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. From a category theoretic point of view, a monoid is a category with one object, and an act is a functor from that category to the category of sets. This immediately provides a generalization to monoid acts on objects in categories other than the category of sets. Another important special case is a transformation semigroup. This is a semigroup of transformations of a set, and hence it has a tautological action on that set. This concept is linked to the more general notion of a semigroup by an analogue of Cayley's theorem. (A note on terminology: the terminology used in this area varies, sometimes significantly, from one author to another. See the article for details.)
Formal definitions Let S be a semigroup. Then a (left) semigroup action (or act) of S is a set X together with an operation • : S × X → X which is compatible with the semigroup operation ∗ as follows:
for all s, t in S and x in X, s • (t • x) = (s ∗ t) • x. This is the analogue in semigroup theory of a (left) group action, and is equivalent to a semigroup homomorphism into the set of functions on X. 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
for all x in X: e • x = x where e is the identity element of M. This correspondingly gives a monoid homomorphism. Right monoid actions are defined in a similar way. A monoid M with an action on a set is also called an operator monoid. 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.
Terminology and notation 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. 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. It is also quite common to work with right acts rather than left acts. 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.
Acts and transformations It is often convenient (for instance if there is more than one act under consideration) to use a letter, such as T {\displaystyle T} , to denote the function
T : S × X → X {\displaystyle T\colon S\times X\to X}
defining the S {\displaystyle S} -action and hence write T ( s , x ) {\displaystyle T(s,x)} in place of s ⋅ x {\displaystyle s\cdot x} . Then for any s {\displaystyle s} in S {\displaystyle S} , we denote by
T s : X → X {\displaystyle T_{s}\colon X\to X}
the transformation of X {\displaystyle X} defined by
T s ( x ) = T ( s , x ) . {\displaystyle T_{s}(x)=T(s,x).}
By the defining property of an S {\displaystyle S} -act, T {\displaystyle T} satisfies
T s ∗ t = T s ∘ T t . {\displaystyle T_{s*t}=T_{s}\circ T_{t}.}
Further, consider a function s ↦ T s {\displaystyle s\mapsto T_{s}} . It is the same as curry ( T ) : S → ( X → X ) {\displaystyle \operatorname {curry} (T):S\to (X\to X)} (see Currying). Because curry {\displaystyle \operatorname {curry} } is a bijection, semigroup actions can be defined as functions S → ( X → X ) {\displaystyle S\to (X\to X)} which satisfy
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