Let $\tilde\rho_X\subseteq\doublep{X}$ be the binary relation on the free semigroup with involution $\doublep X$ defined by $$\tilde\rho_X=\gbra{(ww^{-1}w,w),\ (ww^{-1}vv^{-1},vv^{-1}ww^{-1})\,|\,v,w\in\doublep{X}}.$$ The Wagner congruence on $X$ is the congruence $\rho_X$ generated by $\tilde\rho_X$ , i.e. $\rho_X=(\tilde\rho_X)^\co$ .

A well known result of inverse semigroups theory says that the quotient $$\fis(X)=\doublep{X}/\rho_X$$ is an inverse semigroup. Moreover $\fis(X)$ 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 $\overline\Phi:\fis(X)\rightarrow S$ exists such that the following diagram commutes:

In analogous way, using the free monoid with involution $\doubles{X}$ instead of the free semigroup with involution $\doublep{X}$ , we obtain the inverse monoid $$\fim(X)=\doubles{X}/\rho_X,$$ that is the Free Inverse Monoid on $X$ .

Bibliography

1

N. Petrich, Inverse Semigroups, Wiley, New York, 1984.

2

V.V. Wagner, Generalized Groups, Dokl. Akad. Nauk SSSR 84 (1952), 1119-1122.