On weak convex MV-algebras

Abstract

The aim of this paper is to introduce convex structures on MV-algebras such that the MV-operations are convexity preserving or weak convexity preserving. Therefore, we propose the concepts of paraconvex MV-algebras and weak convex MV-algebras. We give some characterizations of weak convex MV-algebras. Further, we show that the standard MV-algebra endowed with its interval convexity is a weak convex MV-algebra. In particular, a finite MV-chain endowed with a non-trivial convex structure is a weak convex MV-algebra iff the convex structure is precisely its interval convexity. Moreover, the direct product of finite weak convex MV-algebras is still a weak convex MV-algebra. Based on this, we further get that each finite MV-algebra endowed with its interval convexity is a weak convex MV-algebra. By using ideals, we introduce the ideal convexity on an MV-algebra which turns it to be a paraconvex MV-algebra. Finally, we discuss the separation axioms on weak convex MV-algebras. Communicated by Ángel del Río Mateos