Lattice-valued interval operators and its induced lattice-valued convex structures

Abstract

Galois correspondence in category theory plays an important role in establishing the relationships between different types of spatial structures. In this paper, we apply Galois correspondence as a tool to the theory of lattice-valued convex structures. We mainly introduce the concept of lattice-valued interval operators and discuss its relationships with $L$ -fuzzifying convex structures and $L$ -convex structures. It is shown that there is a Galois correspondence between the category of lattice-valued interval spaces and the category of $L$ -fuzzifying convex spaces. In particular, the category of arity 2 $L$ -fuzzifying convex spaces can be embedded in the category of lattice-valued interval spaces as a reflective subcategory. Also, it is proved that there is a Galois correspondence between the category of lattice-valued interval spaces and the category of $L$ -convex spaces. Specially, the category of arity 2 $L$ -convex spaces can be embedded in the category of lattice-valued interval spaces as a reflective subcategory.