On properties not inherited by monoids from their Schutzenberger groups

TL;DR

This paper constructs a monoid example where Schutzenberger groups have finite derivation type, but the monoid itself does not, highlighting limitations in inheritance of rewriting properties from groups to monoids.

Contribution

It provides a counterexample demonstrating that finite derivation type is not inherited by monoids from their Schutzenberger groups, and shows these properties are not preserved under certain extensions.

Findings

Monoid with finitely many ideals has Schutzenberger groups with finite derivation type

Counterexample to inheritance of finite derivation type in monoids

Finite derivation type not preserved under finite Green index extensions

Abstract

We give an example of a monoid with finitely many left and right ideals, all of whose Schutzenberger groups are presentable by finite complete rewriting systems, and so each have finite derivation type, but such that the monoid itself does not have finite derivation type, and therefore does not admit a presentation by a finite complete rewriting system. The example also serves as a counterexample to several other natural questions regarding complete rewriting systems and finite derivation type. Specifically it allows us to construct two finitely generated monoids M and N with isometric Cayley graphs, where N has finite derivation type (respectively, admits a presentation by a finite complete rewriting system) but M does not. This contrasts with the case of finitely generated groups for which finite derivation type is known to be a quasi-isometry invariant. The same example is also used to show that neither of these two properties is preserved under finite Green index extensions.