Fuzzy homomorphisms of algebras

Fuzzy Sets and Systems 160 (2009) 2345 – 2365 www.elsevier.com/locate/fss

Fuzzy homomorphisms of algebras夡 ´ ca,∗ , Stojan Bogdanovi´cb Jelena Ignjatovi´ca , Miroslav Ciri´ a Faculty of Sciences and Mathematics, University of Niš, Višegradska 33, P. O. Box 224, 18000 Niš, Serbia b Faculty of Economics, University of Niš, Trg Kralja Aleksandra 11, 18000 Niš, Serbia

Received 11 July 2007; received in revised form 22 November 2008; accepted 25 November 2008 Available online 6 December 2008

Abstract In this paper we consider fuzzy relations compatible with algebraic operations, which are called fuzzy relational morphisms. In particular, we aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations, called uniform fuzzy relational morphisms, and those which are partially uniform F-functions, called fuzzy homomorphisms. Both uniform fuzzy relations and partially uniform F-functions were introduced in a recent paper by us. Uniform fuzzy relational morphisms are especially interesting because they can be conceived as fuzzy congruences which relate elements of two possibly different algebras. We give various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms, we establish certain relationships between them and fuzzy congruences, and we prove homomorphism and isomorphism theorems concerning them. We also point to some applications of uniform fuzzy relational morphisms. © 2008 Elsevier B.V. All rights reserved.

1. Introduction Fuzzy approach to universal algebraic concepts started with the well-known Rosenfeld’s paper [44] on fuzzy subgroups of a group. That paper led to extensive study of fuzzy subsystems of various algebraic structures (see [36–38], and the references cited there), as well as to study of certain related concepts, such as fuzzy congruences, various kinds of fuzzy homomorphisms, and others (cf. [6–8,22,32,33,35]). Later, these concepts were generalized to arbitrary universal algebras [20,39,40,45–47]. Another fuzzy approach to universal algebras was initiated by Bˇelohlávek and Vychodil [2,3], who studied the so-called algebras with fuzzy equalities and developed a fuzzy equational logic. Except these considerations of fuzzy concepts in ordinary algebraic structures, Demirci in a series of papers dealt with fuzzy operations defined by means of the concept of a fuzzy function developed in his previous papers (cf. [13,15–18]). Also, Yuan and Lee [49] dealt with fuzzy operations defined using Malik and Mordeson’s definition of a fuzzy function [35]. A detailed discussion about Yuan and Lee’s paper was given by Demirci [19]. In contrast to fuzzy subsystems and fuzzy congruences, which were defined in a similar way in different algebraic structures, there were several different fuzzy approaches to homomorphisms. In most cases, the role of fuzzy homomorphisms was assumed by ordinary functions and homomorphisms, which were studied in connection with various fuzzy concepts on algebraic structures. In particular, Bˇelohlávek and Vychodil [2,3] studied ordinary homomorphisms of algebras which are compatible with fuzzy equalities. Demirci [13], as well as Yuan and Lee [49], studied ordinary 夡 Research supported by Ministry of Science, Republic of Serbia, Grant no. 144011. ∗ Corresponding author. Tel.: +381 18224492; fax: +381 18533014.

´ c). E-mail address: [email protected] (M. Ciri´ 0165-0114/$ - see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2008.11.024

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and fuzzy functions which are compatible with fuzzy operations. Other similarly defined concepts of a fuzzy homomorphism can be found [22,32,33]. On the other hand, fuzzy homomorphisms studied by Choudhury et al. [6–8], and by Malik and Mordeson [35], were based on some specific concepts of fuzzy functions and compatibility of these fuzzy functions with algebraic operations. In the fuzzy set theory there were many different approaches to the concept of a fuzzy function. In a number of papers various kinds of fuzzy functions based on fuzzy equivalence relations were studied. In particular, such approach was used in definitions of partial fuzzy functions and fuzzy functions, given by Klawonn [27], strong fuzzy functions and perfect fuzzy functions, given by Demirci [12,15], and other related concepts (see [2,10,14] and the papers cited there). Fuzzy functions based on fuzzy equivalence relations have shown oneself to be very useful in many applications in approximate reasoning, fuzzy control, vague algebra and other fields. In [10] the authors studied a special type of partial and perfect fuzzy functions, called uniform fuzzy relations. These fuzzy relations were introduced as a way for establishing natural relationships between fuzzy partitions of two sets, and it was shown that they can be successfully applied in approximate reasoning, especially in fuzzy control. It is worth noting that uniform fuzzy relations can be conceived as fuzzy equivalences which relate elements of two possibly different sets. The authors in [10] also introduced and studied the concept of a partially uniform F-function, which is defined combining the concept of a partially uniform fuzzy relation, a fuzzy relation between two sets A and B which is not necessarily uniform on the whole A × B, but it is uniform on some subset of A × B, and the concept of an F-function, a fuzzy relation whose kernel is an ordinary crisp function, introduced by Novák [41]. In this paper we consider fuzzy relations compatible with algebraic operations, which are called fuzzy relational morphisms. Fuzzy relational morphisms trace one’s origin to relational morphisms of semigroups, which were introduced by Tilson [21, Chapters 11 and 12] to solve some problems related to the wreath product decomposition of finite semigroups. But it turned out that relational morphisms can be useful in the study of recognizable languages. Namely, the renowned Eilenberg’s variety theorem gives a one-to-one correspondence between varieties of languages and varieties of semigroups. In certain cases, this correspondence can be extended to correspondence between operations on languages and operations on semigroups, and relational morphisms of semigroups have shown oneself to be a very powerful tool in the study of correspondences between these operations (cf. [21,42,43,48]). We aim our attention to those fuzzy relational morphisms which are uniform fuzzy relations, and especially to those which are partially uniform F-functions. Fuzzy relational morphisms which are uniform fuzzy relations are called   uniform fuzzy relational morphisms. We show that the L-fuzzy equivalences E A on A and E B on B induced by a uniform fuzzy relational morphism  ∈ L A×B are fuzzy congruences, and in Theorem 4.2 we characterize uniform fuzzy relational morphisms in terms of these fuzzy congruences. By Theorem 4.3 we determine several necessary and sufficient conditions for existence of a uniform fuzzy relational morphism of A to B which induce a given pair of fuzzy congruences E on A and F on B. It is worth noting that equivalence of conditions (i) and (iv) of Theorem 4.3 can be conceived as the homomorphism theorem for uniform fuzzy relational morphisms. Furthermore, by Theorem 4.4 we construct a uniform fuzzy relational morphism of A to B which induce a given fuzzy congruence F on B, and by Theorem 4.5 we construct a uniform fuzzy relational morphism which induce a given fuzzy congruence E on A. One of the main reasons why we study uniform fuzzy relational morphisms is the fact that they can be conceived as fuzzy congruences which relate elements of two possibly different algebras. On the other hand, they have interesting applications. At the end of Section 4 we give an example which demonstrates an application of uniform fuzzy relational morphisms in study of equivalence of states of deterministic automata with fuzzy sets of terminal states. Similar questions concerning automata with fuzzy transitions and fuzzy outputs will be a subject of our further research. Those fuzzy relational morphisms which are partially uniform F-functions, called fuzzy homomorphisms, are studied in Sections 5 and 6. We describe some basic properties of fuzzy homomorphisms, by Theorem 5.1 we establish a correspondence between fuzzy homomorphisms and fuzzy congruences, analogous to the well-known correspondence between crisp homomorphisms and crisp congruences, and by Theorems 5.3–5.5 we determine necessary and sufficient conditions for existence of a fuzzy homomorphism of an algebra A to an algebra B which induces a given fuzzy congruence on A and a given compatible L-fuzzy equality on B, and we give two constructions of fuzzy homomorphisms. In Section 6 we prove the homomorphism and isomorphism theorems concerning fuzzy homomorphisms and fuzzy congruences. It is worth noting that our concept of a fuzzy homomorphism is closely related to the concept of a homomorphism of algebras with fuzzy equalities, studied by Bˇelohlávek and Vychodil [2,3], and at the end of Section 5 we point out

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similarities and differences between these two concepts. Besides, our work is closely related to that of Murali [40] and Samhan [45]. 2. Preliminaries In this paper we will use complete residuated lattices as the structures of truth values. A residuated lattice is an algebra L = (L , ∧, ∨, ⊗, →, 0, 1) such that (L1) (L , ∧, ∨, 0, 1) is a lattice with the least element 0 and the greatest element 1, (L2) (L , ⊗, 1) is a commutative monoid with the unit 1, (L3) ⊗ and → form an adjoint pair, i.e., they satisfy the adjunction property: for all x, y, z ∈ L, x ⊗ yz

⇔

x  y → z.

(1)

If, in addition (L , ∧, ∨, 0, 1) is a complete lattice, then L is called a complete residuated lattice. The operations ⊗ (called multiplication) and → (called residuum)  are intended for modeling the conjunction and implication of the corresponding logical calculus, and supremum ( ) and infimum ( ) are intended for modeling of the existential and general quantifier, respectively. An operation ↔ defined by x ↔ y = (x → y) ∧ (y → x)

(2)

called biresiduum (or biimplication) is used for modeling the equivalence of truth values. It can be easily verified that with respect to  , ⊗ is isotonic in both arguments, and → is isotonic in the second and antitonic in the first argument. Emphasizing their monoidal structure, in some sources residuated lattices are called integral, commutative and residuated -monoids. For other properties of complete residuated lattices we refer to [2,24]. The most studied and applied structures of truth values, defined on the real unit interval [0, 1] with x ∧ y = min(x, y) and x ∨ y = max(x, y), are the Łukasiewicz structure (x ⊗ y = max(x + y − 1, 0), x → y = min(1 − x + y, 1)), the product structure (x ⊗ y = x · y, x → y = 1 if x  y and = y/x otherwise) and the Gödel structure (x ⊗ y = min(x, y), x → y = 1 if x  y and = y otherwise). More generally, an algebra ([0, 1], ∧, ∨, ⊗, →, 0, 1)is a complete residuated lattice if and only if ⊗ is a left-continuous t-norm and the residuum is defined by x → y = {u ∈ [0, 1]|u ⊗ x  y}. Another important set of truth values is the set {a0 , a1 , . . . , an }, 0 = a0 < · · · < an = 1, with ak ⊗ al = amax(k+l−n,0) and ak → al = amin(n−k+l,n) . A special case of the latter algebras is the two-element Boolean algebra of classical logic with the support {0, 1}. The only adjoint pair on the two-element Boolean algebra consist of the classical conjunction and implication operations. In the further text L will be a complete residuated lattice. An L-fuzzy subset of a set A is any function from A into L. The set of all L-fuzzy subsets of A is denoted by L A . Let f, g ∈ L A . The equality of f and g is defined as the usual equality of functions, i.e., f = g if and only if f (x) = g(x), for every x ∈ A. The inclusion f  g is also defined A pointwise: f  g if and only if f (x) g(x), for every  partial order L forms a complete  x ∈ A. Endowed with this residuated lattice, in which the meet (intersection) i∈I f i and the join (union) i∈I f i of an arbitrary family { f i }i∈I of L-fuzzy subsets of A are functions from A into L defined by         f i (x) = f i (x), f i (x) = f i (x) i∈I

i∈I

i∈I

i∈I

and the product f ⊗ g is an L-fuzzy subset defined by f ⊗ g(x) = f (x) ⊗ g(x), for every x ∈ A. The kernel of an L-fuzzy subset f of A is a crisp subset ker( f ) of A defined by ker( f ) = {x ∈ A| f (x) = 1}. An L-fuzzy subset f of A is normalized (or modal, in some sources) if f (x) = 1 for at least one x ∈ A, i.e., if its kernel is nonempty. An L-fuzzy relation on A is any function from A × A into L, that is to say, any L-fuzzy subset of A × A, and the equality, inclusion, joins, meets and ordering of L-fuzzy relations are defined as for L-fuzzy sets. An L-fuzzy relation R on A is said to be (R) reflexive (or fuzzy reflexive) if R(x, x) = 1, for every x ∈ A; (S) symmetric (or fuzzy symmetric) if R(x, y) = R(y, x), for all x, y ∈ A;

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(T) transitive (or fuzzy transitive) if for all x, a, y ∈ A we have R(x, a) ⊗ R(a, y) R(x, y). A reflexive, symmetric and transitive L-fuzzy relation on A is called an L-fuzzy equivalence. An L-fuzzy equivalence E on A is called an L-fuzzy equality if for any x, y ∈ A, E(x, y) = 1 implies x = y. Let E be an L-fuzzy equivalence on A. For any a ∈ A we define an L-fuzzy subset E a of A by E a (x) = E(a, x), for each x ∈ A, and we call E a an L-fuzzy equivalence class, or just an equivalence class, of E determined by the  on the element a. The set A/E = {E a | a ∈ A} is called the factor set of A with respect to E. An L-fuzzy relation E  factor set A/E defined by E(E x , E y ) = E(x, y), for any x, y ∈ A, is well defined and it is an L-fuzzy equality on A/E. For more information on L-fuzzy equivalences we refer to [2,9,11,14,27,30] and the papers cited there. Let A and B be non-empty sets. Any L-fuzzy subset  ∈ L A×B , i.e., any function  : A × B → L, is called an L-fuzzy relation of A to B. If  is a normalized L-fuzzy subset of A × B, then it is called a normalized L-fuzzy relation. A crisp relation is an L-fuzzy relation which takes values only in the set {0, 1}, and if  is a crisp relation of A to B, then expressions “(x, p) = 1” and “(x, p) ∈ ” will have the same meaning. The inverse of an L-fuzzy relation  ∈ L A×B is an L-fuzzy relation −1 ∈ L B×A defined by −1 ( p, x) = (x, p), for all x ∈ A and p ∈ B. For an L-fuzzy relation  ∈ L A×B , subsets Dom  of A and Im  of B are defined by Dom  = {x ∈ A | (∃ p ∈ B)(x, p) = 1}, Im  = { p ∈ B | (∃x ∈ A)(x, p) = 1}. Clearly, Dom  and Im  are non-empty if and only if  is a normalized L-fuzzy relation. 3. Uniform fuzzy relations and fuzzy functions In this section we recall some notions, notation and results from [10], concerning uniform fuzzy relations, fuzzy functions and related concepts. Let A and B be non-empty sets and let E and F be L-fuzzy equivalences on A and B, respectively. If an L-fuzzy relation  ∈ L A×B satisfies (EX1) (x, p) ⊗ E(x, y) (y, p), for all x, y ∈ A and p ∈ B, then it is called extensional w.r.t. E, and if it satisfies (EX2) (x, p) ⊗ F( p, q)  (x, q), for all x ∈ A and p, q ∈ B, then it is called extensional w.r.t. F. If  is extensional w.r.t. E and F, and it satisfies (PFF) (x, p) ⊗ (x, q)  F( p, q), for all x ∈ A and p, q ∈ B, then it is called a partial fuzzy function w.r.t. E and F. Partial fuzzy functions have been introduced by Klawonn [27], and studied also by Demirci [12,15]. By the adjunction property and symmetry, conditions (EX1) and (EX2) can be also written as (EX1 ) E(x, y) (x, p) ↔ (y, p), for all x, y ∈ A and p ∈ B; (EX2 ) F( p, q) (x, p) ↔ (x, q), for all x ∈ A and p, q ∈ B.  For any L-fuzzy relation  ∈ L A×B we can define an L-fuzzy equivalence E A on A by   (x, p) ↔ (y, p) (3) E A (x, y) = p∈B



for all x, y ∈ A, and an L-fuzzy equivalence E B on B by   (x, p) ↔ (x, q), E B ( p, q) =

(4)

x∈A

for all p, q ∈ B. They will be called L-fuzzy equivalences on A and B induced by . According to (EX1 ) and (EX2 ),   E A and E B are the greatest L-fuzzy equivalences on A and B, respectively, such that  is extensional w.r.t. them.  An L-fuzzy relation  ∈ L A×B is called just a partial fuzzy function if it is a partial fuzzy function w.r.t. E A and  E B [10]. Partial fuzzy functions were characterized in [10] as follows: ´ c et al. [10]). Let A and B be non-empty sets and let  ∈ L A×B be an L-fuzzy relation. Then the Theorem 3.1 (Ciri´ following conditions are equivalent: (i)  is a partial fuzzy function;  (ii) (x, p) ⊗ (x, q) E B ( p, q), for all x ∈ A and p, q ∈ B;

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(iii) (x, p) ⊗ (y, p)  E A (x, y), for all x, y ∈ A and p ∈ B; (iv) (x, p) ⊗ (y, p) ⊗ (x, q) (y, q), for all x, y ∈ A and p, q ∈ B. An L-fuzzy relation  ∈ L A×B is called an L-function if for each x ∈ A there exists p ∈ B such that (x, p) = 1, and it is called surjective if for each p ∈ B there exists x ∈ A such that (x, p) = 1, i.e., if −1 is an L-function. For a surjective L-fuzzy relation  ∈ L A×B we also say that it is an L-fuzzy relation of A onto B. If  is an L-function and it is surjective, i.e., if both  and −1 are L-functions, then  is called a surjective L-function. Let us note that an L-fuzzy relation  ∈ L A×B is an L-function if and only if there exists a function  : A → B such that (x, (x)) = 1, for all x ∈ A (see [15,18]). A function  with this property we will call a crisp description of , and we will denote by C R() the set of all such functions. An L-function which is a partial fuzzy function w.r.t. E and F is called a perfect fuzzy function w.r.t. E and F. Perfect fuzzy functions have been introduced and studied by Demirci [12,15]. An L-fuzzy relation  ∈ L A×B which is a   perfect fuzzy function w.r.t. E A and E B will be called just a perfect fuzzy function. Let A and B be non-empty sets and let E be an L-fuzzy equivalence on B. An ordinary function  : A → B is called E-surjective if for any p ∈ B there exists x ∈ A such that E((x), p) = 1. In other words,  is E-surjective if and only if  ◦ E  is an ordinary surjective function of A onto B/E, where E  : B → B/E is a function given by E  ( p) = E p , for each p ∈ B. It is clear that  is an E-surjective function if and only if its image Im  has a non-empty intersection with every equivalence class of the crisp equivalence ker(E). Let A and B be non-empty sets and let  ∈ L A×B be a partial fuzzy function. If, in addition,  is a surjective L-function, then it will be called a uniform fuzzy relation [10]. In other words, a uniform fuzzy relation is a perfect fuzzy function having the additional property that it is surjective. A uniform fuzzy relation that is also a crisp relation is called a uniform relation. The following characterizations of uniform fuzzy relations obtained in [10] will be used in the further text: ´ c et al. [10]). Let A and B be non-empty sets and let  ∈ L A×B be an L-fuzzy relation. Then the Theorem 3.2 (Ciri´ following conditions are equivalent: (i)  is a uniform fuzzy relation;  (ii)  is an L-function, and for all  ∈ C R(), x ∈ A and q ∈ B we have that  is E B -surjective and 

(x, q) = E B ((x), q);

(5) 

(iii)  is an L-function, and for all  ∈ C R() and x, y ∈ A we have that  is E B -surjective and 

(x, (y)) = E A (x, y).

(6)

Let A and B be non-empty sets, let  ∈ L A×B be a normalized L-fuzzy relation, and let the restriction of  onto Dom  × Im  be denoted by . ¯ Evidently,  ¯ ∈ L Dom ×Im  is a surjective L-function, and if ¯ is a uniform fuzzy relation, then  will be called a partially uniform fuzzy relation [10]. Let A and B be non-empty sets and let  ∈ L A×B be an L-fuzzy relation. Recall that  is called an L-function if for any x ∈ A there exists p ∈ B such that (x, p) = 1. If for any x ∈ A there exists a unique p ∈ B such that (x, p) = 1, then  is said to be an F-function. Such fuzzy relations have been introduced and studied by Novák [41]. It is worth noting that  is an F-function if and only if it is an L-function and its crisp description C R() is a singleton. For an F-function  ∈ L A×B , the unique function from C R() will be denoted by . Evidently, the function  : A → B is the kernel of . An F-function which is a partially uniform fuzzy relation is called a partially uniform F-function, an F-function which is a uniform fuzzy relation is called a uniform F-function, and an F-function which is a perfect fuzzy function is called a perfect F-function [10]. Evidently, uniform F-functions are exactly surjective partially uniform F-functions. Any perfect F-function is a partially uniform F-function, but the converse does not hold. A partially uniform F-function  is called injective if  is an injective function. A partially uniform F-function which is both injective and surjective, i.e., an injective uniform F-function, is called a bijective uniform F-function [10].

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Let E be an L-fuzzy equivalence on a set A, and let an L-fuzzy relation E  ∈ L A×A/E be defined by E  (x, ) = (x), for all x ∈ A and  ∈ A/E. Then E  is a uniform F-function, and it is called the natural uniform F-function of E. It is easy to see that E  (x, E y ) = E(x, y), for all x, y ∈ A. Let A, B and C be non-empty sets, let 1 ∈ L A×B and 2 ∈ L B×C be F-functions and let an L-fuzzy relation 1 • 2 ∈ L A×C be defined by (1 • 2 )(x, p) = 2 ( 1 (x), p), for any x ∈ A and p ∈ C. Then 1 • 2 is also an •  =  ◦  , where ◦ is the ordinary composition of functions. We call 1 • 2 the composition F-function and  1 2 1 2 of F-functions 1 and 2 [10]. Moreover, if 1 and 2 are (partially) uniform F-functions, then 1 • 2 is also a (partially) uniform F-function, and if 1 and 2 are perfect F-functions, then 1 • 2 is a perfect F-function. 4. Uniform fuzzy relational morphisms For undefined algebraic notions and notation used in the sequel we refer to the book by Burris and Sankappanavar [4]. The concepts of a fuzzy subalgebra, fuzzy congruence and fuzzy factor algebra will be defined in the same way as in the papers by Murali [39,40] and Samhan [45]. The only difference is that Murali and Samhan used the Gödel structure as the structure of truth values, but here we use a complete residuated lattice. If  is a type of algebras and n is a non-negative integer, then n will denote the set of all n-ary functional symbols from . Let A and B be algebras of the same type . An L-fuzzy relation  ∈ L A×B will be called a fuzzy relational morphism if for any f ∈ n , n  1, and xi ∈ A, pi ∈ B, 1i n, we have (x1 , p1 ) ⊗ · · · ⊗ (xn , pn ) ( f A (x1 , . . . , xn ), f B ( p1 , . . . , pn ))

(7)

and for any f ∈ 0 we have ( f A , f B ) = 1. If, in addition,  is a uniform fuzzy relation, then it will be called a uniform fuzzy relational morphism. It is worth noting that  is a uniform fuzzy relational morphism if and only if −1 has the same property. If  ∈ L A×B is a fuzzy relational morphism, and it is a crisp relation, then it will be called a relational morphism. It is easy to verify that  is a relational morphism if and only if for any f ∈ n , n 1, and all xi ∈ A, pi ∈ B, 1 i  n, we have (x1 , p1 ) ∈  & · · · & (xn , pn ) ∈  ⇒ ( f A (x1 , . . . , xn ), f B ( p1 , . . . , pn )) ∈ 

(8)

and for any f ∈ 0 we have ( f A , f B ) ∈ . A relational morphism which is also a uniform relation will be called a uniform relational morphism. Evidently, if  is a fuzzy relational morphism (resp. a uniform fuzzy relational morphism), then ker() is a relational morphism (resp. a uniform relational morphism). First we consider some basic properties of fuzzy relational morphisms. Let A and B be non-empty sets, U and V be L-fuzzy subsets of A and B, respectively, and let  ∈ L A×B be an L-fuzzy relation. Then an L-fuzzy subset (U ) of B defined by  U (x) ⊗ (x, p) for any p ∈ B ((U ))( p) = x∈A

is called the image of U under , and an L-fuzzy subset −1 (V ) of A defined by  (−1 (V ))(x) = (x, p) ⊗ V ( p) for any x ∈ A p∈B

is called the inverse image of V under . Let A be an algebra of type . An L-fuzzy subset U of A is called a fuzzy subalgebra of A if for any f ∈ n , n  1, and x1 , . . . , xn ∈ A we have U (x1 ) ⊗ · · · ⊗ U (xn ) U ( f A (x1 , . . . , xn )) and if for any f ∈ 0 we have U ( f A ) = 1 [39]. Theorem 4.1. Let A and B be algebras of type  and  ∈ L A×B a fuzzy relational morphism. Then (a) the image of any fuzzy subalgebra of A under  is a fuzzy subalgebra of B;

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(b) the inverse image of any fuzzy subalgebra of B under  is a fuzzy subalgebra of A. Proof. (a) Let U be a fuzzy subalgebra of A. For the sake of simplicity set (U ) = V . Consider arbitrary f ∈ n , n  1, and p1 , . . . , pn ∈ B. Then ⎛ ⎞ ⎛ ⎞   U (x1 ) ⊗ (x1 , p1 )⎠ ⊗ · · · ⊗ ⎝ U (xn ) ⊗ (xn , pn )⎠ V ( p1 ) ⊗ · · · ⊗ V ( p n ) = ⎝ x1 ∈A



=

xn ∈A

(U (x1 ) ⊗ · · · ⊗ U (xn ) ⊗ (x1 , p1 ) ⊗ · · · ⊗ (xn , pn ))

(x1 , ...,xn )∈An





(U ( f A (x1 , . . . , xn )) ⊗ ( f A (x1 , . . . , xn ), f B ( p1 , . . . , pn )))

(x1 , ...,xn )∈An





(U (x) ⊗ (x, f B ( p1 , . . . , pn ))) = V ( f B ( p1 , . . . , pn )).

x∈A

Moreover, for any f ∈ 0 we have that  U (x) ⊗ (x, f B ) U ( f A ) ⊗ ( f A , f B ) = 1 V ( f B) = x∈A

so V ( f B ) = 1. Therefore, V = (U ) is a fuzzy subalgebra of A. (b) This claim can be proved in a similar way as claim (a).  Let A be an algebra of type . An L-fuzzy relation E on A is called compatible if for any f ∈ n , n  1, and any elements xi , yi ∈ A, 1i n, we have E(x1 , y1 ) ⊗ · · · ⊗ E(xn , yn )  E( f A (x1 , . . . , xn ), f A (y1 , . . . , yn )).

(9)

A compatible L-fuzzy equivalence on A is called a fuzzy congruence. If E is a fuzzy congruence on an algebra A of type , then for any f ∈ n , n 1, we define an n-ary operation f A/E on the factor set A/E by f A/E (E x1 , . . . , E xn ) = E f A (x , ...,x ) n 1

(10)

for all x1 , . . . , xn ∈ A, and for f ∈ 0 we set f A/E = E f A . Then A/E is also an algebra of type , called the factor algebra or quotient algebra of A with respect to E ([40,45]). Let A and B be algebras of type  and let E be an L-fuzzy equivalence on B. A function  : A → B is called an E-homomorphism if for any f ∈ n , n  1, and any xi ∈ A, 1i n, we have E(( f A (x1 , . . . , xn )), f B ((x1 ), . . . , (xn ))) = 1

(11)

and for any f ∈ 0 we have E(( f A ), f B ) = 1. Clearly, if E is an L-fuzzy equality, then  is a fuzzy homomorphism if and only if it is an ordinary homomorphism. Moreover, if E is a fuzzy congruence, then  is an E-homomorphism if and only if ◦E  is an ordinary homomorphism of A into B/E. Recall that E  : B → B/E is a function (homomorphism) given by E  ( p) = E p , for each p ∈ B. If  is both E-surjective and an E-homomorphism, or equivalently, if  ◦ E  is an epimorphism, then  is called an E-epimorphism of A onto B. By the following theorem we show that fuzzy equivalences induced by a uniform fuzzy relational morphism are fuzzy congruences, and we characterize uniform fuzzy relational morphisms in terms of these fuzzy congruences. Theorem 4.2. Let A and B be algebras of type  and let  ∈ L A×B be an L-function. Then the following conditions are equivalent: (i)  is a uniform fuzzy relational morphism;

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(ii) E A is a fuzzy congruence, ker() is a relational morphism, and for all  ∈ C R() and x, y ∈ A we have that   is an E B -epimorphism and 

(x, (y)) = E A (x, y);

(12)





(iii) E B is a fuzzy congruence and for all  ∈ C R(), x ∈ A and q ∈ B we have that  is an E B -epimorphism and 

(x, q) = E B ((x), q).

(13)

Proof. (i)⇒(ii). Let  be a uniform fuzzy relational morphism. As we have already noted, ker() is a uniform relational  morphism. Also, by Theorem 3.2 it follows that every  ∈ C R() is an E B -surjective function and for all x, y ∈ A we have that (12) holds. Next, consider any f ∈ n , n  1, and arbitrary xi ∈ A, 1i  n. First, by (7) we obtain that 1 = (x1 , (x1 )) ⊗ · · · ⊗ (xn , (xn ))( f A (x1 , . . . , xn ), f B ((x1 ), . . . , (xn ))) so ( f A (x1 , . . . , xn ), f B ((x1 ), . . . , (xn ))) = 1. Also, ( f A (x1 , . . . , xn ), ( f A (x1 , . . . , xn ))) = 1, and by Theorem 3.1 we obtain that 

E B (( f A (x1 , . . . , xn )), f B ((x1 ), . . . , (xn ))) ( f A (x1 , . . . , xn ), ( f A (x1 , . . . , xn ))) ⊗ ( f A (x1 , . . . , xn ), f B ((x1 ), . . . , (xn ))) = 1, 



whence, E B (( f A (x1 , . . . , xn )), f B ((x1 ), . . . , (xn ))) = 1. Similarly we prove that E B (( f A ), f B ) = 1, for each   f ∈ 0 . Thus,  is an E B -homomorphism, and consequently, it is an E B -epimorphism. Further, consider any f ∈ n , n  1, and arbitrary xi , yi ∈ A, 1i n. We have that 

E B (( f A (y1 , . . . , yn )), f B ((y1 ), . . . , (yn ))) = 1 what implies that (x, ( f A (y1 , . . . , yn ))) = (x, f B ((y1 ), . . . , (yn ))), for every x ∈ A. By this, and by conditions (12) and (7) it follows that 



E A (x1 , y1 ) ⊗ · · · ⊗ E A (xn , yn ) = (x1 , (y1 )) ⊗ · · · ⊗ (xn , (yn ))



 ( f A (x1 , . . . , xn ), f B ((y1 ), . . . , (y1 ))) = ( f A (x1 , . . . , xn ), ( f A (y1 , . . . , yn )))  = E A ( f A (x1 , . . . , xn ), f A (y1 , . . . , yn )).

Therefore, we have proved that E A is a fuzzy congruence on A.  (ii)⇒(iii). Let E A be a fuzzy congruence, let ker() be a relational morphism, and for all  ∈ C R() and x, y ∈ A  let  be an E B -epimorphism and (12) holds. Then by Theorem 3.3 of [10] we obtain that  is a uniform fuzzy relation and for all x ∈ A and q ∈ B condition (13) holds. Moreover, for all x, y ∈ A we have that 



E B ((x), (y)) = (x, (y)) = E A (x, y). Next, consider any f ∈ n , n  1, and any pi , qi ∈ B, 1 i  n. Then there exist xi , yi ∈ A such that (xi , pi ), (yi , qi ) ∈ ker(), for each i, 1i  n, and since ker() is a relational morphism, we obtain that ( f A (x1 , . . . , xn ), f B ( p1 , . . . , pn )), ( f A (y1 , . . . , yn ), f B (q1 , . . . , qn )) ∈ ker(). 



By this and by Corollary 3.1 of [10] it follows that E B ( pi , qi ) = E A (xi , yi ), for each i, 1i  n, and 



E B ( f B ( p1 , . . . , pn ), f B (q1 , . . . , qn )) = E A ( f A (x1 , . . . , xn ), f A (y1 , . . . , yn )). 

By the hypothesis, E A is a fuzzy congruence, whence 







E B ( p1 , q1 ) ⊗ · · · ⊗ E B ( pn , qn ) = E A (x1 , y1 ) ⊗ · · · ⊗ E A (xn , yn ) 

 E A ( f A (x1 , . . . , xn ), f A (y1 , . . . , yn )) 



= E B ( f B ( p1 , . . . , pn ), f B (q1 , . . . , qn )).

Therefore, we have proved that E B is a fuzzy congruence on B.

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(iii)⇒(i). Let E B be a fuzzy congruence and for all  ∈ C R(), x ∈ A and q ∈ B we have that  is an E B epimorphism and condition (13) holds. According to Theorem 3.2,  is a uniform fuzzy relation.  Next, consider any f ∈ n , n 1, and arbitrary xi ∈ A and pi ∈ B, 1i n. Then we have that E B (( f A (x1 , . . . , xn )), B f ((x1 ), . . . , (xn ))) = 1, what implies that 



E B (( f A (x1 , . . . , xn )), q) = E B ( f B ((x1 ), . . . , (xn )), q) for every q ∈ B, and hence 



E B (( f A (x1 , . . . , xn )), f B ( p1 , . . . , pn )) = E B ( f B ((x1 ), . . . , (xn )), f B ( p1 , . . . , pn )). 

By this, the fact that E B is a fuzzy congruence and condition (13) we obtain that 



(x1 , p1 ) ⊗ · · · ⊗ (xn , pn ) = E B ((x1 ), p1 ) ⊗ · · · ⊗ E B ((xn ), pn ) 

 E B ( f B ((x1 ), . . . , (xn )), f B ( p1 , . . . , pn )) 

= E B (( f A (x1 , . . . , xn )), f B ( p1 , . . . , pn )) = ( f A (x1 , . . . , xn ), f B ( p1 , . . . , pn )). 

Moreover, ( f A , f B ) = E B (( f A ), f B ) = 1, for each f ∈ 0 . Therefore, we have proved that  is a uniform fuzzy relational morphism.  Let A and B be algebras of type , let an L-fuzzy relation  ∈ L A×B be both a fuzzy relational morphism and a perfect fuzzy function, and let  be the restriction of  to A × Im . Then by Theorem 4.1 and Lemma 4.1 of [10]     we have that  is a uniform fuzzy relation, E A = E A and E Im  is the restriction of E B to Im . Thus, according to 



Theorem 4.2, E A and E Im  are fuzzy congruences, but our hypothesis that  is a perfect fuzzy function do not give  enough information to prove that E B is a fuzzy congruence. By the following theorem we determine several necessary and sufficient conditions for existence of a uniform fuzzy relational morphism which induce a given pair of fuzzy congruences. Let us note that equivalence of conditions (i) and (iv) of this theorem can be conceived as the homomorphism theorem for uniform fuzzy relational morphisms. Theorem 4.3. Let A and B be algebras of type , let E be a fuzzy congruence on A and F a fuzzy congruence on B. Then the following conditions are equivalent: (i) there exists a uniform fuzzy relational morphism  ∈ L A×B such that 



E A = E and E B = F;

(14)

(ii) there exists a uniform relational morphism  ∈ 2 A×B such that E(x, y) = F( p, q)

(15)

for all x, y ∈ A and p, q ∈ B such that (x, p), (y, q) ∈ ; (iii) there exists an F-epimorphism  : A → B such that E(x, y) = F((x), (y)) for all x, y ∈ A;

(16)

(iv) there exists an isomorphism  : A/E → B/F such that  x , E y ) = F((E  E(E x ), (E y )) for all x, y ∈ A.

(17)

Proof. (i)⇒(ii). This follows immediately by Theorem 3.4 of [10] and Theorem 4.2. (ii)⇒(iii). Let  ∈ 2 A×B be a uniform relational morphism satisfying condition (15). According to Theorem 3.4 of [10], there exists an F-surjective function  : A → B such that (x, (x)) ∈ , for every x ∈ A, and (16) holds. Thus, it remains to prove that  is an F-homomorphism. 

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Consider any f ∈ n , n  1, and arbitrary xi ∈ A, 1i  n. Then by (xi , (xi )) ∈ , for each i, 1 i  n, it follows that ( f A (x1 , . . . , xn ), f B ((x1 ), . . . , (xn ))) ∈ , and by this, by ( f A (x1 , . . . , xn ), ( f A (x1 , . . . , xn ))) ∈  and by (15) we conclude that F(( f A (x1 , . . . , xn )), f B ((x1 ), . . . , (xn ))) = E( f A (x1 , . . . , xn ), f A (x1 , . . . , xn )) = 1. Similarly we show that F(( f A ), f B ) = 1. Therefore,  is an F-homomorphism. (iii)⇒(iv). Let  : A → B be an F-epimorphism satisfying condition (16), and let  : A/E → B/F be a function defined by (E x ) = F(x) for each x ∈ A. According to Theorem 3.4 of [10],  is a bijective function and (17) holds. Therefore, it remains to prove that  is a homomorphism. Consider any f ∈ n , n  1, and any xi ∈ A, 1i  n. Since  is an F-homomorphism, we have that F( f A (x , ...,x )) = F f B ((x ), ...,(x )) n n 1 1 what implies that ( f A/E (E x1 , . . . , E xn )) = (E f A (x , ...,x ) ) = F( f A (x , ...,x )) n n 1 1 = F f B ((x ), ...,(x )) = f B/F (F(x1 ) , . . . , F(xn ) ) 1

n

= f B/F ((E x1 ), . . . , (E xn )). Similarly, ( f A/E ) = f B/F , for each f ∈ 0 . Thus,  is a homomorphism, and hence, it is an isomorphism of A/E onto B/F. (iv)⇒(i). Let  : A/E → B/F be an isomorphism satisfying condition (17), and let  ∈ L A×B be an L-fuzzy relation defined by  (x, p) = F((E x ), F p ) for all x ∈ A and p ∈ B. Then for arbitrary x, y ∈ A and p, q ∈ B we have that    (x, p) ⊗ (y, p) ⊗ (x, q) = F((E x ), F p ) ⊗ F((E y ), F p ) ⊗ F((E x ), Fq )    ), (E )) ⊗ F((E ), F )  F((E  F((E x y x q y ), Fq ) = (y, q) so  is a partial fuzzy function. Moreover, for any x ∈ A there exists p ∈ B such that (E x ) = F p , what yields (x, p) = 1, and conversely, for any p ∈ B there exists x ∈ A such that (E x ) = F p , and again we obtain that (x, p) = 1. Therefore,  is a surjective L-function, and hence, it is an uniform fuzzy relation. Next, consider any f ∈ n , n  1, and arbitrary xi ∈ A, pi ∈ B, 1i  n. By the hypothesis, F is a fuzzy congruence,  is also a fuzzy congruence, so and we have that F   (x 1 , p1 ) ⊗ · · · ⊗ (xn , pn ) = F((E x1 ), F p1 ) ⊗ · · · ⊗ F((E xn ), F pn ) B/F  ((E x1 ), . . . , (E xn )), F B/F (F p1 , . . . , F pn ))  F( f  f A/E (E x1 , . . . , E xn )), F B ) = F(( f ( p1 , ..., pn )  = F((E f A (x , ...,x ) ), F f B ( p , ..., p ) ) 1

n

1

n

= ( f A (x1 , . . . , xn ), f B ( p1 , . . . , pn )). In a similar way we show that ( f A , f B ) = 1, for each f ∈ 0 . Therefore, we have proved that  is a uniform fuzzy relational morphism.  By the next two theorems we give constructions of uniform fuzzy relational morphisms which induce given fuzzy congruences.

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Theorem 4.4. Let A and B be algebras of type , let F be a fuzzy congruence on B and let  : A → B be an F-epimorphism. Then (a) an L-fuzzy relation E on A defined by E(x, y) = F((x), (y)) for any x, y ∈ A

(18)

is a fuzzy congruence on A; (b) an L-fuzzy relation  ∈ L A×B defined by (x, p) = F((x), p) for any x ∈ A and p ∈ B

(19)

is a uniform fuzzy relational morphism satisfying 



E = E A and F = E B .

(20)

Proof. By Theorem 3.5 of [10], E is an L-fuzzy equivalence and  is a uniform fuzzy relation satisfying (20). By this  and by Theorem 4.2 we obtain that  is a uniform fuzzy relational morphism, and by the same theorem, E = E A is a fuzzy congruence.  Let A and B be non-empty sets and  ∈ 2 A×B . For each x ∈ A we set x = { p ∈ B | (x, p) ∈ }, and for each p ∈ B we set  p = {x ∈ A | (x, p) ∈ }. Theorem 4.5. Let A and B be algebras of type , let E be a fuzzy congruence on A, and let  ∈ 2 A×B be a uniform relational morphism such that E A = ker(E). Then (a) an L-fuzzy relation F on B defined by F( p, q) = E(x, y) for any p, q ∈ B and any x ∈  p and y ∈ q ,

(21)

is a fuzzy congruence on B; (b) an L-fuzzy relation  ∈ L A×B defined by (x, p) = E(x, y) for any x ∈ A, p ∈ B and any y ∈  p

(22)

is a uniform fuzzy relational morphism satisfying 

E = EA,



F = E B and

ker() = .

(23)

Proof. By Theorem 3.5 of [10], F is an L-fuzzy equivalence and  is a uniform fuzzy relation satisfying (23). Since {x }x∈A is a partition of B, according to the axiom of choice for any x ∈ A we can choose an element (x) ∈ x ⊆ B, what determines a function  : A → B given by  : x(x). For each p ∈ B there exists x ∈ A such that p ∈ x , and by (x, p), (x, (x)) ∈  = ker() and Theorem 3.1 we obtain that F( p, (x))(x, p) ⊗ (x, (x)) = 1. Therefore,  is F-surjective. Next, consider any f ∈ n , n  1, and arbitrary xi ∈ A, 1 i  n. Then by (xi , (xi )) ∈ , for each i, 1 i  n, we obtain that ( f A (x1 , . . . , xn ), f B ((x1 ), . . . , (xn ))) ∈  = ker() and also ( f A (x1 , . . . , xn ), ( f A (x1 , . . . , xn ))) ∈  = ker()

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whence it follows that F(( f A (x1 , . . . , xn )), f B ((x1 ), . . . , (xn ))) = 1. In a similar way we prove that F(( f A ), f B ) = 1, for each f ∈ 0 . Hence,  is an F-homomorphism, i.e., it is an F-epimorphism. Finally, for arbitrary x, y ∈ A, by y ∈ (y) and (22) we obtain that (x, (y)) = E(x, y), and according to Theorem 4.2, F is a fuzzy congruence and  is a uniform fuzzy relational morphism.  In the next example we demonstrate an application of uniform fuzzy relational morphisms in study of equivalence of states of deterministic automata (unary algebras) with fuzzy sets of terminal states. Example 4.1. A deterministic automaton is a triple A = ( A, X,  A ), where A and X are non-empty sets, called, respectively, the set of states and the input alphabet of A, and  A : A × X → A is an ordinary crisp function, called the transition function of A. We can extend  A up to a function ∗A : A × X ∗ → A by ∗A (a, e) = a and ∗A (a, ux) =  A (∗A (a, u), x), for all a ∈ A, u ∈ X ∗ and x ∈ X , where X ∗ is the free monoid over X and e is the identity of X ∗ (the empty word). It is well known that the deterministic automaton A can be treated as a unary algebra of type {x | x ∈ X }, where for any x ∈ X the unary operation xA on A is defined by xA (a) =  A (a, x), for every a ∈ A. A quadruple A = ( A, X,  A ,  A ), where ( A, X,  A ) is a deterministic automaton and  A is an L-fuzzy subset of A, is called the terminal deterministic fuzzy automaton, and  A is called the fuzzy set of terminal states. Moreover, for a state a ∈ A, the five-tuple Aa = ( A, a, X,  A ,  A ) is called a deterministic fuzzy recognizer. The fuzzy language recognized by Aa is an L-fuzzy subset L Aa of X ∗ given by L Aa (u) =  A (∗A (a, u)) for each u ∈ X ∗ (cf. [1,26,34]). Let A = (A, X,  A ,  A ) and B = (B, X,  B ,  B ) be two terminal deterministic fuzzy automata. Define an L-fuzzy relation  ∈ L A×B by 

(a, b) =

u∈X ∗

L Aa (u) ↔ L Bb (u) for any a ∈ A and b ∈ B.

The value (a, b) can be interpreted as the degree of equality of fuzzy languages L Aa and L Bb , and it will be called the degree of equivalence of the states a and b. The states a and b are equivalent if L Aa = L Bb , i.e., if (a, b) = 1. The L-fuzzy relation  is a partial fuzzy function and a fuzzy relational morphism, i.e., (a, b)  (xA (a), xB (b)), for all a ∈ A, b ∈ B and x ∈ X . Moreover,  is a uniform fuzzy relation, i.e., it is a uniform fuzzy relational morphism, if and only if any state of A is equivalent to some state of B, and vice versa. In this case we say that A and B are equivalent automata. It is worth noting that if A and B are equivalent automata, that is, if  is a uniform fuzzy relational morphism, then by Theorem 4.2 we obtain that E = E A and F = E B are fuzzy congruences, what means that E(a, a  )  E(xA (a), xA (a  )) and F(b, b ) F(xB (b), xB (b )), for all a, a  ∈ A and b, b ∈ B. They can be represented by E(a, a  ) =

 u∈X ∗

L Aa (u) ↔ L Aa (u),

F(b, b ) =

 u∈X ∗

L Bb (u) ↔ L Bb (u)

for all a, a  ∈ A and b, b ∈ B. We can construct the factor automaton A/E = (A/E, X,  A/E ,  A/E ) by  A/E (E a , x) = E  A (a,x) , and  A/E (E a ) =  A (a), for any a ∈ A and x ∈ X , and similarly we can construct the factor automaton B/F. By Theorem 4.3 it follows that factor automata A/E and B/F are isomorphic, where by an isomorphism of A/E onto B/F we mean a bijective function  : A/E → B/F satisfying ( A/E (, x)) =  B/F ((), x), for all  ∈ A/E and x ∈ X . It can be proved that the automata A/E and B/F are minimal (they have minimal number of states) in the class of all terminal deterministic fuzzy automata equivalent to A and B, and any minimal automaton in this class is isomorphic to them.

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5. Fuzzy homomorphisms Let A and B be algebras of the same type . An L-fuzzy relation  ∈ L A×B is called a fuzzy homomorphism if it is both a fuzzy relational morphism and a partially uniform F-function. An injective fuzzy homomorphism is called a fuzzy monomorphism, a surjective fuzzy homomorphism is a fuzzy epimorphism, and a bijective fuzzy homomorphism is a fuzzy isomorphism. It is worth noting that an L-fuzzy relation  ∈ L A×B is a surjective partially uniform F-function  if and only if it is a uniform fuzzy relation and E B is an L-fuzzy equality (cf. [10]), and consequently,  is a fuzzy  epimorphism if and only if it is a uniform fuzzy relational morphism and E B is an (compatible) L-fuzzy equality. First we consider some basic properties of fuzzy homomorphisms. Lemma 5.1. Let A and B be algebras of type . If  is a fuzzy homomorphism of A into B, then  is a homomorphism of A into B. Proof. Consider any f ∈ n , n  1, and arbitrary xi ∈ A, 1i  n. Then by 1 = (x1 , (x1 )) ⊗ · · · ⊗ (xn , (xn ))( f A (x1 , . . . , xn ), f B ( (x1 ), . . . , (xn ))) we obtain that ( f A (x1 , . . . , xn ), f B ( (x1 ), . . . , (xn ))) = 1, and hence ( f A (x1 , . . . , xn )) = f B ( (x1 ), . . . , (xn )). Moreover, for any f ∈ 0 , by ( f A , f B ) = 1 it follows ( f A ) = f B . Hence, we have proved that  is a homomorphism.  The converse of the previous lemma does not hold, as the following example shows: Example 5.1. Let L be the Gödel structure, let A = {x1 , x2 , x3 , x4 } and B = { p1 , p2 , p3 }, and let binary operations on A and B be given by the following tables: x1 x2 x3 x4

p1 p2 p3

x1 x1 x2 x1 x1

p1 p1 p1 p1

x2 x2 x1 x2 x2

p2 p1 p2 p2

x3 x1 x2 x3 x3

p3 p1 p2 p3

x4 x1 x2 x3 x4

With respect to these operations, A and B are semigroups. Define now an L-fuzzy relation  ∈ L A×B by the following table:  p1 p2 p3 x1 1 0.2 0.3 x2 1 0.2 0.3 x3 0.2 1 0.2 x4 0.3 0.2 1

Then  is a uniform F-function and  is a homomorphism of A onto B, but (x1 , p3 ) ⊗ (x3 , p2 ) = 0.3 ⊗ 1 = 0.3 > 0.2 = (x1 , p2 ) = (x1 x3 , p3 p2 ), so  is not a fuzzy homomorphism. Lemma 5.2. Composition of two fuzzy homomorphisms is also a fuzzy homomorphism. Proof. Let A, B and C be algebras of type  and let 1 ∈ L A×B and 2 ∈ L B×C be fuzzy homomorphisms. Then for any f ∈ n , n 1, and x1 , . . . , xn ∈ A, p1 , . . . pn ∈ C, we have

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(1 • 2 )(x1 , p1 ) ⊗ · · · ⊗ (1 • 2 )(xn , pn ) =  = =

2 ( 1 (x1 ), p1 ) ⊗ · · · ⊗ 2 ( 1 (xn ), pn ) B 1 (x1 ), . . . , 1 (xn )), f C ( p1 , . . . , pn )) 2 ( f ( A 1 ( f (x1 , . . . , xn )), f C ( p1 , . . . , pn )) 2 ( (1 • 2 )( f A (x1 , . . . , xn ), f C ( p1 , . . . , pn ))

and for each f ∈ 0 , 1 ( f A ), f C ) = 2 ( f B , f C ) = 1. (1 • 2 )( f A , f C ) = 2 ( Therefore, 1 • 2 is a fuzzy homomorphism.  It is worth noting that the previous two lemmas remain valid if fuzzy homomorphisms are replaced by L-fuzzy relations which are both F-functions and fuzzy relational morphisms. The next theorem establishes a correspondence between fuzzy homomorphisms and fuzzy congruences, analogous to the well-known correspondence between crisp homomorphisms and crisp congruences. Theorem 5.1. (i) For any algebras A and B of type  and any fuzzy epimorphism  ∈ L A×B , the L-fuzzy equivalences   E A and E B are fuzzy congruences. (ii) For any algebra A of type  and any fuzzy congruence E on A, the natural uniform F-function E  is a fuzzy homomorphism. 



Proof. (i) Let  ∈ L A×B be a fuzzy epimorphism. Then by Theorem 4.2 it follows that E A and E B are fuzzy congruences. (ii) Let E be a fuzzy congruence on an algebra A of type . Consider any f ∈ n , n 1, and arbitrary xi , yi ∈ A, 1 i  n. Then E  (x1 , E y1 ) ⊗ · · · ⊗ E  (xn , E yn ) = E(x1 , y1 ) ⊗ · · · ⊗ E(xn , yn )  E( f A (x1 , . . . , xn ), f A (y1 , . . . , yn )) = E  ( f A (x1 , . . . , xn ), E f A (y , ...,y ) ) 1

n

= E  ( f A (x1 , . . . , xn ), f A/E (E y1 , . . . , E yn )).

Moreover, for any f ∈ 0 we have E  ( f A , f A/E ) = E  ( f A , E f A ) = E( f A , f A ) = 1. Therefore, E  is a fuzzy homomorphism.  The previous theorem indicates an interesting property of L-fuzzy equalities. In the crisp case we always have that the equality is a congruence, but an L-fuzzy equality is not necessary a fuzzy congruence, what the next example shows. Example 5.2. Let L be the Gödel structure, and let B be a semigroup from Example 5.1. An L-fuzzy relation F on B given by F p1 p2 p3 p1 1 0.2 0.3 p2 0.2 1 0.2 p3 0.3 0.2 1

is an L-fuzzy equality, but it is not a fuzzy congruence, since F( p1 , p3 ) ⊗ F( p2 , p2 ) = 0.3 ⊗ 1 = 0.3 > 0.2 = F( p1 , p2 ) = F( p1 p2 , p3 p2 ). 

Let us note that F = E B , where  is a uniform F-function from Example 5.1. The next theorem characterizes fuzzy homomorphisms in terms of fuzzy congruences induced by them.

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Theorem 5.2. Let A and B be algebras of type  and let  ∈ L A×B be a uniform F-function. Then the following conditions are equivalent: (i)  is a fuzzy homomorphism;  (ii)  is a homomorphism and E A is a fuzzy congruence;  (iii)  is a homomorphism and E B is a fuzzy congruence. Proof. The implications (i)⇒(ii) and (i)⇒(iii) follow by Lemma 5.1 and Theorem 4.2, and the implications (ii)⇒(i) and (iii)⇒(i) follow by Theorems 3.2 and 4.2.  Necessary and sufficient conditions for existence of a fuzzy epimorphism which induce a given fuzzy congruence and a given compatible fuzzy equality are determined by the following theorem. Theorem 5.3. Let A and B be algebras of type , let E be a fuzzy congruence on A and F a compatible L-fuzzy equality on B. Then the following conditions are equivalent: (i) there exists a fuzzy epimorphism  ∈ L A×B satisfying 



E = E A and F = E B ;

(24)

(ii) there exists an epimorphism  : A → B such that E(x, y) = F((x), (y)) for all x, y ∈ A;

(25)

(iii) there exists an isomorphism  : A/E → B such that  x , E y ) = F((E x ), (E y )) for all x, y ∈ A. E(E

(26)

Proof. (i)⇒(ii). If  ∈ L A×B is a fuzzy epimorphism satisfying (24), then by Lemma 5.1 and Theorem 4.2 we obtain that  =  is an epimorphism of A onto B satisfying (25). (ii)⇒(i). Let  : A → B be an epimorphism satisfying (25). According to the proof of Theorem 4.3 of [10], an L-fuzzy relation  ∈ L A×B defined by (x, p) = F((x), p) for x ∈ A and p ∈ B, is a uniform fuzzy relation satisfying (24). Clearly,  is a uniform F-function and  = , and by Theorem 5.2 we obtain that  is a fuzzy homomorphism. (ii)⇒(iii). If  : A → B is an epimorphism satisfying (25), then the function  : A/E → B defined by (E x ) = (x), for any x ∈ A, is an isomorphism satisfying (26). (iii)⇒(i). If  : A/E → B is an isomorphism satisfying (26), then a function  : A → B defined by (x) = (E x ), for any x ∈ A, is an epimorphism satisfying (25).  The next two theorems show how we can construct a fuzzy homomorphism which induce a given fuzzy congruence, or a given compatible L-fuzzy equality. Theorem 5.4. Let A and B be algebras of type , let F be a compatible L-fuzzy equality on B and let  : A → B be an epimorphism. Then (a) an L-fuzzy relation E on A defined by E(x, y) = F((x), (y)) for any x, y ∈ A is a fuzzy congruence on A;

(27)

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(b) an L-fuzzy relation  ∈ L A×B defined by (x, p) = F((x), p) for x ∈ A and p ∈ B

(28)

is a fuzzy epimorphism such that 

E = EA,



F = E B and  = .

(29)

Proof. According to Theorem 5.5 of [10], E is an L-fuzzy equivalence and  is a uniform F-function satisfying condition (29). Thus,  is a homomorphism, and by the hypothesis and Theorem 5.2 we obtain that  is a fuzzy homomorphism. Finally, by Theorem 5.1 it follows that E is a fuzzy congruence.  Theorem 5.5. Let A and B be algebras of type , let E be a fuzzy congruence on A and let  : A → B be an epimorphism  such that E A = ker(E). Then (a) an L-fuzzy relation F on B defined by F( p, q) = E(x, y) for any p, q ∈ B and any x ∈  p and y ∈ q

(30)

is a compatible L-fuzzy equality on B; (b) an L-fuzzy relation  ∈ L A×B defined by (x, p) = F((x), p) for x ∈ A and p ∈ B,

(31)

is a fuzzy epimorphism such that 

E = EA,



F = E B and  = .

(32)

Proof. This can be proved similarly as Theorem 5.4, using Theorem 5.5. of [10].  Our concept of a fuzzy homomorphism is closely related to the concept of a homomorphism of algebras with fuzzy equalities, studied by Bˇelohlávek and Vychodil [2,3], and in the sequel we will give several remarks on relationships between these concepts. Roughly speaking, a type of algebras with fuzzy equality is a set  ∪ {E}, where  is an ordinary type of algebras and E is an additional binary fuzzy relation symbol. An algebra with fuzzy equality of type  ∪ {E} is an ordinary algebra A of type , such that E is realized as a compatible L-fuzzy equality E A on A. For strict definitions of algebras with fuzzy equalities and related concepts we refer to [2,3]. If A is an algebra with fuzzy equality of type  ∪ {E}, then a congruence on A is any fuzzy congruence E on A containing the L-fuzzy equality E A (i.e., E A  E). Further, if A and B are algebras with fuzzy equality of the same type  ∪ {E}, then a homomorphism of A into B is defined as an ordinary homomorphism  : A → B satisfying the condition E A (x, y)  E B ((x), (y))

(33)

for all x, y ∈ A. In this case, an L-fuzzy relation  on A defined by  (x, y) = E B ((x), (y))

(34)

for all x, y ∈ A, is a congruence on A, and it is called the kernel of . According to Theorem 5.4 we have that a fuzzy   homomorphism  ∈ L A×B can be constructed so that  = , E A =  , and E B is the restriction of E B to Im . Therefore, any homomorphism of algebras with fuzzy equality is the kernel (crisp part) of some fuzzy homomorphism in the sense of our definition. On the other hand, let A and B be algebras of type  and let  ∈ L A×B be a fuzzy homomorphism. If compatible   L-fuzzy equalities E A on A and E B on B are chosen so that E A  E A and E B ( p, q)  E B ( p, q), for all p, q ∈ Im ,

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then by Theorem 4.2 we obtain that  is a homomorphism of algebras with fuzzy equality, in the sense of Bˇelohlávek and Vychodil. However, compatible L-fuzzy equalities E A and E B can be also chosen so that  is not a homomorphism of algebras with fuzzy equality, as the next example shows. Example 5.3. Let L be the Gödel structure, and let A = {x1 , x2 , x3 , x4 } and B = { p1 , p2 , p3 } be semigroups from Example 5.1. Moreover, let  : A → B be a function given by   x1 x2 x3 x4 = p1 p1 p2 p3 and let E A and E B be L-fuzzy relations on A and B, respectively, given by E A x1 x2 x3 x4 x1 1 0.8 0.5 0.3

E B p1 p2 p3 p1 1 0.3 0.2

x2 0.8 1 0.5 0.3

p2 0.3 1 0.2

x3 0.5 0.5 1 0.3

p3 0.2 0.2 1

x4 0.3 0.3 0.3 1

It is not hard to verify that  is an ordinary homomorphism of semigroups, and E A and E B are compatible L-fuzzy equalities. Define now an L-fuzzy relation  ∈ L A×B by (x, p) = E B ((x), p) for all x ∈ A and p ∈ B. By Theorem 5.4,  is a fuzzy homomorphism of A into B and  = , but E A (x1 , x3 ) = 0.5 > 0.3 = E B ((x1 ), (x3 )) and hence,  is not a homomorphism of algebras with fuzzy equalities, for so chosen L-fuzzy equalities E A on A and E B on B. Therefore, homomorphisms of algebras with fuzzy equalities are kernels (crisp parts) of fuzzy homomorphisms, but fixing compatible fuzzy equalities on algebras the crisp parts of some fuzzy homomorphisms are kept from being homomorphisms of algebras with fuzzy equalities. However, fixing compatible fuzzy equalities on algebras homomorphisms of algebras with fuzzy equalities are enabled to preserve validity of graded identities, and using these homomorphisms Bˇelohlávek and Vychodil proved an analogue of the Birkhoff’s Variety Theorem and more. On the other hand, if we consider semigroups A and B from Example 5.1 as semigroups with L-fuzzy equalities E A and E B given in Example 5.3, then B is an ordinary homomorphic image of A, and the truth degree of the identity x y ≈ x is 0.3 in A, and 0.2 in B, what means that ordinary homomorphisms, as well as fuzzy homomorphisms, do not preserve validity of graded identities. Finally, let us note that Bˇelohlávek and Vychodil [2,3] compared certain concepts concerning algebras with fuzzy equalities, and related concepts studied by Murali [39,40] and Samhan [45]. They gave two general comments on Murali’s and Samhan’s approach. The first comment concerns the Gödel structure of truth values used by Murali and Samhan, which is marked as a very particular structure of truth values, and the second one concerns absence of a concept of a fuzzy homomorphism which is naturally connected with fuzzy congruences. It has been shown here that any of these two shortcomings can be eliminated. Namely, using a complete residuated lattice as the structure of truth values, we have defined all concepts developed by Murali and Samhan in the same way, and also, we have introduced a concept of a fuzzy homomorphism which is naturally connected with fuzzy congruences. 6. Homomorphism and isomorphism theorems In this section we prove homomorphism and isomorphism theorems concerning fuzzy homomorphisms and fuzzy congruences.

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Theorem 6.1 (Homomorphism theorem). Let A and B be algebras of type , let  ∈ L A×B be a fuzzy epimorphism,  and let E = E A . Then there exists a unique fuzzy isomorphism  ∈ L (A/E)×B such that  = E  • . Proof. Define an L-fuzzy relation  ∈ L (A/E)×B by (E x , p) = (x, p) for any x ∈ A and p ∈ B.

(35)

Let x, y ∈ A such that E x = E y . This means that E(x, y) = 1, and by (3) it follows that (x, p) = (y, p), for every p ∈ B. Therefore,  is well defined. For arbitrary x, y ∈ A and p, q ∈ B we have that (E x , p) ⊗ (E y , p) ⊗ (E x , q) = (x, p) ⊗ (y, p) ⊗ (x, q) (y, q) = (E y , q) so we conclude that  is a partial fuzzy function. Seeing that  is a surjective L-function, we have that  is also bicomplete. Therefore,  is a uniform fuzzy relation. Consider x ∈ A and p, q ∈ B such that (E x , p) = (E x , q) = 1. Then (x, p) = (x, q) = 1, so we have that (E x ) = p= (x) = q. Therefore,  is an F-function, and hence, it is a uniform F-function. We also have that (x), for each x ∈ A. (E x ) = (E y ) = p, i.e., (E x , p) = (E y , p) = 1. Then (x, p) = Further, let x, y ∈ A and p ∈ B such that (y, p) = 1, what implies p = (x) = (y). Now 

(y)) = (x, (x)) = 1 E(x, y) = E A (x, y) = (x,  is an injective function, i.e.,  is a bijective uniform F-function. what yields E x = E y . Thus, Consider now any f ∈ n , n  1, E x1 , . . . , E xn ∈ A/E, where x 1 , . . . , xn ∈ A, and p1 , . . . , pn ∈ B. Then (E x1 , p1 ) ⊗ · · · ⊗ (E xn , pn ) = (x1 , p1 ) ⊗ · · · ⊗ (xn , pn )  ( f A (x1 , . . . , xn ), f B ( p1 , . . . , pn )) = (E f A (x , ...,x ) , f B ( p1 , . . . , pn )) n 1 A/E B (E x1 , . . . , E xn ), f ( p1 , . . . , pn )). = ( f Moreover, for any f ∈ 0 we have that ( f A/E , f B ) = (E f A , f B ) = ( f A , f B ) = 1. Therefore, we have proved that  is a fuzzy homomorphism, and hence, it is a fuzzy isomorphism. It is evident that  = E  • , and it can be easily verified that  is the unique fuzzy function of A/E into B having that property.  Theorem 6.2 (Second isomorphism theorem). Let E and F be fuzzy congruences on an algebra A such that E  F. Then there exists a unique fuzzy epimorphism  ∈ L (A/E)×(A/F) such that E  •  = F  . Proof. Define an L-fuzzy relation  ∈ L (A/E)×(A/F) by (E x , Fy ) = F  (x, Fy ) = F(x, y) for any x, y ∈ A. Let x 1 , x2 , y1 , y2 ∈ A such that E x1 = E x2 and Fy1 = Fy2 . Then E  F yields 1 = E(x1 , x2 )  F(x1 , x2 ), so F(x1 , x2 ) = 1, that is Fx1 = Fx2 . Now we have that F(x1 , y1 ) = Fx1 (y1 ) = Fx2 (y1 ) = F(x2 , y1 ) = Fy1 (x2 ) = Fy2 (x2 ) = F(x2 , y2 ) and we conclude that  is a well-defined L-fuzzy relation. Next, consider x, y1 , y2 ∈ A such that (E x , Fy1 ) = (E x , Fy2 ) = 1. Then F(x, y1 ) = F(x, y2 ) = 1, so F(y1 , y2 ) = 1, and hence, Fy1 = Fy2 . Thus,  is an F-function. Moreover, since (E x , Fx ) = F(x, x) = 1, we obtain that (E x ) = Fx , for every x ∈ A, and by this is follows that  is a surjective L-function.

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Further, for arbitrary x1 , x2 , y1 , y2 ∈ A we have that (E x1 , Fy1 ) ⊗ (E x2 , Fy1 ) ⊗ (E x1 , Fy2 ) = F(x1 , y1 ) ⊗ F(x2 , y1 ) ⊗ F(x1 , y2 )  F(x2 , y2 ) = (E x2 , Fy2 ) whence it follows that  is a partial fuzzy function, and since  is a surjective L-function, we conclude that it is a uniform fuzzy relation. Therefore,  is a uniform F-function. Now, consider any f ∈ n , n  1, and x1 , . . . , xn , y1 , . . . , yn ∈ A. Then (E x1 , Fy1 ) ⊗ · · · ⊗ (E xn , Fyn ) = F(x1 , y1 ) ⊗ · · · ⊗ F(xn , yn )  F( f A (x1 , . . . , xn ), f A (y1 , . . . , yn )) = (E f A (x , ...,x ) , F f A (y , ...,y ) ) n n 1 1 = ( f A/E (E x1 , . . . , E xn ), f A/F (Fy1 , . . . , Fyn )). Also, for any f ∈ 0 we have that ( f A/E , f A/F ) = (E f A , F f A ) = F( f A , f A ) = 1. Hence,  is a fuzzy homomorphism, i.e., it is a fuzzy epimorphism. Finally, for any x, y ∈ A we have that  (x), Fy ) = (E x , Fy ) = F(x, y) = F  (x, Fy ) (E  • )(x, Fy ) = ( E so E  •  = F  , and if  is any fuzzy epimorphism of A/E onto A/F such that E  •  = F  , then for arbitrary x, y ∈ A we obtain that  (x), Fy ) = (E  •  )(x, Fy )  (E x , Fy ) =  ( E  (x), Fy ) = (E x , Fy ), = F  (x, Fy ) = (E  • )(x, Fy ) = ( E and hence,  = . This completes the proof of the theorem.  For related results we refer to the papers by Samhan [45] and Kuroki [32], as well as to the papers cited there (see also the book by [37]). 7. Concluding remarks In this paper we considered fuzzy relations compatible with algebraic operations, which are called fuzzy relational morphisms, and in particular, we aimed our attention to those fuzzy relational morphisms which are uniform fuzzy relations, called uniform fuzzy relational morphisms, and those which are partially uniform F-functions, called fuzzy homomorphisms. Both uniform fuzzy relations and partially uniform F-functions were introduced in our previous paper [10]. Uniform fuzzy relations were introduced as a way for establishing natural relationships between fuzzy partitions of two sets, and they can be conceived as fuzzy equivalences which relate elements of two possibly different sets, and from the same point of view, uniform fuzzy relational morphisms can be conceived as fuzzy congruences which relate elements of two possibly different algebras. We gave various characterizations and constructions of uniform fuzzy relational morphisms and fuzzy homomorphisms, we established certain relationships between them and fuzzy congruences, and we proved homomorphism and isomorphism theorems concerning them. Besides, in Section 4 we gave an example which demonstrates an application of uniform fuzzy relational morphisms in study of equivalence of states of deterministic automata with fuzzy sets of terminal states. Similar questions concerning automata with fuzzy transitions and fuzzy outputs will be a subject of our further research. We also pointed to relationships between our concept of a fuzzy homomorphism and the concept of a homomorphism of algebras with fuzzy equalities, studied by Bˇelohlávek and Vychodil [2,3]. Acknowledgments The authors are very grateful to the referees for valuable remarks and suggestions which helped to improve quality of the paper.

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