On the MacNeille Completion of Weakly Dicomplemented Lattices

On the MacNeille completion of weakly dicomplemented lattices Léonard Kwuida1 , Branimir Seselja2 , and Andreja Tepavčević2 1 Universität Bern Mathematisches Institut Sidlerstrasse 5, CH-3012 Bern [email protected] 2 University of Novi Sad Department of Mathematics and Informatics Trg D. Obradovića 4 21000 Novi Sad Abstract The MacNeille completion of a poset (P, ≤) is the smallest (up to isomorphism) complete poset containing (P, ≤) that preserves existing joins and existing meets. It is wellknown that the MacNeille completion of a Boolean algebra is a Boolean algebra. It is also wellknown that the MacNeille completion of a distributive lattice is not always a distributive lattice (see [Fu44]). The MacNeille completion even seems to destroy many properties of the initial lattice (see [Ha93]). Weakly dicomplemented lattices are bounded lattices equipped with two unary operations satisfying the equations (1) to (3’) of Theorem 3. They generalise Boolean algebras (see [Kw04]). The main result of this contribution states that under chain conditions the MacNeille completion of a weakly dicomplemented lattice is a weakly dicomplemented lattice. The needed definitions are given in subsections 1.2 and 1.3. 2000 Mathematics Subject Classification: 06B23 Key words and phrases: MacNeille completion, weakly dicomplemneted lattices, Formal Concept Analysis. 1 1.1 Introduction Motivation Concept algebras are concept lattices enriched by a weak negation and a weak opposition. They should play for Boolean Concept Logic the rôle played by the powerset algebras for Classical Propositional Logic. The class of weakly dicomplemented lattices is a variety defined by some equations valid in all concept algebras. One important and still open problem in this topic is whether every weakly dicomplemented lattice can be embedded into a concept algebra of a suitable context (called concrete embedding problem in [Kw04, Section 1.4]). A promising step is the prime ideal theorem. Using this result on a weakly dicomplemented lattice (L, ∧, ∨,4 ,5 , 0, 1), a canonical context K4 5 (L) has been constructed and a bounded lattice embedding ϕ : L → B(K4 5 (L)) exhibited (see Subsection 1.4), that satisfies ϕ(x5 ) ≤ ϕ(x)5 ≤ ϕ(x)4 ≤ ϕ(x4 ) 4 where in K4 and 5 (L) the weak negation and weak opposition are also denoted by respectively. So the following question arises: does it make any difference if L is assumed to be a complete lattice? To answer this question we first examine the 5 2 L. Kwuida, B. Seselja & A. Tepavčević MacNeille completion L̃ of L, on which we extend the operations 4 and 5 . Of course L̃ embeds into B(K4 5 (L). Is this a weakly dicomplemented lattice embedding of L 4 into A(K5 (L))? Our aim is to prove that the MacNeille completion L̃ of a weakly dicomplemented lattice L is a weakly dicomplemented lattice and that L embeds 4 into A(K4 5 (L)) iff L̃ embeds into A(K5 (L)). Section 2 presents preliminary results for the first claim. The second claim is still not proved. Before that we recall some basic notions of Formal Concept Analysis in Subsection 1.2 and introduce weakly dicomplemented lattices in Subsection 1.3. The proofs of stated results can be found in [GW99] or [Kw04]. 1.2 Formal Concept Analysis Formal Concept Analysis is a mathematical field that aims to support human thinking. It has been introduced by Rudolf Wille in the early 80ies, and is based on the theory of lattices and ordered sets. It is started by formalizing the notions of “concept” and “concept hierarchy”. The notion of concept is rather philosophical. A concept is considered to be determined by its extent and its intent. The extent consists of all entities belonging to the concept and the intent is the set of all common properties shared by all objects of the concept. The hierarchy on concept states that “a concept is more general if it contains more entities”. For this purpose the following notions were adopted. Definition 1 A formal context is a triple (G, M, I) of sets such that I ⊆ G × M . The members of G are called objects and those of M attributes. If (g, m) ∈ I the object g is said to have m as an attribute. For A ⊆ G and B ⊆ M , the derivation operation 0 is defined by A0 := {m ∈ M | ∀g ∈ A gIm} and B 0 := {g ∈ G | ∀m ∈ B gIm}. A formal concept of (G, M, I) is a pair (A, B) with A ⊆ G and B ⊆ M such that A0 = B and B 0 = A. We call A the extent and B the intent of the concept (A, B). The set of all formal concepts of (G, M, I) is denoted by B(G, M, I). For concepts (A, B) and (C, D), we call (A, B) a subconcept of (C, D) provided that A ⊆ C (which is equivalent to D ⊆ B). In this case, (C, D) is a superconcept of (A, B) and we write (A, B) ≤ (C, D). The pair (0 ,0 ) forms a Galois connection between the powersets of G and that of M . The basic theorem on concept lattices says that: Theorem 1 [GW99, Thm. 3] Let (G, M, I) be a formal context. B(G, M, I) is a complete lattice in which infimum and supremum are given by: !00 ! !00 ! ^ \ [ _ [ \ (At , Bt ) = At , Bt and (At , Bt ) = At , Bt . t∈T t∈T t∈T t∈T t∈T t∈T A complete lattice L is isomorphic to B(G, M, I) iff there are mappings γ̃ : G → L and µ̃ : M → L such that γ̃(G) is supremum-dense in L, µ̃(M ) is infimum-dense in L and for all g ∈ G and m ∈ M gIm ⇐⇒ γ̃(g) ≤ µ̃(m). In particular L ∼ = B(L, L, ≤). B(G, M, I) is called the concept lattice of the context (G, M, I). A particular case is the context (P, P, ≤) where (P, ≤) is a poset. Its concept lattice is (isomorphic to) the MacNeille completion of (P, ≤)). In fact MacNeille completion of weakly dicomplemented lattices 3 Theorem 2 [GW99, Thm. 4] For a poset (P, ≤) the map ϕ : P → B(P, P, ≤) x 7→ (↓x, ↑x) is an order embedding of (P, ≤) into B(P, P, ≤) preserving existing suprema and infima. If ψ is another embedding of (P, ≤) into a complete lattice L, then there is an order embedding λ of B(P, P, ≤) into L such that ψ = λ ◦ ϕ. To formalize a negation on concepts, two unary operations are introduced: a weak negation and a weak opposition. We refer to [Wi00] for more details and similar operations encoding a negation. 1.3 Weakly dicomplemented lattices Definition 2 Let K := (G, M, I) be a formal context. For a formal concept (A, B) we define 00 0
its weak negation by (A, B)4 := (G \ A) , (G \ A) 0 00
and its weak opposition by (A, B)5 := (M \ B) , (M \ B) .
A(K) := B (K) ; ∧, ∨,4 ,5 , 0, 1 is called the concept algebra of the context K, where ∧ and ∨ denote the supremum and infimum operations of the concept lattice. Rudolf Wille showed that Theorem 3 ([Wi00]) For each formal context K := (G, M, I), the following properties hold in its concept algebra A (K): (1) x44 ≤ x, (2) x ≤ y =⇒ x4 ≥ y 4 , (3) (x ∧ y) ∨ (x ∧ y 4 ) = x, (1’) x55 ≥ x, (2’) x ≤ y =⇒ x5 ≥ y 5 , (3’) (x ∨ y) ∧ (x ∨ y 5 ) = x. Since we are interested in the equational theory (if there is one) of concept algebras, we define abstract structures using the equations in Theorem 3. Then we try to find whether these are enough to describe concept algebras; i.e., if each equation valid in all concept algebras is also valid in these abstract structures and vice versa. Definition 3 A weakly dicomplemented lattice is an algebra (L; ∧, ∨,4 ,5 , 0, 1) of type (2, 2, 1, 1, 0, 0) for which (L; ∧, ∨, 0, 1) is a bounded lattice satisfying properties (1), (2), (3), (10 ), (20 ) and (30 ) of Theorem 3. x4 is called a weak complement and x5 a dual weak complement of x. The pair (4 ,5 ) is called a weak dicomplementation. All concept lattices are complete lattices. If we expect each weakly dicomplemented lattice to be (isomorphic to) a concept algebra, we should add the completeness property in Definition 3. However, being a complete lattice cannot be expressed by means of equations. 1.4 Canonical context A context can be assigned canonically to each weakly dicomplemented lattice as follows. Let L be a weakly dicomplemented lattice. A subset X of L is called primary filter (resp. primary ideal) if for all x ∈ L we have x ∈ X or x4 ∈ X (resp. x ∈ X or x5 ∈ X). The prime ideal theorem holds for weakly dicomplemented lattices namely, 4 L. Kwuida, B. Seselja & A. Tepavčević for any filter F and any ideal I such that F ∩ I = ∅ there is a primary filter G, a primary ideal J with F ⊆ G, I ⊆ J such that G ∩ J = ∅. [Kw04, Lemma 2.2.1] An immediate consequence is the existence of primary filters and primary ideals. We denote by Fpr (L) the set of primary filters of L and by Ipr (L) the set of primary ideals of L. A relation ∆ ⊆ Fpr (L) × Ipr (L) is defined by F ∆I : ⇐⇒ F ∩ I 6= ∅. We call (Fpr (L), Ipr (L), ∆) the canonical context of L and denote it by K4 5 (L). For x ∈ L, we set Fx := {F ∈ Fpr (L) | x ∈ F } and Ix := {I ∈ Ipr (L) | x ∈ I}. The map ϕ : x 7→ (Fx , Ix ) is a bounded lattice embedding of L into B(K4 5 (L)) such that ϕ(x5 ) ≤ ϕ(x)5 ≤ ϕ(x)4 ≤ ϕ(x4 ). [Kw04, Thm 2.2.4] Is ϕ an embedding of L into A(K4 5 (L))? This question is still open and is known as the concrete representation problem for weakly dicomplemented lattices. 2 MacNeille Completion of weakly dicomplemented lattices In this section we prove that in the MacNeille completion of a weakly dicomplemented lattice L, unary operations can be naturally defined to extend the ones existing on L. The first observation towards this definition is the more general form of the De Morgan laws. For a subset X of L we set X 4 := {x4 | x ∈ X} and X 5 := {x5 | x ∈ X}. Lemma 1 Let L be a weakly dicomplemented lattice and X a subset of L. We have ^ _ (i) ( X)4 = X4 and (ii) ( _ X)5 = ^ X5 Proof. (i) Note that ^ _ _ ( ∅)4 = 14 = 0 = ∅= ∅4 . The equality trivially holds for X = ∅. We consider X to be a nonempty subset and set X := {xi | i ∈ I}. We have ^ ^ _ ( X)4 ≥ x4 for all i ∈ I, and hence ( X)4 ≥ X 4. i Now let t ≥ xV4 t4 ≤ x44 ≤ xi for i for all i ∈ I. Consequently i V V all 4i ∈ I holds as 44 4 well as t4 ≤ X. Therefore t ≥ t ≥ ( X) holds. Thus ( X) is the lowest V W upper bound of X 4 i.e., ( X)4 = X 4 . The equality in (ii) is proved dually. u t We consider L, a weakly lattice and B(L, L, ≤), its MacNeille W dicomplemented V completion. Since L is -dense and -dense in B(L, L, ≤), we take advantage of Lemma 1 (i) to define the weak complement and of Lemma 1 (ii) to define the dual weak complement of each x ∈ B(L, L, ≤). Before that, we have to prove that the definition does not depend on the representation. That is what Lemma 2 says. Lemma 2 Let (L, ∧, ∨,4 ,5 , 0, 1) be a weakly dicomplemented lattice. For any subset X of L we denote by ub(X) (resp. lb(X)) the set of upper bounds (resp. the set of lower bounds) of X. The following statements hold: MacNeille completion of weakly dicomplemented lattices 5 (i) If lb(X1 ) = lb(X2 ) then ub(X14 ) = ub(X24 ). (ii) If ub(X1 ) = ub(X2 ) then lb(X15 ) = lb(X25 ). Proof. To prove (i) we assume that the sets of lower bounds of X1 and that of X2 are equal. Let s ∈ ub(X14 ). We have s ≥ t4 for all t ∈ X1 . Therefore, by (2) in Theorem 3, s4 ≤ t44 ≤ t for all t ∈ X1 . Thus s4 belongs to lb(X1 ), which is by assumption the same as lb(X2 ). It follows that s4 ≤ t for all t ∈ X2 , and hence s44 ≥ t4 for all t ∈ X2 . Therefore t4 ≤ s44 ≤ s for all t ∈ X2 . This means that s belongs to ub(X24 ). Analogously we can prove that ub(X2 ) is contained in ub(X1 ) and get the equality. (ii) can be established dually. t u Lemma 2Vsays on one side that if X1 and X2 are subsets W of L suchWthat the equality V X1 = X2 holds in a completion, then the equality X14 = X24 also holds, W W V V and on the other that if X1 = X2 then X15 = X25 . In the sequel, let (L, ∧, ∨,4 ,5 , 0, 1) be a weakly dicomplemented lattice. We denote by L̃ the MacNeille completion of L. Since the binary operations ∧, ∨ on L are restrictions of the binary operations on the complete lattice L̃, we denote the binary operations on L̃ by the same symbols. We consider the lattice L as a subset of L̃. Therefore, the top and the bottom elements of both lattices (which are the same) are denoted by the same symbols, 0 and 1. By a V well known property, for W each x ∈ L̃ there are subsets X and Y of L such that x = X and x = Y . Using this representation, we define unary operations on L̃: _ ^ x 7→ xN := X 4 and x 7→ xH := Y 5. Lemma 3 Let (L, ∧, ∨,4 ,5 , 0, 1) be a weakly dicomplemented lattice and L̃ its MacNeille completion. The unary operations N and H are well defined, and the equations (1) xNN ≤ x, (2) x ≤ y =⇒ xN ≥ y N , (1’) xHH ≥ x, (2’) x ≤ y =⇒ xH ≥ y H , are satisfied. Proof. The operations x 7→ xN and x 7→ xH are well defined by Lemma 2. Therefore, the subsets X and Y can be chosen systematically. Note that L̃ can be identified with the concept lattice B(L, L, ≤), X with the order filter of L generated by x and Y with the order ideal of L generated by x. We use this identification in the rest of the proof. We proceed as follows: we are going to prove the equations (1) and (2). The others are obtained dually. Let us start with (2). Let x1 and x2 be elements in L with x1 ≤ x2 . Then the order filter generated by x1 , say X1 contains the order filter generated by x2 , say X2 . Therefore _ 4 _ 4 xN X1 ≥ X2 = xN 1 = 2, and (2) is proved. Now let us prove (1). Using (2) we have _ xNN = ( X 4 )N ≤ (t4 )N for all t ∈ X, where X is the order filter generated by x. But _ _ (t4 )N = (↑t4 ∩ L)4 = {a4 | a ≥ t4 , a ∈ L}. Therefore (t4 )N = Thus xNN (1). _ {a4 | a4 ≤ t44 , a ∈ L} ≤ t. V ≤ t for all t ∈ X, and finally xNN ≤ X = x, achieving the proof of t u 6 L. Kwuida, B. Seselja & A. Tepavčević From now on, we consider algebra (L̃; ∧, ∨,N ,H , 0, 1) of type (2, 2, 1, 1, 0, 0). The 3rd equation in the definition of the weakly dicomplemented lattice implies that 4 is a dual semicomplementation (i.e., x ∨ x4 = 1 for all x ∈ L). Before we continue, we verify that it also holds in L̃. It will be useful for later need. Lemma 4 The equation x ∨ xN = 1 holds for all x ∈ L̃. ∼ B(L, L, ≤), the top element is represented by the Proof. In the isomorphism L̃ = W concept (L, {1}). Now let x := (Y, X) ∈ L̃. According to the definition, xN = X 4 . The element xN can also be represented by (↓xN ∩ L, ↑xN ∩ L). To prove that x ∨ xN = 1, it is enough to prove ↑xN ∩ ↑x ∩ L = {1}. W Now let b ∈ ↑xN ∩ ↑x ∩ L. We have b ≥ xN = X 4 , and consequently b ≥ t4 , for all t ∈ X = ↑x ∩ L. Moreover, b ≥ x implies b ∈ X. Thus b ≥ b4 and 1 = b ∨ b4 = b. t u The intended result is immediate for MacNeille completions that are distributive (see [Er82]). In fact, in a distributive lattice, the equation (x ∧ y) ∨ (x ∧ y 4 ) = x is equivalent to y ∨ y 4 = 1. Therefore if L̃ is distributive, we can conclude from Lemma 3 and Lemma 4 that the operation x 7→ (xN , xH ) defines a weak dicomplementation on L̃. Corollary 1 If the MacNeille completion L̃ of a weakly dicomplemented lattice L is distributive then (L̃, ∧, ∨,N ,H , 0, 1) is a weakly dicomplemented lattice. As the following theorem proves, the above result holds also in lattices with enough 1-prime and 0-prime elements. We call an element a 1-prime (resp. 0-prime) if a < 1 = c ∨ d implies a ≤ c or a ≤ d (resp. a > 0 = c ∧ d implies a ≥ c or a ≥ d) for all c, d ∈ L. Theorem 4 Let L be a weakly dicomplemented lattice such that the set J1 (L) (resp. M0 (L)) of 1-prime (resp. 0-prime) elements is supremum (resp. infimum) dense in L. Then the MacNeille completion of L is a weakly dicomplemented lattice. Proof. To prove the theorem, we only need to prove that (x ∧ y) ∨ (x ∧ y N ) = x and its dual hold. Here the calculation is more tedious. We take advantage of the context representation. We are going to prove that in B(J1 (L), M0 (L), ≤) the equation (x ∧ y) ∨ (x ∧ y N ) = x holds. It is enough to prove that (x ∧ y) ∨ (x ∧ y N ) ≥ x. We set x := (Y1 , X1 ), y := (Y2 , X2 ) and y N := (Y3 , X3 ). Note that there is no evidence to write y N = W ((J1 (L) \ Y2 )00 , (J1 (L) \ Y2 )0 ). However y N = X24 and (Y2 ∪ Y3 )00 = J1 (L). By Theorem 1 we have x ∧ y = (Y1 ∩ Y2 , (X1 ∪ X2 )00 ) and x ∧ y N = (Y1 ∩ Y3 , (X1 ∪ X3 )00 ). Hence,   0 (x ∧ y) ∨ (x ∧ y N ) = ((X1 ∪ X2 )00 ∩ (X1 ∪ X3 )00 ) , (X1 ∪ X2 )00 ∩ (X1 ∪ X3 )00 . But 0 (X1 ∪ X2 )00 ∩ (X1 ∪ X3 )00 = ((X10 ∩ X20 ) ∪ (X10 ∩ X30 )) 0 = (X10 ∩ (X20 ∪ X30 )) 0 = (Y1 ∩ (Y2 ∪ Y3 )) MacNeille completion of weakly dicomplemented lattices and 00 (Y1 ∩ (Y2 ∪ Y3 )) ⊆ Y100 ∩ (Y2 ∪ Y3 )00 = Y1 7 (see Lemma 3). 0 Now we want to prove that (Y1 ∩ (Y2 ∪ Y3 )) ⊆ Y10 . Let m ∈ M (L) such that m ∈ 0 (Y1 ∩ (Y2 ∪ Y3 )) ∀g ∈ J(L), g ∈ Y1 ∩ (Y2 ∪ Y3 ) =⇒ g ≤ m. (∗) We should prove that m ≥ h for all h ∈ Y1 . Let h ∈ Y1 . We have h ≤ 1 = y ∨ y N (see Lemma 4). It follows that h ≤ y or h ≤ y N . Thus h ∈ Y2 or h ∈ Y3 . Therefore h ∈ Y1 ∩ (Y2 ∪ Y3 ), and thus implies h ≤ m. t u To capture a larger class of weakly dicomplemented lattices, 1-prime and 0-prime elements can be replaced by primary elements. An element a ∈ L is called ∨-primary in L (resp. ∧-primary) if for all x ∈ L we have a 6≤ x =⇒ a ≤ x4 ( resp. a 6≥ x =⇒ a ≥ x5 ). In particular join-irreducible elements are ∨-primary, and meet-irreducible elements ∧-primary. Lemma 5 Primary elements in L are also primary in L̃. Proof. Let y := (Y2 , X2 ) ∈ L̃ and a a ∨-primary element that a 6≤ y. W in L such V N We should prove that a ≤ y . Note that in L̃ we have Y = y = X2 . Since 2 V g 6≤ X2 , there is some m ∈ X2 ⊆ L such that g 6≤ m. As a is ∨-primary it follows that g ≤ m4 . Therefore _ 4 a≤ X2 = y N . t u There are weakly dicomplemented lattices without irreducible elements that still have enough primary elements. For example the grid L := {0} ⊕ Z × Z ⊕ {1} has no join-irreducible and no meet irreducible elements (apart from 0 and 1). But the unique weakly dicomplemented lattice structure on L is trivial. Here all the elements are ∨-primary and ∧-primary. Since the crucial step in the proof of the theorem is the choice of h, with h ≤ y or h ≤ y N for all y ∈ L̃, we have Corollary 2 The MacNeille completion of a weakly dicomplemented lattice with enough primary elements is a weakly dicomplemented lattice. For the general case we might expect each weakly dicomplemented lattice to have enough ∨-primary and ∧-primary elements. Unfortunately, this is not the case. A typical example is an atomless Boolean algebra. Note that an element a of a Boolean algebra is ∨-primary if and only if it is an atom. Otherwise there would be an element a1 such that a1 < a. But a = (a ∧ a1 ) ∨ (a ∧ a01 ) with a 6≤ a ∧ a1 and a 6≤ a ∧ a01 . It would follow that a ≤ (a ∧ a1 )0 ∧ (a ∧ a01 )0 = a0 , forcing a to be 0, a contradiction. However its MacNeille completion is a Boolean algebra, thus a weakly dicomplemented lattice. To get the result in the general case, we need one more lemma: Lemma 6 Let h ∈ L and y ∈ L̃ such that h 6≤ y and h 6≤ y N . Let G and M be subsets of L such that B(G, M, ≤) is isomorphic to L̃. Then B(G \ {h}, M, ≤) is isomorphic to L̃. Proof. Let (Y, concept to y in (G, M, ≤). We know that W V X) be the W 4 corresponding N N Y = y = X and y = X . From h ≤ 6 y we obtain h 6≤ x4 for all x ∈ X. In V addition h 6≤ y = X implies the existence of an element x0 of X such that h 6≤ x0 . 8 L. Kwuida, B. Seselja & A. Tepavčević Therefore, h 6≤ x0 and h 6≤ x4 0 . But the 3rd axiom of weakly dicomplemented lattice 4 gives h = (h ∧ x0 ) ∨ (h ∧ x4 0 ) with h > h ∧ x0 and h > h ∧ x0 . Therefore 00 h00 = {g ∈ G | g ≤ h ∧ x0 or g ≤ h ∧ x4 0 } and h 6∈ {g ∈ G | g ≤ h ∧ x0 or g ≤ h ∧ x4 0 }. Thus h can be removed from G and the concept lattice of the remaining context is still isomorphic to that of the initial one. u t Recall that a lattice satisfies the (descending and ascending) chain conditions if it has no infinite chains. We can now state the result in the general case. Theorem 5 The MacNeille completion of a weakly dicomplemented lattice satisfying the chain conditions is a weakly dicomplemented lattice. Proof. The proof is similar to that of Theorem 4. The change is to prove (?) replacing J1 (L) with a suitable subset of L. By Lemma 6 such W a subsetWalways exists. In fact Lemma 6 states that Y1 can be replaced by Ỹ1 with Y1 = Ỹ1 such that Ỹ1 ⊆ Y2 ∪ Y3 . This gives the desired equality and concludes the proof. t u Unfortunately we cannot yet remove the chain conditions. The conjecture is that this result holds in general. If this is the case then the MacNeille completion of Boolean algebras can be deduced as a special case. In fact Boolean algebras are determined in the class of weakly dicomplemented lattices by the equation x4 = x5 [Kw04, Thm. 3.3.4]. Therefore it is enough to check this equality for N and H . The 3rd H N axiom of weakly dicomplemented W V lattices gives x ≤ x . Now let y ∈ L̃ represented by (Y, X). Then Y = y = X. For all g ∈ Y and m ∈ X, g ≤ m and by then g 4 ≥ m4 . Thus _ ^ ^ y N = {m4 | m ∈ X} ≤ {g 4 | g ∈ Y } = {g 5 | g ∈ Y } = y H . This would prove that the completion of a Boolean algebra is again a Boolean algebra. 3 Conclusion What is the consequence of Theorem 5 on the concrete representation problem of weakly dicomplemented lattices? First observe that for each x ∈ L, x ”is in” L̃ and xN = x4 as well as xH = x5 . Thus the ϕ : x 7→ (↓x, ↑x) is a weakly dicomplemented 4 lattice embedding of L into L̃. The conjecture is that “L embeds into A(K5 (L)) 4 iff L̃ embeds into A(K5 (L))”. References [Er82] M. Erne. Distributivgesetze und Dedekind’sche Schnitte. Abh. Braunschweig Wiss. Ges. 33, (1982) 117-145. [GW99] B. Ganter & R. Wille. Formal Concept Analysis. Mathematical Foundations Springer Verlag 1999 [Kw04] L. Kwuida. Dicomplemented Lattices. A Contextual Generalization of Boolean algebras Shaker Verlag. 2004 [Fu44] N. Funayama. On the completion by cuts of distributive lattices. Proc. Imp. Acad. Tokyo 20 (1944), 1-2. [Ha93] J. Harding. Any lattice can be regularly embedded into the MacNeille completion of a distributive lattice. Houston J. Math. 19, (1993), No. 1, 39-44. [Wi00] R. Wille. Boolean Concept Logic in B. Ganter & G. W. Mineau (Eds.) Conceptual Structures: Logical, Linguistic, and Computational Issues LNCS 1867 Springer (2000), 317-331. View publication stats