Finite presentability and isomorphism of Cayley graphs of monoids

TL;DR

This paper constructs examples of finitely generated monoids with isomorphic Cayley graphs, demonstrating that finite presentability is not determined solely by the graph structure.

Contribution

It provides the first known examples of finitely generated monoids with isomorphic Cayley graphs where one is finitely presented and the other is not.

Findings

Finitely generated monoids can have isomorphic Cayley graphs regardless of their finite presentability.

The paper shows that Cayley graph isomorphism does not imply similar algebraic properties.

Examples challenge assumptions about the relationship between graph structure and monoid presentation.

Abstract

Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.