We will describe a new notion of the crossed product of a C ∗ -algebra by the action of a semigroup of endomorphisms. This notion extends our earlier notion of crossed product by a single endomorphism for which one has chosen a transfer operator. We will give examples to illustrate the construction, some of which may also be described as groupoid C ∗ -algebras. Time permitting we will also show a curious counter-example based on the theory of cellular automata.