Transformation Semigroup - Cayley Representation

Cayley Representation

In group theory, Cayley's theorem asserts that any group G is isomorphic to a subgroup of the symmetric group of G (regarded as a set), so that G is a permutation group. This theorem generalizes straightforwardly to monoids: any monoid M is a transformation monoid of its underlying set, via the action given by left (or right) multiplication. This action is effective because if ax = bx for all x in M, then by taking x equal to the identity element, we have a = b.

For a semigroup S without a (left or right) identity element, we take X to be the underlying set of the monoid corresponding to S to realise S as a transformation semigroup of X. In particular any finite semigroup can be represented as a subsemigroup of transformations of a set X with |X| ≤ |S| + 1, and if S is a monoid, we have the sharper bound |X| ≤ |S|, as in the case of finite groups.