‘Automatic structures for subsemigroups of Baumslag–Solitar semigroups’
Semigroup Forum, 87, no. 3 (2013), pp. 537–552.
DOI: 10.1007/s00233-013-9490-y. MR: 3128706. ZBL: 1326.20057.

Abstract

This paper studies automatic structures for subsemigroups of Baumslag–Solitar semigroups (that is, semigroups presented by $\langle x,y \mid (yx^m, x^ny)\rangle$, 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.)