Homogeneous Bands

TL;DR

This paper classifies all countable homogeneous bands, showing they are regular bands with a homogeneous structure semilattice, thereby providing a complete understanding of their algebraic structure.

Contribution

It provides the first complete classification of all countable homogeneous bands, linking their homogeneity to regularity and semilattice structure.

Findings

Homogeneous bands are regular bands.

Their structure semilattice is also homogeneous.

Complete classification of countable homogeneous bands.

Abstract

A countable band $B$ is called homogeneous if every isomorphism between finitely generated subbands extends to an automorphism of $B$. In this paper we give a complete classification of all the homogeneous bands. We prove that a homogeneous band belongs to the variety of regular bands, and has a homogeneous structure semilattice.