Let $(X;T)$ be a presentation for the inverse monoid $\mipres{X}{T}$ [resp. inverse semigroup $\sipres{X}{T}$ . In what follows, the argument for inverse semigroups and inverse monoids is exactly the same, so we concentrate on the last one.

Given $m\in \mipres{X}{T}$ let $[m]_{\mathcal{R}}$ be the equivalence class of $m$ with respect to the Right Green relation $\mathcal{R}$ The Right Schützenberger graph of $[m]_{\mathcal{R}}$ with respect to the presentation $(X;T)$ is defined as the $X$ inverse word graph $\schG(X;T;m)$ with vertex and edge set respectively $$\V(\schG(X;T;m))=\gbra{v\in \mipres{X}{T}\,|\,[v]_{\mathcal{R}}= [m]_{\mathcal{R}}},$$ $$\E(\schG(X;T;m))=\gbra{(v_1,x,v_2)\,|\,v_1,v_2\in \V(\schG(X;T;m)),\ x\in\double{X},\ v_2=v_1 \cdot [x]_\tau},$$ where $\tau=(T\cup\rho_X)^\co$ i.e. $\tau$ is the congruence generated by $T$ and the Wagner congruence $\rho_X$ and $[x]_\tau$ is the congruence class of the letter $x\in\double X$ with respect to the congruence $\tau$

This is a good definition, in fact it can be easily shown that given $m,n\in\mipres{X}{T}$ with $[m]_{\mathcal R}=[n]_{\mathcal R}$ we have $\schG(X;T;m)=\schG(X;T;n)$

Analogously we can define the Left Schützenberger graph using the Left Green relation $\mathcal{L}$ instead of the Right Green relation $\mathcal{R}$ but this notion is not used in literature.

Schützenberger graphs play in combinatorial inverse semigroups theory the role that Cayley graphs play in combinatorial group theory. In fact, if $G=\mipres{X}{T}$ happen to be a group (with identity $1_G$ , then the Schützenberger graph $\schG{(X;T;1_G)}$ of its unique $\mathcal{R}$ class is exactly the Cayley graph of the group $G$

Bibliography

1

N. Petrich, Inverse Semigroups, Wiley, New York, 1984.

2

J.B. Stephen, Presentation of inverse monoids, J. Pure Appl. Algebra 63 (1990) 81-112.