L-cones and L-dual cones

Abstract

In this paper, we extend the notions of fuzzy cones and fuzzy dual cones to the lattice-valued case. We show that all convex L-cones form a convex structure and we give the corresponding L-convex hull formula. We prove that each L-dual cone is convex and closed. We study the relationships between the L-dual cone of an L-subset and the L-dual cones of its L-closure, L-interior and L-convex hull. Moreover, we discuss the relationships between α-cuts of L-subsets and their dual cones. Finally, we show that the L-dual cone of the L-dual cone of an L-subset is precisely the minimal closed convex L-cone containing this L-subset.