Proposition 8

Let E be a magma, A a stable subset of E and R an equivalence relation on E compatible with the law on E. The saturation B of A with respect to R (Set

Theory, II, , no. 5) is a stable subset. The equivalence relations $R_A$ and $R_B$ induced by $R$ on $A$ and $B$ respectively are compatible with the induced laws and the mapping derived from the canonical injection of $A$ into $B$ by passing to the quotients is a magma isomorphism of $A / R_A$ onto $B / R_B$

Let $\top$ denote the law on $E$. If $x$ and $y$ are two elements of $B$ there exist two elements $x'$ and $y'$ of $A$ such that $x \equiv x' \pmod{R}$ and $y \equiv y' \pmod{R}$; then $x \top y \equiv x' \top y' \pmod{R}$ and $x' \top y' \in A$, whence $x \top y \in B$. Thus $B$ is a stable subset of $E$ and the other assertions are obvious.

Let $M$ be a magma and $((u_\alpha, v_\alpha))_{\alpha \in I}$ a family of elements of $M \times M$. Consider all the equivalence relations $S$ on $M$ which are compatible with the law on $M$ and such that $u_\alpha \equiv v_\alpha \pmod{S}$ for all $\alpha \in I$. The intersection of the graphs of these relations is the graph of an equivalence relation $R$ which is compatible with the law on $M$ and such that $u_\alpha \equiv v_\alpha \pmod{R}$. Hence $R$ is the finest (Set Theory, III, § 1, nos. 3 and 7) equivalence relation with these two properties. It is called the equivalence relation compatible with the law on $M$ generated by the $(u_\alpha, v_\alpha)$.