Wagner congruence
Let ${\stackrel{~}{\rho}}_{X}\subseteq {\left(X\cup {X}^{-1}\right)}^{+}$ be the binary relation^{} on the free semigroup with involution ${\left(X\cup {X}^{-1}\right)}^{+}$ defined by
$${\stackrel{~}{\rho}}_{X}=\{(w{w}^{-1}w,w),(w{w}^{-1}v{v}^{-1},v{v}^{-1}w{w}^{-1})|v,w\in {\left(X\cup {X}^{-1}\right)}^{+}\}.$$ |
The Wagner congruence on $X$ is the congruence^{} ${\rho}_{X}$ generated by ${\stackrel{~}{\rho}}_{X}$, i.e. ${\rho}_{X}={({\stackrel{~}{\rho}}_{X})}^{\mathrm{c}}$.
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:
$$\text{xymatrix}\mathrm{\&}X\text{ar}{[r]}^{\iota}\text{ar}{[d]}_{\mathrm{\Phi}}\mathrm{\&}\mathrm{FIS}(X)\text{ar}{[dl]}^{\overline{\mathrm{\Phi}}}\mathrm{\&}S\mathrm{\&}$$ |
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 $X$.
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 |
---|---|
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 |