In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using the semigroup operation) is associated with the composite of the two corresponding transformations.
The terminology conveys the idea that the elements of the semigroup are acting as transformations of the set.
From an algebraic perspective, a semigroup action is a generalization of the notion of a group action in group theory.
From the computer science point of view, semigroup actions are closely related to automata: the set models the state of the automaton and the action models transformations of that state in response to inputs.
An important special case is a monoid action or act, in which the semigroup is a monoid and the identity element of the monoid acts as the identity transformation of a set.
Another important special case is a transformation semigroup.
This concept is linked to the more general notion of a semigroup by an analogue of Cayley's theorem.
Right semigroup actions are defined in a similar way using an operation • : X × S → X satisfying (x • a) • b = x • (a ∗ b).
If M is a monoid, then a (left) monoid action (or act) of M is a (left) semigroup action of M with the additional property that where e is the identity element of M. This correspondingly gives a monoid homomorphism.
Right monoid actions are defined in a similar way.
A semigroup action of S on X can be made into monoid act by adjoining an identity to the semigroup and requiring that it acts as the identity transformation on X.
If S is a semigroup or monoid, then a set X on which S acts as above (on the left, say) is also known as a (left) S-act, S-set, S-action, S-operand, or left act over S. Some authors do not distinguish between semigroup and monoid actions, by regarding the identity axiom (e • x = x) as empty when there is no identity element, or by using the term unitary S-act for an S-act with an identity.
[1] The defining property of an act is analogous to the associativity of the semigroup operation, and means that all parentheses can be omitted.
It is common practice, especially in computer science, to omit the operations as well so that both the semigroup operation and the action are indicated by juxtaposition.
In this way strings of letters from S act on X, as in the expression stx for s, t in S and x in X.
[2] However, every right S-act can be interpreted as a left act over the opposite semigroup, which has the same elements as S, but where multiplication is defined by reversing the factors, s • t = t • s, so the two notions are essentially equivalent.
Here we primarily adopt the point of view of left acts.
It is often convenient (for instance if there is more than one act under consideration) to use a letter, such as
is a bijection, semigroup actions can be defined as functions
M-homomorphisms of M-acts, for M a monoid, are defined in exactly the same way.
For a fixed semigroup S, the left S-acts are the objects of a category, denoted S-Act, whose morphisms are the S-homomorphisms.
The corresponding category of right S-acts is sometimes denoted by Act-S. (This is analogous to the categories R-Mod and Mod-R of left and right modules over a ring.)
For a monoid M, the categories M-Act and Act-M are defined in the same way.
If we restrict it to faithful semigroup actions, it has nice properties.
This claim is made precise by the following observation: if we turn
Transformation semigroups are of essential importance for the structure theory of finite-state machines in automata theory.
In particular, a semiautomaton is a triple (Σ,X,T), where Σ is a non-empty set called the input alphabet, X is a non-empty set called the set of states and T is a function called the transition function.
Semiautomata arise from deterministic automata by ignoring the initial state and the set of accept states.
Then the semigroup of transformations of X generated by {Ta : a ∈ Σ} is called the characteristic semigroup or transition system of (Σ,X,T).
It is also sometimes viewed as a Σ∗-act on X, where Σ∗ is the free monoid of strings generated by the alphabet Σ,[note 1] and the action of strings extends the action of Σ via the property Krohn–Rhodes theory, sometimes also called algebraic automata theory, gives powerful decomposition results for finite transformation semigroups by cascading simpler components.