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Zbl 1151.20044
Cain, Alan J.; Robertson, Edmund F.; Ruškuc, Nik
Subsemigroups of groups: presentations, Malcev presentations, and automatic structures. (English)
[J] J. Group Theory 9, No. 3, 397-426 (2006). ISSN 1433-5883; ISSN 1435-4446

Summary: 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 subsemigroups 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 subsemigroups admit finite Malcev presentations is properly contained in the class of coherent groups. Finitely generated subsemigroups of the free product of a free monoid and an Abelian group are asynchronously automatic and therefore have finite Malcev presentations.

MSC 2000:: *20M05 Free semigroups
20F05 Presentations of groups
20M35 Semigroups in automata theory, linguistics, etc.
20F19 Generalizations of solvable and nilpotent groups

Keywords: finitely generated semigroups; virtually nilpotent groups; finite Malcev presentations; asynchronously automatic semigroups; virtually free groups

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