Algebraic Approximations to Partial Group Structures

We are IntechOpen, the world’s leading publisher of Open Access books Built by scientists, for scientists 5,900 144,000 180M Open access books available International authors and editors Downloads Our authors are among the 154 TOP 1% 12.2% Countries delivered to most cited scientists Contributors from top 500 universities Selection of our books indexed in the Book Citation Index in Web of Science™ Core Collection (BKCI) Interested in publishing with us? Contact [email protected] Numbers displayed above are based on latest data collected. For more information visit www.intechopen.com Chapter Algebraic Approximations to Partial Group Structures Özen Özer Abstract In this work, we use ‘Partial Group’ notion and we do further investigations about partial groups. We define ‘Partial Normal Subgroup’ using partial conjugation criteria and we prove few results about partial normal subgroups analogous to normal groups. Also, we define congruence relation for partial groups and via this relation, we state ‘The Quotient of Partial Group or Factor Group’. We give isomorphism theorems for partial groups. Explicitly, this is an analogous concept to group theory and our main is where differences partial groups from groups. Keywords: partial group, partial normal subgroup, partial quotient group, isomorphism theorems for partial groups 1. Introduction It is defined that a group is a set equipped with an operation described on it such that it has some properties as associated elements, an identity element, and inverse elements. Another definition can also be given as algebraic that the group is the set of all the permutations for algebraic expression’s roots that displays the typical that the assembly of the permutations pertains to the set. If questions are “how was group theory developed?, What is the importance of group theory in science or real life? investigated for the mathematical topic group theory, then we can understand easily why we work on the structures of the theory of the many types of groups. As we know from the literature, some primary sources are determined in the development of group theory such as Algebra (Lagrange in the 17. century), Number Theory (Gauss in the 18. century) (Euler’s product formula, Combinatorics, Fermat’s Last Theorem, Class group, Regular primes, Burnside’s lemma), Geometry (Klein, 1874), and Analysis (Lie, Poincaré, Klein in the 18. Century). It seems that three main areas have been described as Number Theory and Algebra (Galois theory, equation with degree 5, Class field theory), Geometry (Torus, Elliptic curves, Toric varieties, Resolution of singularities), and analysis in mathematics. Topology (((co)homology groups, homotopy groups) and algebraic part of it (Eilenberg–MacLane spaces, Torsion subgroups, Topological spaces), the Theory of Manifolds (manifolds with a metric), Algebra, Dynamical systems, Engineering (to create digital holograms), Combinatorial Number Theory, Mathematical Logic, Geometry in Riemannian Space, and Lie Algebra also belongs these three subjects. 1 Coding Theory - Recent Advances, New Perspectives and Applications Group theory is used not just in mathematics but also in computer science, physics, chemistry, engineering, and other sciences. Especially symmetry has a big potential property in the group theory. That is why it is considered as representation theory in physics. For example; mathematical works on quantum mechanics were done by von Neumann, Molecular Orbital Theory. Also, the Standard Model of particle physics, the equations of motion, or the energy eigenfunctions use group theory for their orbitals, classify crystal structures, Raman and infrared spectroscopy, circular dichroism spectroscopy, magnetic circular dichroism spectroscopy or getting periodic tables-gauge theory, the Lorentz group-the Poincaré’s group in modern chemistry or physics. Also, group theory is defined as representation theory in physics. A lot of groups with prime caliber built-in cryptography for elliptic curve do a service for public-key cryptography and Diffie–Hellman key exchange takes advantage of cyclic groups (especially finite) too. Additionally, cryptographic protocols also consider infinite nonabelian groups. We can state the applications of the group theory also in real life as follows: 1. Shopping online (we use our credit card with encryption which is obtained by group structure in RSA algorithm) 2. Music (Elementary group theory is used for the 12-periodicity in the circle of fifths in musical set theory. Transformational theory patterns musical transformations as if they are elements of a mathematical group, cyclic groups create octave and other notions, the musical actions of the dihedral groups.) 3. Medical science (to find out breast cancer) and computer science (robotics computer vision and computer graphics) and material sciences. 4. Machine learning, communication network, signal processing, etc. … 5. Pipeline system, which is described as the Application - Business Object Network Node Layer, is patterned and investigated by the theory of group. These systems are also related with vectors, matrices determined by group structures, and so on. Thus, tools of the group theory are useful for working on applications in many different sciences and also real life as mentioned above. Basic and simple examples can be given for usual groups as follows: • Vector spaces ðV, þÞ have group structures under the addition of vectors with some properties of the scalar multiplication *. • For p primes, elements of p∗ ⊆ p have algebraic structures under multiplication with unit 1. • Assume that F be a number field and m is a natural number. Then S ¼ SLðm, FÞ can be determined to be the set of all regular m  m matrices with logins in F. This is a usual group with a ∗ b described by the multiplication of matrices. We can ask readers “whether or not these examples are partial group”? Sm is demonstrated by symmetric groups such that it includes m! permutations where m objects are taken from a set A. As an illustration, we can give the symmetric group S3 . Supposing that A ¼ fa, b, cg and S3 contains following objects; identity element: 2 Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146  a b c Identity element ¼ and others are; a b c




a b c a b c a b c a b c a b , , , , b a c a c b c b a b c a c a
a b  There are some properties in finite or infinite usual group theory. Some of them can be seen as follows: • Each of the elements in the finite group has finite order. • If H is a finite group, then it is satisfied for each of the subgroups of H: • Assuming that X, Y are groups. Then, the product of them is defined by X  Y ¼ fða, bÞja ∈ X, b ∈ Y g with unit element e ≔ ðeX , eY Þ, where we write eX , eY   are identity element and inverse elements are in the form of a1 , b1 of in X, Y, respectively. • Suppose that X be a usual group and a in X. So, the cyclic group hai becomes a subgroup of X. As an illustration, we can say that ðm , +) is a cyclic group that satisfies h1i ¼ m . • If X be a usual group with order p (p is prime), X is also a cyclic group. • X is abelian usual group iff center of X equals to X: • Let us consider Sm and its two permutations. These are conjugate iff their cycle types are equal/same according to their ordering. • … • …. If literature (briefly, references [1–47]) is investigated, then it is easily seen that partial groups are considered as topological structures more than algebraic structures. It is tried to prepare some new algebraic perspectives/approximations for the partial group. As we know, there are many algebraic infrastructures such as finiteinfinite group, abelian-nonabelian group, quaternion group, symmetric group, cyclic group, simple group, free group, orbits and stabilizers of the group, Lie group, and various kinds of theorems such as Sylow theorems, Cauchy theorem, Lagrange theorem, Cayley theorem, Isomorphism theorems as well as actions of groups for usual group theory. An effect algebra is introduced in the foundations of mechanics [1]. Furthermore, effect algebra subjects are fundamental in fuzzy probability theory [2, 3]. Also, partial group is defined by [4] and used for topological and homological investigations. A pregroup can be defined as following: A pregroup, [5], of a set P containing an element 1, each element p ∈ P has a unique inverse p1 and to each pair of elements p, t ∈ P there is defined at most one product pt ∈ P so that; a. 1 ∗ p = p ∗ 1 ¼ p is always defined, b. p * p1 = p1 * p = 1 is always defined, 3 Coding Theory - Recent Advances, New Perspectives and Applications c. If p ∗ t is defined then t1 ∗ p 1 is defined and equal to (p ∗ t)1. d. If r ∗ p and p ∗ t are defined then either r ∗ ðp ∗ tÞ is defined if and only if ðr ∗ pÞ ∗ t is defined in which case two are equal. e. If q ∗ r, r ∗ p and p ∗ t are defined then either q ∗ ðr ∗ pÞ is defined or r ∗ ðp ∗ tÞ is defined. Every pregroup is a partial group, but the converse is not true in general. In that meaning a partial group definition can be stated as follows: A set P is a partial group in the meaning of ([6], Lemma 4.2.5) if each associated pair ðx, yÞ ∈ P X P there is at most one product x:y so that: 1. There is an element 1 ∈ P satisfying x.1 = 1. x ¼ x for each x ∈ P. 2. For each x ∈ P there exists an element x1 so that x.x1=x1.x=1 3. If x:y ¼ z is defined so is y1.x1 = z1. Inspiring by the groupoid and effect algebra [7] gave an alternative partial group definition as an algebraic style. They introduced partial subgroup, partial group homomorphism, etc. as an analog investigation for group theory. Moreover, readers can learn/consider a lot more structural results on the subject from others [8–47]. In this work, some (remained ones can be considered from readers using our works) fundamental results are given for partial groups. Similarities and differences have been noticed between usual groups and partial groups. Several of them also are described in this work. Also, we do further investigations for partial groups in algebraic style. We define partial normal subgroup and give isomorphism theorems for partial groups. This work is important because it has both topological and algebraic applications. It can be expanded to rings or other algebraic structures. 2. Preliminary results Recently, an algebraic structure named as partial group (also known as Clifford Semigroup is isomorphic to an explicit partial group of partial mappings and it is a semigroup with central idempotents) is investigated with new structures in the literature. A partial group (Clifford semigroup) is a regular semigroup (it means if M is a semigroup of group G then idempotent elements of M exchange with H’s all elements). Another definition can be given for the partial group as “A regular semigroup with its central idempotents is named by Clifford semigroup.” Additionally, several partial algebras such as partial monoid, partial ring, partial group ring, partial quasigroup etc. … have been worked. For example, Jordan Holder Theorem of composition series is known to hold in every abelian category. The classical theory of subnormal series, refinements, and composition series in groups is extended to the class of partial groups which is known to be precisely the classes of Clifford semigroups, or equivalently semilattices of groups. Also, relations among the language theory, words, partial groups, universal group, and homology theory have been considered with an arrow diagram of the partial group. In this chapter, we first state some basic properties of partial groups which are mentioned in several references. 4 Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146 Definition 2.1 [7] Suppose G* is a nonempty set: G* is called as a partial group if the following conditions hold for all x, y, and z ∈ G *: (G1) If xy, ðxyÞz, yz and xðyzÞ are defined, then the equality ðxyÞz ¼ xðyzÞ is valid. (G2) For each x ∈ G*, there exists an e ∈ G* such that xe and ex are defined and the equality xe ¼ ex ¼ x is valid. (G3) For each x ∈ G*, there exists an x’ ∈ G* such that xx’ and x’x are defined and the equality xx’ ¼ x’x ¼ e is valid. The e ∈ G* satisfies (G2) is called the identity element of G* and the X’ ∈ G* satisfies (G3) is called the inverse of x and denoted by x1. Another way, we can give partial definition as follows: Definition 2.2. Let M be a semigroup. It is called a partial group if the followings are held. i. Every k ∈ M has a partial identity ek ii. Every k ∈ M has a partial inverse k1 iii. Mapping eM : M ! M, k↦ek is a semigroup homomorphism iv. Mapping β : M ! M, k↦k is a semigroup antihomomorphism. Note: i. Let M be a semigroup. A partial identity of k, when exists, is unique and idempotent such that denoted by ek . ii. Let M be a semigroup. A partial inverse of k ϵ M, when exists, is uniquely denoted by k1 . Definition 2.3. The regular element of the M semigroup is defined if there exists s ∈M such that ysy = y. Each element of M is regular element also M is named by regular semigroup. Definition 2.4. M semigroup is called as completely regular semigroup for every element s ∈ M ysy = y and ys = sy are satisfied. Note. Unions of groups give us completely regular semigroups which are named by Clifford semigroups. Definition 2.5. Let M be a semigroup. Elements k and s of a semigroup M are said to be inverse of each other if and only if sks = s and ksk = k. Then following theorem can be given from the literature. Theorem 2.1. The following results are equivalent to each other for a semigroup M: i. M is a Clifford semigroup, ii. There exists r ∈ S such that wrw = w and wr = rw for every w ∈M, iii. M is a semilattice of groups, iv. M is a completely regular inverse semigroup. Proposition 2.1. M is a completely regular semigroup iff there are ek and k1 for every k ϵ M. Proposition 2.2 [7] Every group is a partial group and every partial group which is closed under its partial group operation is a group. 5 Coding Theory - Recent Advances, New Perspectives and Applications Proposition 2.3. Assuming that M is a partial group. Then, the following are given: • Every idempotent element in partial group M is its own partial identity and partial inverse, 1 • Let k ϵ M:Then, e1 is hold. k ¼ ek ¼ ðek Þ  1 • Suppose that k ϵ M: k1 ¼ k is satisfied. Example 2.1 [7] Following sets with the given operation, can be seen as an example to the partial groups: 1. Let G ¼ {0,1, … ,n} where n ∈ + and + be known addition operation on Z. Then it is easily seen that G is a partial group but is not a group.  2. Let G ¼ * ∪ n1 : n ∈ Z +}where  *¼   f0g: So it is obvious that G is a partial group but is not a group by the known multiplication on . 3. Let G ¼ ½r, r where r ∈ + and + be known addition operations on . Then it is obvious G is a partial group but is not a group. Definition 2.6 [7] Suppose G* be a partial group, m ∈ Z +, and a ∈ G*. If am is defined and m is the least integer such that am¼ e, the number m is called the order of a. In this case, it is called that a has a finite order element. If there does not exist an m ∈ Z + such that am¼ e, (if only a0 ¼ e); then it is called that a has infinite order. The order of a is denotedby jaj.  Example 2.2 [7] G ¼ 1, 1, i, i, 2i,  2i with the multiplication operation on  is a partial group and ∣i∣ ¼ 4, j2ij ¼ ∞. Definition 2.7 [7]. Suppose G*be a partial group. ðG *Þ ¼ fx ∈ G * ∣If ax and xa are defined for all a ∈ A; ax ¼ xag is called the center of G*. Lemma 2.1 [7] A partial group is called centerless if Z (G*) is trivial i.e., consists of only the identity element. If G* is commutative then G* = Z (G*). Definition 2.8. Supposing that M ¼ Gp be a partial group and  ¼ Hp be a subset of M: is called by sub partial group of M if  is a sub semigroup of M and ek , k1 are in  for all k ∈ . Especially, M and the set of idempotents elements of M are sub partial groups of M. Definition 2.9 [7] Let G* be a partial group and H* be a nonempty subset of G*. If * H is a partial group with the operation in G* then H* is called a partial subgroup of G*. Example 2.3 [7] In Example 2.2 the set G*¼ f0, 1, … , ng where nϵ+ and + be known addition operation on  is a partial group and let H*¼ f0, 1, … , kg where 0 ≤ k ≤ n and k ∈ . Then H* is a partial subgroup of G*. Lemma 2.2 [7] Let G* be a partial group and H* be a nonempty subset of G*.H * is a partial subgroup of G* if and only if the following conditions hold: i. e ∈ H*; ii. a1 ∈ H* for all a ∈ H*. Moreover, Let G * be a partial group and let a be an element of G* such that the elements {ak for all k ∈ } are defined. Denote fak; k ∈ g ¼ < a > It is clear that the 6 Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146 : e a b c e e a b c a a b c e b b c e a c c e d b Table 1. (G) is a partial group even it has not a group structure. set < a > is a partial subgroup of G*. The partial subgroup < a > of G* is called the cyclic partial subgroup generated by a. If there exists an element a in G* such that < a > ¼ G *, then G* is called a cyclic partial group. Example 2.4 [7] Let G ¼ fe, a, b, cg ⊂ S ¼ fe, a, b, c, dg and “:” be a partially defined operation on G as in Table 1. Remark. Note that c:b is undefined. Then G is not a group but it is a partial group. Additionally, in this partial group, < a > ¼ G, G is the cyclic partial group. But all partial subgroups of a cyclic partial group can not be cyclic. For instance, the partial subgroup H ¼ fe, a, cg is not cyclic. But in group theory, if a group is cyclic, all subgroups of it are also cyclic. The partial groups are different from groups in that meaning. Definition 2.10. Assume that M ¼ Gp be a partial group and k ∈ M: Then, we define Mk ¼ fs ∈ M : ek ¼ es g. Theorem 2.2. Suppose that M is a partial group and k ∈ M: Then, Mk is a maximal subgroup M of which has identity ek and M ¼ ⋃fMk : k ∈ Mg. Definition 1.11 [7] Let M and N be partial groups. A function σ : M ⟶ N is called a partial group homomorphism if for all a, b ∈ M such that ab is defined in M, σ ðaÞσ(b) is defined in N and σ ðabÞ ¼ σ ðaÞσ ðbÞ: If σ is injective as a map of sets, σ is said to be a monomorphism. If σ is surjective, σ is called an epimorphism. Definition 2.12. For a partial   group homomorphism σ : M ⟶ N it is defined kerσ ¼ k ∈ M : σ ðkÞ ¼ eσ ðkÞ and Imσ ¼ fσ ðkÞ : k ∈ Mg. Also, σ : M ⟶ N is named isomorphism if it is bijective. As a consequence of the definition the following lemmas can be given: Definition 2.13. Suppose  that σ : M ⟶ Nbe a partial group homomorphism. Then, we define ker σ ¼ k ∈ M : σ ðkÞ ¼ eσ ðkÞ and Imσ ¼ fσ ðkÞ : k ∈ Mg. Theorem 2.3. Assuming that σ : M ⟶ N be a partial group homomorphism and k ∈ M. Then, the following are given. i. σ ðek Þ ¼ eσðkÞ :   ii. σ k1 ¼ σ ðkÞ1 : iii. kerσ is a subpartial group of M: iv. Imσ is a subpartial group of N. v. σ ðMk Þ is a subpartial group of NσðkÞ :   vi. σ 1 Nek is a subpartial group of M. 7 Coding Theory - Recent Advances, New Perspectives and Applications Proposition 2.4 [7] Suppose M, N be partial groups and σ : M ⟶ N be a homomorphism of partial groups. Then the following conditions are satisfied: i. If A is a partial subgroup of M, then σ(A) is a partial subgroup of N. ii. If B is a partial subgroup of N, then σ 1(B) is a partial subgroup of M. Proposition 2.5. Let σ : M ⟶ N be a homomorphism of partial groups. Then, it is obtained that   σ ðek Þ ¼ eσðkÞ , σ k1 ¼ ðσ ðkÞÞ1 for all k ∈ M: Definition 2.14. A sub partial group  ¼ Hp of M ¼ Gp is named wide if the set of idempotent elements of M is a subset of  ¼ Hp and normal, written ⊲M. (if it is wide and kk1 ⊆  for all k ϵ MÞ:It is also trivial that the set of idempotents elements of M is a normal subgroup of M and it is called the set of idempotents elements of M the trivial normal subpartial group of M. Theorem 2.4. Assuming that M be a partial group and k ∈ M, then i. Mk is a maximal subgroup of M with identity ek .  ii. M ¼ ⋃fMk : k ∈ Mg ¼ ⋃ Mek : ek is in the set of idempotents elements of Mg. Theorem 2.5. If M is a partial group, then the set of idempotents elements of M is commutative and central. Definition 2.15. For a partial   group homomorphismσ : M ⟶ N it is defined kerσ ¼ k ∈ M : σ ðkÞ ¼ eσ ðkÞ and Imσ ¼ fσ ðkÞ : k ∈ Mg. Also, σ : M ⟶ N is named isomorphism if it is bijective. Definition 2.16. σ is named as idempotent separating if σ ðek Þ ¼ σ ðes Þ implies that ek ¼ es , where ek , es are in the set of idempotent elements of M for a partial group homomorphism σ : M ⟶ N . 3. Partial normal subgroups In group theory, normal subgroup plays an important role in the classification of groups and gives lots of algebraic results. Now, we will construct an analog definition for partial groups. Throughout Gp will denote the partial group. In this chapter, we should notice that if G is a group, then G is a partial group with fectfeg. Also, [8] in a group every element has a unique inverse, but in partial groups [7] for every element a ∈ G, we have Inv ðaÞ 6¼ 0 because of that reason the identity element of the group differs from the identity element of the partial group. We can continue to work under these assumptions. From here on in, we will use the notation Gp for partial groups. Definition 3.1 (Partial Conjugation Criteria). Let Gp be a partial group, the element gp.xp.gp1 (or gp1.xp.gp) is called partial conjugate of xp by gp for fixed gp,xp ∈ G p. Theorem 3.1. Let Gp be a partial group and Np be a partial subgroup of Gp then following conditions are satisfied: i. Np is normal in Gp if and only if for all xp ∈ Np and gp ∈ Gp we have gp1.xp. gp ∈ N p 8 Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146 ii. N p is normal in Gp if and only if for every element of N p all partial conjugates of that element also lie in N p. Proof: i. It comes from the definition of partial normal subgroup. ii. ():) Let N p is partial normal subgroup in Gp. Then we need to show for every xp ∈ Np and fixed gp ∈ N p the partial conjugates gp1.xp.gp lies in N p. Since gp ∈ Np then gp ∈ Gp. Also, since N p is a partial normal subgroup of Gp gp1.xp.gp ∈ N p, this gives the proof. ((:) Conversely, let for every element of N p all partial conjugates of that element lie in Np: Then it comes directly from partial subgroup definition Conclusion(s). It is preferable to include a Conclusion(s) section which will summarize the content of the book chapter. Remark 3.1. Any partial subgroup Hp ⊆ Gp has right and left congruence (equivalence) class that cannot be the same. But if left and right congruence classes are the same (i.e., for any x ∈ Gp, Hp.x = x.Hp) then Hp is called as normal partial subgroup. Theorem 3.2. Let Gp be a partial group and N p be a partial subgroup of a partial group Gp and so the following conditions are coincided: Proposition 3.1. i. N p is a partial normal subgroup of a partial group Gp. ii. gp.N p=N p.gp for all gp ∈ Gp, iii. gp.N p.gp1 ⊆ N p for all gp ∈ Gp. Proof: (i) )(ii) If N p is a partial normal subgroup of Gp then it is easy that for all xp ∈ N p and gp ∈ Gp we have gp.xp.gp1 ∈ Np and from just before the theorem gp.Np.gp1 ∈ Np. (iii))(ii) Let gp be an element of Gp: We need to see gp.N p¼ N p.gp. Assume that xp ∈ gp.N p. Then xp = gp.n1 is satisfied for n1 ∈ N p. Sincex p.gp1 = gp.np.gp1 and gp.N p.gp1 ⊆ N p we have xpgp1 ∈ N pand so that there exists an n2 ∈ N p such that xp.gp1 = n2. If we product from right with gp then we have the ðxp.gp1Þ.gp = n2.gp equality. Using the associativity property the equality becomes xp.(gp1.gp) = n2.gp and using identity element property and converse element property we get xp = n2. gp ∈ N p.gp. It implies that gp.N p ⊆ Np.gp. In a similar way gp1.N p.gp ⊆ N p and we have N p.gp ⊆ gp.N p. Therefore we obtain gpNp = Np.gp. (ii))(i) Supposing that gp ∈ Gp we have to prove gp.np.gp 1 is contained in N p for all np ∈ N p. Since gp.N p = N p.gp, we can say that gp.np ∈ N p.gp for all np ∈ N p. Then, by associativity gp.np.gp1 is contained in N p.gp.gp1 and then for all np ∈ N p, gp.np.gp1. Lemma 3.1. The center of a partial group Gp is a partial normal subgroup of Gp. Proof: The center of a partial group is defined as below: Ζ (Gp)=fxp ∈ GjIf for every gp ∈ Gp xp.gp and gp.xp are defined, xp.gp¼ gp.xpg. Let gp ∈ Gp and xp ∈ Ζ (Gp) then we need to show gp.xp.gp1 is contained in Ζ ðGpÞ. Since xp ∈ Ζ ðGp) for every gp ∈ Gp if gp.xp and xp.gp is defined then xp. gp = gp.xp. Using this argument gp.xp.gp1=xp.gp.gp1=xp ∈ Ζ ðGp) and then we have Z(Gp) is a partial normal subgroup of Gp. 9 Coding Theory - Recent Advances, New Perspectives and Applications Proposition 3.2. Let φ: Gp! Hp be a partial group homomorphism. Then the kernel of φ is partial normal subgroup of Gp. Proof: Let K = Ker(φÞ. We know that Ker(φ) is a partial subgroup of Gp. Suppose that y ∈ K and gp ∈ Gp. Then using the fact φ is a partial group homomorphism  φ gp :yp :gp 1    ¼ φ gp :φ yp :φ gp 1   ¼ φ gp :eHp :φ gp 1   ¼ φ gp :φ gp 1  ¼ eHp we have φ gp :yp :gp 1 ¼ eHp and we get gp.yp.gp1 ∈ K. So Ker(φÞ is a partial normal subgroup of Gp. Partial normal subgroups Theorem 3.3. Suppose Gp is a partial group and Hp is a partial subgroup of Gp.H is a partial normal subgroup of Gp if and only if (aHp) (bHp) = abHp equality holds p for all a, b ∈ Hp. Proof: ():) Suppose Hp is a partial normal subgroup of Gp. We need to see the equality (aHp) (bHp) = abHp holds. ( ⊆ :) Let x ∈ (aHp) (bHp) then ∃h,h 2 ∈ H such that x = (ah1) (bh2). By using associativity property, we get x ¼ (h1bh2). From the identity element, x = abb1h1bh2 is obtained. Considering associativity property we can write x ¼ ab (b1h1b)h2. Since Hp is a partial normal subgroup of Gp, b1h1b ∈ Hp Then x ∈ abH p and this implies that (aHp) (bH p) ⊆ abH p. ( ⊇ :Þ Conversely, let y ∈ abH p then there exists an h ∈ H p such that y ¼ abh. So we can write y = abh as follows; y ¼ ðaeÞðbhÞ ∈ (aHp) (bHp). It implies that abHp ⊆ (aHp) (bHp). Therefore ðaHp) (bHp)¼ abH p ((:) Let us consider (aHp) (bHp) ¼ abH p for all a, b ∈ Gp. If h ∈ Hp and g ∈ G p then we must see whether or not ghg1 ∈ H: Using associativity property and considering hypothesis; ghg1 = (gh)(g1e) ∈ (gHp)(g1Hp) = gg1Hp = Hp. This implies that ghg1 ∈ Hp. So that Hp is a partial normal subgroup of Gp. Theorem 3.4. Suppose Gp be a partial group and H, K are partial subgroups of Gp. If K is a partial normal subgroup of Gp, then the following cases are satisfied: i. H ∩ K is a partial normal subgroup of H. ii. If K and H are partial normal subgroups of Gp and H ∩ K = feg then hk ¼ kh ðor, HK ¼ KH) for every h ∈ H and every k ∈ K. Proof: i. Since H and K are partial subgroups, H ∩ K is also a partial subgroup of G. By H ∩ K ⊆ K we can conclude that H ∩ K is also a partial subgroup of H. Let consider a ∈ H ∩ K and h ∈ H: Since a, h ∈ H, ha and hah1 can be defined. It gives that hah1 ∈ H Also since K is a partial normal subgroup of Gp we have hah1 ∈ K . It shows that H ∩ K is a partial normal subgroup of H. ii. Let H be a partial normal subgroup of Gp, H ∩ K ¼ feg and h ∈ H and k ∈ K. Since H is a partial normal subgroup, we know that k:h:k1 ∈ H . If h.k.h1. k1 ∈ H then we get ðh:k:h1).k1 ∈ K by K is a partial normal subgroup. So 10 Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146 we have, h.k.h1.k1 ∈ H ∩ K ¼ feg. Then h.k.h1.k1=e and so, hk ¼ kh; for all h ∈ H, k ∈ K. Thus, we prove that HK ¼ KH. Proposition 3.3. Let Gp be a partial group and Hp be a partial subgroup of index 2 in Gp: Then Hp is a partial normal subgroup in Gp. Proof: Let Hp be a partial subgroup of index 2 in Gp and gp be an element of Gp. If gp ∈ Hp, then gpHp = Hpgp is satisfied. If gp is not in Hp, two left cosets must be as Hp and gpHp. Since left cosets are disjoint we know gp.Hp = GpHp. Also, the right cosets are disjoint so we can write Hp.gp = GpHp. Thus gpHp = Hpgp for all gp ∈ Gp. So Hp is normal. Example 3.1. Let Gp be an abelian partial group. Then any subgroup of Gp is a partial normal subgroup of Gp. Proof: If Gp is an abelian partial group and xp.yp ∈ G p then xp.yp = yp.xp for every xp.yp ∈ Gp. If gp.xp.gp1 ∈ N p then N p is a partial subgroup of Gp. Using the hypothesis we get gp :xp :gp 1 ¼ gp :gp 1 :xp ¼ eGp ¼ xp ∈ N p Therefore, the partial subgroup N p is the partial normal subgroup of Gp. Example 3.2. Let Hp and K p be any partial normal subgroup of Gp. Then HpK p is also a partial normal subgroup of GpGp. Proof: HpK p = {np¼(hp,kp) jhp ∈ Hp and kp ∈ K p}. We have to show that for all gp in Gp and np in HpK p gp.np.gp1 is in HpK p.   gp :np ¼ gp : hp , kp  ¼ gp :hp , gp :kp and  gp :np :gp 1 ¼ gp :hp , gp :kp :gp 1  ¼ gp :hp :gp 1 , gp :kp :gp 1 and since Hp and K p are partial normal subgroups of Gp, then gp.hp.gp1 ∈ H p, and gp.kp.gp1 ∈ K p and so that (gp.hp.gp1, gp.kp.gp1) ∈ HpKp. Then the Cartesian product of two partial normal subgroups is also a partial normal subgroup. Theorem 3.5. Let be a partial group and N p be a partial normal subgroup of Gp. The congruence modulo Np is a congruence relation for the partial group operation “:”. Proof. Let xRN py denote that x and y are in the same coset, that is; xRN py ⟺ x:N p ¼ y:N p Let xRN px and yRN py. To demonstrate that RN p is a congruence relation for :, we need to show, reflexivity, symmetry, and transitivity. These axioms are obvious from the definition of relation. Theorem 3.6. If N p is a partial normal subgroup of a partial group Gp and Gp¼ N p is the set of all cosets of N p in Gp, then Gp∕N p is a partial group under the operation given by ðaN pÞðbN pÞ ¼ abN p. 11 Coding Theory - Recent Advances, New Perspectives and Applications Proof. Let aN p, bN p,cN p ∈ Gp∕N p. We must see partial group axioms are satisfied: (G1) If (aN p)(bN p)¼ abN p. ðbN pÞðcNpÞ ¼ bcN p and a.( bcNp) are defined then       a: bcN p ¼ aN p bN p cN p     ¼ ð aN p bN p cN p ¼ ðabÞcN p (G2) For any aN p in Gp∕N p, eN p is a candidate for identity element, i.e.,    aN p eN p ¼ aeN p ¼ aN p and    eN p aN p ¼ eaN p ¼ aN p (G3) Since is a partial group a ∈ Gp has an inverse a ∈ ́ Gp. For every aN p in Gp∕N p a ́N p is a candidate for the inverse of aN p.    aN p a N ́ p ¼ ðaáÞN p ¼ eN p ¼ Np This completes the proof. Definition 3.2. Let Gp be a partial group and Np is a partial normal subgroup of Gp, then the partial group Gp∕N p is called the quotient of the partial group or factor group of Gp by N p. Proposition 3.4. Let f : Gp ⟶ H p be a homomorphism of partial groups, then the kernel of f is a partial normal subgroup Q of Gp. Conversely, if N Qp is a partial normal subgroup of Gp, then the map : Gp ⟶ Gp∕N p given by (ap)¼ aN p is an epimorphism with kernel N p. Proof: Kerf ¼ fx p ∈ Gp jf ðxpÞ ¼ eH pg, we need to show that if xp ∈ Kerf and ap ∈ Gp whether or not apxpap1 ∈ Kerf if and only if f (apxpap1Þ ¼ f ðap)f (xp) f ðap1)¼ eH p So, a pxpQ ap1 ∈ Kerf . Therefore, Kerf is a partial Q normal Q subgroup of Gp. It is trivial that : Gp ⟶ Gp∕N p is surjective. Kerð Þ ¼ {xpj (xp)¼ epN p}¼ N p. Theorem 3.7. Let f : G p! Hp is partial group homomorphism and Np is a partial normal subgroup of Gp contained in the kernel of f , then there is exactly unique homomorphism f : Gp∕N p ⟶ Hp such that f ðaN pÞ ¼ f ðaÞ for all a ∈ Gp. Besides Imf ¼ Imf and kerf ¼ (kerf Þ /N p.f an isomorphism if and only if f is an epimorhism and N p¼ ker f . f f Proof: f : G p! H p, Gp ⟶ Gp∕N p, Gp∕N p ! Hp diagram is commutative. If b ∈ aN p, then b ¼ an p,n p ∈ N and also f (bp)¼ f (anp)¼ f (a)f (np)¼ f ðaÞe ¼ f ðaÞ, since Np ≤ ker f . Therefore, f has the same effect on every element of aNp and the map f : Gp∕N p ⟶ Hp given by f (aN p)¼ f ðaÞ. It is easily seen that f is a well-defined function. Now we need to prove whether or not f is a homomorphism of partial groups.     f aN p :bN p ¼ f abN p ¼ f ðabÞ 12 Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146 ¼ f ðaÞ:f ðbÞ     ¼ f aN p :f bN p So, f is a partial group homomorphism. Imf ¼ Imf and aN p ∈ kerf ⇔f ðaÞ ¼ e ⟺ a ∈ kerf , whence   kerf ¼ aNp ja ∈ kerf ¼ kerf ∕N p f is unique since it is completely determined by f . Also, f is a partial group if and only if f is an epimorphism of partial groups f is a monomorphism if and only if for kerf ¼ ðker f Þ∕N pker f equal to N p. Example 3.3. In Example 2.2, it is stated that G ¼ f0,1,2, … ,n} is a partial group with known addition operation on . We can easily say that the subset N = {0,1,, … ,n  1g of G is a partial subgroup of G. Let us show whether or not N is a partial normal subgroup. If gp1 + np + gp ∈ N for all gp ∈ Gp then N is a partial normal subgroup. gp1¼ gp in this group so that gp + np + gp ∈ N p i.e. np ∈ N and then N is a partial normal subgroup of G. Theorem 3.8 (First Isomorphism Theorem for Partial Groups). Let Gp, Hp be partial groups and f : Gp ⟶ H p be partial group epimorphism then Gp∕Ker f is isomorphic to Hp. Proof: We know that ker f is a partial normal subgroup of Gp. Then Gp∕Ker f is defined. Let show Ker f ¼ K and g: Gp∕K ⟶ H p mapping defined as gððaK ÞðbK ÞÞ ¼ gðabK Þ ¼ f ðabÞ for every aK, bK ∈ Gp∕K. Since gða:bK Þ ¼ f ðabÞ ¼ f ðaÞf ðbÞ ¼gðaK ÞgðbK Þ then g is a homomorphism. Since f is onto there exists a ∈ Gp such that f ðaÞ ¼ h then aK ∈ Gp∕K and gðaK Þ ¼ f ðaÞ ¼ h and g is onto. For one-to-one conditions let aK, bK ∈ Gp∕K and g ðaK Þ ¼ gðbK Þ ifff ðaÞ ¼ f ðbÞ   f ðbÞ1 f ðaÞ ¼ f ðbÞ1 f ðbÞ ifff b1 a ¼ eHp b1 a ∈ K or kerf iffaK ¼ bK Then, g is an isomorphism and Gp∕Ker f ffi Hp. Lemma 3.2. Let Gp be a partial group and Hp be a partial subgroup of Gp and N p be a partial normal subgroup of Gp. If Hp is a partial normal subgroup of Gp; then HpN p is a partial normal subgroup of Gp. Proof: Trivial. Theorem 3.9 (Second Isomorphism Theorem for Partial Groups). Let Gp be a partial group. Hp be a partial subgroup of Gp and Np be a partial normal subgroup of Gp, then the following isomorphism holds:   Hp N p ∕N p ffi Hp Hp ∩ N p Proof: It can be easily seen using First Isomorphism Theorem. So, the proof is left to the reader. Theorem 3.10 (Third Isomorphism Theorem for Partial Groups). Let Gp be a partial group and Hp,N p partial normal subgroups of Gp with Hp ≤ N p Then Hp is also a partial normal subgroup of N p.N∕Hp is a partial normal subgroup of Gp∕Hp and also Gp∕N p is isomorphic to ðGp∕Hp)∕(N p∕Hp). 13 Coding Theory - Recent Advances, New Perspectives and Applications Proof: Let define f : Gp∕Hp ⟶ Gp∕N p, f ðaHpÞ ¼ aN p First let show f is welldefined: If aHp = bHp then we need to prove whether or not f ðaHpÞ ¼ f ðbHpÞ. Since aHp = bHp then ab1 ∈ Hp and Hp ≤ N p,ab 1 ∈ N p. Since ab 1 ∈ N p then aN p = bN p and so that f ðaHpÞ ¼ f ðbHpÞ, i.e., f is well defined. f is homomorphism. And using the First Isomorphism Theorem we can conclude the result. 4. Conclusion There are some papers such as solvable partial groups, topological structures of partial group, Transitivity Theorem-Thompson Theorem of partial groups, k-partial groups, primitive pairs of partial groups so on related with the partial group in the literature. It is known that every group is partial but the converse is not true. That is why some structures are different from each other for the usual group and partial group. In this chapter, some structures of partial groups (Clifford Semigroup) are sought to demonstrate algebraically. At the beginning of the chapter (preliminaries section), several fundamental results of partial groups with some numerical examples are given from the literature. For example, if A and B be two usual groups such that the intersection of them is equal to {1 = e}, then the union of subgroups of A and subgroups of B is a partial group. Partial normal groups and partial quotient groups have introduced an analog of the group theory. By using them, a number of isomorphism theorems are proved for partial groups with several other ideas. All results are obtained using closely group theory as algebraic approximations. Readers also may consider/investigate other structures/properties of the partial groups different from the group as algebraically. Conflict of interest The author declare no conflict of interest. Author details Özen Özer Department of Mathematics, Faculty of Science and Arts, Kırklareli University, Kırklareli, Turkey *Address all correspondence to: [email protected]; [email protected]; [email protected] © 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 14 Algebraic Approximations to Partial Group Structures DOI: http://dx.doi.org/10.5772/intechopen.102146 References [1] Foulis D, Bennett MK. Effect algebras [11] Abd-Allah AM, Aggour AI, Fathy A. and unsharp quantum logics. 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