Exercise 2, § 7

Let $X$ be a set and $M$ one of the magmas $M(X), Mo(X)$ or $N^{(X)}$. Show that every automorphism of $M$ leaves $X$ stable. Deduce an isomorphism of $\mathfrak{S}_X$ onto the group $\mathrm{Aut}(M)$.

Show that the endomorphisms of $M$ correspond bijectively to the mappings of $X$ into $M$.