Topological and differentiable aspects of Clifford semigroups

TL;DR

This paper explores the topological and differentiable properties of Clifford semigroups, providing explicit metrics, subgroup criteria, and rigidity results related to their algebraic and topological structure.

Contribution

It introduces new criteria for subgroup Lie group structure and proves a rigidity theorem linking $C^1$-regularity to the discreteness of the semilattice.

Findings

Constructed a compatible metric for the Bowman topology in compact Hausdorff Clifford semigroups.

Established conditions under which maximal subgroups are Lie groups.

Proved that $C^1$-regularity at idempotents implies the semilattice is discrete.

Abstract

This paper investigates the interplay between algebraic structure, topology, and differentiability in Clifford semigroups. The study is developed along three main themes. First, in the compact Hausdorff setting, we provide an explicit construction of a compatible metric for the Bowman topology. Second, we address Hilbert-fifth-type questions by establishing criteria under which the maximal subgroups are forced to be Lie groups. Finally, we prove a structural rigidity theorem: $C^1$-regularity at the idempotents implies that the idempotent semilattice is discrete.