Variable-basis fuzzy interior operators

TL;DR

This paper develops a generalized theory of variable-basis fuzzy interior operators using categorical and lattice-theoretical methods, extending previous work on fuzzy topologies and interior operators.

Contribution

It introduces a broader framework for fuzzy interior operators based on variable bases, combining categorical and lattice-theoretic approaches.

Findings

Established a new general theory of fuzzy interior operators.

Constructed topological categories within the fuzzy setting.

Extended existing models to non-complemented lattices.

Abstract

For a topological space it is well-known that the associated closure and interior operators provide equivalent descriptions of set-theoretic topology; but it is not generally true in other categories, consequently it makes sense to define and study the notion of interior operators I in the context of fuzzy set theory, where we can find categories in a lattice-theoretical context. Fuzzy interior operators have been studied by U. Hohle, A. Sostak and others, (1999), these works were used to describe L-topologies on a set X. More recently, M. Diker, S. Dost and A. Ugur (2009) present interior and closure operators on texture spaces in the sense of Cech, and F. G. Shi(2009) studies interior operators via L-fuzzy neighborhood systems. The aim of this paper is to propose a more general theory of variable basis fuzzy interior operators, employing both categorical tools and the lattice theoretical foundations investigated by S. E. Rodabaugh (1999), where tha lattices are usually non-complemented. furthermore, we construct some topological categories.