On (L, M)-fuzzy quasi-uniform spaces

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. Sostak 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-Sostak 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.

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