Wagner congruence

Let ρ~X(XX-1)+ be the binary relationMathworldPlanetmath on the free semigroup with involution (XX-1)+ defined by


The Wagner congruence on X is the congruencePlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ρX generated by ρ~X, i.e. ρX=(ρ~X)c.

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


is an inverse semigroup. Moreover FIS(X) is the Free Inverse Semigroup on X, in the sense that it resolve the following universalPlanetmathPlanetmathPlanetmath mapping problem: given an inverse semigroup S and a map Φ:XS, a unique inverse semigroups homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath Φ¯:FIS(X)S exists such that the following diagram commutes:


where ι:XFIS(X) is the projection to the quotient, i.e. ι(x)=[x]ρX. It is well known from universal algebraMathworldPlanetmathPlanetmath that FIS(X) is unique up to isomorphismsMathworldPlanetmathPlanetmath.

In analogous way, using the free monoid with involution (XX-1) instead of the free semigroup with involution (XX-1)+, we obtain the inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath monoid


that is the Free Inverse Monoid on X.


  • 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