Let $\doubles{X}$ be the free monoid with involution on $X$ , and $T\subseteq \doubles X\times \doubles X$ be a binary relation between words. We denote by $T^\e$ [resp. $T^\co$ ] the equivalence relation [resp. congruence] generated by $T$ .

A presentation (for an inverse monoid) is a couple $(X;T)$ . We use this couple of objects to define an inverse monoid $\mipres{X}{T}$ . Let $\rho_X$ be the Wagner congruence on $X$ , we define the inverse monoid $\mipres{X}{T}$ presented by $(X;T)$ as $$\mipres{X}{T}=\doubles{X}/(T\cup\rho_X)^\co.$$

In the previous dicussion, if we replace everywhere $\doubles X$ with $\doublep X$ we obtain a presentation (for an inverse semigroup) $(X;T)$ and an inverse semigroup $\sipres{X}{T}$ presented by $(X;T)$ .

A trivial but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on $X$ , that is usually denoted by $\fim(X)$ [resp. $\fis(X)$ ] and is defined by $$\fim(X)=\mipres{X}{\varnothing}=\doubles{X}/\rho_X,\ \ \mbox{[resp. $\fis(X)=\sipres{X}{\varnothing}=\doublep{X}/\rho_X$]}.$$

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.