Proposition 1

Let $E$ and $F$ be two magmas and $f$ a homomorphism of $E$ into $F$. (i) The image under $f$ of a stable subset of $E$ is a stable subset of $F$. (ii) The inverse image under $f$ of a stable subset $F$ is a stable subset of $E$. (iii) Let $X$ be a subset of $E$. The image under $f$ of the stable subset of $E$ generated by $X$ is the stable subset of $F$ generated by $f(X)$. (iv) If $g$ is a second homomorphism of $E$ from $F$ the set of elements $x$ of $E$ such that $f(x) = g(x)$ is a stable subset of $E$.

Assertions (i), (ii) and (iv) are obvious; we prove (iii). Let $\overline{X}$ be the stable subset of E generated by X and $\overline{f(X)}$ the stable subset of F generated by $f(X)$. By (i) $\overline{f(X)} \subset f(\overline{X})$ and by (ii) $\overline{X} \subset f^{-1}(\overline{f(X)})$, whence $f(\overline{X}) \subset \overline{f(X)}$.