Cayley’s theorem for semigroups
Let be a set.
We can define on ,
the set of functions from to itself,
a structure of semigroup by putting
.
Such semigroup is actually a monoid,
whose identity element
is the identity function of .
Theorem 1 (Cayley’s theorem for semigroups)
For every semigroup
there exist a set
and an injective map
which is a morphism of semigroups from to .
In other words,
every semigroup is isomorphic to
a semigroup of transformations of some set.
This is an extension
of Cayley’s theorem on groups,
which states that every group is isomorphic to
a group of invertible
transformations
of some set.
Proof of Theorem 1.
The argument is similar to the one for Cayley’s theorem on groups.
Let , the set of elements of the semigroup.
First, suppose is a monoid with unit . For define as
(1) |
Then for every we have
so is a homomorphism of monoids,
with .
This homomorphism is injective
,
because if ,
then .
Next, suppose is a semigroup but not a monoid. Let . Construct a monoid by putting and defining
Then is isomorphic to a submonoid of as by (1). For put : then for every , , and is isomorphic to with .
Observe that the theorem remains valid if is defined as . In this case, the morphism is defined by .
Title | Cayley’s theorem for semigroups |
---|---|
Canonical name | CayleysTheoremForSemigroups |
Date of creation | 2013-03-22 19:04:37 |
Last modified on | 2013-03-22 19:04:37 |
Owner | Ziosilvio (18733) |
Last modified by | Ziosilvio (18733) |
Numerical id | 8 |
Author | Ziosilvio (18733) |
Entry type | Theorem |
Classification | msc 20M20 |
Classification | msc 20M15 |
Related topic | CayleysTheorem |