Homotopy bases and finite derivation type for subgroups of monoids

Details Homotopy bases and finite derivation type for subgroups of monoids Homotopy bases and finite derivation type for subgroups of monoids

2009-12-07

arxiv.org/abs/0912.1284v1

Mathematics

Group Theory

24 pages

Scientific paper

Given a monoid defined by a presentation, and a homotopy base for the derivation graph associated to the presentation, and given an arbitrary subgroup of the monoid, we give a homotopy base (and presentation) for the subgroup. If the monoid has finite derivation type (FDT), and if under the action of the monoid on its subsets by right multiplication the strong orbit of the subgroup is finite, then we obtain a finite homotopy base for the subgroup, and hence the subgroup has FDT. As an application we prove that a regular monoid with finitely many left and right ideals has FDT if and only if all of its maximal subgroups have FDT. We use this to show that a regular monoid with finitely many left and right ideals satisfies the homological finiteness condition FP_3 if all of its maximal subgroups satisfy the condition FP_3.

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