ti ) Γ(γ) i=1
γ−1 α,ψ n−1 α,ψ ψ(t) − ψ(a) = λIa+ x (t) + Ia+ p(t), n = 1, 2, 3, . . .. for For n = 1 and λ > 0, if we assume that x is d-increasing, then it follows that x1 (t) ⊖gH Pm m
γ−1 X Ci x(ti ) λ(ψ(t) − ψ(a))α α,ψ Ci x(ti ) ψ(t) − ψ(a) = + Ia+ p(t), Γ(γ) Γ(α + 1) i=1 i=1 t ∈ [a, b]. On the other hand, if we assume that λ < 0 and x is d-decreasing, then it follows that (−1)  Pm
γ−1 Ci x(ti ) ψ(t) − ψ(a) ⊖gH x1 (t) Γ(γ) i=1  = m X i=1 Ci x(ti ) λ(ψ(t) − ψ(a))α α,ψ + Ia+ p(t). Γ(α + 1) For n = 2, we also see that x2 (t) ⊖gH Pm   m
γ−1 X Ci x(ti ) λ[ψ(t) − ψ(a)]α λ2 [ψ(t) − ψ(a)]2α Ci x(ti ) + ψ(t) − ψ(a) = Γ(γ) Γ(α + 1) Γ(2α + 1) i=1 i=1 α,ψ 2α,ψ + Ia+ p(t) + Ia+ p(t). Suppose λ < 0 and x is d-decreasing such that  Pm 
γ−1 i=1 Ci x(ti ) (−1) ψ(t) − ψ(a) ⊖gH x2 (t) Γ(γ)   m X λ2 (ψ(t) − ψ(a))2α λ(ψ(t) − ψ(a))α α,ψ 2α,ψ + Ia+ p(t) + Ia+ p(t). Ci x(ti ) = + Γ(α + 1) Γ(2α + 1) i=1 If we proceed inductively and let n → ∞, we obtain the Pm
γ−1 i=1 Ci x(ti ) ψ(t) − ψ(a) x(t) ⊖gH Γ(γ)
jα Z t ∞ ∞ m X X X λj−1 ψ ′ (τ )(ψ(t) − ψ(τ )) jα−1 λj ψ(t) − ψ(a) Ci x(ti ) = p(τ )dτ + Γ(jα + 1) Γ(jα) a j=1 j=1 i=1
jα Z t ∞ m ∞ X λj ψ ′ (τ )(ψ(t) − ψ(τ )) jα+(α−1) X X λj ψ(t) − ψ(a) p(τ )dτ + = Ci x(ti ) Γ(jα + 1) Γ(jα + α) a j=0 i=1 j=1
jα Z t ∞ m ∞ X λj (ψ(t) − ψ(τ ))jα ′ X X λj ψ(t) − ψ(a) = + ψ (τ )(ψ(t) − ψ(τ ))α−1 p(τ )dτ. Ci x(ti ) Γ(jα + 1) Γ(jα + α) a j=0 i=1 j=1 We see that, λ > 0 and x is d-increasing or λ < 0 and x is d-decreasing, respectively. Then, P∞ xk , α, β > 0, by applying definition of Mittag-Leffler function Eα,β (x) = j=1 Γ(jα+β) if λ > 0 and x is d-increasing then the solution of problem (3.9) is given by FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE Pm m X
γ−1 Ci x(ti ) ψ(t) − ψ(a) Γ(γ) i=1 x(t) ⊖gH = 51 Ci x(ti )Eα,1 (λ(ψ(t) − ψ(a))α ) i=1 + Z t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 Eα,α (λ(ψ(t) − ψ(a))α )p(τ )dτ. a On the other hand, if λ < 0 and x is d-decreasing, then we obtain the solution of problem (3.9) is given by Pm
γ−1 i=1 Ci x(ti ) ψ(t) − ψ(a) x(t) ⊖gH Γ(γ) m X Ci x(ti )Eα,1 (λ(ψ(t) − ψ(a))α ) = i=1 ⊖ (−1) Z t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 Eα,α (λ(ψ(t) − ψ(a))α )p(τ )dτ a Remark. In problem (3.9), suppose that λ > 0 and the solution of (3.9) is d-increasing. We observe that the solution of problem (3.9) admit particular cases as follows: if β = 0, then we obtain the solution of problem (3.9) with the ψ-HFD as follows: Pm
α−1 i=1 Ci x(ti ) ψ(t) − ψ(a) x(t) ⊖gH Γ(α) m X = Ci x(ti )Eα,1 (λ(ψ(t) − ψ(a))α ) i=1 + Z t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 Eα,α (λ(ψ(t) − ψ(a))α )p(τ )dτ. a If the value of ψ(x) = x and taking β = 0, then we obtain the solution of the problem (3.9) with the Caputo fractional derivative as follows: Pm = m X
α−1 Ci x(ti ) t−a Γ(α) i=1 x(t) ⊖gH Ci x(ti )Eα,1 (λ(t − a)α ) i=1 + Z t (t − τ )α−1 Eα,α (λ(t − a)α )p(τ )dτ. a 52 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED In addition, if the value of ψ(x) = log x and taking β = 0 , then we obtain the following solution of problem (3.9) with the Caputo-Hadamard fractional derivative: Pm t

α−1 i=1 Ci x(ti ) log x(t) ⊖gH Γ(α) a m X t
α ) = Ci x(ti )Eα,1 (λ log a i=1 Z t 1 t
α−1 t
α + log Eα,α (λ log )p(τ )dτ. τ a a τ Remark. Suppose that λ < 0 and the solution of (3.9) is d-decreasing. We observe that the solution of problem (3.9) admit the following cases: if β = 0 then the solution (3.9) with the ψ-type Caputo fractional derivative as follows: x(t) ⊖gH = m X Pm
α−1 Ci x(ti ) ψ(t) − ψ(a) Γ(α) i=1 Ci x(ti )Eα,1 (λ(ψ(t) − ψ(τ ))α ) i=1 ⊖ (−1) Z t ′ ψ (τ )(ψ(t) − ψ(τ ))α−1 Eα,α (λ(ψ(t) − ψ(τ ))α )p(τ )dτ. a If the value of ψ(x) = x and taking β = 0, then we obtain the following solution of problem (3.9) with the Riemann-Liouville fractional derivative as follows: Pm
α−1 i=1 Ci x(ti ) x(t) ⊖gH (t − a) Γ(α) m X Ci x(ti )Eα,1 (λ(t − a)α ) = i=1 ⊖ (−1) Z t (t − τ )α−1 Eα,α (λ(t − τ )α )p(τ )dτ. a In addition, if the value of ψ(x) = log x and taking β = 0, then we obtain the following solution of problem (3.9) with Riemann-Hadamard fractional derivative as follows, Pm t

α−1 i=1 Ci x(ti ) x(t) ⊖gH log Γ(α) a m X t
α ) Ci x(ti )Eα,1 (λ log = a i=1 Z t t
α−1 t
α 1 ⊖ (−1) log Eα,α (λ log )p(τ )dτ. τ a a τ 4. C ONCLUDING REMARKS The existence and uniqueness of solutions for a fuzzy differential equations of ψ-Hilfer fractional derivative with nonlocal condition have obtained. Our investigation based on the successive approximation. The acquired results in this paper are more general and cover many of the parallel problems that contain special cases of function ψ, because FDES WITH ψ-HILFER FRACTIONAL DERIVATIVE 53 our proposed system contains a global fractional derivative that integrates many classic fractional derivatives. ACKNOWLEDGEMENT The authors are thankful to the anonymous reviewers and the handling editor for the fruitful comments that made the presentation of the work more interested. AUTHORS CONTRIBUTIONS All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript. R EFERENCES [1] S. Abbas, M. Lazreg, JE. Lazreg, Y. Zhou, A survey on Hadamard and Hilfer fractional differential equations, Anal. Stability, 102 (2017), 47-71. [2] K.M. Furati, M.D. Kassim, N.e-.Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl, 64(6)(2012), 1616-1626. [3] I. Podlubny, Fractional Differential Equation, Academic Press, New York, 1999. [4] R. Hilfer, Applications in Fractional Calculus in Physics, Fractional Time in Evolution, World Scientific, London, 2000. [5] V. Lakshmikantham, S. Leela, J.V. Devi, Theory of Fractional Dynamics Systems, Cambridge Scientific Publishers, 2009. [6] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler functions, Related Topics and Applications, Springer, Verlag Berlin Heidelberg, 2014. [7] KD. Kucche, AD. Mali, C. da, JV. Sousa, On the nonlinear ψ-Hilfer fractional differential equation, Comput. Appl. Math, 38(2)(2019), 73. [8] C. da, JV. Sousa, EC. de Oliveira, On the ψ-Hilfer fractional derivative, Comun. NonLinear Sci. Numer. Simul, (60)2018, 72-91. [9] S. Arshad, On existence and uniqueness of solution of fuzzy fractional differential equations, Iran. J. Fuzzy Syst, 6(10)(2013), 137-151. [10] R.P. Agarwal, V. Lakashmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear. Anal, 72(2010), 2859-2862. [11] R. Alikhani, F. Bahrami, Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations, Commun. Nonlinear Sci.Numer.Simulat, 18(2013), 2007-2017. [12] T. Allahviranloo, Z. Gouyandeh, A. Armand, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, J. intell. Fuzzy Syst, 26(2014), 1481-1490 [13] N.V. Hoa, V. Lupulescu, D.O’Regan, Solving interval-valued fractional initial value problems under Caputo gH-fractional differentiability, Fuzzy Sets. Syst, 309(2017), 1-34. [14] P. Prakash, J.J. Nieto, S. Senthilvelavan, G. Sudha Priya, fuzzy fractional initial value problem, J. Intell. Fuzzy Syst, 28(2015), 2691-2704. [15] S.Salahshour, T. Allahviranloo, S. Abbasbandy and D. Baleanu, Existence and uniqueness results for fractional differential equations with uncertainty, Adv. Differ. Equ, 112(2012), 2012. [16] N.V. Hoa, Existence results for extremal solutions of interval fractional function integro-differential equations, Fuzzy Sets Syst. 347(2018), 29-53. [17] B.Bede, L.Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst. 230(2013), 119-141. [18] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V. Amsterdam, 2006. [19] V. Lakshmikantham, R.N. Mohapatra, Theory of Fuzzy Differential Equations and Applications, Taylor and Francis, London(2003). [20] N.V. Hoa, H. Vu, T.M. Duc, Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative approch, Fuzzy Sets, 375(2019), 70-99. [21] D.S. Oliveria, E. Capelas de Oliveria, Hilfer-Katugampola fractional derivative. Comput. Appl. Math. 37(2019), 3672-3690. 54 K. KANAGARAJAN, R. VIVEK, D. VIVEK AND E. M. ELSAYED [22] Mohammed A. Almalahi, Mohammed S. Abdo, Satish K. Panchal, Existence and Ulam Hyers stability results of a coupled system of ψ-Hilfer sequential fractional differential equations, Result in Appl. Math., 10 (2021), 100142. K. K ANAGARAJAN D EPARTMENT OF M ATHEMATICS , S RI R AMAKRISHNA M ISSION V IDYALAYA C OLLEGE OF A RTS AND S CI ENCE , C OIMBATORE -641020, I NDIA . Email address: [email protected] R. V IVEK D EPARTMENT OF M ATHEMATICS , S RI R AMAKRISHNA M ISSION V IDYALAYA C OLLEGE OF A RTS AND S CI ENCE , C OIMBATORE -641020, I NDIA . Email address: [email protected] D. V IVEK D EPARTMENT OF M ATHEMATICS , PSG C OLLEGE OF A RTS AND S CIENCE , C OIMBATORE -641014, I NDIA . Email address: [email protected] E. M. E LSAYED D EPARTMENT OF M ATHEMATICS , FACULTY OF S CIENCE ,M ANSOURA U NIVERSITY, M ANSOURA 35516, E GYPT. Email address: [email protected] ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 5, Number 1 (2022), 55-62 ISSN: 2582-0818 © http://www.technoskypub.com ON Ⓢ-CLOSED SETS AND SEMI Ⓢ-CLOSED IN NANO TOPOLOGICAL SPACES I. RAJASEKARAN∗ , N. SEKAR AND A. PANDI A BSTRACT. In this article focuss on nano Ⓢ-closed sets and nano Ⓢs -closed sets are introduce and study. Also, we introduce and study nano Ⓢ-continuous functions and nano Ⓢs -continuous functions. Furthermore, we introduce the notions of nano topological spaces called nano Ⓢ-T 1 space and nano Ⓢ-Ts space. 2 1. I NTRODUCTION AND P RELIMINARIES Several idea of nano topology have been generalized by considering the concept of nano semi-open sets due to M. L. Thivagar (2013) instead of nano open sets. The study of nano generalized closed sets in a nano topological space was initiated by K. Bhuvaneshwari (2014) and introduced the class of nano semi-generalized closed (nsg-closed), nano generalized semi-closed (ngs-closed) sets are used them to obtain some properties. In this article focuss on called nano Ⓢ-closed sets and nano Ⓢs -closed sets are introduce and study. Also, we introduce and study nano Ⓢ-continuous functions and nano Ⓢs continuous functions. Furthermore, we introduce the notions of nano topological spaces called nano Ⓢ-T 21 space and nano Ⓢ-Ts space. Definition 1.1. [9] Let U be a non-empty finite set of objects called the universe and R be an equivalence relation on U named as the indiscernibility relation. Elements belonging to the same equivalence class are said to be indiscernible with one another. The pair (U, R) is said to be the approximation space. Let X ⊆ U . (1) The lower approximation of X with respect to R is the set of all objects, which can be for certain classified as X with respect to R and it is denoted by LR (X). That S is, LR (X) = x∈U {R(x) : R(x) ⊆ X}, where τR (x) denotes the equivalence class determined by x. 2010 Mathematics Subject Classification. 54A05, 54A20, 54D10. Key words and phrases. nano Ⓢ-closed sets; nano Ⓢs -closed sets; nano topological spaces. Received: April 24, 2022. Accepted: June 20, 2022. Published: June 30, 2022. *Corresponding author. 55 56 I. RAJASEKARAN, N. SEKAR AND A. PANDI (2) The upper approximation of X with respect to R is the set of all objects, which can be possiblySclassified as X with respect to R and it is denoted by UR (X). That is, UR (X) = x∈U {R(x) : R(x) ∩ X ̸= φ}. (3) The boundary region of X with respect to R is the set of all objects, which can be classified neither as X nor as not - X with respect to R and it is denoted by BR (X). That is, BR (X) = UR (X) − LR (X). Definition 1.2. [7] Let U be the universe, R be an equivalence relation on U and τR (X) = {U, φ, LR (X), UR (X), BR (X)} where X ⊆ U . Then τR (X) satisfies the following axioms: (1) U and φ ∈ τR (X), (2) The union of the elements of any sub collection of τR (X) is in τR (X), (3) The intersection of the elements of any finite sub collection of τR (X) is in τR (X). Thus τR (X) is a topology on U called the nano topology with respect to X and (U, τR (X)) is called the nano topological space. The elements of τR (X) are called nano-open sets (briefly n-open sets). The complement of a n-open set is called n-closed. Through out this paper, we denote a nano topological space by (U, N ), (U ′ , N ′ ) where N = τR (X). The nano- interior and nano-closure of a subset H of U are denoted by In (H) and Cn (H), respectively. Definition 1.3. [7] A subset H of a space (U, N ) is called (1) nano semi-open (resp. ns-open) if H ⊆ Cn (In (H)). (2) nano α-open (resp. nα-open ) if H ⊆ In (Cn (In (H))). (3) nano regular-open (resp. nr-open) if H = In (Cn (H)). The complements of the above mentioned sets are called their respective closed sets. Definition 1.4. A subset H of a nano topological space (U, N ) is called (1) nano generalized closed (resp. ng-closed) [1] if Cn (H) ⊆ G, whenever H ⊆ G and G is n-open. (2) nano semi generalized closed (resp. nsg-closed) [3] if sCn (H) ⊆ G, whenever H ⊆ G and G is ns-open. (3) nano generalized semi closed (resp. ngs-closed) [3] if sCn (H) ⊆ G, whenever H ⊆ G and G is n-open. Definition 1.5. A function f : (U, N ) → (U ′ , N ′ ) is said to be (1) nano continuous (resp. n-continuous) [7] if f −1 (H ′ ) is n-open in (U, N ) for every n-open set H ′ of (U ′ , N ′ ). (2) nano generalized-continuous (resp. ng-continuous) [2] if f −1 (H ′ ) is ng-closed in (U, N ) for every n-closed set H ′ of (U ′ , N ′ ). (3) nano semi generalized-continuous (resp. nsg-continuous) [4] if f −1 (H ′ ) is nsgclosed in (U, N ) for every n-closed set H ′ of (U ′ , N ′ ). (4) nano generalized semi-continuous (resp. ngs-continuous) [10] if f −1 (H ′ ) is ngsclosed in (U, N ) for every n-closed set H ′ of (U ′ , N ′ ). (5) nano semi-continuous (resp. ns-continuous) [5] if f −1 (H ′ ) is ns-open in (U, N ) for every n-open set H ′ of (U ′ , N ′ ). (6) nano contra-continuous (resp. nc-continuous) [8] if f −1 (H ′ ) is n-closed in (U, N ) for every n-open set H ′ of (U ′ , N ′ ). (7) nano perfectly-continuous (resp. np-continuous) [8] if f −1 (H ′ ) is both n-open and n-closed in (U, N ) for every n-open set H ′ of (U ′ , N ′ ). ON Ⓢ-CLOSED SETS AND SEMI Ⓢ-CLOSED IN NANO TOPOLOGICAL SPACES 57 2. O N SOME Ⓢ- CLOSED SETS Definition 2.1. Let (U, N ) be a nanotopological space and H ⊆ U is said to be, (1) nano Ⓢ-closed (resp. nⓈ-closed) if H ⊆ G, G ∈ ns-open =⇒ sCn (H) ⊆ In (G). (2) nano Ⓢs -closed (resp. nⓈs -closed) if H ⊆ G, G ∈ ns-open =⇒ sCn (H) ⊆ In (Cn (G)). Proposition 2.1. A nanotopological space (U, N ), if (1) H is n-open and ns-closed =⇒ H is nⓈ-closed (2) H is nⓈ-closed set =⇒ H is nⓈs -closed and ngs-closed. (1) Let H be a n-open and ns-closed and H ⊆ G, where G is a ns-open. Then, sCn (H) = H = In (H) ⊆ In (H). Hence, H is nⓈ-closed. (2) Let H be a nⓈ-closed and H ⊆ G, where G is ns-open. Then, sCn (G) ⊆ In (G) ⊆ Cn (In (G)). Hence, H is a Ⓢs -closed. To prove the second part, let H be a nⓈ-closed and H ⊆ G, where G is a n-open. Then, sCn (H) ⊆ In (G) ⊆ G. Hence, H is ngs-closed. □ Proof. Remark. In Proposition 2.1, the converses are not necessarily true. (1) Not every nⓈ-closed set is ns-closed. (2) Not every nⓈ-closed set is nⓈs -closed. (3) Not every nⓈ-closed set is ngs-closed. Example 2.2. (1) Let U = {Y1 , Y2 , Y3 , Y4 }, U R = {{Y1 , Y2 }, {Y3 , Y4 }}, X = {Y3 , Y4 } and N = {φ, {Y3 , Y4 }, U }. The subset {Y1 , Y2 , Y3 } is nⓈ-closed but not nsclosed. (2) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } and N = {φ, {Y1 }, U }. The subset {Y1 } is nⓈs -closed but it is neither nⓈ-closed nor ngs-closed. Therefore {Y1 } is not ng-closed. (3) Let U = {Y1 , Y2 , Y3 }, U R = {{Y2 }, {Y1 , Y3 }}, X = {Y2 , Y3 } and N = {φ, {Y2 }, {Y3 }, {Y2 , Y3 }, U }. The subset {Y1 } is ng-closed and ngsclosed but it is neither nⓈs -closed nor nⓈ-closed. Remark. In nano topological spaces, (1) ng-closed and nⓈ-closed are independent. (2) ngs-closed and nⓈs -closed are independent. (3) ng-closed and nⓈs -closed are independent. Example 2.3. By Example 2.2(3), (1) (2) (3) (4) the subset {Y1 } is ng-closed but nⓈ-closed. the subset {Y2 } is nⓈ-closed but not ng-closed. the subset {Y1 , Y2 } is ng-closed but not nⓈs -closed. the subset {Y2 , Y3 } is nⓈs -closed but not ng-closed. Example 2.4. Let U = {Y1 , Y2 , Y3 }, {φ, {Y1 }, U }. U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then N = (1) the subset {Y2 } is ngs-closed but nⓈs -closed. (2) the subset {Y1 } is nⓈs -closed but not ngs-closed. 58 I. RAJASEKARAN, N. SEKAR AND A. PANDI Remark. We have the following relationship between nⓈ-closed sets, nⓈs -closed sets and related sets. n-clopen ↓ n-open and ns-closed ↓ nⓈ-closed nⓈs -closed set ↮ Diagram -I n-closed ↓ ng-closed ↮ ↮ nⓈs -closed ↮ ↮ Diagram -II ng-closed set ↮ → → ns-closed ↓ nsg-closed ↓ ngs-closed nⓈ-closed set ↓ ngs-closed set Proposition 2.2. If a subset H is nⓈ-closed, then sCn (H) − H does not contain a non empty ns-closed set. Proof. Let G be a ns-closed set such that G ⊆ sCn (H) − H. Then G ⊆ sCn (H) and H ⊆ U − G. Since H is nⓈ-closed, then sCn (H) ⊆ In (U − G) = U − Cn (G). Therefore, G ⊆ Cn (G) ⊆ U − sCn (H). Hence, G ⊆ (U − sCn (H)) ∩ sCn (H) = φ. □ Proposition 2.3. If a subset H is nⓈs -closed, then sCn (H) − H does not contain a non empty ns-clopen set. Proof. Let G be ns-clopen such that G ⊆ sCn (H) − H. Then we have that H ⊆ U − G and sCn (H) ⊆ In (Cn (U − G)) = U − Cn (In (G)). Thus we obtain G ⊆ Cn (In (G)) ⊆ U − sCn (H). Therefore, G ⊆ (U − sCn (H)) ∩ sCn (H) = φ. □ Proposition 2.4. If a subset H of (U, N ) is ns-open and nⓈ-closed, then it is ns-closed. Proof. Since H is ns-open and nⓈ-closed, then sCn (H) ⊆ In (H) ⊆ H. Hence, sCn (H) = H and H is ns-closed. □ Theorem 2.5. A subset H of nono topological spaces is nr-open ⇐⇒ H is nα-open and nⓈ-closed. Proof. Suppose H is nα-open and nⓈ-closed set. Then H is ns-open and nⓈ-closed and by Proposition 2.4, H is ns-closed. So, In (Cn (H)) ⊆ H. Since H is nα-open, then H ⊆ In (Cn (In (H))) ⊆ In (Cn (H)). Thus, H = In (Cn (H)) and H is nr-open. Conversely, let H be nr-open, then H is nα-open. Since H is nr-open, n-open and ns-closed. By Proposition 2.1, H is nⓈ-closed. □ Theorem 2.6. Every n-open set is ngs-closed ⇐⇒ nⓈ-closed. Proof. Let H be n-open and ngs-closed set. Assume that H ⊆ G, where G is a ns-open set. Thus H = In (H) ⊆ In (G). Since In (G) is n-open in U and H is ngs-closed, then sCn (H) ⊆ In (G) and H is nⓈ-closed set. Conversely, it is obvious that every nⓈ-closed set is ngs-closed. □ ON Ⓢ-CLOSED SETS AND SEMI Ⓢ-CLOSED IN NANO TOPOLOGICAL SPACES 59 3. O N NANO Ⓢ- CONTINUITY AND NANO Ⓢs - CONTINUITY Definition 3.1. A function f : (U, N ) → (U ′ , N ′ ) is said to be (1) nano Ⓢ-continuous (resp. nⓈ-continuous) if f −1 (H ′ ) is nⓈ-closed in (U, N ) for every n-closed set H ′ of (U ′ , N ′ ). (2) nano Ⓢs -continuous (resp. nⓈs -continuous) if f −1 (H ′ ) is nⓈs -closed in (U, N ) for every n-closed set H ′ of (U ′ , N ′ ). (3) nano Ⓢ-irresolute (resp. nⓈ-irresolute) if f −1 (H ′ ) is nⓈ-closed in (U, N ) for every nⓈ-closed set H ′ of (U ′ , N ′ ). (4) nano ⓈS -irresolute (resp. nⓈs -irresolute) if f −1 (H ′ ) is nⓈs -closed in (U, N ) for every nⓈs -closed set H ′ of (U ′ , N ′ ). Proposition 3.1. In a nano topological spaces, (1) Every nⓈ-continuous function is ngs-continuous. (2) Every nⓈ-continuous function is nⓈs -continuous. (3) Every nc-continuous and ns-continuous function is nⓈ-continuous. Remark. We have the following relationship between nⓈ-closed and related sets. Diagram -III np-cts → n-cts → ↓ ↓ n-contra cts and ns-cts ↮ ng-cts ↔ ↓ nⓈ-cts ↮ nⓈs -cts ↮ nⓈs -cts ↮ Diagram -IV ns-cts ↮ sets, nⓈs -closed sets ns-cts ↓ nsg-cts ↓ ngs-cts nⓈ-cts ↓ nsg-cts Remark. In Proposition 3.1, the converses are not necessarily true. (1) Not every ngs-continuous function is nⓈ-continuous. (2) nⓈs -continuous function need not be nⓈ-continuous. (3) nⓈ-continuous function is not always ns-continuous. Example 3.2. (1) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then ′ ′ N = {φ, {Y1 }, U } and let U = {Y1 , Y2 , Y3 }, UR = {{Y2 }, {Y1 , Y3 }}, X ′ = {Y2 , Y3 } then N ′ = {φ, {Y2 }, {Y1 , Y3 }, U ′ }. Clearly f : (U, N ) → (U ′ , N ′ ) is defined by f (Y1 ) = Y1 , f (Y2 ) = Y2 and f (Y3 ) = Y3 . Thus f is ngs-continuous but not nⓈ-continuous. (2) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then N = {φ, {Y1 }, U } ′ and let U ′ = {Y1 , Y2 , Y3 }, UR = {{Y2 }, {Y1 , Y3 }}, X ′ = {Y1 , Y3 } then N ′ = {φ, {Y1 , Y3 }, U ′ }. Clearly f : (U, N ) → (U ′ , N ′ ) is defined by f (Y1 ) = Y2 , f (Y2 ) = Y3 and f (Y3 ) = Y1 . Thus f is nⓈs -continuous but not nⓈ-continuous. (3) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then N = {φ, {Y1 }, U } ′ ′ and let U = {Y1 , Y2 , Y3 }, UR = {{Y2 }, {Y1 , Y3 }}, X ′ = {Y1 , Y3 } then N ′ = {φ, {Y1 , Y3 }, U ′ }. Clearly f : (U, N ) → (U ′ , N ′ ) is defined by f (Y1 ) = Y1 , f (Y2 ) = Y2 and f (Y3 ) = Y3 . Thus f is ns-continuous but not nⓈ-continuous. Remark. In a nano topological spaces, 60 I. RAJASEKARAN, N. SEKAR AND A. PANDI (1) nⓈs -continuity and ng-continuity are independent. (2) nⓈs -continuity and ngs-continuity are independent. Example 3.3. (1) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then ′ N = {φ, {Y1 }, U } and let U ′ = {Y1 , Y2 , Y3 }, UR = {{Y2 }, {Y1 , Y3 }}, X ′ = {Y2 , Y3 } then N ′ = {φ, {Y2 }, {Y1 , Y3 }, U ′ }. Clearly f : (U, N ) → (U ′ , N ′ ) is defined by f (Y1 ) = Y1 , f (Y2 ) = Y2 and f (Y3 ) = Y3 . Thus f is ng-continuous but not nⓈs -continuous. (2) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then N = {φ, {Y1 }, U } ′ ′ and let U = {Y1 , Y2 , Y3 }, UR = {{Y1 }, {Y2 , Y3 }}, X ′ = {Y2 , Y3 } then N ′ = {φ, {Y2 , Y3 }, U ′ }. Clearly f : (U, N ) → (U ′ , N ′ ) is defined by f (Y1 ) = Y1 , f (Y2 ) = Y2 and f (Y3 ) = Y3 . Thus f is nⓈs -continuous but not ng-continuous. (3) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then N = {φ, {Y1 }, U } ′ and let U ′ = {Y1 , Y2 , Y3 }, UR = {{Y2 }, {Y1 , Y3 }}, X ′ = {Y2 , Y3 } then N ′ = {φ, {Y2 }, {Y1 , Y3 }, U ′ }. Clearly f : (U, N ) → (U ′ , N ′ ) is defined by f (Y1 ) = Y1 , f (Y2 ) = Y2 and f (Y3 ) = Y3 . Thus f is ngs-continuous but not nⓈs continuous. (4) Let U = {Y1 , Y2 , Y3 }, U R = {{Y1 }, {Y2 , Y3 }}, X = {Y1 } then N = {φ, {Y1 }, U } ′ ′ and let U = {Y1 , Y2 , Y3 }, UR = {{Y1 }, {Y2 , Y3 }}, X ′ = {Y2 , Y3 } then N ′ = {φ, {Y2 , Y3 }, U ′ }. Clearly f : (U, N ) → (U ′ , N ′ ) is defined by f (Y1 ) = Y1 , f (Y2 ) = Y2 and f (Y3 ) = Y3 . Thus f is nⓈs -continuous but not ngs-continuous. 4. A PPLICATIONS Definition 4.1. A nano topological space (U, N ) is said to be (1) nⓈ-T 12 if every nⓈs -closed set is ns-closed. (2) nⓈ-Ts if every nⓈs -closed set is n-closed. Proposition 4.1. Let (U, N ) be a nano topological space. (1) For each x ∈ U , {x} is ns-closed or its complement U − {x} is nⓈ-closed. (2) For each x ∈ U , {x} is n-open and ns-closed or its complement U − {x} is nⓈs -closed. (1) Suppose that {x} is not ns-closed. Then U − {x} is not ns-open and the only ns-open set containing U − {x} is U . Therefore, sCn (U − {x}) ⊆ In (U ) = U . So, U − {x} is nⓈ-closed. (2) Suppose that {x} is not ns-closed. Then by (1), U − {x} is nⓈ-closed and then nⓈs -closed. Suppose that {x} is not n-open and let G be a ns-open set such that U − {x} ⊆ G. If G = U , then sCn (U − {x}) ⊆ In (Cn (G)) = G. If G = U − {x}, then we have that In (Cn (G)) = In (Cn (U − {x})) = In (U ) = U . Hence, sCn (U − {x}) ⊆ In (Cn (G)). Therefore, U − {x} is nⓈs -closed. □ Proof. Theorem 4.2. For a nano topological space, the next conditions are equivalent: (1) Every nⓈ-closed set is ns-closed. (2) For each x ∈ U , {x} is ns-open or ns-closed. Proof. (1) (1) =⇒ (2). Suppose that for a point x ∈ U , {x} is not ns-closed. By Proposition 4.1(1), U − {x} is nⓈ-closed. By assumption, U − {x} is ns-closed and hence {x} is ns-open. Therefore, each singleton is ns-open or ns-closed. ON Ⓢ-CLOSED SETS AND SEMI Ⓢ-CLOSED IN NANO TOPOLOGICAL SPACES 61 (2) (2) =⇒ (1). Let H be a nⓈ-closed set. We want to prove that sCn (H) = H. Suppose x ∈ sCn (H). Case 1: {x} is ns-open. Then {x} ∩ H ̸= φ which implies x ∈ H. Case 2: {x} is ns-closed and x ∈ / H. Then sCn (H) − H contains a ns-closed set {x} and this contradicts Proposition 2.2. Hence x ∈ H and H is ns-closed. Therefore, Every nⓈ-closed set is nsemi-closed. □ Theorem 4.3. For a nano topological space, the following properties hold: (1) If (U, N ) is nⓈ- Ts , then for each x ∈ U the singleton {x} is n-open or ns-closed. (2) (U, N ) is nⓈ- T 12 ⇐⇒ for each x ∈ U , {x} is ns-open or ns-closed and n-open. (3) If (U, N ) is nⓈ-Ts , then it is nⓈ-T 21 . Proof. (1) Suppose that for some x ∈ U , {x} is not ns-closed. By Proposition 4.1, U − {x} is nⓈ-closed. Hence, U − {x} is nⓈs -closed. Since (U, N ) is nⓈ-Ts , then U − {x} is n-closed. Thus {x} is n-open. (2) Necessity. Suppose that a singleton {x} is not ns-closed or n-open. By Proposition 4.1, U − {x} is nⓈs -closed. Using the assumption we have that {x} is ns-open. Sufficiency. It follows from the assumption that every subset is ns-open and ns-closed. Then (U, N ) is nⓈ-T 21 . (3) It is straightforward from the definitions of nⓈ-Ts spaces and nⓈ-T 21 spaces. □ 5. C ONCLUSION In this paper we have discussed the concepts of nⓈ-closed and nⓈs -closed. These concepts can be used to derive a new real world applications in future. 6. ACKNOWLEDGEMENT The authors thank the referees for their valuable comments and suggestions for improvement of this paper R EFERENCES [1] K. Bhuvaneshwari and K. Mythili Gnanapriya, Nano Generalizesd closed sets, International Journal of Scientific and Research Publications, 4(5)(2014), 1-3. [2] K. Bhuvaneshwari and K. Mythili Gnanapriya, On nano generalizesd continuous functions in nano topololgical space, International Journal of Mathematical Archive , 6(6)(2015), 182-186. [3] K. Bhuvaneshwari and K. Ezhilarasi, On nano semi generalized and nano generalized semi-closed sets, IJMCAR. 4(3)(2014), 117-124. [4] K. Bhuvaneshwari and K. Ezhilarasi, On nano semi generalized continuous maps in nano topological space, International Research Journal of pure Algebra, 5(9)(2015), 149-155. [5] P. Karthik Sankar, Nano totally continuous functions in nano topological space, International Journal of Science Research & Engineering Trends, 5(1)(2019), 234-236. [6] M. Lellis Thivagar and Carmel Richard, On Nano forms of weakly open sets, International Journal of Mathematics and Statistics Invention,1(1)(2013), 31-37. [7] M. Lellis Thivagar and Carmel Richard, On nano continuity , Math. Theory Model, (7)(2013), 32-37. [8] M. Lellis Thivagar, Saeid Jafari and V. Sutha Devi, On new class of contra continuity in nanotopology, Available on researchgate in https://www.reserchgate.net/publication/315892547. 62 I. RAJASEKARAN, N. SEKAR AND A. PANDI [9] Z. Pawlak, Rough sets, International journal of computer and Information Sciences, 11(5)(1982), 341-356. [10] I. Rajasekaran, O. Nethaji and S. Jackson, On nano gs -closed sets and nano gs -continuous functions, Bull. Int. Math. Virtual Inst., 11(3)(2021), 539-543. [11] I. Rajasekaran, On ⋆ b-open sets and ⋆ b-sets in nano topological spaces, Asia Mathematika, 5(3)(2021), 84-88. [12] I. Rajasekaran and O. Nethaji, On some new subsets of nano topologicalspaces, Journal of New Theory, 16(2017), 52-58. I. R AJASEKARAN D EPARTMENT OF M ATHEMATICS , T IRUNELVELI DAKSHINA M ARA NADAR S ANGAM C OLLEGE , T. K ALLIKULAM 627 113, T IRUNELVELI D ISTRICT, TAMIL NADU , I NDIA Email address: [email protected] N. S EKAR D EPARTMENT OF M ATHEMATICS , A RUMUGAM P ILLAI S EETHAI A MMAL C OLLEGE , T IRUPPATTUR , S IVA GANGAI D ISTRICT, TAMIL NADU , I NDIA Email address: [email protected] A. PANDI D EPARTMENT OF M ATHEMATICS , R ATHINAM T ECHNICAL C AMPUS , C OIMBATORE , TAMIL NADU , I NDIA Email address: [email protected] ANNALS OF COMMUNICATIONS IN MATHEMATICS Volume 5, Number 1 (2022), 63-73 ISSN: 2582-0818 © http://www.technoskypub.com L-FUZZIFYING PROXIMITY, L-FUZZIFYING UNIFORM SPACE AND L-FUZZIFYING STRONG UNIFORM SPACE MOHAMMED M. KHALAF A BSTRACT. In this paper the concept of proximity in L-fuzzifying topology is established and some of its properties are discussed. Furthermore we introduce and study the concepts ofL-fuzzifying uniform space and L-fuzzifying strong uniform space. 1. P RELIMINARIES In 1993, M. Ying [11] introduced and studied the uniformity in [0, 1]−fuzzifying topology as a fuzzy concept, i.e., as a fuzzy subset of P (X × X) for an ordinary set X. In 2003, H. F. Kheder [5], introduced and studied concepts of proximity and strong uniformity in fuzzifying topology as fuzzy concepts. In this paper we introduce and study the concept of proximity, uniformity and strong uniformity in L-fuzzifying topology. In section 2, we extend the concept of fuzzifying proximity due to (Kheder, et al (2003)[5]) into L-fuzzifying setting. Some of basic properties of this extenstion are studied. Section 3, is devoted to extend and study the concept of uniformity in the sense of (Ying (1993)[10])in L–fuzzifying topology. Finally, the notion of fuzzifying strong uniform space (Kheder, et al (2003)[5]) is generalized by introducing the concept of L-fuzzifying strong uniform spaces. Some results concerning this concept are obtained. In the present paper L is assumed to be a completely residuated lattice such that the following conditions are satisfied: (1) L is totally ordered as a poset.( i.e. for each a, b ∈ L, a < b, or b < a. ) (2) L satisfies that , ∧, is disributive over arbitrary joins. Definition 1.2. [9]. A structure (L,∨, ∧,∗, →, ⊥, ⊤) is called a complete residuated lattice iff (1) (L,∨, ∧, ⊥, ⊤) is a complete lattice whose greatest and least element are ⊤, ⊥ respectively, (2) (L,∗, ⊤) is a commutative monoid, i.e., 2010 Mathematics Subject Classification. 54A05, 54A20, 54D10. Key words and phrases. fuzzifying topology; fuzzifying proximity; fuzzifying uniformity. Received: May 24, 2022. Accepted: June 25, 2022. Published: June 30, 2022. 63 MOHAMMED M. KHALAF 64 (a) ∗ is a commutative and associative binary operation on L, and (b) ∀ a ∈ L, a ∗ ⊤ = ⊤ ∗ a = a, (3)(a) ∗ is isotone, (b) → is a binary operation on L which is antitone in the first and isotone in the second variable, (c) → is couple with ∗ as: a ∗ b ≤ c iff a ≤ b → c ∀ a, b, c ∈ L. The basic operations on the family LX of all L-sets on a non-empty set X was defined as follows: Definition 1.3. [1]. A complete lattice L is called completely distributive if the following law is satisfied: ∀{A j |j ∈ J } ⊆WP (L), V W Vwhere P (L) is the power subset of L we have, Aj = ( f (j)). Q j∈J Aj j∈J f∈ j∈J Definition 1.4.(Csa′ sza′ r (1978)[2]). A binary relation δ on P (X) × P (X) is called a proximity on a set X if it satisfies the following conditions: (P1) If (A, B) ∈ δ, then A ̸= φ and B ̸= φ and δ(φ, X) = 0, (P2) If A ∩ B ̸= φ, then (A, B) ∈ δ (P3) If (A1 ∪ A2 , C) ∈ δ, then (A1 , C) ∈ δ or (A2 , C) ∈ δ (P4) If (A, B) ∈ δ, then (B, A) ∈ δ (P4) If (A, B) ∈ / δ, then there exists D such that (A, D) ∈ / δ and (X − D, B) ∈ /δ The pair (X, δ) is said to be a proximity space. The following concepts are given in (Kheder, et. al. ( 2003)[5]). Definition 1.5. Let X be a set and let δ ∈ I (P (X)×P (X)) , i.e., δ : P (X) × P (X) → [0, 1]. Assume that for any A, B, C ∈ P (X) the following axioms are satisfied: (F P 1) |= ¬(X, φ) ∈ δ, (F P 2) |= (A, B) ∈ δ ↔ (B, A) ∈ δ, (F P 3) |= (A, B ∪ C) ∈ δ ↔ (A, B) ∈ δ ∨ (A, C) ∈ δ, (F P 4) for every A, B ⊆ X, there exists C ⊆ X such that |= ((A, C) ∈ δ ∨ (B, X − C) ∈ δ) → (A, B) ∈ δ, (F P 5) |= {x} ≡ {y} ↔ ({x}, {y}) ∈ δ. Then δ is called a fuzzifying proximity on X and (X, δ) is called a fuzzifying proximity space. Theorem 1.1. Let (X, δ) be a fuzzifying proximity space. Then we have (1) |= (A, B) ∈ δ ∧ B ⊆ C → (A, C) ∈ δ, (2) |= (A ∩ B) ̸= φ → (A, B) ∈ δ, (3) |= ¬δ(A, φ). Proposition 1.1. For every α ∈ (0, 1], δα is a proximity on X, where δα is the α-level of δ, i.e., δα = {(A, B) : δ(A, B) ≥ α}. Definition 1.6. Let (X, δ) be a fuzzifying proximity space. For each α ∈ (0, 1], we define the interior operation induced by δα , denoted by S intδα : P (X) → P (X), as follows: intδα (A) = B ∀A ∈ P (X). B∈P (X),(B,X−A)∈δ / α Proposition 1.2. For every α ∈ (0, 1], the family τδα = {A : A ⊆ X and intδα (A) = A} is a topology on X. L-FUZZIFYING PROXIMITY AND L-FUZZIFYING STRONG UNIFORM SPACE 65 Theorem 1.2. Let (X, δ) be a fuzzifying proximity space. The mapping τδ : P (X) → W α is a fuzzifying topology and is called the [0, 1] defined by: τδ (A) = α∈(0,1) ,A∈τδα fuzzifying topology induced by the fuzzifying proximity δ. Definition 1.7. (Csa′ sza′ r (1978)[2]). A uniform structure U on a set X is a family of subsets of X × X, called entourage, which satisfies the following properties: (U 1) If u ∈ U, then △ ⊆ u, where △ is the diagonal: △ = {(x, x) |x ∈ X } (U 2) If v ⊆ u, and v ∈ U then u ∈ U, (U 3) for every u, v ∈ U, u ∩ v ∈ U, (U 4) If u ∈ U, then u−1 ∈ U, where u−1 = {(x, y) |(y, x) ∈ u }. (U 3) for every u ∈ U, there exists v ⊆ U such that v ◦ v ⊆ u, where v ◦ v ⊆ u, where v ◦ u is defined by: v ◦ u = {(x, y)| ∃z ∈ X such that (x, z) ∈ u and (z, y) ∈ u}, ∀x, y ∈ X. The pair (X, U ) is said to be a uniform space. The following results are given in [Ying (1992)[11]). Definition 1.8. Let X be a set and U ∈ I P (X×X) . If for any U, V ⊆ X × X, (U 1) |= (U ∈ U ) →(△ ⊆ U ), (U 2) |= (U ∈ U ) →(U −1 ∈ U), (U 3) |= (U ∈ U ) →(∃V )(V ∈ U) ∧ (V ◦ V ⊆ U ), (U 4) |= (U ∈ U )∧(V ∈ U) → (U ∩ V ⊆ U ), (U 5) |= (U ∈ U)∧(U ⊆ V ) → (V ∈ U ). Then, U is called Fuzzifying uniformity and (X, U ) is called fuzzifying uniform space. Lemma 1.1. Let (X, U ) be a fuzzifying uniform space and ℑ ∈ I P (X) defined by: T ∈ ℑ := (∀x)(x V W∈ T ) → (∃U )((U ∈ U) ∧ (U [x] ⊆ T ))), T ⊆ X i.e., ℑ(T ) := U (U ), T ⊆ X. where U [x] = {y ∈ X : (x, y) ∈ U }. Then ℑ is a x∈T U [x]⊆T fuzzifying topology on X and called the fuzzifying (uniform) topology of U . The following concepts are given in (Kheder, et. al. ( 2003)[5]). Definition 1.9. Let X be a set and let U : P (X × X) → I. Assume that U is normal, i.e. ∃ U ⊆ X × X s.t. U [U ] = 1. If for any U, V ⊆ X × X, (F U 1) |= (U ∈ U ) →(△ ⊆ U ), (F U 2) |= (U ∈ U ) →(U −1 ∈ U), (F U 3)∗ There exists H ❁ P (X × X) s.t. |= (U ∈ U ) →(∃V )(V ∈ H) ∧ (V ∈ U ) ∧ (V ◦ V ⊆ U ),where ❁ stands for ,, a finite subset of,, , (F U 4) |= (U ∈ U )∧(V ∈ U) → (U ∩ V ⊆ U ), (F U 5) |= (U ∈ U)∧(U ⊆ V ) → (V ∈ U ). Then, U is called a strong fuzzifying uniformity and (X, U ) is called a strong fuzzifying uniform space. 66 MOHAMMED M. KHALAF Theorem 1.3. Let (X, U ) be a strong fuzzifying uniform space. Then for each α ∈ (0, 1), the α level of U denoted by Uα is a classical uniformity on X. where Uα = {U ∈ P (X × X) s.t. U (U ) ≥ α}. Theorem 1.4. Let (X, U ) be a strong W fuzzifying uniform space. The fuzzy set τU ∈ α, is a fuzzifying topology. It is called the (F(P (X)), defined by: τU (A) = α∈(0,1],A∈τUα fuzzifying topology induced by the strong fuzzifying uniformity U Theorem 1.5 Let δUα be the proximity induced by the uniformity W Uα . Then the mapping α, is a fuzzifying δU : P (X × X) → [0, 1], defined by δU (A, B) = α∈(0,1],(A,B)∈δUα proximity. It is called the fuzzifying proximity induced by the strong fuzzifying uniformity U. 2. L-fuzzifying proximity space Definition 2.1. The binary crisp predicate CE ∈ {⊥, ⊤}P (X)×P (X) , called crisp equality, is given as follows:  ⊤ if A = B CE(A, B) = ⊥ if A ̸= B Definition 2.2. Let X be a set and let δ ∈ LP (X)×P (X) , i.e., δ : P (X) × P (X) → L. Assume that for every A, B, C ∈ P (X), the following axioms are satisfied: (LF P 1) δ(X, φ) = ⊥, (LF P 2) δ(B, A) = δ(A, B), (LF P 3) δ(A, B ∪ C) = δ(A, B) ∨ δ(A, C), (LF P 4) For every A, B ∈ P (X), ∃C ∈ P (X) s.t. δ(A, B) ≥ δ(A, C) ∨ δ(B, X − C), (LF P 5) δ({x}, {y}) = CE({x}, {y}). Then δ is called an L-fuzzifying proximity on X and (X, δ) is called an L-fuzzifying proximity space. Definition 2.3. The binary crisp predicate ⊆∈ {⊥, ⊤}P (X)×P (X) ,called crisp inclusion, is defined as follows:  ⊤ if A ⊆ B, ⊆ (A, B) = ⊥ if A ⊈ B. Definition 2.4. The binary crisp predicate ∩ ∈ {⊥, ⊤}P (X)×P (X) , called crisp intersection, is defined as follows:  ⊤ if A ∩ B ̸= φ ∩(A, B) = ⊥ if A ∩ B = φ Lemma 2.1. If ⊆ (B, C) = ⊤, then δ(A, B) ≤ δ(A, C) ∀A ∈ P (X). Proof. δ(A, C) = δ(A, B ∪ C) = δ(A, B) ∨ δ(A, C) ≥ δ(A, B). L-FUZZIFYING PROXIMITY AND L-FUZZIFYING STRONG UNIFORM SPACE 67 Theorem 2.1. Let (X, δ) be an L-fuzzifying proximity space. For every A, B, C ∈ P (X),then we have (1) δ(A, C) ≥ δ(A, B)∧ ⊆ (B, C), (2) δ(B, A) ≥ ∩(A, B), (3) δ(A, φ) → ⊥ = ⊤. Proof. (1) If ⊆ (B, C) = ⊥, then δ(A, C) ≥ δ(A, B) ∧ ⊥ and if ⊆ (B, C) = ⊤, from Lemma 2.1 we have δ(A, C) ≥ δ(A, B) ∧ ⊤. Then δ(A, C) ≥ δ(A, B)∧ ⊆ (B, C). (2) If ∩(A, B) = ⊥, the result hold. Let ∩(A, B) = ⊤, i.e., ∃x ∈ A ∩ B. From (LF P 5), δ({x}, {x}) = ⊤. Applying Lemma 2.1 and (LF P 2), δ(A, B) ≥ δ(A, {x}) = δ({x}, A) ≥ δ({x}, {x}) = ⊤. Hence δ(A, B) ≥ ∩(A, B). (3) From Lemma 2.1, δ(A, φ) ≤ δ(X, φ). Then δ(A, φ) → ⊥ ≥ δ(X, φ) → ⊥ = ⊥ → ⊥ = ⊤. Theorem 2.2. For every α ∈ L-{⊥}, δα is a proximity on X, where δα is the α-cut of an L-fuzzifying proximity δ, i.e., δα = {(A, B) : δ(A, B) ≥ α}. Proof. Let α ∈ L − {⊥}. (P 1) From (LF P 1) we have δ(X, φ) = ⊥. Then δ(X, φ) < α. So, (X, φ) ∈ / δα . (P 2) Suppose (A, B) ∈ δα . Then δ(A, B) ≥ α. From (LF P 2), δ(B, A) = δ(A, B) ≥ α. Hence (B, A) ∈ δα . (P 3) Let (A, B ∪ C) ∈ δα then δ(A, B ∪ C) ≥ α. From (LF P 3) we have δ(A, B) ≥ α or δ(A, C) ≥ α and hence (A, B) ∈ δα or (A, C) ∈ δα . (P 4) Let (A, B) ∈ / δα .since L is totally ordered then we have δ(A, B) < α. From (LF P 4) there exists C ∈ P (X) such that δ(A, B) ≥ δ(A, C) ∨ δ(B, X − C). Then δ(A, C) ∨ δ(B, X − C) < α which implies that δ(A, C) < α and δ(B, X−C) < α which implies that (A, C) ∈ / δα and (B, X−C) ∈ / δα . (P 5) Suppose x = y. Then CE({x}, {y}) = ⊤ so that from (LF P 5), δ({x}, {y}) = ⊤ ≥ α. Hence ({x}, {y}) ∈ δα . Definition 2.5. Let (X, δ) be an L- fuzzifying proximity space. For each α ∈ L-{⊥}, the interior operation induced S by δα , denoted by intδα : P (X) → P (X), is defined as follows: intδα (A) = B ∀A ∈ P (X). B∈P (X),(B,X−A)∈δ / α Theorem 2.3. For every α ∈ L-{⊥}, the family τδα = {A : A ⊆ X and intδα (A) = A} is a classical topology on X. Proof. Let α ∈ L-{⊥}. Then: MOHAMMED M. KHALAF 68 S (1)since intδα (X) = B B∈P (X),(B,X)∈δ / α B∈P (X),(B,φ)∈δ / α = φ, then X, φ ∈ τδα . S B = X and intδα (φ) = (2) Let A, C ∈ τδα s.t. intδα (A) = A and intδα (C) = C. S Then intδα (A ∩ C) = B B∈P (X),(B,X−(A∩C))∈δ / α S = S B = B∈P (X),(B,(X−A)∪(X−C))∈δ / α ! S B ∩ B∈P (X),(B,X−A)∈δ / α B B∈P (X),(B,X−C)∈δ / α ! = A ∩ C. So, A ∩ C ∈ τ δα . S (3) Let {Aλ : λ ∈ Λ} ⊆ τ δα . Now Aλ = S int δα (Aλ ) ⊆ int δα ( Aλ ), λ∈Λ λ∈Λ λ∈Λ S S because int δα is monotone (indeed, If A ⊆ C, then int δα (A) = B B∈P (X),(B,X−A)∈δ / α S ⊆ B = int δα (C).). Also, int δα ( S because int δα (A) = B ⊆ B∈P (X),(B,X−A)∈δ / α S Aλ ) ⊆ S Aλ λ∈Λ λ∈Λ B∈P (X),(B,X−C)∈δ / α = S S B B ∈P (X), ∩(B, X−A)=⊥ B = A for any A ∈ P (X) B∈P (X),B ⊆A Then int δα ( S Aλ ) = S Aλ ∈ τ δα . λ∈Λ λ∈Λ λ∈Λ S Aλ . Hence Theorem 2.4. Let (X, δ) be an L- fuzzifying proximity space and let X satisfies the W comα is pletely distributive law. The mapping τδ : P (X) → L defined by: τδ (A) = α∈L−{⊥},A∈τ δα an L- fuzzifying topology and is called the L-fuzzifying topology induced by the Lfuzzifying proximity δ. W Proof. (1) τδ (X) = α = ⊤, τδ (φ) = α∈L−{⊥}, X∈τ W (2) τδ (A ∩ B) = = W α∈L−{⊥},A∈τ α δα ! ∧ δα W α∈L−{⊥},B∈τ α = ⊤. α∈L−{⊥},φ∈τ δα W α≥ α∈L−{⊥},A∩B∈τ W α∈L−{⊥},A∈τ α δα ! α δα ∧B∈τ δα = τδ (A) ∧ τδ (B). δα L-FUZZIFYING PROXIMITY AND L-FUZZIFYING STRONG UNIFORM SPACE (3) Let {Aλ : λ ∈ Λ} ⊆ P (X). Then we have, τδ ( S W ≥ α= W α∈L−{⊥}, S λ∈Λ α= λ∈Λ α∈L−{⊥},Aλ ∈τ δα ,λ∈Λ α∈L−{⊥},Aλ ∈τ V W Aλ ) = λ∈Λ 69 V α Aλ ∈τ δα τδ (Aλ ). λ∈Λ δα 3. L-fuzzifying uniformity and L-fuzzifying strong uniformity Definition 3.1. Let X be a nonempty set and let U ∈ LP (X×X) . Assume that the following statments are satisfied: (LF U 0) There exists U ∈ P (X × X) s.t. U (U ) = ⊤, (LF U 1) For any U ∈ P (X × X), U (U ) > 0, [[△, U [[= ⊤, −1 (LF U 2) For any U ∈ P (X × X), U (U W ) = U (U ), U (V )∧ ⊆ (V ◦ V, U ) ≥ U(U ), (LF U 3) For any U ∈ P (X × X), V ⊆X×X (LF U 4) For any U, V ∈ P (X × X), U (U ∩ V ) ≥ U(U ) ∧ U (V ), (LF U 5) For any U, V ∈ P (X × X), U (V ) ≥ U (U )∧ ⊆ (U, V ). Then U is called an L-fuzzifying uniformity and (X, U ) is called an L-fuzzifying uniform space. P (X) Theorem 3.1. defined V LetW(X, U ) be an L-fuzzifying uniform space . Then τ ∈ L by τ (A) = U (U ) ∀A ∈ P (X). is an L-fuzzifying topology on X and is called x∈A U [x]⊆A the L-fuzzifying topology on X induced by U . Proof. It is clear from (LF U 0) that τ (X) = ⊤. Let A1 , A2 ⊆ X. From (LF U 4) we have, W V τ (A1 ) ∧ τ (A2 ) = ( U (U1 )) ∧ ( x1 ∈A1 U1 [x1 ]⊆A1 W V = W V U (U2 )) x2 ∈A2 U2 [x2 ]⊆A2 (U (U1 ) ∧ U (U2 )) x1 ∈A1 ,x2 ∈A2 U1 [x1 ]⊆A1 , U2 [x2 ]⊆A2 V ≤ x1 ∈A1 ,x2 ∈A2 V ≤ x∈A1 ∩A2 V ≤ x∈A1 ∩A2 we have τ ( ≥ ( V U (U1 ∩ U2 ) U1 [x1 ]⊆A1 , U2 [x2 ]⊆A2 W U (U1 ∩ U2 ) U1 ∩ U2 [x]⊆A1 ∩A2 W U (U ) = τ (A1 ∩ A2 ).Finally, for any Ai ⊆ X (i ∈ I), U [x]⊆A1 ∩A2 S Ai ) = i∈I V W x∈ V S W Ai U [x]⊆ i∈I x∈Ai U [x]⊆Ai U (U )) = U (U ) = Ai i∈I i∈I W S V V ( V W i∈I x∈Ai U [x]⊆ S U (U )) Ai i∈I τ (Ai ). i∈I In [5], the authers introduced a counterexample in [0, 1]-fuzzifying setting to illustrate that there exists some α-cut of the [0, 1]-fuzzifying uniformity in the sense of M. S. Ying [11], which not a uniformity. In the following we introduce the concept of an L-fuzzifying strong uniform space as a generalization of the concept of fuzzifying strong uniform space [5]. MOHAMMED M. KHALAF 70 Definition 3.3. Let X be a nonempty set and let U ∈ LP (X×X) . If the following statments are satisfied: (LF SU 0) There exists U ∈ P (X × X) s.t. U (U ) = ⊤ (LF SU 1) For any U ∈ P (X × X), [[△, U [[= ⊤, (LF SU 2) For any U ∈ P (X × X), U (U ) ≤ U(U −1 ), (LF SU 3) For any U ∈ P (X × X), ∃ V ∈ P (X × X) s.t. V ◦ V ⊆ U and U (V ) ≥ U(U ), (LF SU 4) For every U, V ∈ P (X × X), U (U ∩ V ) ≥ U(U ) ∧ U (V ), and (LF SU 5) For every U, V ∈ P (X ×X), U (V ) ≥ U(U )∧ ⊆ (U, V ), then U is called an L-fuzzifying strong uniform and (X, U ) is called an L-fuzzifying strong uniformity space. Remark 3.1. If L = [0, 1], the condition (LF SU 3) implies the condition (F U 3)∗ in Definition 1.9. Theorem 3.2. Let (X, U ) be an L-fuzzifying strong uniformity space. Then for each α ∈ L − {⊥}, then the α-cut of U denoted by Uα is a uniformity . Proof. Let α ∈ L − {⊥}. (U 0) From (LF SU 0) ∃U ∈ P (X × X) s.t. U (U ) = ⊤ ≥ α so that Uα ̸= φ. (U 1) Let U ∈ Uα . So from condition (LF SU 1), [[△, U [[= ⊤ so that △ ⊆ U. (U 2) Let U ∈ Uα . So from (LF SU 2), U (U −1 ) ≥ U(U ) ≥ α. Then U −1 ∈ Uα . (U 3) Let U ∈ Uα . Then from (LF SU 3), ∃ V ∈ P (X × X) s.t. V ◦ V ⊆ U and U (V ) ≥ U(U ) ≥ α. Hence V ∈ Uα . (U 4) Let U, V ∈ Uα . Then from (LF SU 4), U (U ∩ V ) ≥ U(U ) ∧ U (V ) ≥ α so that U ∩ V ∈ Uα . (U 5) Let U ∈ Uα and U ⊆ V. Then from (LF SU 5), U (V ) ≥ U(U )∧ ⊆ (U, V ) = U (U ) ≥ α. Hence V ∈ Uα . Theorem 3.3. Let (X, U ) be an L-fuzzifying strong uniformity space and let X satisfies theW completely distributive law. The L-fuzzy set τU ∈ LP (X) , defined by: τU (A) = α is an L-fuzzifying topology. It is called the L-fuzzifying topology inα∈L−{⊥}, A∈τUα duced by the L-fuzzifying strong uniformity U . W α = ⊤. Proof. (1) Since X, φ ∈ τU⊤ , then we have that τU (X) = α∈L−{⊥}, X∈τUα W α = ⊤. τU (φ) = and α∈L−{⊥}, φ∈τUα (2) τU (A ∩ B) = W α∈L−{⊥}, A∩B∈τUα α ≥ W (α1 ∧ α2 ) (α1 ∧α2 )∈L−{⊥}, A∈τUα ,B∈τUα 1 2 L-FUZZIFYING PROXIMITY AND L-FUZZIFYING STRONG UNIFORM SPACE W = α∈L−{⊥}, A∈τUα (3) τU ( S V W W Aλ ) = α∈L−{⊥}, V W α= λ∈Λ α∈Mλ λ∈Λ 2 W α≥ S λ∈Λ Mλ = α2 = τU (A) ∧ τU (B). α∈L−{⊥}, B∈τUα 1 λ∈Λ = W α1 ∧ 71 α∈L−{⊥}, Aλ ∈τUα ,λ∈Λ Aλ ∈τUα V W α= V α= λ∈Λ Aλ ∈τUα f∈ W Q V Mλ λ∈Λ λ∈Λ τU (Aλ ). λ∈Λ Where Mλ = {α ∈ L − {⊥} : Aλ ∈ τUα ∀ λ ∈ Λ} . Theorem 3.4. Let δUα be the proximity induced by the uniformity Uα α ∈ L−{⊥}. Then W α is an Lthe mapping δU ∈ LP (X)×P (X) defined by δU (A, B) = α∈L−{⊥}, (A,B)∈δUα fuzzifying proximity. It is called the L-fuzzifying proximity induced by the L-fuzzifying strong uniformity U . W Proof. (LF P 1) δU (X, φ) = α = ⊥. α∈L−{⊥}, (X,φ)∈δUα W (LF P 2) δU (A, B) = α = δU (B, A). α∈L−{⊥}, (B,A)∈δUα α∈L−{⊥}, (A,B)∈δUα W (LF P 3) δU (A, B ∪ C) = W α= α α∈L−{⊥}, (A,B∪C)∈δUα W = α α∈L−{⊥}, (A,B)∈δUα ! ∨ W α α∈L−{⊥}, (A,C)∈δUα ! = δU (A, B) ∨ δU (A, C). / δ Uα (LF P 4) (A, B) ∈ / δUα ⇒ ∃C ∈ P (X) s.t. (A, C) ∈ and (B, X − C) ∈ / δUα . Therefore δU (A, B) = W ≥ α∈L−{⊥}, (A,C)∈δUα =( W α∈L−{⊥}, (A,C)∈δUα C). or α) W α α∈L−{⊥}, (A,B)∈δUα α α∈L−{⊥}, (B,X−C)∈δUα ∨ ( W α ) = δU (A, C) ∨ δU (B, X − α∈L−{⊥}, (B,X−C)∈δUα (LF P 5) Frist suppose that CE({x}, {y}) = ⊤. Then {x} = {y}. So, f (λ) MOHAMMED M. KHALAF 72 ({x}, {y}) ∈ δUα , for any α ∈ L − {⊥}, Therefore δU ({x}, {y}) = W α = ⊤. Second if CE({x}, {y}) = ⊥, α∈L−{⊥}, ({x},{y})∈δUα then x ∈ / {y}. So, ({x}, {y}) ∈ / δUα , for any α ∈ L − {⊥}. Hence δU ({x}, {y}) = W α = ⊥. α∈L−{⊥}, ({x},{y})∈δ / Uα 4. Conclusions in this paper, the notion of fuzzifying strong uniform space (Kheder, et al (2003)[5]) is generalized by introducing the concept of L-fuzzifying strong uniform spaces. Some results concerning this concept are obtained. In the present paper L is assumed to be a completely residuated lattice such that the following conditions are satisfied: (1) L is totally ordered as a poset.( i.e. for each a, b ∈ L, a < b, or b < a. ) (2) L satisfies that , ∧, is disributive over arbitrary joins. In the future, we will study topological notions defined by means of regular open sets when these are planted into the framework of Ying’s fuzzifying topological spaces (in Lukasiewicz fuzzy logic). We used fuzzy logic to introduce almost separation axioms (almost Hausdorff)-, (almost-regular)-and (almost-normal). we gave the relations of these axioms with each other as well as the relations with other fuzzifying separation axioms. 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