Stratified L-convex groups

TL;DR

This paper introduces stratified L-convex groups, exploring their categorical properties, embeddings, and topological structures, thereby expanding the understanding of convexity-preserving group structures.

Contribution

It defines stratified L-convex groups, shows their categorical properties, and embeds convex groups as a reflective subcategory within this framework.

Findings

SLCG forms a concrete category with well-defined structures.

Convex groups embed as a reflective subcategory in SLCG.

SLCG has localization, initial, and final structures, making it topological.

Abstract

This paper investigates a novel structure of stratified L-convex groups, defined as groups possessing stratified L-convex structures, in which the group operations are L-convexity-preserving mappings. It is verified that stratified L-convex groups serve as objects, while L-convexity-preserving group homomorphisms serve as morphisms, together forming a concrete category, denoted as SLCG. As a specific instance of SLCG (i.e., when L=2), the category of convex groups, denoted as CG, is also defined. We show that CG can be embedded within SLCG as a reflective subcategory. In addition, we demonstrate that SLCG possesses well-defined characterizations, localization properties, and initial and final structures, establishing it as a topological category over groups.