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 $X$.

Title	Wagner congruence
Canonical name	WagnerCongruence
Date of creation	2013-03-22 16:11:07
Last modified on	2013-03-22 16:11:07
Owner	Mazzu (14365)
Last modified by	Mazzu (14365)
Numerical id	15
Author	Mazzu (14365)
Entry type	Definition
Classification	msc 20M05
Classification	msc 20M18
Defines	Wagner congruence
Defines	free inverse semigroup
Defines	free inverse monoid