Shteinbuk, Veniamin

Šostak, Alexander

Institute of Mathematics AS CR, v. v. i.

KybernetikaVeniamin Shteinbuk; Alexander P. ŠostakOn endomorphism semigroups of a fuzzily structured setKybernetika, Vol. 28 (1992), No. Suppl, 54--57Persistent URL: http://dml.cz/dmlcz/124196Terms of use:© Institute of Information Theory and Automation AS CR, 1992Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czS U P P L E M E N T T O K Y B E R N E T I K A VOLUME 28 ( 1 9 9 2 ) , P A G E S 5 4 - 5 7 ON ENDOMORPHISM SEMIGROUPS OF A FUZZILY STRUCTURED SET VENIAMIN SHTEINBUK AND ALEXANDER SOSTAK Category FS(C) of £-fuzzily structured sets (fs-sets) (X, L,T) is introduced. FS(C) contains, for appropriately chosen category C of lattices, various categories of fuzzy topological spaces. The problem of definability of fs-sets by means of £-endomorphism semigroups is discussed. However the tool of usual endomorphism semigroups used successfully in topology appears to be completely inadequate for this purpose: there are essentially different "good" fs-sets with isomorphic endomorphism semigroups. This difficulty is overcome by using a richer semigroup SC(X,L,T) defined on the basis of the usual endomorphism semigroup Cc(X, L, r) . Let £ be a category whose objects are complete lattices with 0 and 1 and whose morphisms are mapping of some kind between the lattices. By an (£)-fuzzily s t ructured set (or an fs-set for short) we call a triple (X,L,T) where X is a set, L £ Ob (£) and T C Lx. Let FS(C) be the category, the objects of which are fs-sets and the morphisms are pairs (f,n) : (X\,L\,TX) —> (X2,L2,T2), where / £ Morse t (X\ ,X 2 ) ( i .e . / : X\ —> X2 is a mapping) , n £ Mor,c(L 2 ,Li) and /JO V o f £ TJ for each V £ T2 . Notice tha t (as it will be specified to some extent below) various categories of fuzzy topological spaces considered in [1], [2], [7] e.g. are in fact full subcategories of the categories FS(C) for appropriately chosen £ . Let FT(C) denote the complete subcategory of FS(C) whose objects are fs-sets (X, L,T) where T is an L-fuzzy topology on X [2] ( i .e . (1) 0, 1 £ T, (2) if U, V £ T, then U A V £ r, and (3) if <77 £ T for all 7 € T, then \J 1U-< £ r). For a lattice L € O b ( £ ) let FSL(C) (resp. FTL(C)) denote the complete subcategory of FS(C) (resp. of FT(C)) the objects of which are fs-sets (X, L,T) where L is the given lattice. Extending s tandard topological terminology to the situation under discussion, the morphisms of FS(C) will be called £-continuous mappings. For an fs-set (X,L,T) let Cc(X, L,T) denote the semigroup of all its endomorphisms ( = £-continuous 'mappings of (X,L,T) into itself) in the category FS(C). Two fs-sets are called £-homeomorphic if they are isomorphic as objects of FS(C). We emphasize that the relation of £-homeomorphism essentially depends on the choice of the category £ . Two fs-sets (X\, L{, TJ and (X2,L2,T2) are called quasihomeomorphic if there exists a pair (/,/«) such that / : X\ —> A'j and /( : L2 —> L\ are bijections and / l o l / o / e r , iff V £ T2. The main problem considered in the paper is to reveal the possibility of definability up to £-homeomorphism of an fs-set by means of its £-endomorphism semigroup. We shall restrict ourselves here to two specific categories C = C\ and £ = £ 2 introduced On Endomorphism Semigroups of a Fuzzily Structured Set 55 below. However, the tool of usual endomorphism semigroups which is successfully used in General Topology (see e.g. [4], [9]) appears to be completely inadequate for our purposes: there are many essentially different (in FS(C)) "good" fs-sets with equal endomorphism semigroups. We overcome these difficulties by using a richer semigroup Sc(X, L, T) introduced below instead of the semigroup Cc(X, L, T). By the Plotkin endomorphism semigroup (with respect to the category FS(C)) of an fs-set (X, L,T) we call the product Sc(X, L,T) S Cc(X, L,T) X L X equipped with operation "•" defined as follows ( / i , / " i ,U i ) • (h,l*2,U2) = ( / 2 o / , , /x, o(i2, U2 o / i ) . (A similar semigroup first appeared in [5] in connection with the theory of algebraic au tomata . ) In the sequel we write sometimes Sc(X) instead of Sc(X, L,T). Notice tha t apart from the binary operation "•" there are two additional structures on the semigroup Sc(X, L, T ) . The first one is the subset T of the lattice Lx and the second one is the partial order relation "X" introduced as follows: (f\,fi\,U\) < (f2, H2,U2) iff / i = h, H\ = /̂ 2 and U\ < U2 ( i .e . U\(x) < U2(x) for each x € X). According to these s tructures we consider the following three kinds of isomorphism for Plotkin semigroups. We say tha t Plotkin semigroups Sc(X\, L\,T\) and Sc(X2, L2, r2) are (1) isomorphic, if they are isomorphic in the category of semigroups; (2) T-isomorphic, if there exists an isomorphism a : Sc(X\, L\,T\) —> Sc(X2, L2,T2) such that a(Cc(X\) x Tl) = CC(X2) x T2; (3) ^-isomorphic, if there exists a T-isomorphism a : Sc(X\, L\,T\) — • Sc(X2, L2,T2) such tha t ( / , / / , U\) X (f,n,U2) \ffa(f,ii,U\) < a(f,li,U2). To formulate the main results we have first to specify the category C. Namely, let C\ and £ 2 be categories whose objects are complete lattices with 0 and 1, M o r ( £ ! ) consists of all mappings / : L\ —• L2 preserving arbitrary non-empty suprema and finite infima and M o r ( £ 2 ) consists of identical mappings EL : L —» L only ( i .e . £ 2 is a discrete category). (Here L\, L2, L £ O b ( £ ] ) = O b ( £ 2 ) . ) Notice tha t FT(C\) is in fact a slight enlargement of Rodabaugh's category T [8] (cf. also the category FUZZ from [7]). It is easy to notice also that FTi(C2) is just the category of L-fuzzy topological spaces as they are defined by Goguen [2]; specifically, FTi(C2), where / = [0,1], is the category of Chang fuzzy topological spaces [1] and FTz(C2), where Z ~ {0, 1}, in an obvious way can be identified with the category Top of topological spaces. We shall need also the next notion. An fs-set (X, L, r ) is called laminated if r contains constant mappings a A- : X —• L for all a £ L (cf. Lowen's definition of a fuzzy topology; see e.g. [3]). 56 V. SHTEINBUK AND A. SOSTAK T h e o r e m 1. For laminated fs-sets (Ai ,L] ,T i ) and (X2,L2,T2) the following condi-tions are equivalent: (1) the semigroups Sc\(X\) and S£,(A'2) are ^-isomorphic; (2) the semigroups Sc2(X\) and S£2(A'2) are w-isomorphic; (3) fs-sets ( A i , L i , r i ) and (X2,L2,T2) are £i-homeomorphic . T h e o r e m 2. Laminated fs-sets (X\, L\,T\) and (X2, L2,T2) are quasihomeomorphic iff the semigroups Sc,(X\) and Sc,(X2) are r-isomorphic (i = 1, 2). To restore a laminated fs-set up to £2-homeomorphism by means of its Plotkin endo-morphism semigroup we need the following special kind of ^-isomorphism: A r-isomorphism a : Sc(X\, L,T\) —> S£(A2,L,r2) is called tough, if a (ex,, £L, a) = (^A'2,£L,Cf) for each a £ L. One can prove that each tough isomorphism of laminated fs-sets is an (^-isomorphism. T h e o r e m 3 . Laminated fs-sets (X\,L,T\) and (X2,L,T2) are £ 2 -homeomorphic iff the semigroups S£,(Ai) and Sc,(X2) are toughly isomorphic (i = 1, 2). These theorems immediately imply analogous results for laminated fuzzy topological spaces: T h e o r e m 1'. Laminated fuzzy topological spaces ( A i , L i , T i ) and (A ' 2 ,L 2 , r 2 ) are homeomorphic (in FT(C\)) iff their Plotkin semigroups S£,(Ai) and S£2(A'2) are ui-isomorphic. T h e o r e m 3 ' . Laminated L-fuzzy topological spaces [2] (A],Ti) and ( A 2 , r 2 ) are homeomorphic iff their Plotkin semigroups Sc,(X\,L,T\) and Sc,(X2, L, r2) are toughly isomorphic (i = 1, 2). E x a m p l e 1. T h e c o n d i t i o n of l a m i n a t e d n e s s is of e s s e n c e . Let (A, T) be a topological space such that C(X, T) = {ex} U {ex : c e X}. Thus the semigroup of endomorphisms of X consists only of constant mappings and the identity. (Such a space can be found e.g. in [6].) Fix two constants 0 < a < f3 < 1 and two points a,b £ X. Let M denote the set of all mappings /i : /> —> / preserving non-empty suprema and finite infima such that fi(a) = a, n((3) = //. Define fuzzy sets U, : X —> / , i = 1, 2 as follows. Let U\(x) = a if x ^ a and U\(a) = fi and let U2(x) = a if x ^ a, b and U2(a) = U2(b) = [1. Let r;, i = 1, 2, be the fuzzy topology having TU{/7,} as its subbase. It is easy to notice tha t the semigroups S£,(A, / , r i ) and Sc,(X, I, r2) are w-isomorphic (even toughly isomorphic) but nevertheless the spaces ( A , / , r i ) and ( A , / , r 2 ) are not £ , -homeomorphic , i = 1, 2. On Endomorphism Semigroups of a Fuzzily Structured Set 57 E x a m p l e 2. I n a d e q u a c y of s e m i g r o u p s of c o n t i n u o u s t r a n s f o r m a t i o n s in fuzzy s e t t i n g . Let (X, T) be a topological space. For a constant a £ (0, 1] let Ta be a fuzzy topology on X generated by the subbase oa = {ail : U € T} U {ax • a £ I}. (Obviously, T-> = toT is the set of all lower semicontinuous functions M : (X, T) —* / ; see [3].) It is easy to notice that Cc2(X, 1 ,Ta) = Cc2(X, 1,Ta<) for any a, a' G (0,1] and if a , a ' ^ 1, then Cc1(X, I,ra) and Ccs(X, I,Ta>) are isomorphic. On the other hand, if a ^ a', then the fs-sets (X,I,Ta) and (X, I,Ta') are neither £ 2-homeomorphic, nor £i-hoineomorphic. REFERENCES  [1] C. L. Chang: Fuzzy topological spaces. J . Math . Anal. Appl. 24 (1968), 182-190. [2] J . A. Goguen: The fuzzy Tychonoff theorem. J. Math. Anal. Appl. 43 (1973), 734-742. [3] R. Lowen: Fuzzy topological spaces and fuzzy compactness. J . Math. Anal. Appl. 56 (1976), 6 2 1 -633. [4] K. D. Magill: A survey of semigroups of continuous selfmaps. Semigroup Forum 11 (1975/76), 189-282. [5] B . I . Plotkin: Algebra of au tomata : some problems. Vestnik MGU Ser. Math . Mech. 4 (1980), 96 (in Russian). [6] A. Pul t r and V. Trnkova: Combinatorial , algebraic and topological representation of groups, semi-groups and categories. Prague 1980. [7] S. E. Rodabaugh: A categorical accommodation of various notions of fuzzy topology. Fuzzy Sets Syst. 9 (1983), 241-265. [8] S. E. Rodabaugh : A point-set lattice-theoretic framework T for topology which contains LOC as a subcategory of singlelon spaces.... Preprint , February 1986, Youngstown State University Youngs-town, Ohio. [9] Ye. M. Vechtomov: Problems of definability of topological spaces by algebraic systems of continuous functions. Pogi Nauki i Techn., Ser. Algebra, Topology, Geometry 28 (1990), 3-46. Dr. Veniamin Shteinbuk, Department of Applied Mathematics, Riga Technical University, Riga 226355. Latvia. Dr. Alexander Sostak, Department of Mathematics, Latvian University, Riga 226098. Latvia. 

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On endomorphism semigroups of a fuzzily structured set

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On endomorphism semigroups of a fuzzily structured set

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