Frame-valued assembly and its applications

Abstract

The aim of this paper is to use fuzzy domain theory to study some cartesian closed categories containing the category of stratified L-topological spaces when the truth value table L is a frame. Firstly, we introduce the concept of L-fuzzy assemblies and prove that L-FAssm—the category of L-fuzzy assemblies—is cartesian closed. L-FAssm contains the category of stratified L-topological spaces as a full and faithful subcategory. Secondly, by equipping a stratified $T_0$-L-topological space with an equivalence relation, we introduce the concept of L-equilogical spaces and show that L-Equ—the category of L-equilogical spaces—is also cartesian closed. Then we give two categories which are equivalent to L-Equ. Finally, we show that there is a pair of adjoint functors between the category of L-fuzzy assemblies and the category of L-generalized convergence spaces. Further, we construct two subcategories of L-fuzzy assemblies satisfying certain conditions, which are equivalent to the category of L-Kent convergence spaces and the category of L-limit spaces, respectively.