Characterizations of L-convex spaces via domain theory

摘要

We provide some new characterizations of L-convex spaces by using the way-below relation in domain theory. For this purpose, we first study the notion of domain finiteness in the lattice-valued case, and then introduce three kinds of spaces: algebraic L-closure spaces, restricted L-hull spaces, and L-entailment spaces. These three spaces are shown to be categorically isomorphic to L-convex spaces. Additionally, we introduce the notion of L-polytopes and prove that a subcollection of a dense L-convex structure is a base if and only if it contains all L-polytopes. These results indicate that in the study of the theory of (fuzzy) convex spaces, (fuzzy) domain theory has important applications. (C) 2019 Elsevier B.V. All rights reserved.