McAlister covering theorem

A subset X in an inverse semigroup S is called unitary if for any elements xX and sS, xsX or sxX implies sX.

An inverse semigroup is E-unitary if its semigroupPlanetmathPlanetmath of idempotentsMathworldPlanetmathPlanetmath is unitary.


Let S be an inverse semigroup; then, there exists an E-unitary inverse semigroup P and a surjectivePlanetmathPlanetmath, idempotent-separating homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath θ:PS.

Also, if S is finite, then P may be chosen to be finite as well.

Note that a homomorphism is idempotent-separating if it is injectivePlanetmathPlanetmath on idempotents.


  • 1 M. Lawson, Inverse Semigroups: The Theory of Partial SymmetriesPlanetmathPlanetmath, World Scientific, 1998
Title McAlister covering theorem
Canonical name McAlisterCoveringTheorem
Date of creation 2013-03-22 14:37:19
Last modified on 2013-03-22 14:37:19
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Theorem
Classification msc 20M18
Defines unitary
Defines E-unitary
Defines idempotent-separating