Applied General Topology Universidad Politecnica de Valencia olume No @ V pp Fuzzy functions a fuzzy extension of the category SET and some related categories Ulrich Hohle HansE Porst Alexander P Sostak Abstract In research works where fuzzy sets are used mostly certain usual functions are taken as morphisms On the other hand the aim of this pap er is to fuzzify the concept of a function itself Namely a certain class of Lrelations F X Y ! L is distinguished which could b e considered as fuzzy functions from an Lvalued set X E to an Lvalued set Y E We study X Y basic prop erties of these functions consider some prop erties of the corresp onding category of Lvalued sets and fuzzy functions as well as briey describ e some categories related to algebra and top ology with fuzzy functions in the role of morphisms AMS Classication A A A Keywords Lrelation Lfuzzy function Fuzzy category Fuzzy top ology Fuzzy group Introduction In research works where fuzzy sets are involved in particular in the theory of fuzzy top ological spaces fuzzy algebra fuzzy measure theory etc mostly certain usual functions are taken as morphisms they can b e certain mappings b etween the corresp onding sets or b etween the fuzzy p owersets of these sets etc On the other hand there are only few pap ers where attempts to fuzzify the concept of a function itself are undertaken see eg etc The aim of our work is also to present a p ossible approach to this problem Namely a certain class of Lrelations ie mappings f X Y L is distinguished which seem reasonable to b e viewed as Lfuzzy functions from a set X to a set Y We dene composition of fuzzy functions study images and preimages of Lsets under fuzzy functions intro duce prop erties of injectivity and surjectivity for them describ e products and coproducts in the corresp onding category etc In the last part of the pap er we dene some categories related to top ology and algebra where fuzzy functions play the role of morphisms Ulrich Hohle HansE Porst Alexander P Sostak In conclusion we would like to mention the following two p eculiarities of our approach First the appropriate context for our work is formed not by usual sets or by their Lsubsets ie mappings f X L but rather by Lvalued sets ie sets endowed with an Lvalued equality E X X L see eg and their Lsubsets And second in the result we obtain not a usual category but the so called a fuzzy category a concept intro duced and studied in Prerequisites Let L L b e an innitely distributive GLmonoid cf eg ie a commutative integral divisible clmonoid cf It is well known that every GLmonoid is residuated ie there exists a further binary op eration implication such that L n n We set and further by induction Let and denote resp ectively the top and the b ottom elements of L Following UHohle cf eg by an Lvalued set we call a pair X E where X is a set and E is an Lvalued equality ie a mapping E X X L such that eq E x y E x x E y y x y X eq E x y E y x x y X eq E x y E y y E y z E x z x y z X An Lvalued set X E is called separated if W eq E x x E y y E x y x y x y X An Lvalued equality E is called global if eq E x x x X Further recall that an Lset or more precisely an Lsubset of a set X is just a mapping A X L In case X E is an Lvalued set its L subset A is called strict if Ax E x x x X A is called extensional if X sup Ax E x x E x x Ax x X x By L SET L we denote the category whose ob jects are triples X E A where X E is an Lvalued set and A is its strict extensional Lsubset and morphisms from X E A to Y E B are mappings f X Y which X Y preserve equalities ie E x x E f x f x and resp ect Lsubsets X Y ie A B f Let L SET L stand for the full sub category of the category L SET L determined by global separated Lvalued sets To recall the concept of an Lfuzzy category consider an ordinary classical category C and let ObC L and Mor C L b e Lfuzzy sub classes of its ob jects and morphisms resp ectively Now an Lfuzzy category can b e dened as a triple C satisfying the following axioms cf also in case f X Y X Y ObC and f Mor X Y g f f g whenever the comp osition g f is dened Fuzzy functions a fuzzy extension of the category SET and some related categories e X where e X X is the identity morphism X X Our aim is starting from the category L SET L to dene a fuzzy category L F SET L having the same class of ob jects as L SET L but an essentially wider class of p otential morphisms Fuzzy category L F SET L Category L FSET L We start with dening a usual ie crisp cat egory L FSET L Namely let L FSET L denote the category having the same ob jects as L SET L and whose morphisms called potential fuzzy functions from X E A to Y E B are Lmappings F X Y L such X Y that F x y E x x E y y y Y x X X Y sup Ax E x x F x y B y y Y X x F x y E y y E y y F x y x X y y Y Y Y E x x E x x F x y F x y x x X y Y X X F x y E x x F x y E y y x X y y Y X Y In particular when A and B we write F X E Y E X Y X Y instead of F X E Y E X X Y Y Notice that conditions say that F is a certain Lrelation while axiom sp ecies that the Lrelation F is a function Remark Since F x y E x x and a b a b b a by X divisibility of L we have F x y E x x E x x X X E x x E x x F x y E x x E x x X X X X E x x E x x F x y X X Therefore axiom can b e given in the following equivalent form F x y E x x E x x F x y X X Remark Applying it is easy to establish that F x y F x y F x y E x x F x y X E y y Y E y y E y y Y Y Remark Let F X E Y E b e a fuzzy function X X Y Y X Y 0 0 on X and E and Y b e dened as the restrictions and let the Lequalities E Y X of the equalities E and E resp ectively Then dening a mapping F X X Y Y L by the equality F x y F x y x X y Y a fuzzy function 0 0 F Y is obtained We refer to it as the restriction of F to X E E X Y 0 0 the subspaces X E and Y E X Y Given two fuzzy functions F X E A Y E B and G Y E B X Y Y Z E C we dene their composition G F X E A Z E C by the Z X Z Ulrich Hohle HansE Porst Alexander P Sostak formula G F x z F x y E y y Gy z Y y Y Since by divisibility of L F x y E y y E y y F x y and Y Y Gy z E y y E y y Gy z the comp osition can b e dened Y Y also by the formula G F x z E y y F x y Gy z Y y Y Prop osition G F X E A Z E C is indeed a fuzzy function X Z Proof The pro of of the validity of is straightforward Taking into account divisibility of L strictness of A and axiom for F we get sup Ax E x x G F x z X x sup E x x Ax G F x z X x W E x x Ax F x y E y y Gy z X Y xy W B y E y y Gy z Y y Y C z By axiom for G we have E z z E z z Gy z Gy z y Y z z Z Z Z Then for xed x X y Y and z z Z we have F x y E y y Gy z E z z E z z Y Z Z F x y E y y Gy z Gy z Gy z Y F x y E y y Gy z Y Now taking suprema by y Y on the b oth sides of the inequality we get G F x z E z z E z z G F x z Z Z We prove this axiom in the form Applying for F we have G F x z E x x E x x X X W F x y E y y Gy z E x x E x x Y X X y W F x y E y y Gy z Y y G F x z We have to show that for all x X z z Z G F x z E x x G F x z E z z X Z To establish this inequality we have to show that for any y y Y it holds F x y E y y Gy z Y E x x F x y E y y Gy z X Y E z z Z Fuzzy functions a fuzzy extension of the category SET and some related categories By divisibility of L axiom for F and G and axiom for G we have F x y E y y Gy z Y E x x F x y E y y Gy z X Y F x y E y y Gy z E x x E x x E x x F x y E y y Gy z F x y E y y Gy z Y E x x F x y E y y Gy z F x y E y y Gy z Y F x y E y y E y y Gy z Y Y E y y E y y Gy z E y y Gy z Y Gy z E y y Gy z E z z By a direct verication it is easy to show that the op eration of comp osition is as so ciative given fuzzy functions F X E A Y E B G Y E B X Y Y Z E C and H Z E C T E D it holds H G F H G F Z Z T X E A T E D Further the identity morphism is dened by the cor X T resp onding Lvalued equality E X E A X E A It is easy to X X X verify that it satises the conditions ab ove and that F E E X X and E F E for each fuzzy function F X E A Y E B Thus Y Y X Y L FSET L is indeed a category Remark In case when the equalities E and E on X and Y resp ectively X Y are global the condition b ecomes redundant and the conditions can b e reformulated in the following simpler way sup Ax F x y B y y Y x F x y E y y F x y x X y y Y Y E x x F x y F x y x x X y Y X F x y F x y E y y x X y y Y Y Fuzzy category L F SET L Given a fuzzy function F X E A X Y E B let Y F inf sup F x y x y Thus we dene an Lsub class of the class of all morphisms of L FSET L In case F we refer to F as a fuzzy function If F X E A X Y E B and G Y E B Z E C are fuzzy functions then G F Y Y Z G F Indeed let x X and y Y b e xed Then sup F x y E y y Gy z F x y sup Gy z F x y G Y z z and therefore for a xed x X sup sup F x y E y y Gy z sup F x y G F G Y y z y Since x X is arbitrary we get G F G F Ulrich Hohle HansE Porst Alexander P Sostak Further given an Lvalued set X E let X E E inf E x x x Thus a fuzzy category L F SET L L F SET L is obtained Remark If F X E Y E is the restriction of F X E Y X X Y E see Remark ab ove and F then F .