Automatic structures for subsemigroups of Baumslag--Solitar semigroups

TL;DR

This paper investigates the automatic structures of subsemigroups within Baumslag--Solitar semigroups, revealing conditions under which these subsemigroups are automatic or not, based on parameters m and n.

Contribution

It introduces a geometric approach to determine automaticity of subsemigroups in Baumslag--Solitar semigroups, a novel method in this context.

Findings

Subsemigroups are right-automatic if m > n

Subsemigroups are left-automatic if m < n

Some subsemigroups are not automatic when m = n

Abstract

This paper studies automatic structures for subsemigroups of Baumslag--Solitar semigroups (that is, semigroups presented by $\ < x,y \mid (yx^m, x^ny)\ >$, where $m$ and $n$ are natural numbers). A geometric argument (a rarity in the field of automatic semigroups) is used to show that if $m \gt n$, all of the finitely generated subsemigroups of this semigroup are [right-] automatic. If $m \lt n$, all of its finitely generated subsemigroups are left-automatic. If $m = n$, there exist finitely generated subsemigroups that are not automatic. An appendix discusses the implications of these results for the theory of Malcev presentations. (A Malcev presentation is a special type of presentation for semigroups embeddable into groups.)