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Representation of lattices by fuzzy weak congruence relations
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Representation of lattices by fuzzy weak congruence relations ✩ Branimir Šešelja a , Vanja Stepanovi´c b , Andreja Tepavˇcevi´c a,∗ a Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Serbia b Faculty of Agriculture, University of Belgrade, Serbia
Received 30 December 2012; received in revised form 18 April 2014; accepted 6 May 2014
Abstract Fuzzy (lattice valued) weak congruences of abstract algebras are investigated. For an algebra, the family of all such fuzzy relations is a complete lattice; its structure and cut properties are investigated and fully described. These fuzzy weak congruences are applied in representation of complete and algebraic lattices. A wider class of lattices can be represented in such a fuzzy framework, than in classical algebra. We prove that there is a straightforward representation of any complete lattice, using it as a co-domain. In a more general case, it is proved that several subdirect powers of lattices are also representable by fuzzy weak congruences. © 2014 Elsevier B.V. All rights reserved. Keywords: Fuzzy relations; Algebra; Fuzzy algebra; Fuzzy weak congruence; Complete lattice
1. Introduction We deal with the fuzzy approach to congruences in general algebra, where the set of membership values is a complete lattice (an approach started by Goguen in [18]). Investigating these fuzzy compatible relations on algebraic structures, we apply our results to the classical lattice theory, namely in representation of lattices. Our link to existing results in both fuzzy and classical algebra is as follows. It is known that algebraic structures, like other notions in the fuzzy framework, depend on the membership function. This function offers a variety of properties and possibilities for applications, much more than the classical characteristic function. In addition, in the cut-worthy approach, there is a connection between a fuzzy structure as a mapping and a collection of crisp substructures of the same domain. Therefore, these investigations have a long history. At the beginning there were fuzzy groups (Rosenfeld [28] and Das [13], later Demirci [14] and others), then other structures and applications (Malik and Mordeson e.g., [25]). Investigations of notions from universal algebra were also carried out (e.g., Di Nola, Gerla ✩
Research supported by Ministry of Education and Science, Republic of Serbia, Grant No. 174013 and also by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, Grant “Ordered structures and applications” for the 1st and 3rd author. * Corresponding author. E-mail addresses: [email protected] (B. Šešelja), [email protected] (V. Stepanovi´c), [email protected] (A. Tepavˇcevi´c). http://dx.doi.org/10.1016/j.fss.2014.05.009 0165-0114/© 2014 Elsevier B.V. All rights reserved.
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[15] and the present authors [5,30–34]). In particular, fuzzy congruences, which are the central topic here, have been widely investigated (e.g., Kim and Bae [20], Kuroki [23], Kuraoka and Suzuki [22], Tan [39], Murali [26], recently Louskou-Bozapalidou [24], Ignjatovic et al. [19]), and also lattices of fuzzy congruences (Ajmal and Thomas [1] and others, recently Rainard and Mangalambal [27]; for hyper-structures Bakhshi and Borzooei [2] and Cabrera et al. [7,8]). Concerning our approach to fuzzy relations on fuzzy sets, it appeared in Filep’s paper [17]. A version of weak reflexivity was used long time ago in the paper [40] by Yeh and Bang, and then also in the paper [37]. In fuzzy algebraic investigations the set of membership values is the unit interval, or more generally a complete lattice (in many cases residuated); posets or relational systems are also used [30]. In the recent period, Bˇelohlávek (also together with Vychodil) develops and investigates the most important universal algebraic topics (congruences, subalgebras, products, and also varieties, see books [3,4]). The set of values of fuzzy structures in this approach is usually a complete residuated lattice. Let us mention that the residuation in the structure of multilattice was introduced in [9]. Our motivation is connected to our investigations of weak congruences in universal algebra, (see e.g. papers [10,11,16,29,36], and monograph [35]). Namely, the lattice of all weak congruences of an algebra is algebraic; congruence lattices of all its subalgebras, as well as, up to an isomorphism, the lattice of its subalgebras, come out to be its sublattices. An open problem in universal algebra is a representation of algebraic lattices by weak congruences. Here we investigate the analogue topic in the fuzzy framework. Using the idea from [37], we introduce fuzzy (lattice valued) weak congruences of an algebra and connect them with congruences on its fuzzy subalgebras. We prove that the lattice of fuzzy weak congruences is complete and fully describe its structure in lattice-theoretic terms. In addition, we deal with the representation problem of lattices by fuzzy weak congruences. It turns out that in this fuzzy framework it is possible to represent complete lattices which are not algebraic. We also prove representation theorem for several classes of lattices which are known to be non-representable in the classical way. 2. Preliminaries We deal with fuzzy algebraic structures whose co-domain is a complete lattice. Here we list relevant notions and notation. For theoretical background in lattice theory and universal algebra, we refer to books [12] and [6]. 2.1. Algebras, lattices An algebra (universal algebra) is a pair (A, F ), denoted by A , where A is a nonempty set and F is a set of operations on A. A subalgebra of A is an algebra defined on a subset B ⊆ A, where the operations are restrictions of the operations from F . A subuniverse of A is a subset (which may be empty) closed under operations. An equivalence relation ρ on A which is compatible with all the operations from F , in the sense that xi ρyi , i = 1, . . . , n imply f (x1 , . . . , xn )ρf (y1 , . . . , yn ), is a congruence on A . If A , B are algebras of the same type, then the mapping ϕ : A → B is a homomorphism from A to B if for every n-ary f ∈ F and all x1 , . . . , xn ∈ A, ϕ(f (x1 , . . . , xn )) = f (ϕ(x1 ), . . . , ϕ(xn )); for a nullary operation c, f (cA ) = cB . If A = B , ϕ is an endomorphism. Our main notion is a complete lattice, denoted by (L, ∧, ∨, ). A greatest, top element is denoted by 1, and a smallest, bottom element by 0. We use also the notion of a principal filter generated by a ∈ L, denoted by ↑a: ↑a = {x ∈ L | a x}. A principal ideal generated by a is defined dually: ↓a = {x ∈ L | x a}. For a, b ∈ L, a b the interval [a, b] is defined by: [a, b] = ↑a ∩ ↓b. A sublattice N of L is convex, if a, b ∈ N implies [a, b] ⊆ N . An element a of the lattice L is said to be codistributive if for all x, y ∈ L a ∧ (x ∨ y) = (a ∧ x) ∨ (a ∧ y). Proposition 1. If a is a codistributive element in L, then x → a ∧ x is an endomorphism of L onto ↓a.
2
By the above, if a is a codistributive element in L, then the sublattice ↓a is a retraction in L (i.e., it is a sublattice and at the same time a homomorphic image of L in a homomorphism which fixes it). A complete lattice L is algebraic if it is compactly generated, i.e., if every element in L is supremum of compact elements (see [6]). The following are known facts about complete lattices and their role in universal algebra (see e.g. [6]).
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Every poset with the top element which is closed under arbitrary infima, is a complete lattice. The set SubA of all the subuniverses of an algebra A is an algebraic lattice under inclusion. The set ConA of all the congruences on an algebra A is an algebraic lattice under inclusion. Next we recall the representation theorems for algebraic lattices by subuniverses (by Birkhoff and Frink) and congruences (by Grätzer and Schmidt), respectively. For every algebraic lattice L there is an algebra A , such that L is isomorphic to SubA . For every algebraic lattice L there is an algebra A , such that L is isomorphic to ConA . A weak congruence ρ on A , is symmetric, transitive, compatible and weakly reflexive relation on A . A property of being weakly reflexive is defined in such a way that for every nullary operation (constant) c ∈ A , it holds cρc. The following is known (see [35]). Proposition 2. The collection WConA of all weak congruences of an algebra A is an algebraic lattice. It is a union of congruence lattices of subalgebras of A and contains a subuniverse lattice as a retract. 2 The problem of representation of algebraic lattices by weak congruences is still open (there are some particular solutions, see [29,35,38]). For a deeper insight to universal algebra, see e.g. book [6]. 2.2. Fuzzy sets, structures and relations Here we introduce notions and properties from fuzzy set theory, used throughout the paper. The notions introduced in our papers are also listed. For general approach to fuzziness, see e.g. [21]. Throughout the paper, as the co-domain of fuzzy sets and relations, i.e., as the structure of membership values, we use a complete lattice, usually denoted by (L, ∧, ∨), with 0 and 1 being the bottom and the top element under the order in L. A fuzzy set μ on a nonempty set A is a function μ : A → L. We also call it an L-valued set on A, in particular if we want to emphasize the co-domain lattice L. If A and L are fixed, then by LA we denote the set of all functions from A to L, i.e., the collection of all fuzzy sets on A with membership values in L: LA := {μ | μ : A → L}. A binary relation ⊆ (inclusion of fuzzy sets) is defined on LA componentwise with respect to the lattice order : for any μ, ν : A → L μ⊆ν
if and only if μ(x) ν(x) for every x ∈ A.
Similarly, the union and the intersection of a family of fuzzy sets are defined componentwise by join and meet respectively. For p ∈ L, a cut set, or a p-cut of μ is a subset μp of A defined by: μp = x ∈ A μ(x) p . Obviously, the cut μp is the inverse image of the principal filter ↑p under μ: μp = μ−1 (↑p). If μ : A → L is a fuzzy set, and {μp | p ∈ L} is the family of cuts of this fuzzy set, then μ can be computed by its family of cuts, as follows: μ(x) = {p ∈ L | x ∈ μp }. (1) A fuzzy set on a square of the nonempty set A, i.e., a mapping ρ : A2 → L is a fuzzy (binary) relation on A, or an L-valued relation on A. Since a fuzzy relation is a fuzzy set, the definitions of inclusion, union and intersection are valid also for fuzzy relations. Here are some well known properties of fuzzy relations: ρ is reflexive if ρ(x, x) = 1 for all x ∈ A;
(2)
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ρ is symmetric if ρ(x, y) = ρ(y, x) for all x, y ∈ A; ρ is transitive if ρ(x, y) ρ(x, z) ∧ ρ(z, y) for all x, y ∈ A,
(3) (4)
z∈A
which is equivalent with ρ(x, y) ρ(x, z) ∧ ρ(z, y),
for all x, y, z ∈ A.
A fuzzy relation ρ on A which is reflexive, symmetric and transitive is a fuzzy equivalence on A. Using characteristic functions with values 0 and 1 from L, we also consider the identity relation ΔA , or simply Δ, and the square A2 of A as fuzzy equivalences: These are both mappings from A2 to L, such that 1 if x = y Δ(x, y) = A2 (x, y) = 1 for all x, y ∈ A. 0 else. We say that a fuzzy relation ρ : A2 → L is a diagonal relation if ρ(x, y) = 0 for all x, y ∈ A, x = y. Obviously, the identity relation is a special diagonal relation on A. If A = (A, F ) is an algebra, then a fuzzy subalgebra of A is a mapping μ : A → L satisfying the following inequality: For any operation f from F , f : An → A, n ∈ N, and all a1 , . . . , an ∈ A: n
μ(ai ) μ f (a1 , . . . , an ) ,
i=1
and for a nullary operation (constant) c ∈ F , μ(c) = 1 ∈ L. It is well known that the cuts of fuzzy subalgebras are crisp subalgebras. To be more precise: Lemma 1. A fuzzy set on a crisp algebra is a fuzzy subalgebra if and only if all the cuts are crisp subalgebras. 2 Observe that in the case when there are no nullary operations, also the characteristic function of the empty set
O : A → L, such that for every x ∈ A, O (x) = 0 ∈ L, fulfills the definition of a fuzzy subalgebra. Consequently, the collection of all fuzzy subalgebras, including O if there are no nullary operations, is also called a set of fuzzy subuniverses of A , analogously to the crisp case. This notion we use in the sequel and we denote it by FsA . The intersection of a collection of fuzzy subuniverses of A is again a fuzzy subuniverse of A . In addition, a constant
function I : A → L, such that I (x) = 1 for every x ∈ A is also a fuzzy subuniverse. Therefore, we have the following structure property of the collection of subuniverses. Proposition 3. If A is an algebra and L a complete lattice, then the collection FsA of all L-valued subuniverses of A is a complete lattice under inclusion. 2 A fuzzy relation ρ : A2 → L on an algebra A = (A, F ) is said to be compatible with the operations, if for every n-ary operation f ∈ F and for all a1 , . . . , an , b1 , . . . , bn ∈ A n ρ f (a1 , . . . , an ), f (b1 , . . . , bn ) ρ(ai , bi ).
(5)
i=1
In particular, for n = 1, ρ(f (a1 ), f (b1 )) ρ(a1 , b1 ). The following definition is well known. Let A = (A, F ) be an algebra. A reflexive, symmetric and transitive fuzzy relation on A, which is compatible with operations from F is a fuzzy congruence on algebra A . Cuts of a fuzzy congruence relation are crisp congruence relations. Moreover, the following is true: Lemma 2. Let A be an algebra. Then a fuzzy relation ρ : A → L is a fuzzy congruence if and only if all the cuts are crisp congruences. 2 If A is an algebra and L a complete lattice, then we denote by Fc(A ) the collection of all fuzzy (L-valued) congruences on A . The following proposition is known, though we present the proof which we use in the subsequent sections.
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Proposition 4. Fc(A ) is a complete lattice under inclusion, with the full relation being the top element, and the intersection of fuzzy sets being the infimum. Proof. The full relation is the one that maps all the elements from A × A to the top element of the lattice. It is obvious that this relation is a fuzzy congruence and that it is the greatest such relation (under inclusion). Further, consider a collection of fuzzy congruences: {ρi | i ∈ I }. Its intersection is also a fuzzy congruence on A (the proof is straightforward). Intersection is the infimum under inclusion, hence Fc(A ) is a complete lattice. 2 The following property of a fuzzy relation ρ on an algebra A is a weaker version of reflexivity: A fuzzy relation ρ on A is weakly reflexive on A if ρ(c, c) = 1 for all nullary operations c ∈ F.
(6)
A weakly reflexive, symmetric and transitive fuzzy relation on A, which is compatible with the operations from F , is a fuzzy weak congruence on this algebra. If ρ is a fuzzy weak congruence, by transitivity and symmetry, we have that for every x, y ∈ A, ρ(x, y) ρ(x, x) and ρ(x, y) ρ(y, y). Next we recall the notion of a fuzzy relation on a fuzzy set. Let μ : A → L be a fuzzy set on A. A mapping ρ : A2 → L (i.e., a fuzzy relation on A) is called a fuzzy relation on a fuzzy set μ if ρ(x, y) μ(x) ∧ μ(y) for all x, y ∈ A.
(7)
Some properties of fuzzy relations on a fuzzy set are equivalently defined as those for fuzzy relations on the crisp domain. E.g., a fuzzy relation ρ on μ is symmetric if it satisfies the property (3) and transitive, if it fulfills (4), as a fuzzy relation on A. However, the reflexivity is defined differently. Namely, if ρ is a fuzzy relation on a fuzzy set μ : A → L, then ρ is reflexive on μ
if ρ(x, x) = μ(x) for all x ∈ A.
(8)
Obviously, if μ coincides with A (as a constant characteristic function with value 1 for every x ∈ A), then the present property coincides with (2). To sum up, we deal here with three reflexivity properties for fuzzy relations: two for fuzzy relations on a crisp domain ((2) and its weak version (6)) and reflexivity (8) of fuzzy relations on fuzzy sets. All of these are essential for the results we present in the sequel. A fuzzy relation ρ on μ : A → L which is symmetric, transitive and reflexive on μ, is a fuzzy equivalence on μ. Finally, if μ : A → L is a fuzzy subalgebra of an algebra A , then a relation ρ on μ is a fuzzy congruence on μ if it is a fuzzy equivalence on μ, and it is compatible with operations on A in the sense of (5). Let us mention our notation. Throughout the text, we deal with several lattices: the membership values lattice L and for an algebra A , lattices FcA , WfcA etc. In all these we denote lattice operations and relations in the same way: meet by ∧, join by ∨, orderings by and . Confusions should not appear: which operation and order (in which lattice) are used is always clear from the context. In some cases, however, we use ∧L and ∨L in order to indicate that the corresponding meet and join are computed in the lattice L. 3. Results 3.1. Lattice of fuzzy congruences Our first and introductory result deals with fuzzy congruences on algebras. We present a straightforward representation of complete lattices by means of the mentioned fuzzy relations, which form a complete lattice, by Proposition 4. Namely, having a lattice M, we look for a crisp algebra and a complete lattice L, so that the lattice of fuzzy (L-valued) congruences of A is isomorphic with M. This problem is easily solved in the fuzzy framework, as follows. Observe that in this representation the membership values lattice coincides with the one to be represented. Theorem 1. Every complete lattice L is isomorphic to the lattice of all L-valued congruences of a two-element group.
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Proof. Let L be a complete lattice and G = ({e, a}, ·) a two-element group: · e a
e e a
a a. e
Obviously, each L-valued congruence ρ on G has the form ρ e a
e 1 p
a p, 1
p ∈ L.
The proof is now straightforward.
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Theorem 1 shows that, contrary to the crisp case, the lattice of fuzzy congruences need not be algebraic. 3.2. Fuzzy weak congruences Here we deal with fuzzy weak congruences, which differ from fuzzy congruences in the reflexivity condition, as defined in preliminaries. Proposition 5. Let L be a complete lattice, let A = (A, F ) be an algebra and ρ : A2 → L a fuzzy (L-valued) relation on A. Then ρ is a fuzzy weak congruence relation on A , if and only if all the cut relations are weak congruences on A . Proof. Suppose that ρ is a fuzzy weak congruence, and let p ∈ L. By the definition, a cut relation ρp is a subset of A2 such that (x, y) ∈ ρp if and only if ρ(x, y) p. It is known that ρp is a symmetric, transitive and compatible relation on A . Further, if c ∈ F is a nullary operation from F , then ρ(c, c) = 1 p, which means that (c, c) ∈ ρp . To prove the converse, suppose that all the cuts of ρ are weak congruences. Using the well known formula (1), for fuzzy relations: ρ(x, y) = p ∈ L (x, y) ∈ ρp , we prove that ρ is a fuzzy weak congruence. 2 Next we fix a complete lattice L and an algebra A = (A, F ). We consider a collection of all fuzzy (L-valued) weak congruences on A = (A, F ) under inclusion and denote it by Wfc(A ). The proof of the following proposition is analogous to the one of Proposition 4. Proposition 6. Wfc(A ) is a complete lattice under inclusion, with the full relation being the top element, and the intersection of fuzzy sets being the infimum. 2 Next we analyze properties of lattices of fuzzy weak congruences. Theorem 1 witnesses that the lattice of fuzzy congruences is not an algebraic lattice in general. Analyzing the proof of this theorem in a such way that we consider both elements of the two-element group as constants, we obtain an algebra in which fuzzy weak congruences coincide with fuzzy congruences (for which all the pairs with equal components have the constant value 1). Hence Theorem 1 shows that a lattice of fuzzy weak congruences need not be algebraic. This is a difference between a lattice of fuzzy weak congruences and a lattice of (crisp) weak congruences, since the latter is always an algebraic lattice. In the lattice of weak congruences, the identity relation has a distinguished role. Similar holds with the lattice of fuzzy weak congruences. Therefore, in the following we investigate lattice-theoretic properties of identity relation Δ in a fuzzy weak congruence lattice and we analyze the structure of this lattice comparing it with the lattice of crisp weak congruences. Proposition 7. If A = (A, F ) is an algebra, L a complete lattice and ρ a compatible, weakly reflexive fuzzy relation on A , then a fuzzy set μρ : A → L defined by:
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μρ (x) := ρ(x, x)
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(9)
is a fuzzy subalgebra of A . Proof. This is straightforward, by compatibility, and for nullary operations by weak reflexivity of ρ. 2 According to Proposition 7, we say that a fuzzy relation ρ determines the subalgebra μρ . Observe that in the crisp case we have the analogue situation: every weak congruence ρ on an algebra A determines a subalgebra B of A by the corresponding identity relation i.e., by B = {x ∈ A | xρx}. Now we deal with the converse. Let A = (A, F ) be an algebra, let L be a complete lattice and μ : A → L a fuzzy subalgebra of A . We define a fuzzy relation Δμ : A2 → L by
μ(x) for x = y Δμ (x, y) = (10) 0 for x = y. It is easy to see that Δμ ∈ Wfc(A ), i.e., that Δμ it is a diagonal fuzzy weak congruence. In addition, it is straightforward to check that μΔμ = μ. Particular fuzzy weak congruences in Wfc(A ) that we use in the sequel are diagonal fuzzy relations of the form ρ ∧ Δ, where ρ is a weak fuzzy congruence. For these fuzzy relations the following holds. Proposition 8. Let ρ be a fuzzy weak congruence on A . Then in the lattice Wfc(A ) ρ ∧ Δ = Δμ ρ . Proof. Since the meet in the lattice Wfc(A ) is the set intersection, we have that ρ ∧ Δ is a diagonal relation. Then the proof follows by Proposition 7 and by (10). 2 By Proposition 8, if ρ, θ ∈ Wfc(A ) then μρ = μ θ
if and only if
ρ ∧ Δ = θ ∧ Δ.
In the crisp case, the greatest weak congruence determining (as mentioned after Proposition 7) a subalgebra B , is the square B 2 (considered as a relation on A). In the fuzzy framework, the greatest fuzzy congruence determining the same fuzzy subalgebra as ρ is not the square (represented by its characteristic function as a fuzzy relation) in general, as proven in the sequel. Proposition 9. Let ρ be a fuzzy weak congruence on A . Define ρ : A2 → L by ρ(x, y) := ρ(x, x) ∧L ρ(y, y).
(11)
Then (i) ρ is a fuzzy weak congruence on A . (ii) ρ is the greatest fuzzy weak congruence determining the fuzzy subalgebra μρ . Proof. (i) Let us, for short, denote ρ0 := ρ ∧ Δ. Since ρ0 is a fuzzy weak congruence, we have the following inequality in the codomain lattice L: ρ0 (f (x1 , . . . , xn ), f (x1 , . . . , xn )) ρ0 (x1 , x1 ) ∧L . . . ∧L ρ0 (xn , xn ). By (11), we have ρ(x, y) = ρ0 (x, x) ∧L ρ0 (y, y). Now, ρ is obviously symmetric, transitive, weakly reflexive and also compatible: for an n-ary operation f on A , we have ρ f (x1 , . . . , xn ), f (y1 , . . . , yn ) = ρ0 f (x1 , . . . , xn ), f (x1 , . . . , xn ) ∧L ρ0 f (y1 , . . . , yn ), f (y1 , . . . , yn ) ρ0 (x1 , x1 ) ∧L . . . ∧L ρ0 (xn , xn ) ∧L ρ0 (y1 , y1 ) ∧L . . . ∧L ρ0 (yn , yn ) = ρ0 (x1 , x1 ) ∧L ρ0 (y1 , y1 ) ∧L . . . ∧L ρ0 (xn , xn ) ∧L ρ0 (yn , yn ) = ρ(x1 , y1 ) ∧L . . . ∧L ρ(xn , yn ).
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(ii) For every fuzzy weak congruence ρ, we have ρ(x, x) ρ(x, y) and ρ(y, y) ρ(x, y), thus ρ(x, x) ∧L ρ(y, y) ρ(x, y). By (11), it follows that ρ is the greatest such fuzzy weak congruence on A . 2 Proposition 10. The identity relation Δ of an algebra A is a codistributive element in the fuzzy weak congruence lattice WfcA . Proof. We have to prove that for every two fuzzy weak congruences ρ, θ : A2 → L, Δ ∧ (ρ ∨ θ ) = (Δ ∧ ρ) ∨ (Δ ∧ θ ). Since the inequality in the above formula is fulfilled as a general lattice property, what remains to show is that Δ ∧ (ρ ∨ θ ) (Δ ∧ ρ) ∨ (Δ ∧ θ ). As before, if ρ ∈ WfcA , we put ρ0 = ρ ∧ Δ. In the following we use (11) and the fact that ρ ρ. We have also (again by Proposition 9) ρ ∧ Δ = ρ ∧ Δ. Now we have ρ(x, y) ∨L θ (x, y) ρ(x, x) ∧L ρ(y, y) ∨L θ (x, x) ∧L θ (y, y) = ρ0 (x, x) ∧L ρ0 (y, y) ∨L θ0 (x, x) ∧L θ0 (y, y) ρ0 (x, x) ∨L θ0 (x, x) ∧L ρ0 (y, y) ∨L θ0 (y, y) (ρ0 ∨ θ0 )(x, x) ∧L (ρ0 ∨ θ0 )(y, y) = ρ0 ∨ θ0 (x, y). Thus, ρ ∨ θ ρ0 ∨ θ 0 . Therefore, Δ ∧ (ρ ∨ θ ) Δ ∧ ρ0 ∨ θ0 = Δ ∧ (ρ0 ∨ θ0 ) = Δ ∧ (Δ ∧ ρ) ∨ (Δ ∧ θ ) = (Δ ∧ ρ) ∨ (Δ ∧ θ ), which proves the codistributivity of Δ. 2 Any fuzzy congruence ρ on a fuzzy subalgebra μ of A is, as a fuzzy relation on A , weakly reflexive, since by reflexivity we have that, for any constant operation c, ρ(c, c) = μ(c) = 1. Therefore, ρ is also a fuzzy weak congruence on A . On the other hand, let ρ be a fuzzy weak congruence on A . Since ρ(x, y) ρ(x, x) and ρ(x, y) ρ(y, y), we have that ρ(x, y) ρ(x, x) ∧L ρ(y, y), thus ρ is a fuzzy relation on the fuzzy set μρ : A → L, defined by μρ (x) = ρ(x, x). By Proposition 7, μρ is a fuzzy subalgebra of A . Obviously, ρ is reflexive on μ. Thus, any fuzzy weak congruence ρ on A is a fuzzy congruence of the fuzzy subalgebra μρ of A . Therefore we have the following connection between fuzzy congruences on fuzzy subalgebras of A and fuzzy weak congruences on A . Proposition 11. A fuzzy congruence on a fuzzy subalgebra μ of A is a fuzzy weak congruence on A . Conversely, any fuzzy weak congruence on A is a fuzzy congruence of the fuzzy subalgebra μρ of A . Let us denote by Fc(μ) the set of all fuzzy congruences on a fuzzy subalgebra μ of A . In the lattice WfcA , the principal ideal ↓Δ contains precisely all identity relations, defined by (10). Further, as a consequence of Propositions 10 and 1, we conclude that the mapping ρ → ρ ∧ Δ is a lattice endomorphism from WfcA onto its sublattice ↓Δ. Consequently, the classes (convex sublattices) of the kernel of this homomorphism consist of all fuzzy weak congruences which determine the same fuzzy subalgebra, in the sense of Proposition 7. In other words these are fuzzy congruences on the corresponding fuzzy subalgebra. By Proposition 11, we conclude that for each fuzzy subalgebra μ of A , Fc(μ) is a complete lattice isomorphic to a corresponding kernel class. In particular, the
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sublattice ↑Δ is a lattice of all fuzzy congruences on the whole algebra A . Indeed, filter ↑Δ contains all reflexive fuzzy weak congruences, hence these are fuzzy congruences on A . Let us denote by f Δ the following mapping from the sublattice ↓Δ of Wfc(A ) to the lattice Fs(A ) of all fuzzy subuniverses of A : For ρ ∈ ↓Δ, we take f Δ(ρ) := μρ , where μρ is defined by (9). It is easy to see that f Δ is a bijection, moreover, the lattices ↓Δ and Fs(A ) are isomorphic. Indeed, the meet and the join of two subalgebras in FsA correspond respectively to the meet and the join of the corresponding identity relations. The greatest fuzzy congruence on subalgebra μ, denoted by μ2 , is given by μ2 (x, y) = μ(x) ∧L μ(y). Obviously, for a fuzzy weak congruence ρ on A , we have that ρ = (μρ )2 (see Proposition 9(iii)). The previous analysis actually proves the following. Theorem 2. Let Cwf A be the lattice of all fuzzy weak congruences on an algebra A . Then the following hold. (i) For every fuzzy subalgebra μ of A , Fc(μ) is the interval sublattice [Δμ , μ2 ] of lattice Cwf A , in particular FcA = ↑Δ. (ii) The set Cwf A is a disjoint union of intervals [Δμ , μ2 ] for all μ ∈ FsA . (iii) The lattice FsA of all fuzzy subuniverses of A is isomorphic to the principal ideal ↓Δ, under μ → Δμ . (iv) The mapping mΔ : ρ → ρ ∧ Δ is a retraction from Cwf A onto ↓Δ. 3.3. Representation of complete lattices by fuzzy weak congruences Section 3.1 contains a representation theorem of complete lattices by fuzzy congruence lattices. The problem of representation of lattices by fuzzy weak congruences is much more complicated. This problem is investigated but not fully solved in the crisp case (see e.g. [29,35]) and here we deal with it in the fuzzy framework. Representation problem. Let M be a complete lattice and a ∈ M. Find a lattice L and an algebra A such that the lattice WfcA of fuzzy (L-valued) weak congruences of A is isomorphic to M and that Δ corresponds to a under this isomorphism. The main reason for investigating the above problem is that it is possible to represent a wider class of lattices by fuzzy weak congruence lattices than by classical weak congruence lattices. The first argument is already mentioned: a fuzzy weak congruence lattice need not be algebraic, which is a necessary property in the classical case. Another argument is explained by the following example. Example 1. Let M be a 25 element lattice represented by a diagram in Fig. 1(i), and let a ∈ M, as indicated. It is not possible to represent M by a weak congruence lattice of some algebra (in the classical settings), with the identity element corresponding to element a. Indeed, this element does not fulfill particular necessary conditions for being the identity of an algebra in a representation of M by its weak congruence lattice (a is not Δ-suitable, see [29,35]). However, a representation is possible in the fuzzy framework, as follows. Let G be a 2 element groupoid ({a, b}, ∗) with two subgroupoids {a} and {b}, given by the following table ∗ a b
a a b
b b. b
Let L be a 4 element Boolean lattice, Fig. 1(ii). M is the fuzzy (L-valued) weak congruence lattice of G, with the element a representing its identity relation. 2
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Fig. 1. L-valued weak congruence lattice of G and lattice L.
Continuing our research of the fuzzy weak congruence representation problem, we analyze a decomposition of a lattice. Since a lattice doesn’t have to be algebraic to be representable in the above sense, the question whether the classes under the equivalence x ∼ y ⇔ x ∧ a = y ∧ a have the top elements makes sense. Proposition 12. If a lattice M is fuzzy weak congruence representable, with a ∈ M corresponding to the identity, then its classes under the equivalence x ∼ y ⇔ x ∧ a = y ∧ a have top elements. Proof. Indeed, the classes under the equivalence ∼ coincide in the fuzzy weak congruence lattice of an algebra A with classes of the kernel of the endomorphism ρ → ρ ∧ Δ. By Proposition 11 and Theorem 2, the mentioned classes are actually lattices of fuzzy congruences of subalgebras of A . These lattices do have the top elements, namely the greatest fuzzy congruences on subalgebras, given by (11). 2 In the classical representation of lattices by weak congruences, it is obvious that the top element of the lattice could correspond to the identity relation only if the lattice is a chain with at most 2 elements [35]. In the fuzzy case, the situation here is different. Indeed, let M be a complete lattice. Take a one element algebra with no constant operation and take M as the codomain lattice. Any fuzzy relation ρ on the algebra is defined by ρ(a, a) = m, where a is the only element of the algebra, and m is an element of M. For every m ∈ M this fuzzy relation is also a fuzzy weak congruence. So, the fuzzy weak congruence lattice of this algebra is isomorphic to M, under an isomorphism sending the identity relation to the top element of L. Therefore the following holds. Proposition 13. Any complete lattice is fuzzy weak congruence representable, with the top element corresponding to the identity relation. In some other cases, however, the fuzzy weak congruence representability is equivalent to the weak congruence representability: Proposition 14. Let M be a complete lattice and a an element of M different from the top and from the bottom, such that M = ↓a ∪ ↑a. Then, M is fuzzy weak congruence representable and a corresponds to the identity, if and only if M is weak congruence representable, a corresponding to the identity as well. Proof. If M is weak congruence representable, then M is fuzzy weak congruence representable for a two element codomain lattice. If, on the contrary, M is fuzzy weak congruence representable, then let A = (A, H ) be an algebra and L a codomain lattice, such that WfcA is isomorphic to M under an isomorphism mapping Δ to a.
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Fig. 2. Representable and non-representable lattice.
If A has only one element, Δ is the top element of WfcA and, accordingly, a is the top element of M, which contradicts the assumptions; so, let A have more than one element. If L has only one element, then WfcA has only one element and, accordingly, M has only the top element, which contradicts the assumptions. If L has more than two elements, let p be its element different from the bottom and the top element. Let ρ be a fuzzy relation defined as follows.
1 for all x, y belonging to the smallest subuniverse generated by constants in A ρ(x, y) = p for all other x, y ∈ A. Obviously, if there are no constants in A , then ρ(x, y) = p for all x, y ∈ A. Clearly, ρ is a fuzzy weak congruence, as well as the relation ρ ∪ Δ. Since a is not the bottom of L, the smallest subalgebra generated by constants does not coincide with A . Therefore, relations ρ and ρ ∪ Δ differ. Thus, the class of the homomorphism kernel of ρ in WfcA has more than one element, and accordingly, there exists a class of M, different from ↑a, that has more than one element, which contradicts the fact that M = ↓a ∪ ↑a. So L has two elements, and M is representable in the classical sense. 2 Example 2. In Fig. 2 we have two lattices, M and M , both satisfying the conditions from Proposition 14. The left one is weak congruence representable: let A = (A, ∗), where A = {a, b, c, d} and (∗) is given by the table below; CwA is isomorphic to M, and Δ corresponds to p in the isomorphism; accordingly, it is fuzzy weak congruence representable. The right lattice is not weak congruence representable, since element q does not fulfill a necessary condition to be represented by the identity relation, described in [35], therefore it is not fuzzy weak congruence representable. ∗ a b c d
a a d d b
b c a a a
c b a a a
d c a a a
2
Finally we give an answer to the representation problem in the fuzzy framework, for some subdirect powers of an arbitrary complete lattice. Theorem 3. Let M be an arbitrary complete lattice. Then the following lattices are fuzzy weak congruence representable (i) M1 = {(x, y) ∈ M 2 | x y}, the identity corresponding to a = (0, 1). (ii) M2 = {(x, y, z) ∈ M 3 | x y z}, the identity corresponding to a = (0, 1, 1). (iii) M3 = {(x, y, z) ∈ M 3 | x y ∧ z}, the identity corresponding to a = (0, 1, 1).
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Proof. Let M be an arbitrary complete lattice. In order to prove that the corresponding subdirect powers of M are fuzzy weak representable, we take L = M to be the co-domain lattice in all three cases. Then we proceed as follows. (i) Let (G, ·, −1 , e) be a two-element group with the neutral element e, given by table (a). Each fuzzy weak congruence ρ on G has the form presented by table (b), with 1, p, q ∈ L, where 1 is the top element and p q. (a)
· e a
e e a
a a e
(b)
ρ e a
e 1 p
a p. q
Indeed, the cut-relations of ρ are weak congruences on G, namely, for 0 < p < q, they are represented by tables (c), (d) and (e). (c)
ρ1 e a
e a 1 0, 0 0
ρq e a
(d)
e a 1 0 0 1
(e)
ρp e a
e a 1 1, 1 1
ρ0 = ρ p .
The table of ρ is actually determined by the ordered pair (p, q), p, q ∈ L, p q. Therefore, Wf cG is isomorphic to M1 . The identity relation is determined by the pair (0, 1). (ii) Here we take the same binary operation as in (i), but considered as a two-element groupoid G, without constants. In this case, fuzzy weak congruences have the form given in table (f ). (f )
ρ e a
e r p
a p, q
p q r.
Therefore, each fuzzy weak congruence uniquely corresponds to an ordered triple (p, q, r), p, q, r ∈ L, p q r and Wf cG is isomorphic to M2 . The identity relation, according to the table, corresponds to the triple (0, 1, 1). (iii) This case is a construction with the two-element groupoid (H, ∗) with two subgroupoids {a} and {b}, given by table (g) (actually used in Example 1). Fuzzy weak congruences are also presented in the general form, by table (h). (g)
∗ a b
a a b
b b b
(h)
ρ a b
a q p
b p, r
p q ∧ r.
Since each fuzzy weak congruence is associated to an ordered triple of the form (p, q, r), p, q, r ∈ L, p q ∧ r, Wf cH is isomorphic to M3 , and the identity relation here corresponds to (0, 1, 1). 2 A concrete lattice representation by fuzzy weak congruences, proved in Theorem 3, is given in Example 1. The 25-element lattice in Fig. 1 is a subdirect power of the four-element Boolean lattice (same figure) described in (iii) of this theorem. 4. Conclusion Our paper uses advantages of fuzzy algebraic structures and techniques in order to deal with problems in classical lattice theory. A still unsolved problem of representation of algebraic lattices by weak congruences is here situated in the fuzzy framework. What we obtained, is a possibility to represent a wider class of lattices, this time with fuzzy weak congruences. In addition, our results point to some additional constructions of lattices which could be represented by these fuzzy relations, and we see this as our future task. References [1] [2] [3] [4] [5] [6]
N. Ajmal, K.V. Thomas, A complete study of the lattices of fuzzy congruences and fuzzy normal subgroups, Inf. Sci. 82 (1995) 197–218. M. Bakhshi, R.A. Borzooei, Lattice structure on fuzzy congruence relations of a hypergroupoid, Inf. Sci. 177 (2007) 3305–3313. R. Bˇelohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Publishers, New York, 2002. R. Bˇelohlávek, V. Vychodil, Fuzzy Equational Logic, Studies in Fuzziness and Soft Computing, vol. 186/2005, Springer, 2005. B. Budimirovi´c, V. Budimirovi´c, A. Tepavˇcevi´c, Fuzzy ε-subgroups, Inf. Sci. 180 (2010) 4006–4014. S. Burris, H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, NY, 1981.
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Representation of lattices by fuzzy weak congruence relations ✩ Branimir Šešelja a , Vanja Stepanovi´c b , Andreja Tepavˇcevi´c a,∗ a Department of Mathematics and Informatics, Faculty of Science, University of Novi Sad, Serbia b Faculty of Agriculture, University of Belgrade, Serbia
Received 30 December 2012; received in revised form 18 April 2014; accepted 6 May 2014
Abstract Fuzzy (lattice valued) weak congruences of abstract algebras are investigated. For an algebra, the family of all such fuzzy relations is a complete lattice; its structure and cut properties are investigated and fully described. These fuzzy weak congruences are applied in representation of complete and algebraic lattices. A wider class of lattices can be represented in such a fuzzy framework, than in classical algebra. We prove that there is a straightforward representation of any complete lattice, using it as a co-domain. In a more general case, it is proved that several subdirect powers of lattices are also representable by fuzzy weak congruences. © 2014 Elsevier B.V. All rights reserved. Keywords: Fuzzy relations; Algebra; Fuzzy algebra; Fuzzy weak congruence; Complete lattice
1. Introduction We deal with the fuzzy approach to congruences in general algebra, where the set of membership values is a complete lattice (an approach started by Goguen in [18]). Investigating these fuzzy compatible relations on algebraic structures, we apply our results to the classical lattice theory, namely in representation of lattices. Our link to existing results in both fuzzy and classical algebra is as follows. It is known that algebraic structures, like other notions in the fuzzy framework, depend on the membership function. This function offers a variety of properties and possibilities for applications, much more than the classical characteristic function. In addition, in the cut-worthy approach, there is a connection between a fuzzy structure as a mapping and a collection of crisp substructures of the same domain. Therefore, these investigations have a long history. At the beginning there were fuzzy groups (Rosenfeld [28] and Das [13], later Demirci [14] and others), then other structures and applications (Malik and Mordeson e.g., [25]). Investigations of notions from universal algebra were also carried out (e.g., Di Nola, Gerla ✩
Research supported by Ministry of Education and Science, Republic of Serbia, Grant No. 174013 and also by the Provincial Secretariat for Science and Technological Development, Autonomous Province of Vojvodina, Grant “Ordered structures and applications” for the 1st and 3rd author. * Corresponding author. E-mail addresses: [email protected] (B. Šešelja), [email protected] (V. Stepanovi´c), [email protected] (A. Tepavˇcevi´c). http://dx.doi.org/10.1016/j.fss.2014.05.009 0165-0114/© 2014 Elsevier B.V. All rights reserved.
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[15] and the present authors [5,30–34]). In particular, fuzzy congruences, which are the central topic here, have been widely investigated (e.g., Kim and Bae [20], Kuroki [23], Kuraoka and Suzuki [22], Tan [39], Murali [26], recently Louskou-Bozapalidou [24], Ignjatovic et al. [19]), and also lattices of fuzzy congruences (Ajmal and Thomas [1] and others, recently Rainard and Mangalambal [27]; for hyper-structures Bakhshi and Borzooei [2] and Cabrera et al. [7,8]). Concerning our approach to fuzzy relations on fuzzy sets, it appeared in Filep’s paper [17]. A version of weak reflexivity was used long time ago in the paper [40] by Yeh and Bang, and then also in the paper [37]. In fuzzy algebraic investigations the set of membership values is the unit interval, or more generally a complete lattice (in many cases residuated); posets or relational systems are also used [30]. In the recent period, Bˇelohlávek (also together with Vychodil) develops and investigates the most important universal algebraic topics (congruences, subalgebras, products, and also varieties, see books [3,4]). The set of values of fuzzy structures in this approach is usually a complete residuated lattice. Let us mention that the residuation in the structure of multilattice was introduced in [9]. Our motivation is connected to our investigations of weak congruences in universal algebra, (see e.g. papers [10,11,16,29,36], and monograph [35]). Namely, the lattice of all weak congruences of an algebra is algebraic; congruence lattices of all its subalgebras, as well as, up to an isomorphism, the lattice of its subalgebras, come out to be its sublattices. An open problem in universal algebra is a representation of algebraic lattices by weak congruences. Here we investigate the analogue topic in the fuzzy framework. Using the idea from [37], we introduce fuzzy (lattice valued) weak congruences of an algebra and connect them with congruences on its fuzzy subalgebras. We prove that the lattice of fuzzy weak congruences is complete and fully describe its structure in lattice-theoretic terms. In addition, we deal with the representation problem of lattices by fuzzy weak congruences. It turns out that in this fuzzy framework it is possible to represent complete lattices which are not algebraic. We also prove representation theorem for several classes of lattices which are known to be non-representable in the classical way. 2. Preliminaries We deal with fuzzy algebraic structures whose co-domain is a complete lattice. Here we list relevant notions and notation. For theoretical background in lattice theory and universal algebra, we refer to books [12] and [6]. 2.1. Algebras, lattices An algebra (universal algebra) is a pair (A, F ), denoted by A , where A is a nonempty set and F is a set of operations on A. A subalgebra of A is an algebra defined on a subset B ⊆ A, where the operations are restrictions of the operations from F . A subuniverse of A is a subset (which may be empty) closed under operations. An equivalence relation ρ on A which is compatible with all the operations from F , in the sense that xi ρyi , i = 1, . . . , n imply f (x1 , . . . , xn )ρf (y1 , . . . , yn ), is a congruence on A . If A , B are algebras of the same type, then the mapping ϕ : A → B is a homomorphism from A to B if for every n-ary f ∈ F and all x1 , . . . , xn ∈ A, ϕ(f (x1 , . . . , xn )) = f (ϕ(x1 ), . . . , ϕ(xn )); for a nullary operation c, f (cA ) = cB . If A = B , ϕ is an endomorphism. Our main notion is a complete lattice, denoted by (L, ∧, ∨, ). A greatest, top element is denoted by 1, and a smallest, bottom element by 0. We use also the notion of a principal filter generated by a ∈ L, denoted by ↑a: ↑a = {x ∈ L | a x}. A principal ideal generated by a is defined dually: ↓a = {x ∈ L | x a}. For a, b ∈ L, a b the interval [a, b] is defined by: [a, b] = ↑a ∩ ↓b. A sublattice N of L is convex, if a, b ∈ N implies [a, b] ⊆ N . An element a of the lattice L is said to be codistributive if for all x, y ∈ L a ∧ (x ∨ y) = (a ∧ x) ∨ (a ∧ y). Proposition 1. If a is a codistributive element in L, then x → a ∧ x is an endomorphism of L onto ↓a.
2
By the above, if a is a codistributive element in L, then the sublattice ↓a is a retraction in L (i.e., it is a sublattice and at the same time a homomorphic image of L in a homomorphism which fixes it). A complete lattice L is algebraic if it is compactly generated, i.e., if every element in L is supremum of compact elements (see [6]). The following are known facts about complete lattices and their role in universal algebra (see e.g. [6]).
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Every poset with the top element which is closed under arbitrary infima, is a complete lattice. The set SubA of all the subuniverses of an algebra A is an algebraic lattice under inclusion. The set ConA of all the congruences on an algebra A is an algebraic lattice under inclusion. Next we recall the representation theorems for algebraic lattices by subuniverses (by Birkhoff and Frink) and congruences (by Grätzer and Schmidt), respectively. For every algebraic lattice L there is an algebra A , such that L is isomorphic to SubA . For every algebraic lattice L there is an algebra A , such that L is isomorphic to ConA . A weak congruence ρ on A , is symmetric, transitive, compatible and weakly reflexive relation on A . A property of being weakly reflexive is defined in such a way that for every nullary operation (constant) c ∈ A , it holds cρc. The following is known (see [35]). Proposition 2. The collection WConA of all weak congruences of an algebra A is an algebraic lattice. It is a union of congruence lattices of subalgebras of A and contains a subuniverse lattice as a retract. 2 The problem of representation of algebraic lattices by weak congruences is still open (there are some particular solutions, see [29,35,38]). For a deeper insight to universal algebra, see e.g. book [6]. 2.2. Fuzzy sets, structures and relations Here we introduce notions and properties from fuzzy set theory, used throughout the paper. The notions introduced in our papers are also listed. For general approach to fuzziness, see e.g. [21]. Throughout the paper, as the co-domain of fuzzy sets and relations, i.e., as the structure of membership values, we use a complete lattice, usually denoted by (L, ∧, ∨), with 0 and 1 being the bottom and the top element under the order in L. A fuzzy set μ on a nonempty set A is a function μ : A → L. We also call it an L-valued set on A, in particular if we want to emphasize the co-domain lattice L. If A and L are fixed, then by LA we denote the set of all functions from A to L, i.e., the collection of all fuzzy sets on A with membership values in L: LA := {μ | μ : A → L}. A binary relation ⊆ (inclusion of fuzzy sets) is defined on LA componentwise with respect to the lattice order : for any μ, ν : A → L μ⊆ν
if and only if μ(x) ν(x) for every x ∈ A.
Similarly, the union and the intersection of a family of fuzzy sets are defined componentwise by join and meet respectively. For p ∈ L, a cut set, or a p-cut of μ is a subset μp of A defined by: μp = x ∈ A μ(x) p . Obviously, the cut μp is the inverse image of the principal filter ↑p under μ: μp = μ−1 (↑p). If μ : A → L is a fuzzy set, and {μp | p ∈ L} is the family of cuts of this fuzzy set, then μ can be computed by its family of cuts, as follows: μ(x) = {p ∈ L | x ∈ μp }. (1) A fuzzy set on a square of the nonempty set A, i.e., a mapping ρ : A2 → L is a fuzzy (binary) relation on A, or an L-valued relation on A. Since a fuzzy relation is a fuzzy set, the definitions of inclusion, union and intersection are valid also for fuzzy relations. Here are some well known properties of fuzzy relations: ρ is reflexive if ρ(x, x) = 1 for all x ∈ A;
(2)
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ρ is symmetric if ρ(x, y) = ρ(y, x) for all x, y ∈ A; ρ is transitive if ρ(x, y) ρ(x, z) ∧ ρ(z, y) for all x, y ∈ A,
(3) (4)
z∈A
which is equivalent with ρ(x, y) ρ(x, z) ∧ ρ(z, y),
for all x, y, z ∈ A.
A fuzzy relation ρ on A which is reflexive, symmetric and transitive is a fuzzy equivalence on A. Using characteristic functions with values 0 and 1 from L, we also consider the identity relation ΔA , or simply Δ, and the square A2 of A as fuzzy equivalences: These are both mappings from A2 to L, such that 1 if x = y Δ(x, y) = A2 (x, y) = 1 for all x, y ∈ A. 0 else. We say that a fuzzy relation ρ : A2 → L is a diagonal relation if ρ(x, y) = 0 for all x, y ∈ A, x = y. Obviously, the identity relation is a special diagonal relation on A. If A = (A, F ) is an algebra, then a fuzzy subalgebra of A is a mapping μ : A → L satisfying the following inequality: For any operation f from F , f : An → A, n ∈ N, and all a1 , . . . , an ∈ A: n
μ(ai ) μ f (a1 , . . . , an ) ,
i=1
and for a nullary operation (constant) c ∈ F , μ(c) = 1 ∈ L. It is well known that the cuts of fuzzy subalgebras are crisp subalgebras. To be more precise: Lemma 1. A fuzzy set on a crisp algebra is a fuzzy subalgebra if and only if all the cuts are crisp subalgebras. 2 Observe that in the case when there are no nullary operations, also the characteristic function of the empty set
O : A → L, such that for every x ∈ A, O (x) = 0 ∈ L, fulfills the definition of a fuzzy subalgebra. Consequently, the collection of all fuzzy subalgebras, including O if there are no nullary operations, is also called a set of fuzzy subuniverses of A , analogously to the crisp case. This notion we use in the sequel and we denote it by FsA . The intersection of a collection of fuzzy subuniverses of A is again a fuzzy subuniverse of A . In addition, a constant
function I : A → L, such that I (x) = 1 for every x ∈ A is also a fuzzy subuniverse. Therefore, we have the following structure property of the collection of subuniverses. Proposition 3. If A is an algebra and L a complete lattice, then the collection FsA of all L-valued subuniverses of A is a complete lattice under inclusion. 2 A fuzzy relation ρ : A2 → L on an algebra A = (A, F ) is said to be compatible with the operations, if for every n-ary operation f ∈ F and for all a1 , . . . , an , b1 , . . . , bn ∈ A n ρ f (a1 , . . . , an ), f (b1 , . . . , bn ) ρ(ai , bi ).
(5)
i=1
In particular, for n = 1, ρ(f (a1 ), f (b1 )) ρ(a1 , b1 ). The following definition is well known. Let A = (A, F ) be an algebra. A reflexive, symmetric and transitive fuzzy relation on A, which is compatible with operations from F is a fuzzy congruence on algebra A . Cuts of a fuzzy congruence relation are crisp congruence relations. Moreover, the following is true: Lemma 2. Let A be an algebra. Then a fuzzy relation ρ : A → L is a fuzzy congruence if and only if all the cuts are crisp congruences. 2 If A is an algebra and L a complete lattice, then we denote by Fc(A ) the collection of all fuzzy (L-valued) congruences on A . The following proposition is known, though we present the proof which we use in the subsequent sections.
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Proposition 4. Fc(A ) is a complete lattice under inclusion, with the full relation being the top element, and the intersection of fuzzy sets being the infimum. Proof. The full relation is the one that maps all the elements from A × A to the top element of the lattice. It is obvious that this relation is a fuzzy congruence and that it is the greatest such relation (under inclusion). Further, consider a collection of fuzzy congruences: {ρi | i ∈ I }. Its intersection is also a fuzzy congruence on A (the proof is straightforward). Intersection is the infimum under inclusion, hence Fc(A ) is a complete lattice. 2 The following property of a fuzzy relation ρ on an algebra A is a weaker version of reflexivity: A fuzzy relation ρ on A is weakly reflexive on A if ρ(c, c) = 1 for all nullary operations c ∈ F.
(6)
A weakly reflexive, symmetric and transitive fuzzy relation on A, which is compatible with the operations from F , is a fuzzy weak congruence on this algebra. If ρ is a fuzzy weak congruence, by transitivity and symmetry, we have that for every x, y ∈ A, ρ(x, y) ρ(x, x) and ρ(x, y) ρ(y, y). Next we recall the notion of a fuzzy relation on a fuzzy set. Let μ : A → L be a fuzzy set on A. A mapping ρ : A2 → L (i.e., a fuzzy relation on A) is called a fuzzy relation on a fuzzy set μ if ρ(x, y) μ(x) ∧ μ(y) for all x, y ∈ A.
(7)
Some properties of fuzzy relations on a fuzzy set are equivalently defined as those for fuzzy relations on the crisp domain. E.g., a fuzzy relation ρ on μ is symmetric if it satisfies the property (3) and transitive, if it fulfills (4), as a fuzzy relation on A. However, the reflexivity is defined differently. Namely, if ρ is a fuzzy relation on a fuzzy set μ : A → L, then ρ is reflexive on μ
if ρ(x, x) = μ(x) for all x ∈ A.
(8)
Obviously, if μ coincides with A (as a constant characteristic function with value 1 for every x ∈ A), then the present property coincides with (2). To sum up, we deal here with three reflexivity properties for fuzzy relations: two for fuzzy relations on a crisp domain ((2) and its weak version (6)) and reflexivity (8) of fuzzy relations on fuzzy sets. All of these are essential for the results we present in the sequel. A fuzzy relation ρ on μ : A → L which is symmetric, transitive and reflexive on μ, is a fuzzy equivalence on μ. Finally, if μ : A → L is a fuzzy subalgebra of an algebra A , then a relation ρ on μ is a fuzzy congruence on μ if it is a fuzzy equivalence on μ, and it is compatible with operations on A in the sense of (5). Let us mention our notation. Throughout the text, we deal with several lattices: the membership values lattice L and for an algebra A , lattices FcA , WfcA etc. In all these we denote lattice operations and relations in the same way: meet by ∧, join by ∨, orderings by and . Confusions should not appear: which operation and order (in which lattice) are used is always clear from the context. In some cases, however, we use ∧L and ∨L in order to indicate that the corresponding meet and join are computed in the lattice L. 3. Results 3.1. Lattice of fuzzy congruences Our first and introductory result deals with fuzzy congruences on algebras. We present a straightforward representation of complete lattices by means of the mentioned fuzzy relations, which form a complete lattice, by Proposition 4. Namely, having a lattice M, we look for a crisp algebra and a complete lattice L, so that the lattice of fuzzy (L-valued) congruences of A is isomorphic with M. This problem is easily solved in the fuzzy framework, as follows. Observe that in this representation the membership values lattice coincides with the one to be represented. Theorem 1. Every complete lattice L is isomorphic to the lattice of all L-valued congruences of a two-element group.
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Proof. Let L be a complete lattice and G = ({e, a}, ·) a two-element group: · e a
e e a
a a. e
Obviously, each L-valued congruence ρ on G has the form ρ e a
e 1 p
a p, 1
p ∈ L.
The proof is now straightforward.
2
Theorem 1 shows that, contrary to the crisp case, the lattice of fuzzy congruences need not be algebraic. 3.2. Fuzzy weak congruences Here we deal with fuzzy weak congruences, which differ from fuzzy congruences in the reflexivity condition, as defined in preliminaries. Proposition 5. Let L be a complete lattice, let A = (A, F ) be an algebra and ρ : A2 → L a fuzzy (L-valued) relation on A. Then ρ is a fuzzy weak congruence relation on A , if and only if all the cut relations are weak congruences on A . Proof. Suppose that ρ is a fuzzy weak congruence, and let p ∈ L. By the definition, a cut relation ρp is a subset of A2 such that (x, y) ∈ ρp if and only if ρ(x, y) p. It is known that ρp is a symmetric, transitive and compatible relation on A . Further, if c ∈ F is a nullary operation from F , then ρ(c, c) = 1 p, which means that (c, c) ∈ ρp . To prove the converse, suppose that all the cuts of ρ are weak congruences. Using the well known formula (1), for fuzzy relations: ρ(x, y) = p ∈ L (x, y) ∈ ρp , we prove that ρ is a fuzzy weak congruence. 2 Next we fix a complete lattice L and an algebra A = (A, F ). We consider a collection of all fuzzy (L-valued) weak congruences on A = (A, F ) under inclusion and denote it by Wfc(A ). The proof of the following proposition is analogous to the one of Proposition 4. Proposition 6. Wfc(A ) is a complete lattice under inclusion, with the full relation being the top element, and the intersection of fuzzy sets being the infimum. 2 Next we analyze properties of lattices of fuzzy weak congruences. Theorem 1 witnesses that the lattice of fuzzy congruences is not an algebraic lattice in general. Analyzing the proof of this theorem in a such way that we consider both elements of the two-element group as constants, we obtain an algebra in which fuzzy weak congruences coincide with fuzzy congruences (for which all the pairs with equal components have the constant value 1). Hence Theorem 1 shows that a lattice of fuzzy weak congruences need not be algebraic. This is a difference between a lattice of fuzzy weak congruences and a lattice of (crisp) weak congruences, since the latter is always an algebraic lattice. In the lattice of weak congruences, the identity relation has a distinguished role. Similar holds with the lattice of fuzzy weak congruences. Therefore, in the following we investigate lattice-theoretic properties of identity relation Δ in a fuzzy weak congruence lattice and we analyze the structure of this lattice comparing it with the lattice of crisp weak congruences. Proposition 7. If A = (A, F ) is an algebra, L a complete lattice and ρ a compatible, weakly reflexive fuzzy relation on A , then a fuzzy set μρ : A → L defined by:
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μρ (x) := ρ(x, x)
7
(9)
is a fuzzy subalgebra of A . Proof. This is straightforward, by compatibility, and for nullary operations by weak reflexivity of ρ. 2 According to Proposition 7, we say that a fuzzy relation ρ determines the subalgebra μρ . Observe that in the crisp case we have the analogue situation: every weak congruence ρ on an algebra A determines a subalgebra B of A by the corresponding identity relation i.e., by B = {x ∈ A | xρx}. Now we deal with the converse. Let A = (A, F ) be an algebra, let L be a complete lattice and μ : A → L a fuzzy subalgebra of A . We define a fuzzy relation Δμ : A2 → L by
μ(x) for x = y Δμ (x, y) = (10) 0 for x = y. It is easy to see that Δμ ∈ Wfc(A ), i.e., that Δμ it is a diagonal fuzzy weak congruence. In addition, it is straightforward to check that μΔμ = μ. Particular fuzzy weak congruences in Wfc(A ) that we use in the sequel are diagonal fuzzy relations of the form ρ ∧ Δ, where ρ is a weak fuzzy congruence. For these fuzzy relations the following holds. Proposition 8. Let ρ be a fuzzy weak congruence on A . Then in the lattice Wfc(A ) ρ ∧ Δ = Δμ ρ . Proof. Since the meet in the lattice Wfc(A ) is the set intersection, we have that ρ ∧ Δ is a diagonal relation. Then the proof follows by Proposition 7 and by (10). 2 By Proposition 8, if ρ, θ ∈ Wfc(A ) then μρ = μ θ
if and only if
ρ ∧ Δ = θ ∧ Δ.
In the crisp case, the greatest weak congruence determining (as mentioned after Proposition 7) a subalgebra B , is the square B 2 (considered as a relation on A). In the fuzzy framework, the greatest fuzzy congruence determining the same fuzzy subalgebra as ρ is not the square (represented by its characteristic function as a fuzzy relation) in general, as proven in the sequel. Proposition 9. Let ρ be a fuzzy weak congruence on A . Define ρ : A2 → L by ρ(x, y) := ρ(x, x) ∧L ρ(y, y).
(11)
Then (i) ρ is a fuzzy weak congruence on A . (ii) ρ is the greatest fuzzy weak congruence determining the fuzzy subalgebra μρ . Proof. (i) Let us, for short, denote ρ0 := ρ ∧ Δ. Since ρ0 is a fuzzy weak congruence, we have the following inequality in the codomain lattice L: ρ0 (f (x1 , . . . , xn ), f (x1 , . . . , xn )) ρ0 (x1 , x1 ) ∧L . . . ∧L ρ0 (xn , xn ). By (11), we have ρ(x, y) = ρ0 (x, x) ∧L ρ0 (y, y). Now, ρ is obviously symmetric, transitive, weakly reflexive and also compatible: for an n-ary operation f on A , we have ρ f (x1 , . . . , xn ), f (y1 , . . . , yn ) = ρ0 f (x1 , . . . , xn ), f (x1 , . . . , xn ) ∧L ρ0 f (y1 , . . . , yn ), f (y1 , . . . , yn ) ρ0 (x1 , x1 ) ∧L . . . ∧L ρ0 (xn , xn ) ∧L ρ0 (y1 , y1 ) ∧L . . . ∧L ρ0 (yn , yn ) = ρ0 (x1 , x1 ) ∧L ρ0 (y1 , y1 ) ∧L . . . ∧L ρ0 (xn , xn ) ∧L ρ0 (yn , yn ) = ρ(x1 , y1 ) ∧L . . . ∧L ρ(xn , yn ).
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(ii) For every fuzzy weak congruence ρ, we have ρ(x, x) ρ(x, y) and ρ(y, y) ρ(x, y), thus ρ(x, x) ∧L ρ(y, y) ρ(x, y). By (11), it follows that ρ is the greatest such fuzzy weak congruence on A . 2 Proposition 10. The identity relation Δ of an algebra A is a codistributive element in the fuzzy weak congruence lattice WfcA . Proof. We have to prove that for every two fuzzy weak congruences ρ, θ : A2 → L, Δ ∧ (ρ ∨ θ ) = (Δ ∧ ρ) ∨ (Δ ∧ θ ). Since the inequality in the above formula is fulfilled as a general lattice property, what remains to show is that Δ ∧ (ρ ∨ θ ) (Δ ∧ ρ) ∨ (Δ ∧ θ ). As before, if ρ ∈ WfcA , we put ρ0 = ρ ∧ Δ. In the following we use (11) and the fact that ρ ρ. We have also (again by Proposition 9) ρ ∧ Δ = ρ ∧ Δ. Now we have ρ(x, y) ∨L θ (x, y) ρ(x, x) ∧L ρ(y, y) ∨L θ (x, x) ∧L θ (y, y) = ρ0 (x, x) ∧L ρ0 (y, y) ∨L θ0 (x, x) ∧L θ0 (y, y) ρ0 (x, x) ∨L θ0 (x, x) ∧L ρ0 (y, y) ∨L θ0 (y, y) (ρ0 ∨ θ0 )(x, x) ∧L (ρ0 ∨ θ0 )(y, y) = ρ0 ∨ θ0 (x, y). Thus, ρ ∨ θ ρ0 ∨ θ 0 . Therefore, Δ ∧ (ρ ∨ θ ) Δ ∧ ρ0 ∨ θ0 = Δ ∧ (ρ0 ∨ θ0 ) = Δ ∧ (Δ ∧ ρ) ∨ (Δ ∧ θ ) = (Δ ∧ ρ) ∨ (Δ ∧ θ ), which proves the codistributivity of Δ. 2 Any fuzzy congruence ρ on a fuzzy subalgebra μ of A is, as a fuzzy relation on A , weakly reflexive, since by reflexivity we have that, for any constant operation c, ρ(c, c) = μ(c) = 1. Therefore, ρ is also a fuzzy weak congruence on A . On the other hand, let ρ be a fuzzy weak congruence on A . Since ρ(x, y) ρ(x, x) and ρ(x, y) ρ(y, y), we have that ρ(x, y) ρ(x, x) ∧L ρ(y, y), thus ρ is a fuzzy relation on the fuzzy set μρ : A → L, defined by μρ (x) = ρ(x, x). By Proposition 7, μρ is a fuzzy subalgebra of A . Obviously, ρ is reflexive on μ. Thus, any fuzzy weak congruence ρ on A is a fuzzy congruence of the fuzzy subalgebra μρ of A . Therefore we have the following connection between fuzzy congruences on fuzzy subalgebras of A and fuzzy weak congruences on A . Proposition 11. A fuzzy congruence on a fuzzy subalgebra μ of A is a fuzzy weak congruence on A . Conversely, any fuzzy weak congruence on A is a fuzzy congruence of the fuzzy subalgebra μρ of A . Let us denote by Fc(μ) the set of all fuzzy congruences on a fuzzy subalgebra μ of A . In the lattice WfcA , the principal ideal ↓Δ contains precisely all identity relations, defined by (10). Further, as a consequence of Propositions 10 and 1, we conclude that the mapping ρ → ρ ∧ Δ is a lattice endomorphism from WfcA onto its sublattice ↓Δ. Consequently, the classes (convex sublattices) of the kernel of this homomorphism consist of all fuzzy weak congruences which determine the same fuzzy subalgebra, in the sense of Proposition 7. In other words these are fuzzy congruences on the corresponding fuzzy subalgebra. By Proposition 11, we conclude that for each fuzzy subalgebra μ of A , Fc(μ) is a complete lattice isomorphic to a corresponding kernel class. In particular, the
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sublattice ↑Δ is a lattice of all fuzzy congruences on the whole algebra A . Indeed, filter ↑Δ contains all reflexive fuzzy weak congruences, hence these are fuzzy congruences on A . Let us denote by f Δ the following mapping from the sublattice ↓Δ of Wfc(A ) to the lattice Fs(A ) of all fuzzy subuniverses of A : For ρ ∈ ↓Δ, we take f Δ(ρ) := μρ , where μρ is defined by (9). It is easy to see that f Δ is a bijection, moreover, the lattices ↓Δ and Fs(A ) are isomorphic. Indeed, the meet and the join of two subalgebras in FsA correspond respectively to the meet and the join of the corresponding identity relations. The greatest fuzzy congruence on subalgebra μ, denoted by μ2 , is given by μ2 (x, y) = μ(x) ∧L μ(y). Obviously, for a fuzzy weak congruence ρ on A , we have that ρ = (μρ )2 (see Proposition 9(iii)). The previous analysis actually proves the following. Theorem 2. Let Cwf A be the lattice of all fuzzy weak congruences on an algebra A . Then the following hold. (i) For every fuzzy subalgebra μ of A , Fc(μ) is the interval sublattice [Δμ , μ2 ] of lattice Cwf A , in particular FcA = ↑Δ. (ii) The set Cwf A is a disjoint union of intervals [Δμ , μ2 ] for all μ ∈ FsA . (iii) The lattice FsA of all fuzzy subuniverses of A is isomorphic to the principal ideal ↓Δ, under μ → Δμ . (iv) The mapping mΔ : ρ → ρ ∧ Δ is a retraction from Cwf A onto ↓Δ. 3.3. Representation of complete lattices by fuzzy weak congruences Section 3.1 contains a representation theorem of complete lattices by fuzzy congruence lattices. The problem of representation of lattices by fuzzy weak congruences is much more complicated. This problem is investigated but not fully solved in the crisp case (see e.g. [29,35]) and here we deal with it in the fuzzy framework. Representation problem. Let M be a complete lattice and a ∈ M. Find a lattice L and an algebra A such that the lattice WfcA of fuzzy (L-valued) weak congruences of A is isomorphic to M and that Δ corresponds to a under this isomorphism. The main reason for investigating the above problem is that it is possible to represent a wider class of lattices by fuzzy weak congruence lattices than by classical weak congruence lattices. The first argument is already mentioned: a fuzzy weak congruence lattice need not be algebraic, which is a necessary property in the classical case. Another argument is explained by the following example. Example 1. Let M be a 25 element lattice represented by a diagram in Fig. 1(i), and let a ∈ M, as indicated. It is not possible to represent M by a weak congruence lattice of some algebra (in the classical settings), with the identity element corresponding to element a. Indeed, this element does not fulfill particular necessary conditions for being the identity of an algebra in a representation of M by its weak congruence lattice (a is not Δ-suitable, see [29,35]). However, a representation is possible in the fuzzy framework, as follows. Let G be a 2 element groupoid ({a, b}, ∗) with two subgroupoids {a} and {b}, given by the following table ∗ a b
a a b
b b. b
Let L be a 4 element Boolean lattice, Fig. 1(ii). M is the fuzzy (L-valued) weak congruence lattice of G, with the element a representing its identity relation. 2
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Fig. 1. L-valued weak congruence lattice of G and lattice L.
Continuing our research of the fuzzy weak congruence representation problem, we analyze a decomposition of a lattice. Since a lattice doesn’t have to be algebraic to be representable in the above sense, the question whether the classes under the equivalence x ∼ y ⇔ x ∧ a = y ∧ a have the top elements makes sense. Proposition 12. If a lattice M is fuzzy weak congruence representable, with a ∈ M corresponding to the identity, then its classes under the equivalence x ∼ y ⇔ x ∧ a = y ∧ a have top elements. Proof. Indeed, the classes under the equivalence ∼ coincide in the fuzzy weak congruence lattice of an algebra A with classes of the kernel of the endomorphism ρ → ρ ∧ Δ. By Proposition 11 and Theorem 2, the mentioned classes are actually lattices of fuzzy congruences of subalgebras of A . These lattices do have the top elements, namely the greatest fuzzy congruences on subalgebras, given by (11). 2 In the classical representation of lattices by weak congruences, it is obvious that the top element of the lattice could correspond to the identity relation only if the lattice is a chain with at most 2 elements [35]. In the fuzzy case, the situation here is different. Indeed, let M be a complete lattice. Take a one element algebra with no constant operation and take M as the codomain lattice. Any fuzzy relation ρ on the algebra is defined by ρ(a, a) = m, where a is the only element of the algebra, and m is an element of M. For every m ∈ M this fuzzy relation is also a fuzzy weak congruence. So, the fuzzy weak congruence lattice of this algebra is isomorphic to M, under an isomorphism sending the identity relation to the top element of L. Therefore the following holds. Proposition 13. Any complete lattice is fuzzy weak congruence representable, with the top element corresponding to the identity relation. In some other cases, however, the fuzzy weak congruence representability is equivalent to the weak congruence representability: Proposition 14. Let M be a complete lattice and a an element of M different from the top and from the bottom, such that M = ↓a ∪ ↑a. Then, M is fuzzy weak congruence representable and a corresponds to the identity, if and only if M is weak congruence representable, a corresponding to the identity as well. Proof. If M is weak congruence representable, then M is fuzzy weak congruence representable for a two element codomain lattice. If, on the contrary, M is fuzzy weak congruence representable, then let A = (A, H ) be an algebra and L a codomain lattice, such that WfcA is isomorphic to M under an isomorphism mapping Δ to a.
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Fig. 2. Representable and non-representable lattice.
If A has only one element, Δ is the top element of WfcA and, accordingly, a is the top element of M, which contradicts the assumptions; so, let A have more than one element. If L has only one element, then WfcA has only one element and, accordingly, M has only the top element, which contradicts the assumptions. If L has more than two elements, let p be its element different from the bottom and the top element. Let ρ be a fuzzy relation defined as follows.
1 for all x, y belonging to the smallest subuniverse generated by constants in A ρ(x, y) = p for all other x, y ∈ A. Obviously, if there are no constants in A , then ρ(x, y) = p for all x, y ∈ A. Clearly, ρ is a fuzzy weak congruence, as well as the relation ρ ∪ Δ. Since a is not the bottom of L, the smallest subalgebra generated by constants does not coincide with A . Therefore, relations ρ and ρ ∪ Δ differ. Thus, the class of the homomorphism kernel of ρ in WfcA has more than one element, and accordingly, there exists a class of M, different from ↑a, that has more than one element, which contradicts the fact that M = ↓a ∪ ↑a. So L has two elements, and M is representable in the classical sense. 2 Example 2. In Fig. 2 we have two lattices, M and M , both satisfying the conditions from Proposition 14. The left one is weak congruence representable: let A = (A, ∗), where A = {a, b, c, d} and (∗) is given by the table below; CwA is isomorphic to M, and Δ corresponds to p in the isomorphism; accordingly, it is fuzzy weak congruence representable. The right lattice is not weak congruence representable, since element q does not fulfill a necessary condition to be represented by the identity relation, described in [35], therefore it is not fuzzy weak congruence representable. ∗ a b c d
a a d d b
b c a a a
c b a a a
d c a a a
2
Finally we give an answer to the representation problem in the fuzzy framework, for some subdirect powers of an arbitrary complete lattice. Theorem 3. Let M be an arbitrary complete lattice. Then the following lattices are fuzzy weak congruence representable (i) M1 = {(x, y) ∈ M 2 | x y}, the identity corresponding to a = (0, 1). (ii) M2 = {(x, y, z) ∈ M 3 | x y z}, the identity corresponding to a = (0, 1, 1). (iii) M3 = {(x, y, z) ∈ M 3 | x y ∧ z}, the identity corresponding to a = (0, 1, 1).
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Proof. Let M be an arbitrary complete lattice. In order to prove that the corresponding subdirect powers of M are fuzzy weak representable, we take L = M to be the co-domain lattice in all three cases. Then we proceed as follows. (i) Let (G, ·, −1 , e) be a two-element group with the neutral element e, given by table (a). Each fuzzy weak congruence ρ on G has the form presented by table (b), with 1, p, q ∈ L, where 1 is the top element and p q. (a)
· e a
e e a
a a e
(b)
ρ e a
e 1 p
a p. q
Indeed, the cut-relations of ρ are weak congruences on G, namely, for 0 < p < q, they are represented by tables (c), (d) and (e). (c)
ρ1 e a
e a 1 0, 0 0
ρq e a
(d)
e a 1 0 0 1
(e)
ρp e a
e a 1 1, 1 1
ρ0 = ρ p .
The table of ρ is actually determined by the ordered pair (p, q), p, q ∈ L, p q. Therefore, Wf cG is isomorphic to M1 . The identity relation is determined by the pair (0, 1). (ii) Here we take the same binary operation as in (i), but considered as a two-element groupoid G, without constants. In this case, fuzzy weak congruences have the form given in table (f ). (f )
ρ e a
e r p
a p, q
p q r.
Therefore, each fuzzy weak congruence uniquely corresponds to an ordered triple (p, q, r), p, q, r ∈ L, p q r and Wf cG is isomorphic to M2 . The identity relation, according to the table, corresponds to the triple (0, 1, 1). (iii) This case is a construction with the two-element groupoid (H, ∗) with two subgroupoids {a} and {b}, given by table (g) (actually used in Example 1). Fuzzy weak congruences are also presented in the general form, by table (h). (g)
∗ a b
a a b
b b b
(h)
ρ a b
a q p
b p, r
p q ∧ r.
Since each fuzzy weak congruence is associated to an ordered triple of the form (p, q, r), p, q, r ∈ L, p q ∧ r, Wf cH is isomorphic to M3 , and the identity relation here corresponds to (0, 1, 1). 2 A concrete lattice representation by fuzzy weak congruences, proved in Theorem 3, is given in Example 1. The 25-element lattice in Fig. 1 is a subdirect power of the four-element Boolean lattice (same figure) described in (iii) of this theorem. 4. Conclusion Our paper uses advantages of fuzzy algebraic structures and techniques in order to deal with problems in classical lattice theory. A still unsolved problem of representation of algebraic lattices by weak congruences is here situated in the fuzzy framework. What we obtained, is a possibility to represent a wider class of lattices, this time with fuzzy weak congruences. In addition, our results point to some additional constructions of lattices which could be represented by these fuzzy relations, and we see this as our future task. References [1] [2] [3] [4] [5] [6]
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