Wagner-Preston representation theorem
Let be an inverse semigroup and a set. An inverse semigroup homomorphism , where denotes the symmetric inverse semigroup, is called a representation of by bijective partial maps on . The representation is said to be faithful if is a monomorphism, i.e. it is injective.
Given , we define as the bijective partial map with domain
and defined by
Then the map is a representation called the Wagner-Preston representation of . 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 inverse semi-groups, J. London Math. Soc. 29 (1954), 411-419.
|Title||Wagner-Preston representation theorem|
|Date of creation||2013-03-22 16:11:16|
|Last modified on||2013-03-22 16:11:16|
|Last modified by||Mazzu (14365)|
|Defines||representation by bijective partial maps|