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.
Read more about this topic: Monoid
Famous quotes containing the word acts:
“I fear I agree with your friend in not liking all sermons. Some of them, one has to confess, are rubbish: but then I release my attention from the preacher, and go ahead in any line of thought he may have started: and his after-eloquence acts as a kind of accompaniment—like music while one is reading poetry, which often, to me, adds to the effect.”
—Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)