Definition 1

Let M be a monoid, with law written multiplicatively and identity element denoted by e, and E a set. An action $\alpha \mapsto f_\alpha$ of M on E is called a left (resp. right) operation of M on E if $f_e = \mathrm{Id}_E$ and $f_{\alpha \beta} = f_\alpha \circ f_\beta$ (resp. $f_{\alpha \beta} = f_\beta \circ f_\alpha$) for all $\alpha, \beta \in M$.

In other words, a left (resp. right) operation of a monoid M on a set E is a monoid homomorphism of M into the monoid $E^E$ (resp. the opposite monoid of $E^E$) with composition of mappings. If the law of action corresponding to the action of M is denoted by left (resp. right) multiplication, the fact that this action is a left (resp. right) operation may be expressed by the formulae

(1)

(resp. $x.e = x; \quad (x.\alpha).\beta = x.(\alpha \beta) \quad \text{for } \alpha, \beta \in M \text{ and } x \in E$).

Under these conditions, it is also said that M operates on E on the left (resp. right) and that the corresponding laws of action are laws of left (resp. right) operation of the monoid M on E.

Let M be a monoid; a set E with a left (resp. right) operation of M on E is called a left (resp. right) M-set. The monoid M is said to operate on the left (resp. right) faithfully if the mapping $\alpha \mapsto f_\alpha$ of M into $E^E$ is injective.