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Fuzzy Sets and Systems 160 (2009 2979 2988 www.elsevier.com/locate/fss Algebras of fuzzy sets I. Bošnjak, R. Madarász, G. Vojvodić Department of Mathematics and Informatics, University of Novi Sad, Trg Dositeja Obradovića 4, 21000 Novi Sad, Serbia Received 16 September 2008; received in revised form 23 March 2009; accepted 17 April 2009 Available online 5 May 2009 Abstract In this paper we investigate two kinds of algebras of fuzzy sets, which are obtained by using Zadeh s extension principle. We give conditions under which a homomorphism between two algebras induces a homomorphism between corresponding algebras of fuzzy sets. We prove that if the structure of truth values is a complete residuated lattice, the induced algebra of a subalgebra of an algebra A can be embedded into the induced algebra of fuzzy sets of A. For direct products we give conditions under which the direct product of algebras of fuzzy sets could be embedded into the algebra of fuzzy sets of the direct product. In the case of homomorphisms and direct products, the two kinds of algebras of fuzzy sets behave in different ways. 2009 Elsevier B.V. All rights reserved. Keywords: Algebra; Fuzzy set; Zadeh s extension principle; Residuated lattice; Homomorphism; Subalgebra; Direct product 1. Introduction In general, a fuzzy set in a universe X is a mapping μ : X L,whereL is the support of an appropriate structure L of truth values: the real unit interval [0,1], a complete lattice, a complete residuated lattice or some other structure. In this paper, for L we will use complete residuated lattices, but in some cases, we will also discuss the consequences of changing the structure of truth values L. Fuzzy approaches to various universal algebraic concepts started with Rosenfeld s fuzzy groups. Since then, many fuzzy algebraic structures have been studied (vector spaces, rings, etc.. Also, some authors proposed a general approach to the theory of fuzzy algebras (for example [2,3,5,8,9,10,15,16,18,21]. Although all these papers have similar aims to unify and generalize the earlier results concerning particular algebras, different authors propose different approaches to achieve this. In papers [18,21,23], the authors start with an ordinary universal algebra, and investigate the so-called fuzzy subalgebras (whose universe is a fuzzy subset of the original algebra, fuzzy congruences and the corresponding quotient algebras, and prove fuzzy versions of some fundamental theorems from universal algebra. In papers [15,16], the authors investigate the lattice of fuzzy subalgebras of an algebra: they study what first order sentences transfer from the ordinary lattice of all subalgebras to the lattice of all fuzzy subalgebras. Somewhat different approach could be found in [8], where the notion of a fuzzy homomorphism is defined and used in investigation of groups. In [10] we can find another general approach to the theory of fuzzy algebras. In [9] the authors consider fuzzy functions of arity 2 instead of ordinary binary operations. In [5] the theory of algebras with fuzzy equalities is studied. These structures Research supported by Ministry of Science and Environmental Protection, Republic of Serbia, Grant nos. 144011 and 144013. Corresponding author. Tel.: +38121455290. E-mail address: rozi@im.ns.ac.yu (R. Madarász. 0165-0114/$ - see front matter 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.04.005

2980 I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 have two parts: the functional part, which is an ordinary algebra and the relational part, which is the carrier set of the algebra, equipped with a fuzzy equality which is compatible with all of the fundamental operations of the ordinary algebra. Algebras with fuzzy equalities are structures for the equational fragment of fuzzy logic and they are only special cases of more general fuzzy structures (see [1]. Our aim is to investigate the so-called algebra of fuzzy sets induced by an arbitrary universal algebra. Explicitly, these algebras are first defined in [17], where the set of truth values is the real unit interval [0,1]. The induced algebra of fuzzy sets also appear in [11], where the authors use completely distributive lattice L as the set of truth values for fuzzy logic. In our paper we extend the notion of the algebra of fuzzy sets induced by an algebra for the more general case, when the structure of the truth values is a complete lattice or a complete residuated lattice. In the case of complete residuated lattices, depending on implementation of Zadeh s extension principle, we obtain two different kinds of algebras of fuzzy sets. We study these induced algebras from the universal-algebraic point of view, and obtain results concerning homomorphisms, subalgebras and direct products of algebras of fuzzy sets. The structure of the paper is the following. In Section 2, we recall basic notions on fuzzy sets, fuzzy relations and residuated lattices. The definition of two kinds of algebra of fuzzy sets is given in Section 3. Here we also point out that algebras of fuzzy sets are special kinds of algebras with fuzzy equalities, in the sense of [1]. The main results of the paper are in Section 4. In this section we give a motivation for investigating algebras of fuzzy sets. Namely, in the crisp case algebras of fuzzy sets are in fact power algebras, which are studied and utilized in many areas of mathematics. We show that for any homomorphism between two algebras, the induced mapping is also a homomorphism between corresponding F -algebras, while it is not generally true for F -algebras. In the latter case homomorphisms will be preserved if L is a completely distributive lattice. For both kinds of algebras of fuzzy sets we obtain that the induced algebra of a subalgebra of A can be embedded into the induced algebra of A. For direct products, among other things, we investigate under what conditions the direct product of induced algebras could be embedded into the induced algebra of the direct product. It follows from the results that the two kinds of algebras of fuzzy sets do not behave in the same way. 2. Basic notions on residuated lattices and fuzzy sets Residuated lattices were introduced by Ward and Dilworth in [24], in ring theory. Complete residuated lattices as a structure of truth values were introduced into the context of fuzzy sets and fuzzy logic by Goguen [12]. Goguen thoroughly discussed fuzzy approach and outlined a way to develop logic using fuzzy approach. Goguen s proposal was formally developed by Pavelka [20]. Later on, various logical calculi were investigated using residuated lattices or special types of residuated lattices. A thorough information about role of residuated lattices in fuzzy logic can be found in [13,14,19]. Definition 1. A residuated lattice is an algebra L L,,, 0, 1,, where: (i L,,, 0, 1, is a lattice with the least element 0 and the greatest element 1. (ii L,, 1 is a commutative monoid. (iii and satisfy the adjointness property, i.e. x y z iff x y z holds. If the lattice L,,, 0, 1, is complete, then L is a complete residuated lattice. The operations (called multiplication and (residuum are intended for modeling the conjunction and implication of the corresponding logical calculus, while supremum (and infimum ( are intended for modeling of the existential ( and universal ( quantifier. The operation defined by (x y (x y (y x called biresiduum is used for modeling the equivalence of truth values. Definition 2. Let L L,,, 0, 1,, be a complete residuated lattice, X a nonempty set. An L-fuzzy set ( or simply fuzzy set on X is a mapping μ : X L.ThesetofallL-fuzzy sets on X we denote by F L (X, or simply F(X if the structure L is fixed or clear from the context.

I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 2981 A fuzzy relation on A is any mapping η : A n L. A binary fuzzy relation on A is fuzzy equivalence on A if it is: reflexive: η(x, x 1, for all x A, symmetric: η(x, y η(y, x, for x, y A and transitive: η(x, y η(y, z η(x, z, for all x, y, z, A. A fuzzy equivalence relation η is fuzzy equality if from η(x, y 1 it follows x y. Usually, if η is a binary fuzzy relation on A, then instead of η(x, y we write xηy. There is a natural fuzzy equality defined on the set F(A ofall fuzzy sets on A: (η μ {η(x μ(x x A}. (1 Sometimes this fuzzy equality is called a similarity. 3. Extension principle and the algebra of fuzzy sets In [25], Zadeh proposed a so-called extension principle which became an important tool in fuzzy set theory and applications. The idea is that each function f : X Y induces a corresponding function f : L X L Y defined by f (μ(y {μ(x x X, f (x y}. The function f is said to be obtained from f by the extension principle. This principle has been studied and applied in fuzzy arithmetic and engineering problems, and has also been used by many authors for the analysis of discrete dynamical systems, for the study of fuzzy fractals, for fuzzy transportation problems and so on. Let L be a complete lattice or a complete residuated lattice. The extension principle could be formulated also to extend a function f : X 1 X n Y into f : L X 1 L X n L Y in the following way: f (μ 1,..., μ n (y x i X i f (x 1,...,xny μ 1 (x 1 μ n (x n. (2 This is the case for fuzzy arithmetic when the arithmetic operations on fuzzy numbers are defined using this extension principle. In the sequel we suppose that we deal with (universal algebras of an arbitrary, but fixed type Ω. If f Ω n is an n-ary operational symbol, in order to simplify the notation, let us denote the appropriate fundamental operations in algebra also by f (instead of more precise f A. Definition 3. Let L be a complete lattice or a complete residuated lattice and A A, { f f Ω} be a universal algebra. If f Ω is an n-ary fundamental operation of A,define f : F(A n F(A in the following way: f (μ 1,..., μ n (y x i A f (x 1,...,xny μ 1 (x 1 μ n (x n. If f Ω is a constant, then we define f so that f (x 1ifx f and f (x 0 otherwise. The algebra F (A F(A, { f f Ω} will be called the -algebra of fuzzy sets induced by A. Now, if we have residuated lattice as the structure of truth values, the extension of a function f : X 1 X n Y to a function f : L X 1 L X n L Y couldbealsodefinedinanalternative way, using

2982 I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 instead of : f (μ 1,..., μ n (y x i X i f (x 1,...,xn y μ 1 (x 1 μ n (x n. (3 So, we obtain the second kind of induced algebra of fuzzy sets: Definition 4. Let L be a complete residuated lattice and A A, { f f Ω} be a universal algebra. If f Ω is an n-ary fundamental operation of A, define f : F(A n F(A in the following way: f (μ 1,..., μ n (y μ 1 (x 1 μ n (x n. x i A f (x 1,...,xny If f Ω is a constant, then we define f so that f (x 1ifx f and f (x 0 otherwise. The algebra F (A F(A, { f f Ω} will be called the -algebra of fuzzy sets induced by A. Of course, if L is a complete Heyting algebra, or more specially, if L {0, 1}, the two kinds of induced algebras of fuzzy sets coincide. In the literature, the concept of the algebra of fuzzy sets induced by a universal algebra appeared for the first time in [17], under the name fuzzy universal algebra. In this paper the structure of the truth values is the real unit interval, and the fuzzy universal algebra in fact corresponds to our -algebra of fuzzy sets. The author introduces this notion to define the concept of a fuzzy subalgebra of a universal algebra in an elegant way: if A A, F is an algebra and L [0, 1], then μ L X is a fuzzy subalgebra of A if f (μ, μ,..., μ μ, forall f Ω, where f is obtained by the extension principle (2. In [11] the concept of algebra of fuzzy sets over an algebra is used to prove Kleene theorem for fuzzy tree languages. In this paper the structure of truth values is a completely distributive lattice, and the corresponding algebra of fuzzy sets is our -algebra of fuzzy sets. Among other things, it is proved that any linear regular identity that holds in an algebra holds in the induced algebra of fuzzy sets. There is one more fact that justifies the investigation of algebras of fuzzy sets in more depth: these structures are algebras with fuzzy equalities in the sense of [1]. Definition 5. Let L be a complete residuated lattice, A, Ω be a universal algebra and a fuzzy equality on A.Then the structure A A, Ω, is an algebra with fuzzy equality if each operation f Ω is compatible with, i.e. for any n-ary f Ω, foralla 1,..., a n, b 1,..., b n A we have (a 1 b 1 (a n b n f (a 1,..., a n f (b 1,..., b n. In [1] it is shown that functions obtained by both of extension principles (2 and (3 preserves similarity of fuzzy sets, namely: Theorem 6 (Bělohlávek [1]. Let A be a nonempty set, L a complete residuated lattice and the similarity relation defined on F(A by (1. Then for any f : A n A, and for all η 1,..., η n, μ 1,..., μ n F(A we have and (η 1 μ 1 (η n μ n f (η 1,..., η n f (μ 1,..., μ n (η 1 μ 1 (η n μ n f (η 1,..., η n f (μ 1,..., μ n. Corollary 7. Let L be a complete residuated lattice, A a universal algebra, and the similarity relation defined on F(A L A by (1. Then the structures F (A, and F (A, are algebras with fuzzy equality.

I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 2983 4. Homomorphisms, subalgebras and direct products of algebras of fuzzy sets It is not hard to see that in the crisp case (i.e. for L {0, 1} both kinds of induced algebras of fuzzy sets become the ordinary power algebra of A (often called algebra of complexes or global of A. Definition 8. Let A be a nonempty set, P(AthesetofallsubsetsofA and f : A n A.Wedefine f + : P(A n P(A in the following way: f + (X 1,..., X n {f (x 1,..., x n x 1 X 1,..., x n X n }. If A A, { f f Ω} is an algebra, the power algebra (or complex algebra, orglobal P(A isdefinedas P(A P(A, { f + f Ω}. Proposition 9. Let A be any universal algebra, and L the usual two element Boolean algebra. Then both -algebra and the -algebra of L-fuzzy sets induced by A coincide with the power algebra of A, i.e. F (A F (A P(A. Beside group theory and semigroup theory, power operations are implicitly used in some other fields. For instance, the set of ideals of a distributive lattice L again forms a lattice, and meets and joins in the new lattice are precisely the power operations of meets and joins in L. In the formal language theory the product of two languages is simply the power operation of concatenation of words. Universal-algebraic constructions as homomorphisms, subalgebras direct products and quotient algebras, in the context of power algebras, are discussed by various authors. For an overview, the reader is referred to [7] or [6]. Among other things, in [7] it is proved that the power construction is compatible with homomorphisms and subalgebras, while in [22] the same is implicitly proved for direct products. In this section we will study similar problems for algebras of fuzzy sets. First we present a few results regarding homomorphisms. The following statements could be proved easily: Proposition 10. Let L be a lattice or a residuated lattice. Then: (a If α : A B, β : B C, then (β α β α. (b If α : A B is a bijection, then α : F(A F(B is also a bijection. Now we will study the following problem. Let α : A B be a homomorphism. Will the induced mapping α : F(A F(B be a homomorphism from F (A tof (B, and from F (A tof (B? We will show that the two kinds of induced algebras of fuzzy sets do not behave in the same way. Theorem 11. Let L be a complete residuated lattice, A and B are two algebras of type Ω. If α : A B is a homomorphism, then α : F (A F (B is also a homomorphism. Proof. Let η 1,..., η n F(A. We have to prove α( f (η 1,..., η n f (α(η 1,..., α(η n. First we will transform the right-hand side of the above identity. Let z B.Then ( f (α(η 1,..., α(η n (z α(η 1 (y 1 α(η n (y n f (y 1,...,y n z η 1 (a 1 η n (a n f (y 1,...,y n z α(a 1 y 1 α(a n y n {η1 (a 1 η n (a n α(a 1 y 1,..., α(a n y n } f (y 1,...,y n z

2984 I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 (η 1 (a 1 η n (a n f (α(a 1,...,α(a n z (η 1 (a 1 η n (a n. α( f (a 1,...,a n z Now we treat the left-hand side of the identity: (α( f (η 1,..., η n (z f (η 1,..., η n (y This proves the identity. α(yz α(yz {η1 (a 1 η n (a n f (a 1,..., a n y} α( f (a 1,...,a n z (η 1 (a 1 η n (a n. If L is a complete lattice, the other kind of induced algebra of fuzzy sets F (A does not behave in the same way in respect to homomorphisms. Example 12. We will construct two algebras A and B and give an example of a homomorphism α : A B, such that the induced mapping α : F (A F (B is not a homomorphism. The algebras A and B will be groupoids given, respectively, by their Cayley s tables: u 1 u 2 v 1 v 2 u 1 v 1 v 1 u 1 u 2 u 2 v 1 v 1 u 2 u 2 v 1 v 2 v 2 v 1 v 1 v 2 v 2 v 2 v 1 v 1 ab a ba b bb Let L be the pentagon (with 0 < r < q < 1, 0 < p < 1, p not being comparable to r or qandα : A B is defined by α(u 1 α(u 2 a, α(v 1 α(v 2 b. It is not hard to prove that α is a homomorphism from A to B. Let us show that α : F (A F (B is not a homomorphism. Let η μ ( u1 u 2 v 1 v 2 q q p r. We will prove that α(η μ α(η α(μ, i.e. α(η η α(η α(η. It is easy to compute that ( u1 u η η 2 v 1 v 2, 0 r 1 1 which implies ( ab α(η η. r 1 On the other hand, from ( ab α(η q 1 we obtain ( ab α(η α(η, q 1 which shows that α(η η α(η α(η.

I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 2985 Theorem 13. Let L be a completely distributive lattice, A and B are two algebras of type Ω. If α : A B is a homomorphism, then α : F (A F (B is also a homomorphism. Proof. The proof is analogous to the proof of Theorem 2. Theorem 14. Let L be a complete lattice or a complete residuated lattice, and A a subalgebra of algebra B. Then F (A can be embedded into the algebra F (B. Proof. Let us define the mapping φ : F(A F(B in the following way: φ(η η B,where { η(a ifa A, η B (a 0 ifa B \ A. Clearly, φ is an injective mapping. We will prove that φ is a homomorphism from F (A tof (B. Let f Ω n, η 1,..., η n F(A. We need to show that φ( f (η 1,..., η n f (φ(η 1,..., φ(η n. Let a A. The left-hand side of the above equality now gives (φ( f (η 1,..., η n (a ( f (η 1,..., η n B (a f (η 1,..., η n (a. The right-hand side is f (φ(η 1,..., φ(η n (a f (η 1 B,..., η n B (a {η 1 B (x 1 η n B (x n x 1,..., x n B and f (x 1,..., x n a} {η 1 (x 1 η n (x n x 1,..., x n A and f (x 1,..., x n a} f (η 1,..., η n (a. Suppose now that b B \ A and take a look at the both sides of the equality. By the definition, the left-hand side will be f (η 1,..., η n B (b 0. The right-hand side is f (η 1 B,..., η n B (b {η 1 B (x 1 η n B (x n x 1,..., x n B and f (x 1,..., x n b}. Let us notice that if f (x 1,..., x n b, then there exist at least one x i which belongs to B \ A. Otherwise, b would be in A, sincea is a subuniverse. For this x i it holds η i B (x i 0, which implies {η1 B (x 1 η n B (x n x 1,..., x n B and f (x 1,..., x n b} 0. Now we conclude that φ is an embedding from F (A tof (B. Theorem 15. Let L be a complete residuated lattice, and A a subalgebra of algebra B. Then F (A can be embedded into the algebra F (B. Proof. The proof is analogous to the proof of the previous theorem except that we use the identity x 0 0 x 0. Definition 16. Let L be a complete lattice. The direct product of the family of fuzzy subsets η i F(A i i I is the fuzzy subset η i : A i L defined in the following way: if x A i,wherex i x(i fori I,then ( ηi i I (x {η i (x i i I }.

2986 I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 Theorem 17. Let L be a completely distributive lattice and A i i I be a family of algebras of the type Ω. The mapping φ : i I F(A i F( i I A i definedbyφ(η η(i is a homomorphism from i I F (A i to F ( i I A i. Proof. To keep the proof less technical, we will demonstrate the calculation only for the operations of arity 2. For f Ω 2, let us prove that φ is compatible with f. In the sequel, we will write a b instead of f (a, b. If η, μ i I F (A i, we need to show that φ(η μ φ(η φ(μ. As usual, let η(i η i, μ(i μ i,fori I.Fora i I A i, the left-hand side is ( φ(η μ(a (η μ(i (a (η i I i μ i (a i η i (b i μ i (c i i I i I b i c i a i { {η i (b i μ i (c i i I } ( j I (b j, c j A j & b j c j a j } {( i I η i (b i ( i I μ i (c i ( j I (b j, c j A j & b j c j a j } {( ( η i I i (b μ i I i (c b, c } A i & b c a i I (( ( (a (φ(η φ(μ(a. i I η i i I μ i It is not hard to see that the above defined mapping φ : i I F(A i F( i I A i is not necessarily injective. Example 18. Take L [0, 1], I {1, 2, 3} and η 1, η 2, η 3, η 4 : A L such that η 1 (x 0.1 forall x A, η 2 (x 0.1 forall x A, η 3 (x 0.9 forall x A, η 4 (x 0.8 forall x A. Then φ((η 1, η 2, η 3 φ((η 1, η 2, η 4 μ, whereμ(x 0.1forallx A. Definition 19. The set F + (A of all normalized fuzzy subsets of A is defined by F + (A {η : A L ( x Aη(x 1}. Theorem 20. Let L be a complete lattice and A i i I be a family of sets. Then the mapping φ + : i I F +(A i F + ( i I A i defined by φ + (η i I η(i is injective. Proof. φ + is obviously the restriction of the mapping φ (defined in Theorem 17on i I F +(A i. We need to prove that φ + is an injection, and that for arbitrary η i I F +(A i it holds φ + (η F + ( i I A i. The latter is easy, because if η i η(i, for i I, we know that for all i I there exist a i A i such that η i (a i 1. Now for a i I A i such that a(i a i, i I,wehave ( φ + (η(a i I η i (a η i (a i 1. i I It is left to prove that φ is an injection. Let η, μ i I F +(A i such that φ + (η φ + (μ, i.e. i I η i i I μ i.

I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 2987 Let us take an arbitrary b A j and show that η j (b μ j (b. We know that there exist x i A i for i I, i j,such that η i (x i 1. Then we have η j (b η j (b 1 η j (b ( η i (x i (x, i j,i I i I η i where x( j b, x(i x i, i j. This gives us ( η j (b (x μ i (x i μ j (b μ j (b. i j i I μ i Thus we obtain that for an arbitrary b A j it holds η j (b μ j (b. Analogously we get η j (b μ j (b, which implies η j μ j.sincejwas an arbitrary index, in the same way we can show that for all i I it holds η i μ i, which means η μ. This proves that φ + is an injective mapping. Remark 21. If L is a complete distributive lattice, then the previous theorem can also be proved for a weaker definition of a normalized fuzzy subset, namely for F + (A {η : A L {η(x x A} 1}. Theorem 22. Let A be an algebra of type Ω. Then F + (A is a subuniverse of the algebras F (A and F (A. Proof. Let η 1,..., η n F + (A, f Ω n.foranyi 1,..., n there exist x i A such that η i (x i 1. Take x f (x 1,..., x n. Then f (η 1,..., η n (x η 1 (y 1 η n (y n η 1 (x 1 η n (x n 1. Similarly, f (η 1,..., η n (x f (y 1,...,y n x f (y 1,...,y n x η 1 (y 1 η n (y n η 1 (x 1 η n (x n 1. Therefore f (η 1,..., η n F + (A. The subalgebra of F (A with the carrier set F + (A we will denote by F + (A. Similarly, the subalgebra of F (A with the carrier set F + (A we will denote by F + (A. Theorem 23. Let L be a completely distributive lattice and A i i I be a family of algebras of type Ω. Then the mapping φ + : i I F +(A i F + ( i I A i defined by φ + (η i I η(i is an embedding of the algebra i I F + (A i into the algebra F+ ( i I A i. Proof. It follows from Theorems 17, 20 and 22. In the end, we will show that the two kind of algebras of fuzzy sets behave in different way in respect to direct products. Example 24. We will construct two algebras A 1 and A 2 such that the mapping φ : F(A 1 F(A 2 F(A 1 A 2 defined by φ( η, μ i {1,2} η(i is not a homomorphism from F (A 1 F (A 2 tof (A 1 A 2. The algebras A 1 and A 2 will be groupoids given, respectively, by their Cayley s tables: ab a ba b bb cd c cd d cc

2988 I. Bošnjak et al. / Fuzzy Sets and Systems 160 (2009 2979 2988 Let L be the standard residual lattice on the real unit interval [0,1] with the product structure, i.e. is the usual product of real numbers. Let us take η η 1, η 2, μ μ 1, μ 2,whereη 1 (a 0.6, η 2 (c 0.5, μ 1 (b 0.7, μ 2 (d 0.8, and prove that (φ(η μ( a, d (φ(η φ(μ( a, d. We have the following: (φ(η μ( a, d (φ(η 1 μ 1, η 2 μ 2 ( a, d (η 1 μ 1 (a (η 2 μ 2 (d ( η 1 (x μ 1 (y η 2 (x μ 2 (y (η 1 (a μ 1 (b (η 2 (c μ 2 (d x ya x yd min(0.6 0.7, 0.5 0.8 0.4. On the other hand, (φ(η φ(μ( a, d (φ( η 1, η 2 φ( μ 1, μ 2 ( a, d φ(η 1, η 2 ( x, y φ(μ 1, μ 2 ( u,v References x,y u,v a,d φ(η 1, η 2 ( a, c φ(μ 1, μ 2 ( b, d (η 1 (a η 2 (c (μ 1 (b μ 2 (d min(0.6, 0.5 min(0.7, 0.8 0.5 0.7 0.5 0.7 0.35. [1] R. Bělohlávek, Fuzzy Relational Systems, Foundations and Principles, Kluwer Academic, Plenum Publishers, Dordrecht, New York, 2002. [2] R. Bělohlávek, Fuzzy equational logic, Arch. Math. Logic 41 (2002 83 90. [3] R. Bělohlávek, Birkhoff variety theorem and fuzzy logic, Arch. Math. Logic 42 (2003 781 790. [4] R. Bělohlávek, A note on the extension principle, J. Math. Anal. Appl. 248 (2000 678 682. [5] R. Bělohlávek, V. Vychodil, Algebras with fuzzy equalities, Fuzzy Sets and Systems 157 (2006 161 201. [6] I. Bošnjak, R. Madarász, On power structures, Algebra Discrete Math. 2 (2003 14 35. [7] C. Brink, Power structures, Algebra Universalis 30 (1993 177 216. [8] A.B. Chakraborty, S.S. Khare, Fuzzy homomorphism and algebraic structures, Fuzzy Sets and Systems 59 (1993 211 221. [9] M. Demirci, J. Recasens, Fuzzy groups, fuzzy functions and fuzzy equivalence relations, Fuzzy Sets and Systems 144 (2004 441 458. [10] A. Di Nola, G. Gerla, Lattice valued algebras, Stochastica 11 (1987 137 150. [11] Z. Ésik, G. Liu, Fuzzy tree automata, Fuzzy Sets and Systems 158 (2007 1450 1460. [12] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967 145 174. [13] S. Gottwald, A Treatise on Many-valued Logics, Research Studied Press, Baldock, Hertfordshire, England, 2001. [14] P. Hájek, Metamathematics of Fuzzy Logic, Kluwer, Dordrecht, 1998. [15] T. Kuraoka, Formulas on the lattice of fuzzy subalgebras in universal algebra, Fuzzy Sets and Systems 158 (2007 1767 1781. [16] T. Kuraoka, N.Y. Suzuki, Lattice of fuzzy subalgebras in universal algebra, Algebra Universalis 47 (2002 223 237. [17] V. Murali, Lattice of fuzzy subalgebras and closure systems in I X, Fuzzy Sets and Systems 41 (1991 101 111. [18] V. Murali, Fuzzy congruence relations, Fuzzy Sets and Systems 41 (1991 359 369. [19] V. Novák, I. Perfilieva, J. Močkoř, Mathematical Principles of Fuzzy Logic, Kluwer, Dordrecht, 1999. [20] J. Pavelka, On fuzzy logic I, II, III, Z. Math. Logik Grundl. Math. 25 (1979 45 52 119 139, 447 464. [21] M.A. Samhan, Fuzzy quotient algebras and fuzzy factor congruences, Fuzzy Sets and Systems 73 (1995 269 277. [22] A. Szendrei, The operation ISKP on classes of algebras, Algebra Universalis 6 (1976 349 353. [23] G. Vojvodić, B. Šešelja, On fuzzy quotient algebras, Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 13 (1983 279 288. [24] M. Ward, R.P. Dilworth, Residuated lattices, Trans. Amer. Math. Soc. 45 (1939 335 354. [25] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning, I, II, III, Inform. Sci. 8 (3 (1975 199 251 301 357, 9 (1975 43 80.