On [formula omitted]-fuzzy quasi-uniform spaces

Abstract

An ( L , M ) -fuzzy topology is a graded extension of topological spaces handling M-valued families of L-fuzzy subsets of a referential, where L and M are completely distributive lattices. When M reduces to the set 2 = { 0 , 1 } , a ( 2 , M ) -fuzzy topology is called a fuzzifying topology after Ying. Šostak introduced the notion ( L , M ) -fuzzy uniform spaces. The aim of this paper is to study the relationship between ( 2 , M ) -fuzzy quasi-uniform spaces and ( L , M ) -fuzzy quasi-uniform spaces as well as the relationship between ( 2 , M ) -fuzzy quasi-uniform spaces and pointwise ( L , M ) -fuzzy quasi-uniform spaces—the extension of Shi's L-quasi-uniform space in a Kubiak–Šostak sense. It is shown that the category of ( 2 , M ) -fuzzy quasi-uniform spaces can be embedded in the category of stratified ( L , M ) -fuzzy quasi-uniform spaces as a both reflective and coreflective full subcategory; and the former category can also be embedded in the category of pointwise ( L , M ) -fuzzy quasi-uniform spaces.