presentation of inverse monoids and inverse semigroups

Let ${\left(X\coprod X^{-1}\right)}^{\ast }$ be the free monoidwith involution on $X$, and $T\subseteq {\left(X\coprod X^{-1}\right)}^{\ast }\times {\left(X\coprod X^{-1}\right)}^{\ast }$ be a binary relation between words. We denote by $T^{\mathrm{e}}$ [resp. $T^{\mathrm{c}}$] 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 ${\mathrm{Inv}}^1⟨X|T⟩$. Let ${\rho }_X$ be the Wagner congruence on $X$, we define the inverse monoid ${\mathrm{Inv}}^1⟨X|T⟩$ presented by $(X;T)$ as

$${\mathrm{Inv}}^1⟨X|T⟩={\left(X\coprod X^{-1}\right)}^{\ast }/{(T\cup {\rho }_X)}^{\mathrm{c}}.$$

In the previous dicussion, if we replace everywhere ${\left(X\coprod X^{-1}\right)}^{\ast }$ with ${\left(X\coprod X^{-1}\right)}^{+}$ we obtain a presentation (for an inverse semigroup) $(X;T)$ and an inverse semigroup $\mathrm{Inv}⟨X|T⟩$ 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 $\mathrm{FIM}(X)$ [resp. $\mathrm{FIS}(X)$] and is defined by

$$\mathrm{FIM}(X)={\mathrm{Inv}}^1⟨X|\mathrm{\varnothing }⟩={\left(X\coprod X^{-1}\right)}^{\ast }/{\rho }_X,\text{[resp. }\mathrm{FIS}(X)=\mathrm{Inv}⟨X|\mathrm{\varnothing }⟩={\left(X\coprod X^{-1}\right)}^{+}/{\rho }_X\text{]}.$$