Subsemigroups of groups: presentations, Malcev presentations, and automatic structures

All finitely generated subsemigroups of virtually nilpotent groups admit finite Malcev presentations. (A Malcev presentation is a presentation of a special type for a semigroup that can be embedded in a group.) All automatic or asynchronous automatic semigroups embeddable into groups admit finite Malcev presentations. Finitely generated subsernigroups of virtually free groups are automatic. A finitely generated subsemigroup of the free product of a free group and an abelian group that fails to have a finite Malcev presentation is exhibited. Therefore the class of groups all of whose finitely generated subsernigroups admit finite Malcev presentations is properly contained in the class of coherent groups. Finitely generated subsernigroups of the free product of a free monoid and an abelian group are asynchronously automatic and therefore have finite Malcev presentations.