TY - JOUR

T1 - Homotopy bases and finite derivation type for subgroups of monoids

AU - Malheiro, António José Mesquita da Cunha Machado

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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 ﬁnitely presented regular monoid with ﬁnitely many left and right ideals satisﬁes the homological ﬁniteness condition FP3 if all of its maximal subgroups satisfy the condition FP3.

AB - 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 ﬁnitely presented regular monoid with ﬁnitely many left and right ideals satisﬁes the homological ﬁniteness condition FP3 if all of its maximal subgroups satisfy the condition FP3.

KW - Finitely presented groups and monoids

KW - Finite derivation type

KW - Finiteness conditions

KW - Rewriting systems

KW - Homotopy bases

U2 - 10.1016/j.jalgebra.2014.03.035

DO - 10.1016/j.jalgebra.2014.03.035

M3 - Article

SN - 0021-8693

VL - 410

SP - 53

EP - 84

JO - Journal of Algebra

JF - Journal of Algebra

IS - NA

ER -