transformation semigroup

Another important special case is a transformation semigroup.

Any transformation semigroup can be turned into a semigroup action by the following construction.

Any full transformation semigroup is regular.

A transformation semigroup of a set has a tautological semigroup action on that set.

A transformation semigroup can be made into an operator monoid by adjoining the identity transformation.

Conversely, for any semigroup action of on , define a transformation semigroup .

For the transition functions, this monoid is known as the transition monoid, or sometimes the transformation semigroup.

An analogue of Cayley's theorem shows that any semigroup can be realized as a transformation semigroup of some set.

In algebra, a transformation semigroup (or composition semigroup) is a collection of functions from a set to itself which is closed under function composition.

Analogous to a permutation group having elements that are permutations, a transformation semigroup has elements that are transformations.

Obviously any transformation semigroup S determines a transformation monoid M by taking the union of S with the identity transformation.

If (X,S) is a transformation semigroup then X can be made into a semigroup action of S by evaluation:

If s and t are two different functions of the transformation semigroup, then the semigroup product follows trivially from function composition, so that one defines as .

A transformation semigroup or transformation monoid is a pair consisting of a set Q (often called the "set of states") and a semigroup or monoid M of functions, or "transformations", mapping Q to itself.

Since the notion of functions acting on a set always includes the notion of an identity function, which when applied to the set does nothing, a transformation semigroup can be made into a monoid by taking its union with the identity function.

Here, however, the exponent n no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit: the monoid of the Picard sequence (cf. transformation semigroup) has generalized to a full continuous group.