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

Abstract

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.