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

KEYWORDS:

• Convex structure
• ideal convexity
• MV-algebra
• paraconvex MV-algebra
• separation axiom
• weak convex MV-algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 06D35
• 03G20
• 52A01

Acknowledgments

The authors would like to thank the unknown reviewers for their valuable comments and suggestions.

Disclosure statement

The authors have no potential conflict of interest to declare.

Additional information

Funding

This project is supported by the National Natural Science Foundation of China (11871097, 12271036, 12071033), Beijing Institute of Technology Science and Technology Innovation Plan Cultivation Project (No. 2021CX01030).