The further study on the category of T-convergence groups

TL;DR

This paper advances the theory of T-convergence groups by providing new characterizations, simplifying definitions, demonstrating uniformization, and establishing category properties, thereby deepening the understanding of this mathematical structure.

Contribution

It introduces a novel characterization via the $igodot$-product of T-filters, simplifies the subcategory definition, and proves uniformization and power object existence.

Findings

T-convergence groups are localizable and determined by convergence at the identity.

The subcategory of topological T-convergence groups can be defined without the topological condition.

Every T-convergence group can be reconstructed from a T-uniformly convergent space.

Abstract

T-convergence groups is a natural extension of lattice-valued topological groups, which is a newly introduced mathematical structure. In this paper, we will further explore the theory of T-convergence groups. The main results include: (1) It possesses a novel characterization through the $\odot$-product of T-filters, and it is localizable, meaning that each T-convergence group is uniquely determined by the convergence at the identity element of the underlying group. (2) The definition of its subcategory, the topological T-convergence groups, can be simplified by removing the topological condition (TT). (3) It exhibits uniformization, which means that each T-convergence group can be reconstructed from a T-uniformly convergent space. (4) It possesses a power object, indicating that it has good category properties.