Closure Operators in Lattice-valued Interval Spaces

Abstract

In this paper, initially we introduce the quotient structure in the category $L$-$\textbf{IS}$, and characterize the local $\overline{T_{0}}$, and local $T_{1}$ objects in it. Furthermore, we explicitly characterize closedness of a point and closed (resp. strongly closed) subobject of an object in $L$-$\textbf{IS}$ and examine their relationship with local $\overline{T_{0}}$, and local $T_{1}$. Using these notions of closedness, we introduce six new closure operators in the category of lattice-valued interval spaces and $L$-IP mappings which enjoy idempotent, productive and (weakly) hereditary properties and examine their relationship with Hausdorff lattice-valued interval spaces. Finally, we show that each of these closure operators are quotient reflective subcategories of $L$-$\textbf{IS}$ and examine their mutual relationships, and compare our results with the closure operators of some well-known topological categories.