A new approach to lattice-valued convergence groups via ⊤-filters

In this paper, choosing a complete residuated lattice L as the lattice background, we introduce the concept of ⊤-convergence groups, which is a group equipped with a ⊤-convergence structure such that the group operations are continuous with respect to the ⊤-convergence space. Then we make further investigations on ⊤-convergence groups, including (1) We provide many nice properties of ⊤-convergence groups and two characterization theorems; (2) Considering ⊤-neighborhood groups, we propose topological ⊤-convergence groups and establish their relationships with topological ⊤-neighborhood groups; (3) We further introduce the notion of ⊤-limit groups by equipping a limit condition on ⊤-convergence groups and then investigate the uniformizability of ⊤-limit groups by means of ⊤-uniform limit spaces.