Hull operators and interval operators in (L,M)-fuzzy convex spaces

Considering L being a continuous lattice and M being a completely distributive De Morgan algebra, several basic notions with respect to ( L , M ) -fuzzy convex structures in the sense of Shi and Xiu are introduced and their relationship with ( L , M ) -fuzzy convex structures are studied. Firstly, an equivalent form of ( L , M ) -fuzzy convex structures in the sense of Shi and Xiu is provided. Secondly, two types of fuzzy hull operators are introduced, which are called ( L , M ) -fuzzy hull operators and ( L , M ) -fuzzy restricted hull operators, respectively. It is shown that they can be used to characterize ( L , M ) -fuzzy convex structures. Finally, fuzzy counterparts of interval operators in the ( L , M ) -fuzzy case are proposed, which are called ( L , M ) -fuzzy interval operators. It is proved that there is a Galois correspondence between the category of ( L , M ) -fuzzy interval spaces and that of ( L , M ) -fuzzy convex spaces and further the category of arity 2 ( L , M ) -fuzzy convex spaces can be embedded in the category of ( L , M ) -fuzzy interval spaces as a fully reflective subcategory.