Wagner congruence

Let ${\tilde{\rho }}_X\subseteq {\left(X\cup X^{-1}\right)}^{+}$ be the binary relation on the free semigroup with involution ${\left(X\cup X^{-1}\right)}^{+}$ defined by

$${\tilde{\rho }}_X=\{(w{w}^{-1}w,w),(w{w}^{-1}v{v}^{-1},v{v}^{-1}w{w}^{-1})|v,w\in {\left(X\cup X^{-1}\right)}^{+}\}.$$

The Wagner congruence on $X$ is the congruence ${\rho }_X$ generated by ${\tilde{\rho }}_X$, i.e. ${\rho }_X={({\tilde{\rho }}_X)}^{\mathrm{c}}$.

A well known result of inverse semigroups theory says that the quotient

$$\mathrm{FIS}(X)={\left(X\cup X^{-1}\right)}^{+}/{\rho }_X$$

is an inverse semigroup. Moreover $\mathrm{FIS}(X)$ is the Free Inverse Semigroup on $X$, in the sense that it resolve the following universal mapping problem: given an inverse semigroup $S$ and a map $\mathrm{\Phi }:X\to S$, a unique inverse semigroups homomorphism $\overline{\mathrm{\Phi }}:\mathrm{FIS}(X)\to S$ exists such that the following diagram commutes:

where $\iota :X\to \mathrm{FIS}(X)$ is the projection to the quotient, i.e. $\iota (x)={[x]}_{{\rho }_X}$. It is well known from universal algebra that $\mathrm{FIS}(X)$ is unique up to isomorphisms.

In analogous way, using the free monoid with involution ${\left(X\cup X^{-1}\right)}^{\ast }$ instead of the free semigroup with involution ${\left(X\cup X^{-1}\right)}^{+}$, we obtain the inverse monoid

$$\mathrm{FIM}(X)={\left(X\cup X^{-1}\right)}^{\ast }/{\rho }_X,$$

that is the Free Inverse Monoid on $X$.