Example

(1) Let E be a set; the canonical action of $E^E$ on E (§ 3, no. 1, Example 3) is a left operation.

(2) Let M be a monoid. The left (resp. right) action of M on itself derived from the law on M (§ 3, no. 3, Example 7) is a left (resp. right) operation of M on itself. When considering this operation, we say that M operates on itself by left (resp. right) translation.

Let E be a left (resp. right) M-set and $M^0$ the opposite monoid to M. Under the same action, the monoid $M^0$ operates on E on the right (resp. left). The $M^0$-set obtained is called opposite to the M-set E. The definitions and results relating to left M-sets carry over to right $M^0$-sets when passing to the opposite structures.

In the rest of this paragraph, we shall consider, unless otherwise stated, only left M-sets which we shall call simply M-sets. Their law of action will be denoted by left multiplication.

Let E be a set. Let G be a group operating on E. For all $\alpha$ in G, the element of $E^E$ defined by $\alpha$ is a permutation of E ($\S$ 2, no. 3, Example 2). Being given an operation of G on E therefore amounts to being given a homomorphism of G into $\mathfrak{S}_E$.

In conformity with $\S$ 3, no. 3, we make the following definition: