Overlap functions, as a typical kind of binary aggregation functions, have been widely investigated from both of theoretical and applied viewpoints. In this article, we will continue on the theoretical research on overlap functions. First, we will define$O$-inclusion subsethoods to describe the inclusion degrees between fuzzy sets based on the residuum induced from an overlap function$O$. Second, by means of$O$-inclusion subsethoods, we will propose the concepts of$O$-convexities, algebraic$O$-closure operators and$O$-hull operators, and show that they are one-to-one corresponding. Finally, we will establish the relationship between$O$-convexities and fuzzy interval operators. These results will not only present a new perspective for theoretical research on overlap functions, but also provide a new approach to fuzzifications of convexities.

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.