Homotopy bases and finite derivation type for Schutzenberger groups of monoids

TL;DR

This paper develops a method to derive homotopy bases and presentations for Schutzenberger groups of finitely presented monoids, extending known properties under specific finiteness conditions.

Contribution

It provides a new construction for homotopy bases of Schutzenberger groups based on the monoid’s homotopy base, under certain finiteness assumptions.

Findings

Constructs homotopy bases for Schutzenberger groups from monoid bases.

Shows finiteness of presentations is preserved under specific conditions.

Extends understanding of algebraic properties of monoid subgroups.

Abstract

Given a finitely presented monoid and a homotopy base for the monoid, and given an arbitrary Schutzenberger group of the monoid, the main result of this paper gives a homotopy base, and presentation, for the Schutzenberger group. In the case that the R-class R' of the Schutzenberger group G(H) has only finitely many H-classes, and there is an element s of the multiplicative right pointwise stabilizer of H, such that under the left action of the monoid on its R-classes the intersection of the orbit of the R-class of s with the inverse orbit of R' is finite, then finiteness of the presentation and of the homotopy base is preserved.