Variable-Basis Fuzzy Filters

TL;DR

This paper develops a categorical framework for variable-basis fuzzy filters, generalizing fixed-basis fuzzy filters and exploring their relation to fuzzy topological spaces within the context of complete quasi-monoidal lattices.

Contribution

It introduces the category of variable-basis fuzzy filters and examines its connections to fuzzy topological spaces, extending prior fixed-basis fuzzy filter theories.

Findings

Categorical framework for variable-basis fuzzy filters established

Relations between fuzzy filters and fuzzy topological spaces analyzed

Generalization from fixed-basis to variable-basis filters achieved

Abstract

U. H{\"o}hle and A. {\v{S}}ostak have developed in \cite{HS} the category of complete quasi-monoidal lattices; S. E. Rodabaugh in \cite{RO} proposed its opposite category and with a subcategory $\mathbf{C}$ of the latter, he define grounds of the form $\mathbf{SET\times C}$. In this paper, for each ground category of the form $\mathbf{SET\times C}$, we study categorical frameworks for variable-basis fuzzy filters, particularly the category $\mathbf{C-FFIL}$ of variable-basis fuzzy filters, as a natural generalization of the category of fixed-basis fuzzy filters which was introduced in \cite{LO}. In addition, we get some relations between the category of variable-basis fuzzy filters and the category of variable-basis fuzzy topological spaces.