A fresh look at the category of ⊤-convergence groups

Abstract

⊤-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 ⊤-convergence groups. The main results include: (1) It possesses a novel characterization through the ⊙-product of ⊤-filters, and it is localizable, meaning that each ⊤-convergence group is uniquely determined by the convergence at the identity element of the underlying group. (2) The definition of its subcategory, the topological ⊤-convergence groups, can be simplified by removing the topological condition (TT). (3) It exhibits uniformization, which means that each ⊤-convergence group can be reconstructed from a ⊤-uniformly convergent space. (4) There exists a natural group structure and ⊤-convergence structure on the function space, which makes it a ⊤-convergence group.