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\(L\)-fuzzy closure systems. (English) Zbl 1195.54015

The paper is a contribution to the theory of fuzzy closure operators and systems initiated in the sixties of the 20. century by Appert and Ky Fan (see e.g. [A. Appert and Ky-Fan, Espaces topologiques intermédiaires. Problème de la distanciation. Préface de M. Fréchet. Actualités scientifiques et industrielles. 1121. Exposés d’analyse générale, XV. Paris: Hermann&Cie. 160 p. (1951; Zbl 0045.43901)]). In this paper the authors follow the line of reasoning proposed by R. Bělohlávek (e.g. [J. Math. Anal. Appl. 262, No. 2, 473–489 (2001; Zbl 0989.54006)]) who proposed the generalization of these concepts by using the structure of the residuated lattices. The main difference is that instead of \(L_k\) closure system defined as a classical family of fuzzy subsets, the notion of \(L\)-fuzzy closure systems is based on \(L\)-subsets of powersets (see e.g. [U. Höhle, J. Math. Anal. Appl 201, No. 3, 786–826 (1996; Zbl 0860.03038)]). The authors also introduce the concepts of strong \(L\)-fuzzy closure systems and operators and find relations between them. The last section of the paper is devoted to the categorical aspects of \(L\)-fuzzy closure spaces and \(L\)-fuzzy closure system spaces and links between them.

MSC:

54A40 Fuzzy topology
03E72 Theory of fuzzy sets, etc.
54A05 Topological spaces and generalizations (closure spaces, etc.)
54D80 Special constructions of topological spaces (spaces of ultrafilters, etc.)
06B35 Continuous lattices and posets, applications
06B30 Topological lattices
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