  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.)
