The Geometry of Special Monoids

Abstract: A monoid is said to be special if it admits a presentation in which all defining relations are of the form $w = 1$. Groups are familiar examples of special monoids. This article studies the geometric and structural properties of the Cayley graphs of finitely presented special monoids, building on work by Zhang and Gray-Steinberg. It is shown that the right Cayley graph $\Gamma$ of a special monoid $M$ is a context-free graph, in the sense of Muller & Schupp, if and only if the group of units of $M$ is virtually free. This generalises the geometric aspect of the well-known Muller-Schupp Theorem from groups to special monoids. Furthermore, we completely characterise when the monadic second order theory of $\Gamma$ is decidable: this is precisely when the group of units is virtually free. This completely answers for the class of special monoids a question of Kuske & Lohrey from 2006. As a corollary, we obtain that the rational subset membership problem for $M$ is decidable when the group of units of $M$ is virtually free, extending results of Kambites & Render. We also show that the class of special monoids with virtually free group of units is the same as the class of special monoids with right Cayley graph quasi-isometric to a tree as undirected graphs. The above results are proven by developing two general constructions for graphs which preserve context-freeness, of independent interest. The first takes a context-free graph and constructs a tree of copies of this graph. The second is a bounded determinisation of the resulting tree of copies.