# 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

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

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 $\Phi:X\rightarrow S$, a unique inverse semigroups homomorphism $\overline{\Phi}:\mathrm{FIS}(X)\rightarrow S$ exists such that the following diagram commutes:

 $\xymatrix{&X\ar[r]^{\iota}\ar[d]_{\Phi}&\mathrm{FIS}(X)\ar[dl]^{\overline{\Phi% }}\\ &S&}$

where $\iota:X\rightarrow\mathrm{FIS}(X)$ 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 $X$.

## References

• 1 N. Petrich, Inverse Semigroups, Wiley, New York, 1984.
• 2 V.V. Wagner, Generalized Groups, Dokl. Akad. Nauk SSSR 84 (1952), 1119-1122.
Title Wagner congruence WagnerCongruence 2013-03-22 16:11:07 2013-03-22 16:11:07 Mazzu (14365) Mazzu (14365) 15 Mazzu (14365) Definition msc 20M05 msc 20M18 Wagner congruence free inverse semigroup free inverse monoid