On fuzzification of algebraic and topological structures

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1 On fuzzfcaton of algebrac and topologcal structures Sergejs Solovjovs Department of Mathematcs, Unversty of Latva 19 Rana Blvd., Rga LV-1586, Latva Abstract Gven a concrete category (A, U) over X, the paper presents a fuzzfcaton scheme of algebrac and topologcal structures over the category X(A) of A-valued X-objects wth a fxed bass A. Keywords: L-set, topologcal category, fntary quasvarety, functor-structured category. 1 Introducton The noton of a fuzzy set ntroduced by Zadeh n [18] and generalzed by Goguen n [5] gave rse to fuzzfcaton of dfferent mathematcal structures. In partcular, [3, 7, 9, 13] consder fuzzfed versons of topologcal spaces and groups. All aforesad approaches are fxed-based and use mplctly Goguen category Set(L) of L-sets for a sutable partally ordered set L (see, e.g., [5, 6]). A varable-bass approach over the category of sem-quantales SQuant (whch s good enough snce the categores of fuzzfed structures are topologcal over ther ground categores) s consdered n [10, 11]. Gven a concrete category (A, U) over X, we ntroduced n [15] the category X(A) of A-valued objects of the category X as a varable-bass generalzaton of Goguen category Set(L). Our next step wll be to fnd a good fuzzfcaton machnery over the category X(A). Followng the hstorcal move we start wth the fxed-bass approach and therefore consder a subcategory X(A) (studed thoroughly n [17]) of the category X(A) for a fxed A-object A. In ths paper (a somewhat short verson of our manuscrpt [14]) we present a fuzzfcaton machnery over the category X(A) usng the categores of generalzed algebrac and topologcal structures Alg(T ) and Spa(T ) (see, e.g., [1]). As a result the categores A τ -Alg(T ) and A P - Spa(T ) of A-valued algebrac and topologcal structures arse. Our man concern les n studyng basc propertes of these categores. In partcular, we show a suffcent condton for both of them to be topologcal over ther ground categores. The necessary categorcal background can be found n [1, 2], however t s expected from the reader to be acquanted wth basc concepts of category theory. For convenence sake the frst two sectons recall basc facts about the categores X(A) and X(A). 2 The category X(A) Let (A, U) be a concrete category over X such that the followng condtons are fulflled: (2CAT) The category A s a 2-category (see, e.g., [2]). (ADJ) The functor U has a left adjont (choose an adjont stuaton (η, ɛ) : F U : A X). For shortness sake we wrte f f exsts a 2-cell A τ B. g τ = g nstead of there Let F be the class of all structured arrows wth doman n X,.e., of all trples (X,, A) wth X Ob(X), A Ob(A) and X UA Mor(X). We ntroduce the followng relaton on F. Defnton 2.1. Let X β (ADJ) one has two A-morphsms F X ɛ A F ( ). Defne β ff UA be elements of F. By τ = β. β A wth ( ) = We wll use the followng propertes of relaton. Lemma 2.2. Let X f Y β UA Uϕ UB be X- morphsms. If β, then f β f and Uϕ Uϕ β.

2 Proof. f = ɛ A F ( f) = ɛ A F F f = F f = τ β F f = β f and Uϕ = ɛ B F (Uϕ ) = (ɛ B F Uϕ) F = (ϕ ɛ A ) F = ϕ = τ ϕ β = Uϕ β. Lemma 2.3. Let A ϕ τ = ψ mples Uϕ Uψ. ϕ B be A-morphsms. Then ψ Proof. Use the fact that Uϕ = ϕ ɛ A. Wth the help of Defnton 2.1 we ntroduce the category X(A). Defnton 2.4. X(A) s the concrete category over X A, the objects of whch (called A-valued X- objects) are elements of F. Morphsms (X,, A) (f,ϕ) (f, ϕ) (Y, β, B) are X A-morphsms (X, A) (Y, B) such that Uϕ β f. The underlyng functor (f, ϕ) to X A s gven by (X,, A) (Y, β, B) = (f, ϕ) (X, A) (Y, B). We llustrate Defnton 2.1 by two examples used later on n the paper (other examples can be found n [15]). Example 2.5. Let CSLat( ) be the category of complete lattces and jon-preservng maps; let Set be the category of sets and maps. One gets a construct (CSLat( ), U). Snce free objects are just powersets and the obvous 2-cell structure s nduced by the order structure of the lattces one gets the category Set(CSLat( )) of lattce-valued subsets of sets consdered n [16]. Gven two maps X β UA, t follows that β ff (x) β(x) for every x X. Example 2.6. Let SetRel be the category of sets (as objects) and relatons (as morphsms). The forgetful functor SetRel U Set s gven by U(X ρ Y ) = P(X) f ρ P(Y ) where P(X) s the power-set of X and f ρ (S) = {y Y there exsts x S such that xρy}. Snce free objects are just the same sets and the obvous 2-cell structure s nduced by set-theoretc ncluson of relatons one gets the category Set(SetRel). Gven two maps X β UY, t follows that β ff (x) β(x) for all x X. For the later use we recall the necessary and suffcent condtons for X(A) to be topologcal over X A. Notce that the property requres the exstence of ntal (resp. fnal) lfts. Accordngly we ntroduce two sets of requrements (one for each type of lfts). The frst set conssts of requrements (CLX), (MPX) and (FST) lsted below. (CLX) For every X A-object (X, A), (X(X, UA), ) s a complete lattce. (MPX) For every X-morphsm Y f X and every A- object A, the functon X(X, UA) f X(Y, UA) s meet-preservng. (FST) There exsts a functor A ( ) X op (X op s the dual category of X) such that: A = UA and B = UB; 1 UA ϕ Uϕ; Uϕ ϕ 1 UB ; for every A-morphsm B ϕ A and every X- object X, X(X, UA) ϕ X(X, UB) s an order-preservng map. The second set of requrements s n a sense a dualzed verson of the frst one. Start wth the followng defnton. Defnton 2.7. A category C s a complete quaslattce provded that the quotent category of C w.r.t. the congruence f g ff dom(f) = dom(g) and cod(f) = cod(g) s a complete lattce consdered as a category. The jons of the lattce are called quasjons. Notce that each complete quaslattce s a skeletal category,.e., somorphc objects are dentcal. Wth the aforesad defnton n mnd ntroduce the second set of requrements whch contans requrements (CQLA), (QJPA) and (FCR) lsted below. (CQLA) For every X A-object (X, A), A(F X, A) s a complete quaslattce. (QJPA) For every A-morphsm A ϕ B and every X- object X, the map A(F X, A) ϕ A(F X, B) s quasjon-preservng. (FCR) There exsts a functor X ( ) A op such that: X = F X and Y = F Y ; F f f τ = 1 F Y ; υ 1 F X = f F f; for every X-morphsm X f Y and every f A-object A, A(F X, A) A(F Y, A) s a quasjon-preservng map. The followng theorem s due to [15]. Theorem 2.8. Suppose (2CAT), (ADJ) hold. Equvalent are: () X(A) s topologcal over X A;

3 () (CLX), (MPX), (FST) hold; () (CQLA), (QJPA), (FCR) hold. Proof. For the lack of space we recall only that part of the proof whch wll be used later. () = (): Show the exstence of ntal lfts. Let S = ((X, A) (f, ϕ ) (X,, A ) ) be a -structured source. Straghtforward computatons show that = (ϕ f ) s the requred structure. () = (): Show the exstence of fnal lfts. Let S = ( (X,, A ) (f, ϕ ) (X, A)) be a -costructured snk. Straghtforward computatons show that = (ϕ f ) s the requred structure. As an mmedate consequence one gets that both categores Set(CSLat( )) and Set(SetRel) are topologcal. The requred functors are gven n the followng examples where one should recall that gven a set map X f Y one has two maps (the notaton s due to [12]): P(X) f P(Y ) : S {f(x) Y x S}; P(Y ) f P(X) : S {x X f(x) S}. Example 2.9. In the category Set(CSLat( )) the functor CSLat( ) ( ) Set op s defned by (A ϕ B) = UB ϕ UA wth ϕ (b) = ϕ ( b). functor Set ( ) CSLat( ) op s defned by (X Y ) = P(Y ) f P(X). The f Example In the category Set(SetRel) the ( ) functor SetRel Set op s defned by (X ρ Y ) = P(Y ) f P(X) wth f (S) = {x X {y Y xρy} S}. The functor Set ( ) SetRel op s defned by (X f Y ) = Y f X wth yf x ff f(x) = y. 3 The category X(A) Defnton 3.1. Fx an object A of the category A. The (nonfull) subcategory X(A) of the category X(A) s the concrete category over X, the objects of whch (called A-valued X-objects) are X(A)-objects (X,, A) for shortness sake denoted by (X, ). Morphsms (X, ) (Y, β) are X-morphsms X f Y f (f, 1 A ) such that (X,, A) (Y, β, A) s an X(A)- morphsm. The underlyng functor to X s gven by (X, ) f (Y, β) = X f Y. We llustrate Defnton 3.1 by the followng example. Example 3.2. Let A be a CSLat( )-object. The subcategory Set(A) of the category Set(CSLat( )) concdes wth the category of A-sets ntroduced by Goguen n [5]. From now on we consder an arbtrary (but fxed) subcategory X(A) of the category X(A). In the prevous secton we gave the necessary and suffcent condtons for the category X(A) to be topologcal over X A (Theorem 2.8). It was crucal to have the functors A ( ) X op and X ( ) A op. The stuaton s much smpler n case of the category X(A). Introduce the followng weak versons of requrements (CLX), (MPX). (CLXA) For every X-object X, (X(X, U A), ) s a complete lattce. (MPXA) For every X-morphsm Y f X, the map X(X, UA) f X(Y, U A) s meet-preservng. Theorem 3.3. Suppose (CLXA), (MPXA) hold. Equvalent are: () X(A) s topologcal over X; () (CLXA), (MPXA) hold. Proof. For the lack of space we recall only that part of the proof whch wll be used later. () = (): Show the exstence of ntal lfts. Let S = (X f (X, ) ) be a -structured source. Smlar to Theorem 2.8 one shows that = ( f ) s the desred structure. Theorems 2.8 and 3.3 mply the followng result. Corollary 3.4. Suppose (2CAT), (ADJ) hold. Equvalent are: () X(A) s topologcal over X A; () (FST) holds and every subcategory X(A) of the category X(A) s topologcal over X. By Corollary 3.4 every subcategory Set(A) of the category Set(CSLat( )) or Set(SetRel) s topologcal. From now on assume that the category X(A) (and thus by Corollary 3.4 every subcategory of the form X(A)) s topologcal. 4 Category A τ -Alg(T ) Ths secton consders a fuzzfcaton scheme for algebrac structures over the category X(A). In order not to restrct ourselves to a partcular class of algebras we

4 use the objects of the category Alg(T ) as a suffcent generalzaton of the noton of abstract algebra. For convenence sake we recall ts defnton from [1]. Defnton 4.1. Let X T X be a functor. Alg(T ) s the concrete category over X, the objects of whch (called T -algebras) are pars (X, x) wth X an X- object and T X x X an X-morphsm. Morphsms (X, x) f (X, x ) (called T -homomorphsms) are X- morphsms X f X such that x T f = f x. The underlyng functor to X s gven by (X, x) f (X, x ) = X f X. From now on we consder an arbtrary (but fxed) category of the form Alg(T ). The next defnton ntroduces a fuzzfcaton scheme of Alg(T ) over the category X(A). For the sake of generalty nstead of the whole category Alg(T ) we consder ts subcategory B. Defnton 4.2. Let B be a subcategory of the category Alg(T ). For a fxed Alg(T )-object (UA, τ), the category A τ -B s the concrete category over B, the objects of whch (called A τ -(T -algebras)) are trples (X, x, ) wth (X, x) a B-object and (X, ) an X(A)- object such that τ T x. Morphsms (X, x, ) f (Y, y, β) (called A τ -(T -homomorphsms)) are X-morphsms X f X wth (X, x) f (Y, y) a B-morphsm and (X, ) f (Y, β) an X(A)-morphsm. The underlyng functor to B s gven by (X, x, ) f (Y, y, β) = (X, x) f (Y, y). We show a suffcent condton for A τ -B to be topologcal over B. Introduce the followng requrement: τ T ( ) (OPR) The map X(X, U A) X(T X, U A) s order-preservng for every X-object X. Theorem 4.3. Suppose (OPR) holds. Then A τ -B s topologcal over B. Proof. Let S = ((X, x) f (X, x, ) ) be a - structured source. Defne = ( f ). By Theorem 3.3 t s enough to show that (X, x, ) s a A τ -B-object. Snce τ T ( f ) = τ T T f x T f = f x for I, t follows that τ T = τ T ( ( f )) τ T ( f ) ( f x) = ( f ) x = x. We are gong to consder a partcular category A τ -B and therefore ntroduce the followng requrement. (CFP) The category X has coproducts as well as fnte products (denoted by ((µ ), X ) and ( X, (π ) ) respectvely). Snce X has fnte products and X(A) s topologcal over X, the followng result holds. Proposton 4.4. The category X(A) has fnte products constructed as follows. Let ((X, )) be a fnte famly of X(A)-objects and let ( X, (π X ) ) be a product of the famly (X ) n X. Then the source (( X, ) π X (X, )) wth = ( π X ) s a product of the famly ((X, )). Consder an example for Defnton 4.2 (cf. [4]). Example 4.5. Suppose (CFP) holds. Let Ω = (n ) be a famly of natural numbers ndexed by a set I. For I let X ( )n X : X f Y X n f n Y n. ( ) n Let X T Ω X = X X and get the category Alg(T Ω ). Take the Alg(T Ω )-object (UA, τ Ω ) where [ 1 UA ] T UA τω n UA = T UA UA s defned by the commutatvty of the trangle (UA) n 1 UA n µ (UA) n UA [ n 1 UA ] for every I. One has the category A τω -Alg(T Ω ) of categorcal representatons of unversal algebras. We want to study basc propertes of Example 4.5. For our nvestgaton we need some new notons. Start by extendng the relaton from elements of F to arbtrary U-structured snks. Defnton 4.6. Let S = (X UA) and T = β (X UA) be U-structured snks. Then S T provded that β for I. Wth the help of Defnton 4.6 ntroduce a modfcaton of the noton of ep-snk. f Defnton 4.7. A snk S = (X Y ) n X s sad to be a -ep-snk provded that for every two elements Y β UA of F, S β S mples β. We contnue wth the followng notaton (recall requrement (FCR)): gven a snk S = (X n X let (F S S ) = (F f f ). f X) Defnton 4.8. Let X(A) be topologcal over X A f and let S = (X X) be a snk n X. Say that S s -compatble provded that (F S S ) = 1 F X.

5 Straghtforward computatons (omtted for the lack of space) show the followng proposton. Proposton 4.9. Suppose (CFP) holds and let X(A) be topologcal wth -compatble coproducts. If B s a subcategory of Alg(T Ω ), then the category A τω -B s topologcal over B. Recall from [1] that constructs that are concretely somorphc to mplcatonal subconstructs of Alg(Ω) (whch s smply Alg(T Ω ) wth X = Set) for some Ω are called fntary quasvaretes. In partcular, the constructs Vec, R-Mod, Ab, Grp, Mon, Sgr, Rng, Lat of vector spaces, left R-modules, abelan groups, groups, monods, semgroups, rngs and lattces are fntary quasvaretes. Corollary Let Set(A) be topologcal wth - compatble coproducts. Let A be an A-object and let B be a fntary quasvarety. Then A τω -B s topologcal over B. Corollary For the categores Set(CSLat( )) and Set(SetRel), each of the followng categores s topologcal: A τω -Vec, A τω -(R-Mod), A τω -Ab, A τω -Grp, A τω -Mon, A τω -Sgr, A τω -Rng, A τω -Lat. 5 Category A P -Spa(T ) Ths secton consders a fuzzfcaton scheme for topologcal structures over the category X(A). In order not to restrct ourselves to a partcular class of structures we use the objects of the functor-structured category Spa(T ) as a suffcent generalzaton of the noton of abstract topologcal structure. For convenence sake we recall ts defnton from [1]. Defnton 5.1. Let X T Set be a functor. Spa(T ) s the concrete category over X, the objects of whch (called T -spaces) are pars (X, ) wth T X. Morphsms (X, ) f (Y, β) (called T -maps) are X- morphsms X f Y such that (T (f)) () β. The underlyng functor to X s gven by (X, ) f (Y, β) = X f Y. From now on we consder an arbtrary (but fxed) category of the form Spa(T ). The next defnton ntroduces a fuzzfcaton scheme of Spa(T ) over the category X(A). For the sake of generalty nstead of the whole category Spa(T ) we consder ts subcategory B. Defnton 5.2. Let B be a subcategory of the category Spa(T ). For a fxed Spa(T )-object (UA, P ), the category A P -B s the concrete category over B, the objects of whch (called A P -(T -spaces)) are trples (X, S, ) where (X, S) s a B-object and (X, ) s an X(A)-object such that (X, S) (UA, P ) s a Spa(T )-morphsm. Morphsms (X, S, ) f (Y, Q, β) (called A P -(T -maps)) are X-morphsms X f Y wth (X, S) f (Y, Q) a B-morphsm and (X, ) f (Y, β) an X(A)-morphsm. The underlyng functor to B s gven by (X, S, ) f (Y, Q, β) = (X, S) f (Y, Q). As n the prevous secton we are gong to show a suffcent condton for A P -B to be topologcal over B. Introduce a new requrement (recall condton (CLXA)): (MC) For every object (X, S) of the category B, the hom-set Spa(T )((X, S), (UA, P )) s closed under the formaton of meets. Theorem 5.3. Suppose (MC) holds. Then A P -B s topologcal over B. Proof. Let S = ((X, S) f (X, S, ) ) be a - structured source. Defne = ( f ). It s enough to show that (X, S, ) s an A P -B-object and that follows from the fact that (X, S) f (UA, P ) s a Spa(T )-morphsm for I together wth (MC). Consder an example of the category Spa(T ). Example 5.4. Suppose X has fnte products. Let Ω = (n ) be a famly of natural numbers ndexed by a set I and let (X ) be a famly of X-objects. For I let X ( )n X : X f Y X n f n Y n and let X T Set = X ( )n hom(x, ) X Set where hom(x, ) s the covarant hom-functor. Let X TΩ T Set = X Set and get the category Spa(T Ω ). We are nterested n basc propertes of Example 5.4. Let P Ω (T Ω U)(A) = hom(x, (UA) n ) be closed under the formaton of meets n such a way that the followng condton s fulflled: (A) For arbtrary I and arbtrary subfamly h j (X (UA) n ) j J (π UA h j) j J X P Ω, the X-morphsm (UA) n defned by the commutatvty of the trangle X (π UA h j ) j J (UA) n πua UA (π UA h j ) j J for every projecton π UA, belongs to P Ω. One gets the category A PΩ -Spa(T Ω ). The next proposton shows a property of (A) (we omt straghtforward computatons agan).

6 Proposton 5.5. In case of the functor T Ω, (A) mples (MC). Consder an nterestng specal case of Example 5.4. Example 5.6. Let X(A) be Set(CSLat( )) and let X = { } for I. Then T (X) = hom({ }, Xn ) = X n. One can easly see that Spa(TΩ ) s concretely somorphc to the category Rel(Ω) wth objects all pars (X, (ρ ) ) where X s a set and ρ X n for I and morphsms all maps (X, (ρ ) ) f (Y, (σ ) ) where (f n ) (ρ ) σ for I. Let B be a subcategory of Spa(TΩ ) (e.g., f Ω = (2) consder the category Rel of sets and relatons, the category Prost of preordered sets or the category Pos of partally ordered sets) and let (UA, (ρ ) ) be a Spa(TΩ )-object such that ρ s closed under the formaton of meets for I,.e., a j ρ b j for j J mply ( a j )ρ ( b j ). By Proposton 5.5 A P Ω -B s topologcal over j J j J B. For Ω = (2) Example 5.6 gves the category A ρ -Pos of fuzzfed posets (topologcal over Pos). The objects of the category are trples (X,, ), where (X, ) s a poset, (X, ) s an A-set and, moreover, x y mples (x)ρ (y). The fuzzfcaton dffers from the concept of fuzzy partal order (see, e.g., [8]). At the end of the paper we brefly consder the case of functor-costructured categores of [1],.e., categores of the form (Spa(T )) op for functors X op T Set concrete over X = (X op ) op. In a natural way the category A P op-b arses (see Defnton 5.2). One can easly restate Theorem 5.3 for the case of the category A P op-b. Let Set Q Set be the contravarant powerset functor. By Example 22.7(2) n [1] the category Top of topologcal spaces s a full subcategory of (Spa(Q)) op. Ths motvates our next example. Example 5.7. Suppose X(A) s Set(CSLat( )). Let A be a CSLat( )-object and let (UA, P ) be a (Spa(Q)) op -object fulfllng the next condton: (B) Every V P has the followng propertes: () V = V ; () S V mples S V for every nonempty subset S A. One gets the category A P op-top. For nstance, let P = {A\ a a A}. Then (UA, P ) satsfes (B). The objects of the category A P op-top are trples (X, O(X), ) wth (X, O(X)) a topologcal space, (X, ) an A-set and ( ) ( P ) O(X). Proposton 5.8. A P op-top s topologcal over Top. Proof. Take a Top-object (X, O(X)) and consder a famly of (Spa(Q)) op -morphsms ((X, O(X)) (UA, P )). Defne = and show that (X, O(X)) (UA, P ) s a (Spa(Q)) op -morphsm (then use Theorem 5.3). If I =, then := A and for every V P, (V ) s ether X for V or s otherwse. Suppose I and take V P. Show that (V ) = (V ) (the latter set s an element of O(X)). If x (V ), then (x) V and therefore there exsts 0 I such that 0 (x) V by (B).(). Conversely, x (V ) gves 0 I wth 0 (x) V and therefore (x) V by (B).(). 6 Concluson In the paper we presented a fuzzfcaton machnery for algebrac and topologcal structures. It wll be the topc of our forthcomng studes to dsclose ts relatons to other fuzzfcaton machneres n the lterature. One can see, however, at once that the aforesad category A P op-top dffers from the category L-Top of L-topologcal spaces for a sem-quantale L (see, e.g., [10]) where objects are pars (X, τ) wth X a set and τ L X closed under and arbtrary. On the other hand, the objects of the category A τω -Grp concde wth fuzzy subgroups ntroduced by Rosenfeld n [13]. Thus the algebrac sde of our paper s more consstent wth the lterature. Acknowledgement Ths research was supported by the European Socal Fund. References [1] Jří Adámek, Horst Herrlch, and George E. Strecker, Abstract and Concrete Categores: The Joy of Cats, John Wley & Sons, [2] Francs Borceux, Handbook of Categorcal Algebra 1: Basc Category Theory, Cambrdge Unversty Press, [3] C. L. Chang, Fuzzy topologcal spaces, J. Math. Anal. Appl. 24 (1968) [4] Paul M. Cohn, Unversal Algebra, D. Redel Publ. Comp., [5] J. A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) [6] J. A. Goguen, Categores of V -sets, Bull. Am. Math. Soc. 75 (1969)

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