Cayley’s theorem for semigroups
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.
First, suppose is a monoid with unit . For define as
Then for every we have
Next, suppose is a semigroup but not a monoid. Let . Construct a monoid by putting and defining
Observe that the theorem remains valid if is defined as . In this case, the morphism is defined by .
|Title||Cayley’s theorem for semigroups|
|Date of creation||2013-03-22 19:04:37|
|Last modified on||2013-03-22 19:04:37|
|Last modified by||Ziosilvio (18733)|