Automatic structures for subsemigroups of Baumslag--Solitar semigroups

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This paper studies automatic structures for subsemigroups of Baumslag--Solitar semigroups (that is, semigroups presented by < x,y (yxᵐ, xⁿy)>, 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 n, all of the finitely generated subsemigroups of this semigroup are [right-] automatic. If m 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.)