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Applied General TopologyUniversidad Polit�ecnica de ValenciaVolume 1, No. 1, 2000pp. 115 - 127

Fuzzy functions: a fuzzy extension of thecategory SET and some related categoriesUlrich H�ohle, Hans-E. Porst, Alexander P. �SostakAbstract. In research works where fuzzy sets are used, mostlycertain usual functions are taken as morphisms. On the otherhand, the aim of this paper is to fuzzify the concept of a functionitself. Namely, a certain class of L-relations F : X � Y ! L isdistinguished which could be considered as fuzzy functions froman L-valued set (X;EX) to an L-valued set (Y;EY ). We studybasic properties of these functions, consider some properties ofthe corresponding category of L-valued sets and fuzzy functionsas well as brie
y describe some categories related to algebra andtopology with fuzzy functions in the role of morphisms.2000 AMS Classi�cation: 04A72, 18A05, 54A40Keywords: L-relation, L-fuzzy function, Fuzzy category, Fuzzy topology,Fuzzy group 1. IntroductionIn research works where fuzzy sets are involved, in particular, in the theoryof fuzzy topological spaces, fuzzy algebra, fuzzy measure theory, etc., mostlycertain usual functions are taken as morphisms: they can be certain mappingsbetween the corresponding sets, or between the fuzzy powersets of these sets,etc. On the other hand, there are only few papers where attempts to fuzzifythe concept of a function itself are undertaken (see e.g. [11, 12], etc). The aimof our work is also to present a possible approach to this problem. Namely,a certain class of L-relations (i.e. mappings f : X � Y ! L) is distinguishedwhich seem reasonable to be viewed as (L-)fuzzy functions from a set X to a setY . We de�ne composition of fuzzy functions; study images and preimages ofL-sets under fuzzy functions; introduce properties of injectivity and surjectivityfor them; describe products and coproducts in the corresponding category, etc.In the last part of the paper we de�ne some categories related to topology andalgebra where fuzzy functions play the role of morphisms.



116 Ulrich H�ohle, Hans-E. Porst, Alexander P. �SostakIn conclusion, we would like to mention the following two peculiarities of ourapproach.First, the appropriate context for our work is formed not by usual sets, orby their L-subsets (i.e. mappings f : X ! L), but rather by L-valued sets (i.e.sets, endowed with an L-valued equality E : X � X ! L, see e.g. [6, 7]) andtheir L-subsets. And second, in the result we obtain not a usual category, butthe so called a fuzzy category - a concept introduced and studied in [14, 15].2. PrerequisitesLet L = (L;�;^;_;�) be an in�nitely distributive GL-monoid (cf. e.g. [6],[7]), i.e. a commutative integral divisible cl-monoid (cf. [1]). It is well knownthat every GL-monoid is residuated, i.e. there exists a further binary operation- implication \7�!" such that� � � � 
 () � � � 7�! 
 8�;�;
 2 L:We set �2 = � � � and further by induction: �n = �n�1 � �. Let > and ?denote respectively the top and the bottom elements of L.Following U.H�ohle (cf e.g. [7]) by an L-valued set we call a pair (X;E) whereX is a set and E is an L-valued equality, i.e. a mapping E : X � X ! L suchthat(1eq) E(x;y) � E(x;x) ^ E(y;y) 8x;y 2 X;(2eq) E(x;y) = E(y;x) 8x;y 2 X;(3eq) E(x;y) � �E(y;y) 7�! E(y;z)� � E(x;z) 8x;y;z 2 X.An L-valued set (X;E) is called separated if(4eq) E(x;x)WE(y;y) � E(x;y) () x = y 8x;y 2 X.An L-valued equality E is called global if(5eq) E(x;x) = > 8x 2 X.Further, recall that an L-set, or more precisely, an L-subset of a set Xis just a mapping A : X ! L. In case (X;E) is an L-valued set, its L-subset A is called strict, if A(x) � EX(x;x)8x 2 X; A is called extensional ifsupx A(x) � �E(x;x) 7�! E(x;x0)� � A(x0);8x0 2 X.By L � SET(L) we denote the category whose objects are triples (X;E;A)where (X;E) is an L-valued set and A is its strict extensional L-subset, andmorphisms from (X;EX;A) to (Y;EY ;B) are mappings f : X ! Y whichpreserve equalities (i.e. EX(x1;x2) � EY (fx1;fx2)) and "respect L-subsets",i.e. A � B �f. Let L�SET 0(L) stand for the full subcategory of the categoryL � SET(L) determined by global separated L-valued sets.To recall the concept of an L�fuzzy category [14, 15], consider an ordinary(classical) category C and let ! : Ob(C) ! L and � : Mor(C) ! L be L-fuzzysubclasses of its objects and morphisms respectively. Now, an L-fuzzy categorycan be de�ned as a triple (C;!;�) satisfying the following axioms ([15], cf. also[14] in case � = ^):10 �(f) � !(X) ^ !(Y ) 8X;Y 2 Ob(C) and 8f 2 Mor(X;Y );20 �(g � f) � �(f) � �(g) whenever the composition g � f is de�ned;



Fuzzy functions: a fuzzy extension of the category SET and some related categories 11730 �(eX) = !(X) where eX : X ! X is the identity morphism.Our aim is, starting from the category L�SET(L), to de�ne a fuzzy categoryL�FSET(L) having the same class of objects as L�SET(L) but an essentiallywider class of "potential" morphisms.3. Fuzzy category L � FSET(L).3.1. Category L � FSET(L). We start with de�ning a usual (i.e. crisp) cat-egory L � FSET(L). Namely, let L � FSET(L) denote the category havingthe same objects as L�SET(L) and whose morphisms, called (potential) fuzzyfunctions, from (X;EX;A) to (Y;EY ;B) are L-mappings F : X � Y ! L suchthat(0�) F(x;y) � EX(x;x) ^ EY (y;y) 8y 2 Y;8x 2 X;(1�) supx A(x) � �EX(x;x) 7�! F(x;y)� � B(y) 8y 2 Y ;(2�) F(x;y) � �EY (y;y) 7�! EY (y;y0)� � F(x;y0) 8x 2 X;8y;y0 2 Y ;(3�) EX(x;x0) � �EX(x;x) 7�! F(x;y)� � F(x0;y) 8x;x0 2 X;y 2 Y ;(4�) F(x;y) � �EX(x;x) 7�! F(x;y0)� � EY (y;y0) 8x 2 X;8y;y0 2 Y ;In particular, when A = >X and B = >Y we write F : (X;EX) ! (Y;EY )instead of F : (X;EX>X) ! (Y;EY >Y ).Notice that conditions (0�) - (3�) say that F is a certain L-relation, while axiom(4�) speci�es that the L-relation F is a function.Remark 3.1. Since F(x;y) � EX(x;x), and a � b =) a = b � (b 7�! a) (bydivisibility of L), we haveF(x;y) � (EX(x;x) 7�! EX(x;x0))= EX(x;x) � (EX(x;x) 7�! F(x;y)) � (EX(x;x) 7�! EX(x;x0))= EX(x;x0) � (EX(x;x) 7�! F(x;y)):Therefore axiom (3�) can be given in the following equivalent form:(30�) F(x;y) � �EX(x;x) 7�! EX(x;x0)� � F(x0;y).Remark 3.2. Applying (4�) it is easy to establish thatF(x;y1) � F(x;y2) � F(x;y1) � �EX(x;x) 7�! F(x;y2)�� EY (y1;y2)� EY (y1;y1) 7�! EY (y1;y2):Remark 3.3. Let F : (X;EX) ! (Y;EY ) be a fuzzy function, X0 � X Y 0 � Y ,and let the L-equalities EX0 and EY 0 on X0 and Y 0 be de�ned as the restrictionsof the equalities EX and EY respectively. Then de�ning a mapping F 0 : X0 �Y 0 ! L by the equality F 0(x;y) = F(x;y) 8x 2 X0;8y 2 Y 0 a fuzzy functionF 0 : (X0;EX0) ! (Y 0;EY 0) is obtained. We refer to it as the restriction of F tothe subspaces (X0;EX0) and (Y 0;EY 0).Given two fuzzy functions F : (X;EX;A) ! (Y;EY ;B) and G : (Y;EY ;B) !(Z;EZ;C) we de�ne their composition G � F : (X;EX;A) ! (Z;EZ;C) by the



118 Ulrich H�ohle, Hans-E. Porst, Alexander P. �Sostakformula (G � F)(x;z) = _y2Y �F(x;y) � �EY (y;y) 7�! G(y;z)��:Since, by divisibility of L, F(x;y) = EY (y;y) � �EY (y;y) 7�! F(x;y)� andG(y;z) = EY (y;y) � �EY (y;y) 7�! G(y;z)�, the composition can be de�nedalso by the formula(G � F)(x;z) = _y2Y ��EY (y;y) 7�! F(x;y)� � G(y;z)�:Proposition 3.4. G � F : (X;EX;A) ! (Z;EZ;C) is indeed a fuzzy function.Proof. The proof of the validity of (0�) is straightforward.(1�): Taking into account divisibility of L, strictness of A and axiom (1�)for F we get:supx �A(x) � �EX(x;x) 7�! (G � F)(x;z)��= supx�EX(x;x) 7�! A(x)� � (G � F)(x;z)= Wx;y�EX(x;x) 7�! A(x)� � F(x;y) � �EY (y;y) 7�! G(y;z)�� Wy2Y B(y) � �EY (y;y) 7�! G(y;z)�� C(z):(2�): By axiom (2�) for G we haveEZ(z;z) 7�! EZ(z;z0) � G(y;z) 7�! G(y;z0) 8y 2 Y;8z;z0 2 Z:Then for �xed x 2 X, y 2 Y and z;z0 2 Z we haveF(x;y) � �EY (y;y) 7�! G(y;z)� � �EZ(z;z) 7�! EZ(z;z0)�� F(x;y) � �EY (y;y) 7�! G(y;z)� � �G(y;z) 7�! G(y;z0)�� F(x;y) � �EY (y;y) 7�! G(y;z0)�:Now taking suprema by y 2 Y on the both sides of the inequality we get:(G � F)(x;z) � �EZ(z;z) 7�! EZ(z;z0)� � (G � F)(x;z0):(3�) (We prove this axiom in the form (30�)): Applying (30�) for F we have(G � F)(x;z) � �EX(x;x) 7�! EX(x;x0)�= Wy F(x;y) � �EY (y;y) 7�! G(y;z)� � �EX(x;x) 7�! EX(x;x0)�� Wy F(x0;y) � �EY (y;y) 7�! G(y;z)�= (G � F)(x0;z)(4�): We have to show that for all x 2 X, z;z0 2 Z(G � F)(x;z) � �EX(x;x) 7�! (G � F)(x;z0)� � EZ(z;z0):To establish this inequality we have to show that for any y;y0 2 Y it holds:�F(x;y) � �EY (y;y) 7�! G(y;z)����EX(x;x) 7�! �F(x;y0) � �EY (y0;y0) 7�! G(y0;z0)���� EZ(z;z0):



Fuzzy functions: a fuzzy extension of the category SET and some related categories 119By divisibility of L, axiom (4�) for F and G and axiom (3�) for G, we have:�F(x;y) � �EY (y;y) 7�! G(y;z)����EX(x;x) 7�! �F(x;y0) � �EY (y0;y0) 7�! G(y0;z0)���= F(x;y) � �E(y;y) 7�! G(y;z)���E(x;x) 7�! E(x;x)��E(x;x) 7�! F(x;y0)� � �E(y0;y0) 7�! G(y0;z0)��= �F(x;y) � �EY (y;y) 7�! G(y;z)����E(x;x) 7�! F(x;y0)� � �E(y0;y0) 7�! G(y0;z0)�� F(x;y) � �EY (y;y) 7�! G(y;z)���F(x;y) 7�! EY (y;y0)� � �EY (y0;y0) 7�! G(y0;z0)�� EY (y;y0) � �E(y;y) 7�! G(y;z)� � �E(y0;y0) 7�! G(y0;z0)�� G(y0;z) � �E(y0;y0) 7�! G(y0;z0)�� E(z;z0):By a direct veri�cation it is easy to show that the operation of composition is as-sociative: given fuzzy functions F : (X;EX;A) ! (Y;EY ;B), G : (Y;EY ;B) !(Z;EZ;C), and H : (Z;EZ;C) ! (T;ET ;D) it holds (H�G)�F = H�(G�F) :(X;EX;A) ! (T;ET ;D). Further, the identity morphism is de�ned by the cor-responding L-valued equality: EX : (X;EX;A) ! (X;EX;A). It is easy toverify that it satis�es the conditions (0�) - (4�) above and that F � EX = EXand EY � F = EY for each fuzzy function F : (X;EX;A) ! (Y;EY ;B). ThusL � FSET(L) is indeed a category. �Remark 3.5. In case when the equalities EX and EY on X and Y respectively,are global, the condition (0�) becomes redundant and the conditions (1�) - (4�)can be reformulated in the following simpler way:(1�) supx A(x) � F(x;y) � B(y) 8y 2 Y ;(2�) F(x;y) � EY (y;y0) � F(x;y0) 8x 2 X;8y;y0 2 Y ;(3�) EX(x;x0) � F(x;y) � F(x0;y) 8x;x0 2 X;8y 2 Y ;(4�) F(x;y) � F(x;y0) � EY (y;y0) 8x 2 X;8y;y0 2 Y .3.2. Fuzzy category L�FSET(L). Given a fuzzy function F : (X;EX;A) !(Y;EY ;B) let �(F) = infx supy F(x;y):Thus we de�ne an L-subclass � of the class of all morphisms of L�FSET(L).In case �(F) � � we refer to F as a fuzzy �-function. If F : (X;EX;A) !(Y;EY ;B) and G : (Y;EY ;B) ! (Z;EZ;C) are fuzzy functions, then �(G�F) ��(G) � �(F). Indeed, let x 2 X and y 2 Y be �xed. Thensupz F(x;y) � �EY (y;y) 7�! G(y;z)� � F(x;y) � supz G(y;z) � F(x;y) � �(G);and therefore for a �xed x 2 Xsupy supz F(x;y) � �EY (y;y) 7�! G(y;z)� � supy F(x;y) � �(G) � �(F) � �(G):Since x 2 X is arbitrary, we get �(G � F) � �(G) � �(F).



120 Ulrich H�ohle, Hans-E. Porst, Alexander P. �SostakFurther, given an L-valued set (X;E) let!(X;E) := �(E) = infx E(x;x):Thus a fuzzy category L �FSET(L) = (L � FSET(L);!;�) is obtained.Remark 3.6. If F 0 : (X0;E0X) ! (Y;EY ) is the restriction of F : (X;EX) !(Y;EY ) (see Remark 3.3 above) and �(F) � �, then �(F 0) � �. However,generally the restriction F 0 : (X0;EX0) ! (Y 0;EY 0) of F : (X;EX) ! (Y;EY )may fail to satisfy the condition �(F 0) � �.3.3. Some (fuzzy) subcategories of the fuzzy category L � FSET(L).For a �xed � let L�F�SET(L) consist of all objects of L�FSET(L) and itsfuzzy �-morphisms. In case � is idempotent, L �F�SET(L) is a usual (crisp)category. In particular, it is a crisp category for � = >.If L1;L2;L3 � L, then by L1�FSET(L2;L3) we denote the (fuzzy) subcate-gory of L�FSET(L), whose objects (X;E;A) satisfy the conditions A(X) � L1and E(X � X) � L2, and whose morphisms satisfy the condition F(X �Y ) � L3. By specifying the sets L1, L2 and L3 some known and new (fuzzy)categories related to L-sets can be characterized as (fuzzy) subcategories ofL1 � FSET(L2;L3)-type or of L1 � FSET 0(L2;L3)-type.4. Elementary properties of fuzzy functions. Special types offuzzy functions.4.1. Images and preimages of L-sets under fuzzy functions. Assumethat the GL-monoid (L;^;_;�) is equipped with an additional operation �which is distributive over arbitrary joins and meets and is dominated by �, i.e.(�1 � �1) � (�2 � �2) � (�1 � �1) � (�2 � �2). In particular, ^ can be taken as�. Another option: in case when (L;^;_;�) is an MV -algebra, the originalconjunction � can be taken as �. Given a fuzzy function F : (X;EX) ! (Y;EY )and L-subsets A : X ! L and B : Y ! L of X and Y respectively, wede�ne the fuzzy set F!(A) : Y ! L (the image of A under F) by the equalityF!(A)(y) = Wx F(x;y)�A(x) and the fuzzy set F (B) : X ! L (the preimageof B under F) by the equality F (B)(x) = Wy F(x;y) � B(y).Note that if A 2 LX is extensional, then F!(A) 2 LY is extensional (by(2�)) and if B 2 LY is extensional, then F (B) 2 LX is extensional (by(30�)).Proposition 4.1 (Basic properties of images and preimages of L-sets underfuzzy functions).(1) F!(Wi2I(Ai)) = Wi2I F!(Ai) 8fAi : i 2 Ig � LX;(2) F!(A1 VA2) � F!(A1)VF!(A2) 8A1;A2 2 LX;(3) F (Vi2I(Bi)) � Vi2I F (Bi) 8fBi : i 2 Ig � LY .(30) In case L is completely distributive(î2I F (Bi))5 � F (î2I(Bi)) � î2I F (Bi) 8fBi : i 2 Ig � LY ;in particular,



Fuzzy functions: a fuzzy extension of the category SET and some related categories 121(30̂ ) (Vi2I F (Bi))3 � F (Vi2I(Bi)) � Vi2I F (Bi) 8fBi : i 2 Ig �LX, in case � = ^ and(30̂^) Vi2I F (Bi) = F (Vi2I(Bi)) 8fBi : i 2 Ig � LY , in case � = � =^;(4) F (Wi2I(Bi)) = Wi2I F (Bi) 8fBi : i 2 Ig � LY ;(50�) In case L is completely distributive and � = �, F (F (B)) � B.Proof. (1): �Wi F!(Ai)�(y) = Wi Wx�F(x;y) � Ai(x)�= Wx Wi�F(x;y) � Ai(x)�= Wx(F(x;y) � (Wi Ai)(x))= F!(_iAi)(y):The validity of (2) follows from the monotonicity of F .To prove (3) notice that(Vi F (Bi))(x) = Vi�Wy F(x;y) � Bi(y)�� Wy(Vi(F(x;y) � Bi(y))� Wy(F(x;y) � (Vi Bi(y)))= F (Vi Bi)(x):Assume now that L is completely distributive. Recall that complete distribu-tivity of a lattice L means that the way-below relation � in L is approximative(i.e. � = Wf� 2 L : � � �g for every � 2 L) and every element � is asupremum of coprimes way-below � (see e.g. [3]). Let( î F (Bi))(x) = î _y F(x;y) � Bi(y) := �:Then 8� � �;8i 2 I;9yi 2 Y such that F(x;yi) � Bi(yi) � �:In particular, this means that F(x;yi) � � for every i 2 I. We �x some i0 2 Iand let yi0 := y0. Further, notice that by Remark 3.2�2 � F(x;yi) � F(x;y0) � E(yi;yi) 7�! E(yi;y0);and hence for every i 2 I[F(x;yi) � Bi(yi)] � �4 � �F(x;yi) � (EY (yi;yi) 7�! EY (yi;y0))���Bi(yi) � (EY (yi;yi) 7�! EY (yi;y0))�� F(x;y0) � Bi(x;y0):Therefore �5 � Vi�F(x;yi) � Bi(yi)� � �4� Vi�F(x;y0) � Bi(y0)�= F(x;y0) � Vi Bi(y0)� F (Vi Bi)(x)and, since this holds for any � � �, by complete distributivity we obtainF (Vi Bi)(x) � �5 and hence(î2I F (Bi))5 � F (î2I(Bi)):



122 Ulrich H�ohle, Hans-E. Porst, Alexander P. �SostakIn case � = ^ in the above proof it is su�cient to multiply by �2 instead of�4, and therefore the resulting inequality is(î2I F (Bi))3 � F (î2I(Bi)):Finally, in case � = � = ^ by idempotency we get the equalityî2I F (Bi) = F (î2I(Bi)):The proof of (4) is similar to the proof of (1) and is therefore omitted.To prove (5) assume that F (F (B))(y0) � �, for some y0 2 Y;� 2 L, thenfor each � � � there exist x0;y1 2 Y such that F(x0;y0)�F(x0;y1)�B(y1) � �.Therefore, by extensionality of B:B(y0) � (E(y1;y1) 7�! E(y1;y0)) � B(y1)� F(x0;y0) � F(x0;y1) � B(y1)� �;and hence, since L is completely distributive, it follows thatB(y0) � F (F (B))(y0)and thus B � F (F (B)). �4.2. Injectivity, surjectivity and bijectivity of fuzzy functions. A fuzzyfunction F : (X;EX;A) ! (Y;EY ;B) is called injective, if(inj) F(x;y) � (EY (y;y) 7�! F(x0;y)) � EX(x;x0) 8x;x0 2 X;8y 2 Y .Notice that injective fuzzy functions satisfy the following condition(inj#) F(x;y) � F(x0;y) � (EX(x;x) _ EX(x0;x0)) 7�! EX(x;x0) 8x;x0 2X;8y 2 Y .Indeed, F(x;y) � F(x0;y) � F(x;y) � (E(y;y) 7�! F(x0;y))� E(x;x0)� (E(x;x) 7�! E(x;x0)):Notice, that in case when EY is global, then (inj) just means that F(x;y) �F(x0;y) � EX(x;x0).A fuzzy function F : (X;EX;A) ! (Y;EY ;B) is called �-surjective if itsatis�es the following two conditions:(sur1�) infy supx F(x;y) � �(sur2) F!(A) � B � �.In case F is injective and �-surjective, it is called �-bijective.Remark 4.2. Notice that in case A = >X the second condition in the de�nitionof �-surjectivity (for any B 2 LY , in particular, for B = >Y ) follows from the�rst one. Moreover, in case A = >X, B = >Y and if > acts as a unit withrespect to �, the both conditions become equivalent.



Fuzzy functions: a fuzzy extension of the category SET and some related categories 123Remark 4.3. Let (X;EX);(Y;EY ) be L-valued sets and (X0;EX0), (Y 0;EY 0)be their subspaces. Obviously, the restriction F 0 : (X0;EX0) ! (Y 0;EY 0) of aninjection F : (X;EX) ! (Y;EY ) is an injection. The restriction F 0 : (X;EX) !(Y 0;EY 0) of an �-surjection F : (X;EX) ! (Y;EY ) is an �-surjection. Onthe other hand, generally the restriction F 0 : (X0;EX0) ! (Y 0;EY 0) of an �-surjection F : (X;EX) ! (Y;EY ) may fail to be an �-surjection.A fuzzy function F : (X;EX;A) ! (Y;EY ;B) de�nes a fuzzy relation F�1 :(Y;EY ;B) ! (X;EX;A) by setting F�1(y;x) = F(x;y) 8x 2 X;8y 2 Y .Proposition 4.4 (Basic properties of injections, �-surjections and �-biject-ions).(1) F�1 is a fuzzy function i� F is injective (actually F�1 satis�es (4�) i�F satis�es (inj))(2) F is �-bijective i� F�1 is �-bijective.(3) if L is completely distributive and F satis�es (inj#), then( î F!(Ai))5 � F!( î Ai) � î F!(Ai) 8fAi : i 2 Ig � LX:In particular,(3^) (Vi F!(Ai))3 � F!(Vi Ai) � Vi F!(Ai) if � = ^ and;(3^̂) F!(Vi Ai) = Vi F!(Ai) in case � = ^ = �;(4) If F is >-surjective, then F (F (B)) � B 8B 2 LY ; and hence,in particular, F (F (B)) = B in case � = � and L is completelydistributive.Proof. The validity of (1) and (2) is obvious.To show (3) �x y 2 Y and let (^iF!(Ai))(y) � �. Then for each coprime� � � and each 2 I one can �nd xi 2 X such that F(xi;y) � Ai(xi) � �and hence, in particular, F(xi;y) � �. We �x some i0 and denote xi0 := x0,Ai0;= A0. By (inj#) it is easy to conclude that EX(xi;xi) 7�! EX(x0;xi) � �2.Now, by extensionality of all Ai we get�5 � Vi�(F(xi;y) � Ai(xi)) � �4�� (F(x0;y) � A0(x0)) ^ �Vi6=i0�(F(xi;y) � �2� � (Ai(xi) � �2)�� (F(x0;y) � A0(x0))^�Vi6=i0 �F(xi;y) � (E(xi;xi) 7�! E(xi;x0))���Ai(xi) � (E(xi;xi) 7�! E(xi;x0))��� (F(x0;y) � A0(x0)) ^ �Vi6=i0�(F(x0;y)� � Ai(x0)�= Vi F(x0;y) � Ai(x0)= F(x0;y) � Vi Ai(x0)� F�^iAi�(y):Since this holds for any � � � and L is completely distributive we get( î F!(Ai)))5 � F!( î Ai) � î F!(Ai):



124 Ulrich H�ohle, Hans-E. Porst, Alexander P. �SostakTo show (4) let B(y0) � �. ThenF (F (B))(y0) = Wx�F(x;y0) � �F (B)�(x)= Wx Wy�F(x;y0) � F(x;y) � B(y)�� Wx�F(x;y0) � F(x;y0) � B(y0)�:Now, by >-surjectivity of F we complete the proof noticing thatF (F (B))(y0) � > �> � B(y0) � B(y0): �Proposition 4.5. Let F : X � Y ! L be a fuzzy function and �(F) � �.Then for each coprime � � � there exists Z � Y such that the restrictionG := F jX�Z: X � Z ! L is a �-surjection and �(G) � �.Proof. Given coprime � � �, let Z := fy j 9x 2 X such that (x;y) � �g, andlet G := F : X � Z ! L be the restriction of F to X � Z.To show that �(G) � � assume that, contrary, infx supy2Z F(x;y) = �(G) 6��. Then there would exist x0 2 X such that F(x;y) 6� � for each y 2 Z.On the other hand, from �(F) � � � �, it follows that for each x 2 X, inparticular, for x0 there exists y0 2 Y such that F(x0;y0) � �. Besides, byde�nition of Z it is clear that y0 2 Z. The obtained contradiction implies that�(G) � �.To show that G is �-surjective, assume that infy2Z supx2X G(x;y) 6� �.It follows from here that there exists y0 2 Z such that supx2X G(x;y0) =supx2X F(x;y0) 6� �. However, this contradicts the de�nition of Z. Thus the�rst condition of the de�nition of �-surjectivity holds. To conclude the proofit is su�cient to apply Remark 4.2. �Problem 4.6. Is it true (at least in the case of a completely distributive latticeL), that given a fuzzy function F : X � Y ! L where �(F) � � there existsZ � Y such that the restriction G := F jX�Z: X � Z ! L is an �-surjectionand �(G) � �?5. Constructions in the fuzzy category L � FSET(L)5.1. Products. Let L�FSET}(L) be the subcategory of L�FSET(L) havingthe same potential objects as L�FSET(L) and only such potential morphismsF : X � Y ! L from L � FSET(L) which satisfy the following additionalcondition (a certain counterpart of the axiom of strictness and the weaken formof the axiom of preservation of equalities; see e.g. [6]):(}) F(x;y) 6= 0 =) E(x;x) = E(y;y).Let Y = f(Yi;Ei;Bi) : i 2 Ig be a family of L-valued sets, Y0 = f(yi)i2I 2�iYi j Ei(yi;yi) = Ej(yj;yj)8i;j 2 Ig, let B0 be the restriction of B = Qi2I Bito Y0, and let E(y;y0) = Vi Ei(yi;y0i)8y = (yi);y0 = (y0i) 2 Y . Further, let�i : Y0 ! Yi be the restriction of the projection pi : Qi Yi ! Yi to Y0.The pair (Y;E) thus de�ned is the product of the family Y in the category



Fuzzy functions: a fuzzy extension of the category SET and some related categories 125L � FSET}(L). Indeed let Fi : (X;EX;A) ! (Yi;EYi;Bi), i 2 I, be a fam-ily of fuzzy functions in L � FSET}(L) and let F := �iFi : (X;EX;A) !(Y0;EY ;B), be de�ned by F(x;y) = Vi Fi(x;yi). Then F is a fuzzy function.Indeed, the validity of (0�), (1�), (3�) and (4�) is easy to verify directly apply-ing the corresponding axiom for all Fi, while the validity of (2�) is guaranteedby the condition (}) for all Fi; i 2 I. Besides, it is clear that Fi = �i � F andthat �(F) = Vi �(Fi). Thus, (Y0;EY ;B) is indeed the product of the family(Yi;EYi;Bi) in L � FSET}(L). Notice, that the condition } obviously holdsfor the subcategory L � FSET 0(L) of L � FSET(L). Moreover, if all (Yi;Ei)are taken from L � FSET 0(L), then Y0 = Qi Yi.5.2. Coproducts. Let X = f(Xi;Ai;Ei) : i 2 Ig be a family of L-valued sets,let X0 = SXi be the disjoint sum of sets Xi and let A0 2 LX be de�ned byA0(x) = Ai(x) whenever x 2 Xi. Further, let qi : Xi ! X0 be the inclusionmap. We introduce the L-equality on X0 by setting E(x;x0) = Ei(x;x0) if(x;x0) 2 Xi � Xi for some i 2 I and E(x;x0) = 0 otherwise (cf [6]). Then(X0;A0;E) is the coproduct of X in L � FSET(L) (and hence also in L �FSET}(L)).Indeed, let Fi : (Xi;Ai;Ei) ! (Y;B;EY ), i 2 I, be a family of fuzzy functionsin L � FSET(L) and let F := �iFi : �(Xi;Ai;Ei) ! Y;B;EY ) be de�ned byF(x;y) = Fi(xi;y) whenever x = xi 2 Xi. Then the direct veri�cation showsthat F is a fuzzy function, Fi = F � qi and �(F) = ^i�(Fi).Theorem 5.1 (Factorization of a family of �-morphisms). LetFi : (X;E;A) ! (Yi;Ei;Bi)be a family of fuzzy �-functions in L � FSET}(L). Then for every � � �there exists a fuzzy �-surjective �-function G : (X;E;A) ! (Z;EZ;C) and afamily of usual functions �i : (Z;C;EZ) ! (Yi;Bi;Ei) separating points suchthat Fi = G � �i for every i 2 I.Proof. Indeed, let (Y;EY ) = Qi2I(Yi;Ei) be the product in L � FSET}(L)and let F = 4i2IFi : X � Qi2I Yi ! L. Further, given � � �, let Z � Y andG : X�Z ! L have the same meaning as in Proposition 4.1 and let C := G(A).Thus, by Proposition 4.1 we conclude that G : (X;A;EX) ! (Z;C;EZ) is a �-surjective fuzzy function and �(G) � �. To complete the proof it is su�cient tonotice that the mappings �i : Z ! Yi de�ned as the restrictions of projectionspi : Y ! Yi separate points of Z and that Fi = �i � G. �6. Fuzzy categories related to algebra and topology with fuzzyfunctions as morphisms.On the basis of L�FSET(L) some fuzzy categories related to topology andalgebra can be naturally de�ned. Here are three examples:De�nition 6.1 (Fuzzy category FTOP(L)). Let (X;EX) be an L-valued setand let �X � LX be the (Chang-Goguen) L-topology on X, [2], [4], [5]; see also[9]. A fuzzy function F : (X;EX;�X) ! (Y;EY ;�Y ) is called continuous if



126 Ulrich H�ohle, Hans-E. Porst, Alexander P. �SostakF(V ) 2 �X for all V 2 �Y . L-topological spaces and continuous fuzzy mappingsbetween them form the fuzzy category FTOP(L).De�nition 6.2 (Fuzzy category FFTOP(L)). Let (X;EX) be an L-valued setand let TX : LX ! L be the L-fuzzy topology on X, [16], [9]. A fuzzy functionF : (X;EX;TX) ! (Y;EY ;TY ) is called continuous if TX(F(V )) � TY (V ) forall V 2 LY . L-fuzzy topological spaces and continuous fuzzy mappings betweenthem form the fuzzy category FFTOP(L).De�nition 6.3 (A fuzzy category L �FGr(L)). Let X be a group and EX bean L-valued equality on X such that EX(x �y;x0 �y0) � EX(x;x0)�EX(y;y0) forall x;x0;y;y0 2 X. Further, let GX : X ! L be an (extensional) L-subgroupof X (see e.g. [10], [13]). A fuzzy function F : (X;EX;GX) ! (Y;EY ;GY )is called a fuzzy homomorphism if F(x � x0;y � y0) � F(x;y) � F(x0;y0) for allx;x0 2 X, y;y0 2 Y . L-subgroups of groups endowed with L-valued equalities,and fuzzy homomorphisms between them form a fuzzy category L � FGr(L).These and some other fuzzy categories with fuzzy functions in the role offuzzy morphisms will be studied elsewhere.Acknowledgements. The authors are thankful to Tomasz Kubiak (Universityof Poznan, Poznan, Poland) for reading the manuscript carefully and makingvaluable comments. References[1] G. Birkho�, Lattice Theory, 3rd ed., AMS Providence, RI, 1967.[2] C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190.[3] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, \A Com-pendium of Continuous Lattices". Springer Verlag, Berlin, Heidelberg, New York, 1980.[4] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl., 18 (1967) 145-174.[5] J.A. Goguen, The fuzzy Tychono� theorem, J. Math. Anal. Appl., 43 (1973) 737-742.[6] U.H�ohle, Commutative, residuated l-monoids, In: Non-classical Logics and Their Ap-plications to Fuzzy Subsets: A Handbook of the Mathematical Foundations of FuzzySubsets, E.P. Klement and U. H�ohle eds., Kluwer Acad. Publ., 1994, 53-106.[7] U.H�ohle, M-valued sets and sheaves over integral commutative cl-monoids, In: Applica-tions of Category Theory to Fuzzy Sets., S.E. Rodabaugh, E.P. Klement and U. H�ohleeds., Kluwer Acad. Publ., 1992, pp. 33-72.[8] U.H�ohle, L. Stout, Foundations of fuzzy sets, Fuzzy Sets and Syst., 40 (1991), 257-296.[9] U. H�ohle, A. �Sostak, Axiomatics of �xed-basis fuzzy topologies, In: Mathematics of FuzzySets: Logic, Topology and Measure Theory , U. H�ohle, S.E. Rodabaugh eds. - HandbookSeries, vol.3. Kluwer Academic Publisher, Dordrecht, Boston. -1999. pp. 123 - 273.[10] C.V. Negoita, D.A. Ralescu, \Application of Fuzzy Sets to System Analysis", John Wiley& Sons, New York, 1975.[11] A. Pultr, Fuzzy mappings and fuzzy sets, Comm. Math. Univ. Carolinae, 17 (1976),441-459.[12] A. Pultr, Closed categories of fuzzy sets, Vortr�age zur Automaten un Algorithmentheorie16 (1976), 60-68.[13] A. Rosenfeld, Fuzzy groups, J.Math.Anal.Appl., 35 (1971), 512-517.[14] A. �Sostak, On a concept of a fuzzy category, In: 14th Linz Seminar on Fuzzy Set Theory:Non-classical Logics and Applications. Linz, Austria, 1992, pp. 62-66.



Fuzzy functions: a fuzzy extension of the category SET and some related categories 127[15] A. �Sostak, Fuzzy categories versus categories of fuzzily structured sets: Elements of thetheory of fuzzy categories, In: Mathematik-Arbeitspapiere, Universit�at Bremen, vol 48(1997), pp. 407-437.[16] A. �Sostak, On a fuzzy topological structure, Suppl. Rend. Circ. Matem. Palermo, Ser II,,11 (1985), 89-103.[17] A. �Sostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Math.Surveys, 44: 6 (1989), 125-186.[18] A. �Sostak, Towards the concept of a fuzzy category, Acta Univ. Latviensis (ser. Math.),562, (1991), 85-94. Received March 2000
Ulrich H�ohleBergische Universit�atD-42097, WuppertalGermanyE-mail address: Ulrich.Hoehle@math.uni-wuppertal.de
Hans-E. PorstUniversity of BremenD-28334, BremenGermanyE-mail address: porst@math.uni-bremen.de
Alexander P. �SostakUniversity of LatviaLV-1586, RigaLatviaE-mail address: sostaks@com.latnet.lv