 A semigroup \(H\) is a subsemigroup of a semigroup \(G\) when \(H\) is a subset of \(G\) and the multiplication of \(G\) restricted to \(H\) coincides with that of \(H\). Equivalently a subsemigroup of \(G\) is a subset \(H\) of \(G\) such that for all \(x, y\) in \(H, xy\) is in \(H\).