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1999 Summer Conference on Topology and its Applications
August 4-7, 1999
C.W. Post Campus of Long Island University
Brookville, NY, USA

Organizers
Sheldon Rothman, Ralph Kopperman

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Fuzzy topologies related to topological algebra
by
Alexander Sostak
Department of Mathematics, University of Latvia, Riga, Latvia

Let L = (L, <= , /\ , \/ , *) be an infinitly distributive GL-monoid (cf e.g. [1], [2]) with thetop element 1 and the bot element 0. To recall the concept of an L-fuzzy category [3], consider an ordinary (classical) category C and let \omega: Ob(C) --> L and \mu: Mor( C) --> L be L-fuzzy subclasses of its objects and morphisms respectively. Now, an L-fuzzy category can be defined as a triple (C, \omega, \mu) satisfying the following axioms:

  1. \mu(f) <= \omega(X) /\ \omega(Y) for all X, Y in Ob( C) and for all f in Mor(X, Y);

  2. \mu(g o f) >= \mu(f) * \mu(g) whenever the composition g o f is defined;

  3. \mu(eX) = \omega(X) where eX : X --> X is the identity morphism.
The aim of this talk is to introduce some fuzzy categories related to topological algebra and to discuss basic properties of these fuzzy categories and their relations to classical categories of topological algebra.

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] U. Höhle, A. Sostak, Axiomatics of fixed-basis fuzzy topologies, Chapter 3, pp. 123 - 273; in: Mathematics of Fuzzy Sets: Logic, Topology and Measure Theory , U. Höhle, S.Rodabaugh eds. - Handbook of Fuzzy Sets Series, vol.3. Kluwer Academic Publisher, Dordrecht, Boston - 1999.
[3] A. Sostak, Fuzzy categories versus categories of fuzzily structured sets: Elements of the theory of fuzzy categories, In: Mathematik-Arbeitspapiere, Universität Bremen, vol 48 (1997), pp. 407-437.

Date received: June 9, 1999

Copyright © 1999 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 # cacl-68.