presentation of inverse monoids and inverse semigroups

Let (XX-1) be the free monoid with involution on X, and T(XX-1)×(XX-1) be a binary relationMathworldPlanetmath between words. We denote by Te [resp. Tc] the equivalence relationMathworldPlanetmath [resp. congruencePlanetmathPlanetmathPlanetmathPlanetmath] generated by T.

A presentationMathworldPlanetmathPlanetmathPlanetmath (for an inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath monoid) is a couple (X;T). We use this couple of objects to define an inverse monoid Inv1X|T. Let ρX be the Wagner congruence on X, we define the inverse monoid Inv1X|T presented by (X;T) as


In the previous dicussion, if we replace everywhere (XX-1) with (XX-1)+ we obtain a presentation (for an inverse semigroup) (X;T) and an inverse semigroup InvX|T presented by (X;T).

A trivial but important example is the Free Inverse Monoid [resp. Free Inverse Semigroup] on X, that is usually denoted by FIM(X) [resp. FIS(X)] and is defined by

FIM(X)=Inv1X|=(XX-1)/ρX,[resp. FIS(X)=InvX|=(XX-1)+/ρX].


  • 1 N. Petrich, Inverse Semigroups, Wiley, New York, 1984.
  • 2 J.B. Stephen, Presentation of inverse monoids, J. Pure Appl. AlgebraMathworldPlanetmathPlanetmathPlanetmath 63 (1990) 81-112.
Title presentation of inverse monoids and inverse semigroups
Canonical name PresentationOfInverseMonoidsAndInverseSemigroups
Date of creation 2013-03-22 16:11:01
Last modified on 2013-03-22 16:11:01
Owner Mazzu (14365)
Last modified by Mazzu (14365)
Numerical id 10
Author Mazzu (14365)
Entry type Definition
Classification msc 20M05
Classification msc 20M18
Synonym presentation
Synonym generators and relators