General properties? of divisibility of semigroups

Question

Referencing Grillet's Semigroups pp 151.

Problem Prop. V.3 3.2 Grillet asserts that when $A,S$ are semigroups, then $A$ and $S$ both divide $A\times S$. (He offers no explanation, it's supposedly trivial)

Nothing to see here.

Answer 1

Suppose that $S_1$ and $S_2$ are nonempty. Then the projections from $S_1 \times S_2$ to $S_1$ and to $S_2$ are both surjective morphisms. Thus $S_1$ and $S_2$ divide $S_1 \times S_2$.