Asia Mathematika, Vol 9, issue 3, pages: 25--42.

Research on neighborhood-based characterization and separation axioms of \((L, M)\)-fuzzy convex spaces

Hu Zhao
School of Science, Xi'an Polytechnic University, Xi'an, China.
Run-Mei Shang
School of Science, Xi'an Polytechnic University, Xi'an, China.
Bing-Nan Zhang
School of Science, Xi'an Polytechnic University, Xi'an, China.
Xia Hu
School of Science, Xi'an Polytechnic University, Xi'an, China.

Received: 14 Oct 2025 | Accepted: 15 Dec 2025 | Final Version: 31 Dec 2025

Abstract

This study focuses on the construction of concave \((L, M)\)-fuzzy neighborhood operators and a systematic analysis of the separation axiom system in \((L, M)\)-fuzzy convex spaces. First, concave \((L, M)\)-fuzzy neighborhood operators are formally defined, and a bidirectional induction mechanism between these operators and \((L, M)\)-fuzzy convex structures is established via \((L, M)\)-fuzzy concave structures. The isomorphism between the category of \((L, M)\)-fuzzy convex spaces and that of concave \((L, M)\)-fuzzy neighborhood spaces is proved, establishing a theoretical connection between local neighborhood characterizations and global convex structures. Second, in the study of separation axioms, the validity of \(S_3\) and \(S_4\) axioms in \((L, M)\)-fuzzy convex spaces is verified, particularly the hereditary property of \(S_3\) in subspaces and, under certain conditions, the preservation of the \(S_3\) axiom under products are established, enriching the property system of separation axioms within the framework of fuzzy convexity.

Keywords:
(L,M)-fuzzy convex spaces, concave (L, M)-fuzzy neighborhood operators, subspace and product of (L,M)-fuzzy convex spaces, \(S_3\) and \(S_4\) separation axioms, isomorphic categories

1. Introduction

Abstract convexity theory \cite{Soltan0, Van0}, a subfield of mathematics, boasts extensive connections with other mathematical disciplines and has found applications in diverse research fields, including metric spaces, topological spaces, graphs, and lattices. With the development of fuzzy mathematics, fuzzy set theory has been integrated into numerous mathematical structures—such as fuzzy convergence structures and fuzzy topology ...

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