Characterizations of L-concavities and L-convexities via derived relations

This paper is to characterize L-concavities and L-convexities via some derived forms of relations and operators. Specifically, notions of L-concave derived internal relation space and L-concave derived hull space are introduced. It is proved that the category of L-concave derived internal relation spaces and the category of L-concave derived hull spaces are isomorphic to the category of L-concave spaces. Also, notions of L-convex derived enclosed relation space and L-convex derived hull space are introduced. It is proved that the category of L-convex derived enclosed relation spaces and the category of L-convex derived hull spaces are isomorphic to the category of L-convex spaces.