Lattice-Valued Interval Operators and Its Induced Lattice-Valued Convex Structures

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.