Wagner-Preston representation theorem

Let S be an inverse semigroup and X a set. An inverse semigroup homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ϕ:S(X), where (X) denotes the symmetric inverse semigroup, is called a representation of S by bijectiveMathworldPlanetmathPlanetmath partial maps on X. The representation is said to be faithful if ϕ is a monomorphismMathworldPlanetmathPlanetmath, i.e. it is injectivePlanetmathPlanetmath.

Given sS, we define ρs(S) as the bijective partial map with domain


and defined by


Then the map sρs is a representation called the Wagner-Preston representation of S. The following result, due to Wagner and Preston, is analogous to the Cayley representation theorem for groups.

Theorem 1 (Wagner-Preston representation theorem)

The Wagner-Preston representation of an inverse semigroup is faithful.


  • 1 N. Petrich, Inverse Semigroups, Wiley, New York, 1984.
  • 2 G.B. Preston, Representation of inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath semi-groups, J. London Math. Soc. 29 (1954), 411-419.
Title Wagner-Preston representation theorem
Canonical name WagnerPrestonRepresentationTheorem
Date of creation 2013-03-22 16:11:16
Last modified on 2013-03-22 16:11:16
Owner Mazzu (14365)
Last modified by Mazzu (14365)
Numerical id 10
Author Mazzu (14365)
Entry type Theorem
Classification msc 20M18
Defines representation by bijective partial maps
Defines faithful representation
Defines Wagner-Preston representation