On ordered categories as a framework for fuzzification of algebraic and topological structures

Fuzzy Sets and Systems 160 (2009) 2910 – 2925 www.elsevier.com/locate/fss

On ordered categories as a framework for fuzzification of algebraic and topological structures夡 Sergey A. Solovyov∗ Department of Mathematics, University of Latvia, Zellu iela 8, LV-1002 Riga, Latvia Received 17 December 2005; received in revised form 10 February 2009; accepted 11 February 2009 Available online 25 February 2009

Abstract Using the framework of ordered categories, the paper considers a generalization of the fuzzification machinery of algebraic structures introduced by Rosenfeld as well as provides a new approach to fuzzification of topological structures, which amounts to fuzzifying the underlying “set” of a structure in a suitably compatible way, leaving the structure itself crisp. The latter machinery allows the so-called “double fuzzification”, i.e., a fuzzification of something that is already fuzzified. © 2009 Elsevier B.V. All rights reserved. MSC: 03E72; 08A72; 54A40 Keywords: L-set; L-group; L-topological space; Fuzzification of algebraic (topological) structures; Ordered category; Adjoint functors; Topological category; Variety of algebras; Functor-(co)structured category; Scott open set; Semi-quantale

1. Introduction The notion of fuzzy set introduced by Zadeh in 1965 [40] and generalized by Goguen in 1967 [11] induced many lines of research to study fuzzification of different mathematical structures. In particular, a significant push was done by Goguen himself when he considered the category Set(L) of L-sets for a (suitable) partially ordered set L as a fuzzification of the category Set of sets (see, e.g., [11–13]). Later on, the pioneering papers of Chang [5] and Lowen [19] started the theory of fixed-basis (stratified) fuzzy topological spaces implicitly using the category Set(L). In 1983 Rodabaugh introduced the category FUZZ of variable-basis fuzzy topological spaces [26]. Since then it is known as the category C-Top of variable-basis lattice-valued topological spaces (see, e.g., [30]). Both fixed- and variable-basis topologies induced many researchers to study their properties [15–17,27–30,32,24,25]. On the other hand, the famous paper of Rosenfeld on fuzzy groups [33] (“embarrassingly well-cited” as the author himself says) gave rise to the study of fuzzification of algebraic structures (notice that the notion of fuzzy group was redefined in 1979 by Anthony and Sherwood [2] using the concept of triangular norm (or t-norm) [34]; the new framework incorporates the notion of Rosenfeld as the case of the minimum t-norm). In particular, fuzzy commutative algebra (e.g., abelian L-groups, L-rings, L-modules, L-fields, etc.) is considered in [23] which contains an up to date 夡 This research was supported by the European Social Fund. ∗ Tel.: +371 67033728; fax: +371 67033701.

E-mail address: [email protected]. 0165-0114/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2009.02.009

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state of the field. As an example of a non-commutative approach one can mention [39], where the author introduces the notion of L-category using a GL-monoid L. It is important to notice that all these fuzzifications are fixed-basis and use implicitly the same category Set(L). All the above-mentioned procedures are included in a (probably) more promising framework (which unites both fixed- as well as variable-basis approaches) provided by internal structures in categories (see, e.g., [7–9,20] for internal groups, [4] for internal categories and [14,21,22] for internal topological spaces). The approach yields a fuzzification machinery for algebraic as well as topological structures in any sufficiently good category (existence of some limits is required for algebra and the property of being a “topos-like” category for topology), e.g., in the already mentioned category Set(L) (fixed-basis approach) or in the category Set(CSLat( )) of lattice-valued sets with bases in the  category CSLat( ) of complete-lattices and join-preserving maps [37] (variable-basis approach). In particular, a slight modification in the definition of fuzzy group of Rosenfeld (i.e., the requirement that the unit of a group is always mapped to the top element of the respective lattice) implies the result that fuzzy groups are precisely the groups in the category Set(I ) (I being  the unit interval [0, 1]). Notice, however, that in general neither the category Set(L) nor the category Set(CSLat( )) is a topos, the former one being just a quasitopos in case of L being a frame (see, e.g., [38]) and therefore one can easily get stuck while doing internal topology in those categories. It is the aim of this paper to provide a (possibly) more appropriate categorical fuzzification procedure, i.e., such which • is sufficiently general; • incorporates the above-mentioned machineries in case of algebra; • does not require any topos-like property in case of topology. To achieve the goal we will use our previous results in the field of categorical fuzzy sets restricted (for the sake of simplicity) to the case of ordered categories (i.e., categories whose hom-sets are partially ordered with composition of morphisms preserving order in both variables). To be more precise, our steps will be as follows. Given a concrete category (A, U ) over X, we introduced in [36] (see also [35]) the category X(A) of A-valued X-objects as a categorical generalization of the concept of fuzzy set. This paper aims at introducing a “good” fuzzification machinery over the above-mentioned category. Following the already mentioned historical move we start with the fixed-basis approach and therefore consider a subcategory X( A) of the category X(A) for a fixed A-object A. It is important to notice that the category X( A) is a generalization of the category Set(L) [36] (see also [38] where some properties of X( A) are considered). In this paper we present a fuzzification machinery over the category X( A) using the categories of generalized algebraic and topological structures Alg(T ) and Spa(T ) (see, e.g., [1]; notice as well that the definitions do not impose any specific properties on the categories in question). As a result one obtains (concrete) categories ( A, )-Alg(T ) and (A, P)-Spa(T ) of A-valued algebraic and topological structures (the meaning of the indices  and P is explained in the paper). To provide the new categories with some “nice” properties we show the sufficient conditions for both of them to be topological over their ground categories. As follows from the examples given in the paper  (which include both the categories Set(L) and Set(CSLat( ))), the former category generalizes the approach used by Rosenfeld in [33] (moreover, it generalizes the approach through internal algebraic structures in categories), the latter category, however, falls out of the usual fuzzification schemes mentioned above. The reason is that unlike the classical ones our procedure amounts to fuzzifying the underlying “set” of a structure in a suitably compatible way, leaving the structure itself crisp. It is important to notice, however, that our machinery allows one to make a “double fuzzification”, i.e., to fuzzify something that is already fuzzified. It will be the topic of our forthcoming paper to consider a similar fuzzification procedure for the whole category X(A). The necessary categorical background can be found in [1]. Although we tried to make the paper as much selfcontained as possible, it is expected from the reader to be acquainted with basic concepts of category theory. For convenience of the reader we recall basic facts about the category X(A) from [36].

2. Categories of lattice-valued sets as categories of arrows In this section we provide a simplified version of the (concrete) category X(A) which will be the main object of our study (the original definition can be found in [36]). We also recall the necessary and sufficient conditions for X(A) to be topological over its ground category.

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Let (A, U ) be a concrete category over X such that the following conditions are fulfilled. (OCAT) The category A is an ordered category, i.e., its hom-sets are partially ordered with composition of morphisms preserving order in both variables. (ADJ) The functor U has a left adjoint. The order on hom-sets of the category A will be denoted (as usual) by “ ”. Following (ADJ) we choose an adjoint situation (, ) : Let F be the class of all structured arrows with domain in X, i.e., of all triples (X, , A), where X is an X-object, A  is an A-object and X − → U A is an X-morphism. Introduce the following relation “” on F . 







− → − → → U A be elements of F . By (ADJ) there exist two A-morphisms F X − → A where (−) = Definition 1. Let X −  A ◦ F(−). Define    iff  . The following lemmas provide some properties of relation “” which will be used throughout the paper. 

U f → − − U A −−→ U B be X-morphisms. If   , then  ◦ f   ◦ f and U  ◦   U  ◦ . →Y → Lemma 2. Let X −



Proof.  ◦ f =  A ◦ F( ◦ f ) =  A ◦ F ◦ F f =  ◦ F f   ◦ F f =  ◦ f and U  ◦  =  B ◦ F(U  ◦ ) = ( B ◦ FU ) ◦ F = ( ◦  A ) ◦ F =  ◦    ◦  = U  ◦ .  

− → → B be A-morphisms. Then   implies U   U . Lemma 3. Let A − 

Proof. Use the fact that U  =  ◦  A .  With the help of Definition 1 one can introduce the category X(A) as follows. Definition 4. X(A) is the concrete category over X×A, the objects of which (called A-valued X-objects) are elements of F . Morphisms (X, , A) that U  ◦    ◦ f , i.e.,

( f,)

−−−→

(Y, , B)

X × A-morphisms (X, A)

are

( f, )

( f, )

−−−→

(Y, B)

such

( f, )

The underlying functor to X × A is given by |(X, , A) −−−→ (Y, , B)| = (X, A) −−−→ (Y, B). Definition 4 can be illustrated by two examples used later on in the paper (other examples can be found in [35,36]).  Example 5. Consider the construct (CSLat( ), U ) of complete lattices and join-preserving maps. Since the left  P → CSLat( ) (Set is the category of sets and maps) defined by adjoint of U is provided by the power-set functor Set − f

P( f )

P(X − →Y ) = P(X) −−−→ P(Y ), where P(X) is the power-set of X and (P( f ))(S) = { f (s) | s ∈ S}, and, moreover,  CSLat( ) is an ordered category (point-wise order on hom-sets is obvious), one obtains the category Set(CSLat( ))  → − of lattice-valued subsets of sets [37]. Given two maps X − → U A,    iff (x) (x) for every x ∈ X . 

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Example 6. Consider the category SetRel of sets (as objects) and relations (as morphisms). The forgetful functor

U

f

SetRel − → Set is given by the formula U (X − → Y ) = P(X ) − → P(Y ) where f (S) = {y ∈ Y | there exists x ∈ f

f

F

S such that x y}. Since the left adjoint of U is provided by the functor Set − → SetRel given by F(X − → Y ) = X −→ Y where x f y iff f (x) = y, and, moreover, SetRel is an ordered category (use set-theoretic inclusion of relations), one 

− → → U Y ,    iff (x) ⊆ (x) for every x ∈ X . obtains the category Set(SetRel). Given two maps X − 

 Notice that the forgetful functor of Example 6 goes actually to the category CSLat( ). For later use we recall from [35,36] the necessary and sufficient conditions for X(A) to be topological over X × A. Notice that the property requires the existence of initial (resp. final) lifts. Accordingly we introduce two sets of requirements, one for each type of lifts. The first set consists of requirements (CLX), (MPX) and (FST) listed below. (CLX) For every X × A-object (X, A), (X(X, U A), ) is a complete lattice.

−◦ f

f

(MPX) For every X-morphism Y − → X and every A-object A, X(X, U A) −−→ X(Y, U A) is a meet-preserving map. (−)∗

(FST) There exists a functor A −−→ Xop (Xop is the dual category of X) such that: • A∗ = U A for every A-object A;  • for every A-morphism A − → B, 1U A  ∗ ◦ U  and U  ◦ ∗  1U B ;

∗ ◦ −



• for every A-morphism B − → A and every X-object X, X(X, U A) −−−→ X(X, U B) is an order-preserving map. The second set of requirements is in a sense a dualized version of the first one. (CLA) For every X × A-object (X, A), A(F X, A) is a complete lattice. 

◦−

(JPA) For every A-morphism A − → B and every X-object X, A(F X, A) −−→ A(F X, B) is a join-preserving map. (−)◦

(FCR) There exists a functor X −−→ Aop such that: • X ◦ = F X for every X-object X; f

• for every X-morphism X − → Y , F f ◦ f ◦  1 F Y and 1 F X  f ◦ ◦ F f ;

−◦ f ◦

f

• for every X-morphism X − → Y and every A-object A, A(F X, A) −−−→ A(FY, A) is a join-preserving map. Theorem 7 (Solovyov [36]). Suppose (OCAT), (ADJ) hold. Equivalent are: (i) X(A) is topological over X × A; (ii) (CLX), (MPX), (FST) hold; (iii) (CLA), (JPA), (FCR) hold. Proof. We recall only that part of the proof which will be used later.

( f i , i )

(ii) =⇒ (i): Show the existence of initial lifts. Let S = ((X, A) −−−−→ |(X i , i , Ai )|)i∈I be a | − |-structured source. Straightforward computations show that  = i∈I (∗i ◦ i ◦ f i ) is the required structure. ( f i , i )

(iii)=⇒(i): Show the existence of final lifts. Let  S = (|(X i , i , Ai )| −−−−→ (X, A))i∈I be a | − |-costructured sink. Straightforward computations show that  = i∈I (i ◦ i ◦ f i◦ ) is the required structure.   As an immediate consequence it follows that both categories Set(CSLat( )) and Set(SetRel) are topological. The f

required functors are given in the following examples, where one should recall that given a set map X − → Y one has the following image and preimage operators (the notation is taken from [29–32] and is fairly standard now): f→

• P(X) −−→ P(Y ) : S{ f (x) ∈ Y | x ∈ S}; f←

• P(Y ) −−→ P(X ) : S{x ∈ X | f (x) ∈ S}. Also notice that given a partially ordered set (A,  ), every a ∈ A provides the (lower) set ↓ a = {b ∈ A | b a}.

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  (−)∗  ∗ Example 8. In the category Set(CSLat( )) the functor CSLat( ) −−→ Setop is defined by ( A − → B)∗ = U B −→   f f◦ (−)◦ U A where ∗ (b) = ← (↓ b). The functor Set −−→ CSLat( )op is defined by (X − → Y )◦ = P(Y ) −→ P(X ) where f ◦ (S) = f ← (S). (−)∗

f∗



→ Y )∗ = P(Y ) −→ P(X ) Example 9. In the category Set(SetRel) the functor SetRel −−→ Setop is defined by (X − (−)◦

f

f◦

where f ∗ (S) = {x ∈ X | {y ∈ Y | x y} ⊆ S}. The functor Set −−→ SetRelop is defined by (X − → Y )◦ = Y −→ X ◦ where y f x iff f (x) = y. 3. A generalization of Goguen’s category Set(L) In this section we consider a particular subcategory of the category X(A). We start by recalling its definition from [38] and then show the necessary and sufficient conditions for it to be topological over its base category. Definition 10. Fix an object A of the category A. The (non-full) subcategory X( A) of the category X(A) is the concrete category over X, the objects of which (called A-valued X-objects) are X(A)-objects (X, , A) for the sake of brevity f

f

( f, 1 A )

denoted by (X, ). Morphisms (X, ) − → (Y, ) are X-morphisms X − → Y such that (X, , A) −−−→ (Y, , A) is an f

f

X(A)-morphism. The underlying functor to X is given by |(X, ) − → (Y, )| = X − → Y. The following example illustrates Definition 10.   Example 11. Let A be a CSLat( )-object. The subcategory Set( A) of the category Set(CSLat( )) is the category of A-sets introduced and studied by Goguen [11–13]. From now we fix a subcategory X( A) of the category X(A). In the previous section we gave the necessary and sufficient conditions for the category X(A) to be topological over X × A (Theorem 7). It was crucial to have the (−)∗

(−)◦

functors A −−→ Xop and X −−→ Aop . The situation is much simpler in case of the category X( A). Introduce the following weak versions of requirements (CLX), (MPX). (CLXA) For every X-object X, (X(X, U A), ) is a complete lattice. f

−◦ f

(MPXA) For every X-morphism Y − → X , X(X, U A) −−→ X(Y, U A) is a meet-preserving map. Theorem 12. Suppose (OCAT), (ADJ) hold. Equivalent are: (i) X( A) is topological over X; (ii) (CLXA), (MPXA) hold. Proof. We recall only that part of the proof which will be used later. fi

(ii)=⇒(i): Show the existence of → |(X i , i )|)i∈I be a | − |-structured source. Similar to initial lifts. Let S = (X − Theorem 7 one proves that  = i∈I (i ◦ f i ) is the desired structure.  Theorems 7 and 12 together imply the following result. Corollary 13. Suppose (OCAT), (ADJ) hold. Equivalent are: (i) X(A) is topological over X × A; (ii) (FST) holds and every subcategory X( A) of the category X(A) is topological over X.  By Corollary 13 every subcategory Set( A) of the categories Set(CSLat( )) and Set(SetRel) is topological. From now on assume that the category X(A) is topological. By Corollary 13 every subcategory of the form X( A) is topological as well.

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4. Fuzzification of algebraic structures In this section we introduce a fuzzification scheme for algebraic structures over the category X( A). In order not to restrict ourselves to a particular class of algebras we will use the objects of the category Alg(T ) as a sufficient generalization of the notion of abstract algebra (e.g., T can represent an algebraic signature). For convenience of the reader we recall the definition of the category Alg(T ) from [1]. T

Definition 14. Let X − → X be a functor. Alg(T ) is the concrete category over X, the objects of which (called f

x

T-algebras) are pairs (X, x) with X an X-object and T X − → X an X-morphism. Morphisms (X, x) − → (X  , x  ) f

(called T-homomorphisms) are X-morphisms X − → X  such that the diagram

f

f

commutes. The underlying functor to X is given by |(X, x) − → (X  , x  )| = X − → X . From now on we fix a category of the form Alg(T ). The following definition introduces a fuzzification scheme of Alg(T ) over the category X( A). For the sake of generality, we consider a subcategory of the category Alg(T ). Definition 15. Let B be a subcategory of the category Alg(T ). For a fixed Alg(T )-object (U A, ), the category ( A, )-B is the concrete category over B, the objects of which (called ( A, )-(T-algebras)) are triples (X, x, ) with (X, x) a B-object and (X, ) an X( A)-object such that  ◦ T    ◦ x, i.e.,

f

f

f

Morphisms (X, x, ) − → (Y, y, ) (called ( A, )-(T-homomorphisms)) are X-morphisms X − → Y with (X, x) − → (Y, y) f

f

a B-morphism and (X, ) − → (Y, ) an X( A)-morphism. The forgetful functor to B is given by |(X, x, ) − → (Y, y, )| = f

(X, x) − → (Y, y). We show a sufficient condition for ( A, )-B to be topological over B (notice that the property provides (A, )-B with many “nice” properties of its ground category B) and therefore introduce the following requirement. ◦T (−)

(OPR) For every X-object X, X(X, U A) −−−−→ X(T X, U A) is an order-preserving map. Theorem 16. If (OPR) holds, then ( A, )-B is topological over B.  fi Proof. Let S = ((X, x) − → |(X i , x i , i )|)i∈I be a | − |-structured source. Define  = i∈I (i ◦ f i ). By the proof of Theorem 12 it will be enough to show f i ) =  ◦ T i ◦ T f i   that (X, x, ) is an ( A, )-B-object.Since  ◦ T (i ◦  i ◦x i ◦T f i = i ◦ f i ◦x, ◦T  = ◦T ( i∈I (i ◦ f i ))  i∈I ◦T (i ◦ f i )  i∈I (i ◦ f i ◦x) = i∈I (i ◦ f i )◦x = ◦x.  In the following we provide an example of the category ( A, )-B which, apart from illustrating Definition 21, will show that our fuzzification machinery generalizes the respective one of Rosenfeld [33]. Start by introducing the following requirement. (CFPX) The category X has coproducts and finite products.

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Since X has finite products and X( A) is topological over X (recall our assumption at the end of the last section), the following result follows. Propostion 17. If (CFPX) holds, then the category X( A) has finite products which can be constructed as follows. Let ((X i , i ))i∈I be a finite family of X( A)-objects and let ( i∈I X i , ( i )i∈I ) be a product of the family (X i )i∈I in X. The

j   source (( i∈I X i ,  i ) −→ (X j ,  j )) j ∈I with  i = i∈I (i ◦ i ), is a product of the family ((X i , i ))i∈I . i∈I

i∈I

The next two lemmas contain some useful properties of the operation . fi

 → X ) Lemma 18. Suppose (CFPX) holds. If ((X i i∈I are finite families of X( A)-objects and  i , i ))i∈I and (X i − X-morphisms, respectively, then (  i ) ◦ ( i∈I f i ) =  (i ◦ f i ). i∈I

i∈I

Proof. The following diagram:

implies (  i ) ◦ (  (i ◦

i∈I

i∈I f i ).

 i∈I

fi ) = (



i∈I (i

◦ i )) ◦ (

 i∈I

fi ) =



j ∈ J ( j

◦ j ◦ (

 i∈I

f i )) =



j ∈ J ( j

◦ f j ◦ j ) =



Lemma 19. Suppose (CFPX) holds. If ((X i , i ))i∈I and ((X i , i ))i∈I are finite families of X( A)-objects such that i  i for i ∈ I , then  i   i . i∈I

i∈I

T

→ X (cf., e.g., [3, Chapter VI] as well as [6] where the notion of universal Consider the following example of X − algebra is considered). Example 20. Suppose (CFPX) holds. Let  = (n ) ∈ be a set-indexed family of natural numbers. For every ∈  S

f n

f

T

∈ S

→ X by S (X − → Y ) = X n −−→ Y n . Setting X −→ X = X −−−−−→ X (i.e., T is just the coproduct define X − of the functors S ) yields the category Alg(T ) (notice that we rely strongly on the fact that  is a set). Take the [  1U A ]



n

Alg(T )-object (U A,  ) with T U A −→ U A = T U A −−−−→ U A, where the latter morphism is defined by the commutativity of the triangle

for every  ∈ (notice that   are coproduct injections). For every subcategory B of the category Alg(T ) Example 20 provides the category ( A,  )-B of categorical representations of universal algebras. We are interested in studying properties of such categories and therefore from now on we fix a category ( A,  )-B. To introduce some new notions we return for a moment to the category X(A). Start by extending the relation “” from the elements of F to arbitrary U-structured sinks. i

i

Definition 21. Let S = (X i − → U B)i∈I and T = (X i − → U B)i∈I be U-structured sinks. Define S  T iff i  i for i ∈ I .

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With the help of Definition 21 introduce the following notion. 

fi → − − U A, → Y )i∈I in X is said to be a -sink provided that for every F-elements Y → Definition 22. A sink S = (X i −



 ◦ S   ◦ S implies   . The following lemma shows a relation between -sinks and epi-sinks (recall our assumption (ADJ)). 

→ U F is a mono-transformation (i.e., every  X is an X-monomorphism), then every -sink is an Lemma 23. If 1X − epi-sink. Proof. If f ◦ S = g ◦ S, then cod( f ) ◦ f  cod(g) ◦ g, cod(g) ◦ g  cod( f ) ◦ f and therefore cod( f ) ◦ f = cod(g) ◦ g. Now use the assumption of the lemma.  Propostion 24. Suppose (CFPX) holds. If coproducts in X are -sinks, then  ◦ T (−) is order-preserving. 

 − → → U A with    and let S = (( ) ∈ , ∈ X n ) be a coproduct of (X n ) ∈ in X. Commutativity Proof. Take X − 

of the diagram

 for every  ∈ and Lemma 18 imply  ◦ T () = [ 1U A ] ◦ ( ∈ n ) = [( 1U A ) ◦ n  ] = [n  ] and n  n 



similar for . Since n   n  for ∈ by Lemma 19, [n ] ◦ S  [n ] ◦ S and therefore [n ]  [n ]. 

In the following we present the necessary and sufficient condition for a sink in X to be a -sink. Start with the following notation (recall requirement (FCR)). fi

F fi ◦ f ◦

Definition 25. Given a sink S = (X i − → X )i∈I in X we define FS ◦ S ◦ = (X −−−−→ X )i∈I and  ◦ i∈I (F f i ◦ fi ).

 (FS ◦ S ◦ ) =

fi

→ X )i∈I in X the following are equivalent: Propostion 26. For a sink S = (X i − (i) S is a -sink; (ii) (FS ◦ S ◦ ) = 1 F X .  Proof. (i)=⇒(ii): Since F f i ◦ f i◦  1 F X for i ∈ I , (FS ◦ S ◦ )  1 F X . On the other hand,  X ◦ f j = U F f j ◦  X j = U F f j ◦ U 1 F X j ◦  X j  U F f j ◦ U ( f j◦ ◦ F f j ) ◦  X j = U F f j ◦ U f j◦ ◦ U F f j ◦  X j = U F f j ◦ U f j◦ ◦  X ◦ f j     ◦ ◦ ◦ ( i∈I U F f i ◦ U f i ) ◦ ◦ X ◦ f j = ( i∈I U (F f i ◦ f i)) ◦  X ◦ ◦f j  U ( i∈I F f i ◦ f i ) ◦  X ◦ f j by Lemma 3. Then  X  U ( i∈I F fi ◦ f i ) ◦  X and therefore 1 F X  (FS ◦ S ).    → − → U A be such that  ◦ S   ◦ S. Then  A ◦ F =  A ◦ F ◦ ( FS ◦ S ◦ ) =  A ◦ F ◦ FS ◦ (ii)=⇒(i): Let X −   S ◦   A ◦ F ◦ FS ◦ S ◦ =  A ◦ F and therefore    by Definition 1.  Proposition 26 can be illustrated by the following result.

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 fi Propostion 27. Consider the categories Set(CSLat( )) and Set(SetRel). For a sink S = (X i − → X )i∈I in Set the following are equivalent: (i) S is an epi-sink; (ii) S is an -sink. Proof. To show (recall that epi-sinks (i)=⇒(ii) proceed as follows  in Set are jointly surjective).  Set(CSLat( )): If S ∈ F X = P(X), then (FS ◦ S ◦ )(S) = i∈I fi→ ( f i← (S)) = S and therefore (FS ◦ S ◦ ) = 1F X .  Set(SetRel): If i ∈ I , then F f i ◦ f i◦ = fi ◦ −1 (FS ◦ S ◦ ) = f i = {(x, x) | f i (x i ) = x for an x i ∈ X i } and then  −1 i∈I f i ◦ f i = {(x, x) | x ∈ X } = 1 F X .  X To show (ii)=⇒(i) notice that the universal arrows  are given by X −→ P(X ) : x{x} for both CSLat( ) and SetRel and therefore the condition of Lemma 23 is satisfied.  The following provides the main result of the section. Theorem 28. Suppose (CFPX) holds. If coproducts in X are -sinks, then the category ( A,  )-B is topological over B. Proof. Follows from Theorem 16 and Proposition 24.  We are going to make Example 20 more particular and therefore assume that X = Set. Introduce the following definition. Definition 29. Let M (resp. E) be the class of T -homomorphisms with injective (resp. surjective) underlying maps. A variety of T -algebras is a full subcategory of Alg(T ) closed under the formation of products, M-subobjects (subalgebras) and E-quotients (homomorphic images). The objects (resp. morphisms) of a variety are called algebras (resp. homomorphisms). In particular, the constructs Vec, R-Mod, Ab, Grp, Mon, Sgr, Rng and Lat of (real) vector spaces, (left) modules over a ring R, abelian groups, groups, monoids, semi-groups, rings and lattices are varieties. On the other hand, the constructs Frm, SFrm and SQuant of frames, semi-frames and semi-quantales (popular in lattice-valued topology) are not varieties in our sense. The problem is that neither the family of operations on objects of the above-mentioned categories is a set nor the operations themselves are finite. Theorem 28 yields the following result (recall our assumption X = Set). Corollary 30. Let coproducts in Set be -sinks. For every A-object A and every variety B, ( A,  )-B is topological over B.  Corollary 30 and Proposition 27 together imply that in case of the category Set(CSLat( )) (resp. Set(SetRel)) each of the categories (A,  )-Vec, ( A,  )-(R-Mod), ( A,  )-Ab, ( A,  )-Grp, ( A,  )-Mon, ( A,  )-Sgr as well as ( A,  )-Rng, ( A,  )-Lat is topological. Consider the following continuation of Example 20. Example 31. Let B be a subcategory of Alg(T ). If (X, x, ) is an ( A,  )-B-object, then [ ] = [( 1U A ) ◦ n ] = n n  [ 1U A ] ◦ ( ∈ n ) =  ◦ T ()   ◦ x and therefore   = [ ] ◦    ◦ x ◦  for ∈ , where n n n   n n  X − →  ∈ X is the coproduct injection. Consider two important cases: (i) If n = 0, then  ◦ x ◦  = where is the top element of the lattice X(X n , U A). (ii) If n = 1, then    ◦ x ◦  . Suppose (x ◦  ) ◦ (x ◦  ) = 1 X . Then  ◦ x ◦    ◦ x ◦  ◦ x ◦  =  implies  =  ◦ x ◦  .

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Example 31 generalizes Proposition 5.4 in [33] which says: let  be a fuzzy subgroup of S; then (x −1 ) = (x) and (x) (e) for all x ∈ S, where e is the identity element of S. It is time now to show  that our category ( A,  )-Grp generalizes the respective category of Rosenfeld. Suppose X(A) = Set(CSLat( )),  = (2, 1, 0) and A = I = [0, 1] (the unit interval) in Example 20. Then T -algebras are sets equipped with a binary, an unary and a nullary operation (i.e., tuples (X, ·, (−)−1 , e)) and T -homomorphisms f

f

(X, ·, (−)−1 , e X ) − → (Y, ·, (−)−1 , eY ) are maps X − → Y preserving the operations. Straightforward computations (and Proposition 27) show that (X, ·, (−)−1 , e, ) is an (I,  )-Grp-object iff the following inequalities (presented in diagrammatical form) hold:

In the ordinary (point-wise) notation this means that (x) ∧ (y) (x · y) and (x) (x −1 ) for every x, y ∈ X as well as (e) = , which is the definition of fuzzy group of Rosenfeld with the only exception that he does not require f

the last equality. The morphisms of the category (I,  )-Grp (X, ·, (−)−1 , e X , ) − → (Y, ·, (−)−1 , eY , ) are precisely the group homomorphisms such that (x)  ◦ f (x) for every x ∈ X . As follows from the results obtained in the paper the category (I,  )-Grp is topological over Grp and therefore is both complete and cocomplete (as well as has other “nice” properties of Grp which can be found in, e.g., [1]). The above-mentioned generalization is not an accident. As was already mentioned in the introductory section the category of fuzzy groups of Rosenfeld (with a slight modification) is isomorphic to the category of internal groups in Set(I ). In our case one can easily state the following theorem. Theorem 32. Suppose (CFPX) holds. Let V be a variety in the sense of Definition 29, i.e., a category of algebras which have a certain algebraic signature and which satisfy certain identities. Let B be the respective subcategory of Alg(T ) of Example 20 obtained by incorporating the signature into the functor T and translating the identities into the monoidal category (X, ×, TX ). If coproducts in X are -sinks, then the category ( A,  )-B is isomorphic to the category Vint of internal algebraic structures (from here the index “int”) in the monoidal category (X( A), ×, TX(A) ) determined by V. Proof. Since the translated equations of V hold in X for both Alg(T ) and Vint , the required isomorphism provides f

F

f

the functor Vint − → ( A,  )-B defined by F(((X, ), ( X ) ∈ ) − → ((Y, ), (Y ) ∈ )) = (X, x, ) − → (Y, y, ), where  X

( X ) ∈ is the family of operations on (X, ) (i.e., (X, )n −−→ (X, ) is an X( A)-morphism for every ∈ ) and x

→ X is the X-morphism defined by the commutativity of the triangle T X −

for every  ∈ (notice that   are coproduct injections). To show that (X, x, ) is an (A,  )-B-object, i.e., to obtain

from

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which by Lemma 18 are precisely

we use our requirement on coproducts in X.  We illustrate the claim by one simple example. Let Grpd be the variety of groupoids. Then Grpdint is the category of · groupoids in X( A). Its objects are pairs (actually triples) ((X, ), ·), where (X, ) is an X( A)-object and (X, )×(X, ) − → (X, ) is an X( A)-morphism, i.e.,

The last diagram is precisely the next one (by Lemma 18):

f

f

Grpdint -morphisms ((X, ), ·) − → ((Y, ), ·) are X( A)-morphisms (X, ) − → (Y, ) such that the following diagram commutes:

In other words, we obtained an isomorphism between the categories Grpdint and ( A,  )-Alg(T ) where  = (2) and T is the product functor (notice that in our case the category of internal groupoids in X is precisely the category Alg(T )). Thus our fuzzification machinery generalizes the well-known fuzzification through internal structures in categories. Moreover, our requirement on coproducts of X and Theorem 28 yields the following corollary. X(A)

X (resp. V Corollary 33. Suppose (CFPX) holds. Let V be a variety and let Vint int ) be the category of internal algebraic structures in the monoidal category (X, ×, TX ) (resp. (X( A), ×, TX(A) )) determined by V. If coproducts in X(A) |−|

X are -sinks, then the (obvious) forgetful functor Vint

X is topological. −→ Vint

5. Fuzzification of topological structures In this section we consider a fuzzification scheme for topological structures over the category X( A). In order not to restrict ourselves to a particular class of structures we will use the objects of the functor-structured category Spa(T ) as a sufficient generalization of the notion of abstract topological structure. For convenience of the reader we recall the definition of the category Spa(T ) from [1]. T

Definition 34. Let X − → Set be a functor. Spa(T ) is the concrete category over X, the objects of which (called f

f

T-spaces) are pairs (X, ) with  ⊆ T X . Morphisms (X, ) − → (Y, ) (called T-maps) are X-morphisms X − → Y such

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f

f

that (T ( f ))→ () ⊆ . The underlying functor to X is given by |(X, ) − → (Y, )| = X − → Y . Concrete categories of the form Spa(T ) are called functor-structured categories. From now on we fix a category of the form Spa(T ). The following definition introduces a fuzzification scheme of Spa(T ) over the category X( A). For the sake of generality, we consider a subcategory of the category Spa(T ). Definition 35. Let B be a subcategory of the category Spa(T ). For a fixed Spa(T )-object (U A, P), the category ( A, P)-B is the concrete category over B, the objects of which (called (A, P)-(T-spaces)) are triples (X, S, ), where  (X, S) is a B-object and (X, ) is an X( A)-object such that (X, S) − → (U A, P) is a Spa(T )-morphism (i.e., (T )→ (S) ⊆ f

f

f

P). Morphisms (X, S, ) − → (Y, Q, ) (called ( A, P)-(T-maps)) are X-morphisms X − → Y with (X, S) − → (Y, Q) a Bf

f

morphism and (X, ) − → (Y, ) an X( A)-morphism. The underlying functor to B is given by |(X, S, ) − → (Y, Q, )| = f

(X, S) − → (Y, Q). Notice that there exists the following generalized version of Definition 34 (see, e.g., [1]).  T → CSLat( ). For each topological theory T in X denoted Definition 36. A topological theory in X is a functor X − by Top(T ) the concrete category over X whose objects are pairs (X, t) with X an X-object and t ∈ T (X ), and whose f

f

morphisms (X, t) − → (Y, s) are those X-morphisms X − → Y that satisfy (T f )(t) s.  G P One can easily show that if X − → Set is a functor and Set − → CSLat( ) is the covariant powerset functor, then P ◦ G is a topological theory in X and Top(P ◦ G) = Spa(G). By analogy with Definition 35 one obtains the category ( A, )-Top(T ) replacing P with an element  ∈ T U A, f

i.e., the objects of the new category are triples (X, t, ) such that (T )(t)  ; morphisms (X, t, ) − → (Y, s, ) are f

→ Y such that (T f )(t)  s and   f ◦  (notice that to be more general one can again consider a X-morphisms X − subcategory of the category Top(T )). The categories of the form Top(T ) are considered in [32] in connection with powerset theories which generate topological theories. One could ask a natural question about a relation between the category ( A, )-Top(T ) and topological powerset theories from [32]. Unfortunately, the answer is still unclear to us. As in the previous section we are going to present a sufficient condition for ( A, P)-B to be topological over B. Introduce the following requirement (recall condition (CLXA)). (MC) For every B-object (X, S), the set Spa(T )((X, S), (U A, P)) is closed under the formation of meets. Theorem 37. Suppose (MC) holds. Then ( A, P)-B is topological over B. fi

Proof. Let S = ((X, S) − → |(X i , Si , i )|)i∈I be a | − |-structured source. Define  =



i∈I (i

◦ f i ). By the proof of i ◦ f i

Theorem 12 it will be enough to show that (X, S, ) is an ( A, P)-B-object. The fact that (X, S) −−−→ (U A, P) is a  Spa(T )-morphism for i ∈ I and (MC) imply that (X, S) − → (U A, P) is a Spa(T )-morphism.  The next example is a modified version of the functor of Example 20 suitable for our context. Example 38. Suppose (CFPX) holds. Let  = (n ) ∈ be a set-indexed family of natural numbers and let TX be a S f f n terminal object in X (recall that X has finite products). Given ∈ define X − → X by S (X − → Y ) = X n −−→ Y n . T

S

hom(TX , −)

T

LetX − → Set = X − → X −−−−−−→ Set, where hom(TX , −) is the covariant hom-functor, and set X −→ Set = ∈ T

X −−−−−→ Set.

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Example 38 yields the category Spa(T ). Suppose P ⊆ (T ◦ U )( A) = the formation of meets in such a way that the following condition is fulfilled:



∈ hom(TX , (U A)

n )

is closed under

  i∈I ( ◦h i )

hi

(A) For every ∈ and every subfamily (TX − → (U A)n )i∈I ⊆ P , the X-morphism TX −−−−−−−→ (U A)n defined by the commutativity of the diagram



for every projection morphism (U A)n − → U A, belongs to P . Given a subcategory B of Spa(T ), the above-mentioned considerations provide the category ( A, P )-B. The next proposition shows an important property of condition (A). Propostion 39. In case of the functor T , (A) implies (MC). 

i

i∈I

i

→ (U A, P ))i∈I of Spa(T )-morphisms show that (X, S) −−−−→ (U A, P ) is a Spa(T )Proof. For a family ((X, S) − h

n

→ X n is an X-morphism for some ∈ and i ◦ h = (T i )(h) ∈ P for i ∈ I . morphism. If h ∈ S, then TX − Consider the following diagram:

     n Since (i ◦ X ◦h) = U A ◦ i∈I (i ◦ X◦h) implies ( i∈I i )n ◦h  U A ◦( i∈I i ) ◦h =( i∈I i )◦ X ◦h  = i∈I  n =  i∈I (i ◦ X ◦ h), T ( i∈I i )(h) = ( i∈I i )n ◦ h =  i∈I (i ◦ X ◦ h) =  i∈I ( U A ◦ i ◦ h) ∈ P by (A).  We are going to consider one particular case of Example 38.  Example 40. Let X(A) be Set(CSLat( )). Then TSet {} and therefore T (X ) = hom({}, X n ) X n . One can easily see that Spa(T ) is concretely isomorphic to the category Rel() (recall that  is a set of natural numbers) defined as follows: the objects are pairs (X, ( ) ∈ ), where X is a set and ⊆ X n for ∈ ; the morphisms are f

maps (X, ( X ) ∈ ) − → (Y, ( Y ) ∈ ) such that ( f n )→ ( X ) ⊆ Y for ∈ . Let B be a subcategory of Spa(T ) (for example, if  = (2) one can consider the category of sets and relations Rel, the category of preordered sets Prost or the category of partially ordered sets Pos) and let (U A, (   ) ) ∈ ) be a Spa(T  object such that is closed under the formation of meets for ∈ , i.e., ai bi for i ∈ I imply ( i∈I ai ) ( i∈I bi ). Then ( A, P )-B is topological over B by Proposition 39 and Theorem 37. In particular, Example 40 gives the category ( A, )-Pos of fuzzified posets (topological over Pos), where ⊆ U A × U A is a relation on UA. The objects of the category are triples (X,  , ), where (X, ) is a poset, (X, ) is an A-set and, moreover, x  y implies (x) (y). Notice that our fuzzification differs from the concept of fuzzy partial order (see, e.g., [18, Section 3]). The point is that we fuzzify the underlying set of a structure in a suitably compatible way, but leave the structure itself crisp. At the end of the paper we briefly consider the case of functor-costructured categories (see, e.g., [1]). Given a functor T

Xop − → Set, there exists the category (Spa(T ))op considered as a concrete category over X = (Xop )op and called the functor-costructured category. In a natural way the category ( A, R)-B arises (see Definition 35). One can easily restate

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Theorem 37 for the case of the category ( A, R)-B. Since the paper is aiming at providing a new fuzzification machinery for topological spaces we consider the following example. Q

Example 41. Let Set − → Set be the contravariant powerset functor. The category Top of topological spaces  and continuous maps is a full subcategory of (Spa(Q))op . Using the framework of the category Set(CSLat( )) one obtains the category ( A, R)-Top (concrete over Top), the objects of which (called (A, R)-topological spaces) are triples (X, O(X), ), where (X, O(X )) is a topological space, (X, ) is an A-set and (← )→ (R) ⊆ O(X ). Morphisms f

f

(X, O(X), ) − → (Y, O(Y ), ) (called ( A, R)-continuous maps) are maps X − → Y which are continuous Set( A)morphisms. We are going to show a sufficient condition for ( A, R)-Top to be topological over Top and therefore we introduce the following requirement. (B) Every V ∈ R has the following properties (cf. Scott open sets of Definition II-1.3 in [10]): (i) ↓ V = V ; (ii) S ∈ V implies S ∩ V  ∅ for every non-empty subset S ⊆ A. As an illustration of the introduced notions we provide the following examples. (1) If A is the one-element lattice {}, then the category ( A, R)-Top is isomorphic to Top for every R (which is one of ∅, {∅}, {{}}, {∅, {}}). (2) If R = { A\ ↑ a | a ∈ A}, then (U A, R) satisfies (B). In particular, if A is a chain (i.e., the partial order is linear), then R is the family of all sets {b ∈ A | b < a} and then the definition of ( A, R)-topological spaces says that the topology should contain all anti-level-sets (recall that given an A-set (X, ) and a ∈ A, an a-level set is the set {x ∈ X | a (x)}). Furthermore, if A is a two-element lattice {⊥, }, then R = {∅, {⊥}} and therefore every ({⊥, }, {∅, {⊥}})-topological space (X, O(X ), ) contains ← ({⊥}) in its topology. For example, for every anti-discrete space (X, {∅, X}), the ({⊥, }, {∅, {⊥}})-fuzzification is the object (X, {∅, ← ({⊥}), X }, ). The reason for adding new sets to the topology under the fuzzification is simple. Suppose we have an anti-discrete Y

topological space (X, {∅, X}). A subset Y of X amounts to the characteristic function X −→ {⊥, }, i.e., to a fuzzification Y

(X, {∅, X})  {⊥, }, which yields something like (Y, {∅, X }) that does not make sense. By adding X \Y = ← Y ({⊥}) to the topology we provide some means of distinguishing between the existent points (of Y) and the non-existent ones (of X\Y ). Notice that the former set is closed and the latter one is open. Propostion 42. If (B) holds, then ( A, R)-Top is topological over Top. i

Proof. Given a topological space (X, O(X)) and a family of (Spa(Q))op -morphisms ((X, O(X )) − → (U A, R))i∈I   define  = i∈I i . By Theorem 37 it will be enough to show that (X, O(X )) − → (U A, R) is a (Spa(Q))op -morphism. If I = ∅, then  ≡ (the top element of the lattice A) and therefore for every V ∈ R:  ← (V ) =

X, ∈ V , ∅, ∈ / V.

 Suppose I  ∅ and take some V ∈ R. Show that ← (V ) = i∈I ← i (V ) (notice that the latter set is an element of  O(X)). If x ∈ ← (V ), then  (x) ∈ V and therefore there exists i0 ∈ I  such that i0 (x) ∈ V by (B)(ii). On the i i∈I  other hand, x ∈ i∈I ← (V ) gives i ∈ I such that  (x) ∈ V and therefore 0 i 0 i∈I i (x) ∈ V by (B)(i).  i Notice that the above-mentioned fuzzification procedure differs from the respective ones of [5,17,30,32] since our machinery amounts to fuzzifying the underlying “set” of a structure in a suitably compatible way, leaving the structure itself crisp.

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We close the section with a slightly embarrassing example of a “double fuzzification”. Start by recalling from [32] the definition of the category L-Top of L-topological spaces over a unital semi-quantale L. ⊗

Definition 43. Let L be a unital semi-quantale (i.e., a complete lattice equipped with a binary operation L × L − →L which has the unit e). L-Top is the concrete category over Set, the objects of which are pairs (X, ), where X is a set and  ⊆ L X (L X is the L-powerset of X) is closed under ⊗ and arbitrary joins as well as contains the constant map e. f

f L←

f

→ (Y, ) are maps X − → Y such that ( f L← )→ () ⊆  (L Y −−→ L X is the Zadeh preimage operator Morphisms (X, ) − f

f

defined by f L← ( p) = p ◦ f ). The underlying functor to Set is given by |(X, ) − → (Y, )| = X − → Y. With the help of Definition 43 we introduce the following example. QL

Example 44. Let L be a unital semi-quantale and let Set −−→ Set be the contravariant L-powerset functor induced op by the Zadeh preimage operator.  Then the category L-Top is a full subcategory of (Spa(Q L )) . Using the framework of the category Set(CSLat( )) one obtains the category ( A, R)-(L-Top) (concrete over L-Top), the objects of which (called ( A, R)-(L-topological spaces)) are triples (X, , ), where (X, ) is an L-topological space, (X, ) is an A-set f

f

→ and (← → (Y, , ) (called ( A, R)-(L-continuous maps)) are maps X − → Y which L ) (R) ⊆ . Morphisms (X, , ) − are L-continuous Set( A)-morphisms.

The meaning of the phrase “double fuzzification” should be clear now since we fuzzify the category of L-topological spaces which themselves present a fuzzification of the usual topological spaces. We are going to show a sufficient condition for ( A, R)-(L-Top) to be topological over L-Top and therefore we introduce the following requirement (cf. requirement (B)).   (C) Every V ∈ R takes meets into joins, i.e., V ( S) = V → (S) for every S ⊆ U A (recall that R ⊆ L U A ). Propostion 45. If (C) holds, then ( A, R)-(L-Top) is topological over L-Top. i

→ (U A, R))i∈I define Proof. Given an L-topological space (X, ) and a family of (Spa(Q L ))op -morphisms ((X, ) −   op  = i∈I i . By Theorem 37 it will be enough to show that (X, ) → − (U A, R) is a (Spa(Q L )) -morphism. For    ← ← every x ∈ X,  ( (V ))(x) = (V ◦ )(x) = V ( i∈I i (x)) = i∈I (V ◦ i )(x) = ( i∈I (i (V )))(x) and therefore (← (V )) = i∈I (← i (V )) ∈ .  Notice that the proof of Proposition 45 is a little bit simpler than the respective one of Proposition 42. 6. Conclusion In the paper we presented a fuzzification machinery for algebraic as well as topological structures which amounts to fuzzifying the underlying “set” of a structure in a suitably compatible way, leaving the structure itself crisp. Our results show that the algebraic approach generalizes the existing procedures of Rosenfeld as well as of internal structures in categories. The topological one, however, falls out of the usual fuzzification machineries mentioned in the literature. The reason is that in contrast to [5,17,30,32] we do not fuzzify the concepts themselves but just present a modified version of a given topological structure. In particular, our category ( A, R)-Top introduced above differs from the above-mentioned category L-Top of L-topological spaces [32]. On the other hand, our procedure allows one to make a “double fuzzification”, i.e., to fuzzify something that is already fuzzified and therefore it is in line with the Principle of Fuzzification of Goguen [11] which says that “a fuzzy (or L-fuzzy or L-) something is an L-set of somethings (i.e., an L-fuzzy set on the set of somethings)”. Acknowledgements The author is grateful to the Department of Mathematics and Statistics of the Faculty of Science of the Masaryk University in Brno, Czech Republic (especially to Prof. J. Rosický) for the opportunity of spending three months at the university during which period the revised version of the manuscript was prepared.

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References [1] J. Adámek, H. Herrlich, G.E. Strecker, Abstract and concrete categories: the joy of cats, Repr. Theory Appl. Category 17 (2006) 1–507. [2] J.M. Anthony, H. Sherwood, Fuzzy groups redefined, J. Math. Anal. Appl. 69 (1979) 124–130. [3] G. Birkhoff, Lattice theory, in: American Mathematical Society Colloq. Publications XXV, third ed., American Mathematical Society (AMS), Providence, RI, 1979. [4] F. Borceux, Handbook of Categorical Algebra, Volume 1: Basic Category Theory, Cambridge University Press, Cambridge, 1994. [5] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl. 24 (1968) 182–190. [6] P.M. Cohn, Universal Algebra, D. Reidel Publ. Comp., Dordrecht, 1981. [7] B. Eckmann, P.J. Hilton, Group-like structures in general categories I: multiplications and comultiplications, Math. Ann. 145 (1962) 227–255. [8] B. Eckmann, P.J. Hilton, Group-like structures in general categories II: equalizers, limits, lengths, Math. Ann. 151 (1963) 150–186. [9] B. Eckmann, P.J. Hilton, Group-like structures in general categories III: primitive categories, Math. Ann. 150 (1963) 165–187. [10] G. Gierz, K.H. Hofmann, et al., Continuous Lattices and Domains, Cambridge University Press, Cambridge, 2003. [11] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145–174. [12] J.A. Goguen, Categories of V-sets, Bull. Am. Math. Soc. 75 (1969) 622–624. [13] J.A. Goguen, Concept representation in natural and artificial languages: axioms, extensions and applications for fuzzy sets, Int. J. Man-Mach. Stud. 6 (1974) 513–561. [14] R. Goldblatt, Topoi. The Categorical Analysis of Logic, Rev. Ed., Dover Publications, Inc., Mineola, New York, 2006. [15] U. Höhle, Fuzzy sets and sheaves. Part I: basic concepts, Fuzzy Sets Syst. 158 (11) (2007) 1143–1174. [16] U. Höhle, Fuzzy sets and sheaves. Part II: sheaf-theoretic foundations of fuzzy set theory with applications to algebra and topology, Fuzzy Sets Syst. 158 (11) (2007) 1175–1212. [17] U. Höhle, A.P. Šostak, Axiomatic foundations of fixed-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Handbook of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999, pp. 123–272. [18] H. Lai, D. Zhang, Fuzzy preorder and fuzzy topology, Fuzzy Sets Syst. 157 (2006) 1865–1885. [19] R. Lowen, Fuzzy topological spaces and fuzzy compactness, J. Math. Anal. Appl. 56 (1976) 621–633. [20] S. Mac Lane, Categories for the Working Mathematician, second ed., Springer, Berlin, 1998. [21] S. Mac Lane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory, Springer, Berlin, 1992. [22] I. Moerdijk, Spaced spaces, Compos. Math. 53 (1984) 171–209. [23] J.N. Mordeson, D.S. Malik, Fuzzy Commutative Algebra, World Scientific, Singapore, 1998. [24] A. Pultr, S.E. Rodabaugh, Category theoretic aspects of chain-valued frames: part I: categorical foundations, Fuzzy Sets Syst. 159 (2008) 501–528. [25] A. Pultr, S.E. Rodabaugh, Category theoretic aspects of chain-valued frames: part II: applications to lattice-valued topology, Fuzzy Sets Syst. 159 (2008) 529–558. [26] S.E. Rodabaugh, A categorical accommodation of various notions of fuzzy topology, Fuzzy Sets Syst. 9 (1983) 241–265. [27] S.E. Rodabaugh, Point-set lattice-theoretic topology, Fuzzy Sets Syst. 40 (2) (1991) 297–345. [28] S.E. Rodabaugh, Categorical frameworks for stone representation theories, in: S.E. Rodabaugh, E.P. Klement, U. Höhle (Eds.), Applications of Category Theory to Fuzzy Subsets, Theory and Decision Library: Series B: Mathematical and Statistical Methods, Vol. 14, Kluwer Academic Publishers, Dordrecht, 1992, pp. 177–231. [29] S.E. Rodabaugh, Powerset operator based foundation for point-set lattice-theoretic (poslat) fuzzy set theories and topologies, Quaest. Math. 20 (3) (1997) 463–530. [30] S.E. Rodabaugh, Categorical foundations of variable-basis fuzzy topology, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, The Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999, pp. 273– 388. [31] S.E. Rodabaugh, Powerset operator foundations for poslat fuzzy set theories and topologies, in: U. Höhle, S.E. Rodabaugh (Eds.), Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory, Handbooks of Fuzzy Sets Series, Vol. 3, Kluwer Academic Publishers, Dordrecht, 1999, pp. 91–116. [32] S.E. Rodabaugh, Relationship of algebraic theories to powerset theories and fuzzy topological theories for lattice-valued mathematics, Int. J. Math. Math. Sci. 2007 (2007), Article ID 43645, 71 pages, doi:10.1155/2007/43645. [33] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971) 512–517. [34] B. Schweizer, A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, 1983. [35] S. Solovjovs, On a Categorical Generalization of the Concept of Fuzzy Set: Basic Definitions, Properties, Examples, VDM Verlag Dr. Müller, 2008. [36] S. Solovyov, Categories of lattice-valued sets as categories of arrows, Fuzzy Sets Syst. 157 (6) (2006) 843–854. [37] S. Solovyov, On the category Set(JCPos), Fuzzy Sets Syst. 157 (3) (2006) 459–465. [38] S. Solovyov, On a generalization of Goguen’s category Set(L), Fuzzy Sets Syst. 158 (4) (2007) 367–385. [39] A. Šostak, Fuzzy categories related to algebra and topology, Tatra Mt. Math. Publ. 16 (1) (1999) 159–185. [40] L.A. Zadeh, Fuzzy sets, Inf. Control 8 (1965) 338–365.