A fuzzy category on the basis of the category L-TOP of L-topological spaces
by
Alexander Sostak
University of Latvia, Riga, Latvia

Our aim is to define a fuzzy category [4] on the basis of the category L-TOP of (Chang-Goguen) L-topological spaces ([2], cf. also [3]) in which measure of continuity of mappings can be efficiently defined and investigated.

Let L = (L, <= , /\ , \/ , *) be an infinitely distributive GL-monoid (cf. e.g. [1]), i.e. a commutative integral divisible cl-monoid and let --> be the corresponding implication (i.e.  \alpha*\beta <=      <===>  \alpha <= \beta --> \gamma    for all\alpha, \beta, \gamma in L.) By an L-kernel operator on a set X we mean a mapping K: L^X --> L^X such that K(A) <= A   for allA in L^X  and  A <= B ===> K(B)   for allA, B in L^X. A pair (X, K) will be referred to as an L-kernel space (cf. e.g. [3]). Given an L-kernel operator K: L^X --> L^X let

\omegasa(K) : = Ù A, B in LX  æ è K(A) /\ K(B) --> K(A /\ B) ö ø ,

\omegaid(K) : = Ù A in LX  æ è K(A) --> K(K(A)) ö ø   and  \omegagl(K) : = 1X --> K(1X).

Obviously, \omega_sa, \omega_id and \omega_gl measure the degree to which K is (sub)additive, idempotent and global. In particular, an L-kernel operator K is an L-interior operator (and thus, actually is an L-topology), if and only if \omega(K) = T, where \omega(K) : = \omega_id (K) /\ \oo_gl (K) /\ \omega_sa(K). Further, given two L-kernel spaces (X, K_X), (Y, K_Y) and a mapping f:X --> Y let

\nu(f) : = Ù A in LY  æ è f-1 (KY (A)) --> KX (f-1 (A)) ö ø

and \mu(f) : = \mu(f) /\ \omega(K_X) /\ \omega(K_Y). Thus we come to a fuzzy (L-valued) category (L-Ker_SET, \omega, \mu) (cf. [4]) having L-kernel spaces and all mappings between the corresponding sets as potential objects and potential morphisms respectively, and L-valued subclasses of objects and morphisms determined by \omega and \mu respectively. Basic properties of this and some related categories will be discussed.

References

[1] U.Höhle, Commutative, residuated l-monoids, In: Non-classical Logics and Their Applications to Fuzzy Subsets,  E.P. Klement and U. Höhle eds., Kluwer Acad. Publ., 1994, 53-106.

[2] C.Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24(1968), 182-190.

[3] U. Höhle, A. Sostak, Axiomatics of fixed-basis fuzzy topologies, In: Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory,  U. Höhle, S.E. Rodabaugh eds. - Handbook Series, vol.3. Kluwer Academic Publisher, Dordrecht, Boston. -1999. pp. 123 - 273.

[4] A.  Sostak, Fuzzy categories versus categories of fuzzily structured sets: Elements of the theory of fuzzy categories, In: Mathematik-Arbeitspapiere,  H.-E. Porst ed., Universität Bremen, vol 48 (1997), pp. 407-437.

Date received: May 29, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeq-18.