Automatic structures and Malcev presentations for subsemigroups of Baumslag–Solitar semigroups Cain, Alan J.

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A Baumslag–Solitar semigroup is presented by <x, y | (yxm, xny)>, where m and n are natural numbers. If m > n, all of its finitely generated subsemigroups are [right-] automatic. If m < n, all of its finitely generated subsemigroups are left-automatic. Thus if m and n are unequal, all finitely generated semigroups admit finite Malcev presentations. (A Malcev presentation is a special type of presentation for semigroups embeddable into groups.) If m = n, there exist finitely generated subsemigroups that are not automatic. Some remarks are made about whether all finitely generated semigroups admit finite Malcev presentations in this case.