Definition 2

Let M be a monoid and E and E' M-sets. A mapping f of E into E' such that, for all $x \in E$ and all $\alpha \in M$, $f(\alpha.x) = \alpha.f(x)$ is called an M-set homomorphism (or M-morphism, or mapping compatible with the operations of M).

The identity mapping of an M-set is an M-morphism. The composition of two M-morphisms is an M-morphism. For a mapping of one M-set into another to be an isomorphism, it is necessary and sufficient that it be a bijective M-morphism and the inverse mapping is then an M-morphism.

Let $(E_i)_{i \in I}$ be a family of M-sets and let E be the product set of $E_i$. The monoid M operates on E by $\alpha.(x_i)_{i \in I} = (\alpha.x_i)_{i \in I}$ and E, with this action, is an M-set; let E' be an M-set; a mapping f of E' into E is an M-morphism if and only if $pr_i \circ f$ is an M-morphism of E' into $E_i$ for all $i \in I$.

Let E be an M-set and F a stable subset of E under the action of M; with the induced law, F is an M-set and the canonical injection $F \to E$ is an M-morphism.

Let E be an M-set and R an equivalence relation on E compatible with the action of M; the quotient $E/R$ with the quotient action is an M-set and the canonical mapping $E \to E/R$ is an M-morphism.

Let $\phi : M \to M'$ be a monoid homomorphism, E an M-set and E' an $M'$-set. A mapping f of E into E' such that, for all $x \in E$ and $\alpha \in M$,

is called a $\phi$-morphism of E into E' (cf. $\S$ 3, no. 1).

Extension of a law of operation. Given (for example) three sets $F_1, F_2, F_3$, permutations $f_1, f_2, f_3$ of $F_1, F_2, F_3$ respectively and an echelon F on the base sets $F_1, F_2, F_3$ (Set Theory, IV, $\S$ 1, no. 1), we can define, proceeding step by step on the construction of the echelon F, a permutation of F called the canonical extension of $f_1, f_2, f_3$ to F (Set Theory, IV, $\S$ 1, no. 2); we shall denote it by $\phi_F(f_1, f_2, f_3)$.

Then let G be a group and $h_i$ a homomorphism of G into the symmetric group of $F_i$ ($i = 1, 2, 3$), in other words an operation of G on $F_i$. The mapping $x \mapsto x_F = \phi_F(h_1(x), h_2(x), h_3(x))$ is a homomorphism of G into $\mathfrak{S}_F$, in other words an operation of G on F, called the extension of $h_1, h_2, h_3$ to F. Let P be a subset of F such that, for all $x \in G$, $x_F(P) = P$; let $x_P$ be the restriction of $x_F$ to

P; then the mapping $x \mapsto x_P$ is an operation of G on P, also called the extension of $h_1, h_2, h_3$ to P.

For example, let K and L be two echelons on $F_1, F_2, F_3$; take F to be the set of subsets of $K \times L$ and P to be the set of mappings of K into L, identified with their graphs. If $w \in P$ and $x \in G$, $x_P(w)$ is the mapping $k \mapsto x_L(w(x_K^{-1}(k)))$ of K into L.