Fixed and variable-basis fuzzy closure operators

TL;DR

This paper develops a more general theory of fuzzy closure operators using categorical and lattice-theoretic tools, extending previous work beyond complemented lattices to include non-complemented lattices, and constructs related topological categories.

Contribution

It introduces a unified framework for fixed and variable-basis fuzzy closure operators employing non-complemented lattices and categorical methods, expanding the theoretical foundation.

Findings

Established a general theory of fuzzy closure operators

Constructed topological categories for the new framework

Extended the scope beyond complemented lattices

Abstract

Closure operators are very useful tools in several areas of classical mathematics and in general category theory. In fuzzy set theory, fuzzy closure operators have been studied by G. Gerla (1966). These works generally define a fuzzy subset as a mapping from a set X to the real unit interval, as a complete and complemented lattice. More recently, Y. C. Kim (2003), F. G. Shi (2009), J. Fang and Y. Yue (2010) propose theories of fuzzy closure systems and fuzzy closure operators in a more general settings, but still using complemented lattices. The aim of this paper is to propose a more general theory of fixed and variable-basis fuzzy closure operators, employing both categorical tools and the lattice theoretical fundations investigated by S. E. Rodabaugh (1999), where the lattices are usually non-complemented. Besides, we construct topological categories in both cases.