Monoid - Acts and Operator Monoids

Acts and Operator Monoids

Let M be a monoid, with the binary operation denoted by “•” and the identity element denoted by e. Then a (left) M-act (or left act over M) is a set X together with an operation ⋅ : M × X → X which is compatible with the monoid structure as follows:

• for all x in X: e ⋅ x = x;
• for all a, b in M and x in X: a ⋅ (b ⋅ x) = (a • b) ⋅ x.

This is the analogue in monoid theory of a (left) group action. Right M-acts are defined in a similar way. A monoid with an act is also known as an operator monoid. Important examples include transition systems of semiautomata. A transformation semigroup can be made into an operator monoid by adjoining the identity transformation.