Wagner-Preston representation theorem
Let $S$ be an inverse semigroup and $X$ a set. An inverse semigroup homomorphism^{} $\varphi :S\to \Im (X)$, where $\Im (X)$ denotes the symmetric inverse semigroup, is called a representation of $S$ by bijective^{} partial maps on $X$. The representation is said to be faithful if $\varphi $ is a monomorphism^{}, i.e. it is injective^{}.
Given $s\in S$, we define ${\rho}_{s}\in \Im (S)$ as the bijective partial map with domain
$$\mathrm{dom}({\rho}_{s})=S{s}^{-1}=\{t{s}^{-1}|t\in S\}$$ |
and defined by
$${\rho}_{s}(t)=ts,\forall t\in \mathrm{dom}({\rho}_{s}).$$ |
Then the map $s\mapsto {\rho}_{s}$ 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.
References
- 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 |
---|---|
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 |