Presentations for $${\mathbb {P}}^K$$

It is a classical result that the direct product $$A\times B$$ of two groups is finitely generated (finitely presented) if and only if A and B are both finitely generated (finitely presented). This is also true for direct products of monoids, but not for semigroups. The typical (counter) example is when A and B are both the additive semigroup $${\mathbb {P}}=\{1,2,3,\ldots \}$$ of positive integers. Here $${\mathbb {P}}$$ is freely generated by a single element, but $${\mathbb {P}}^2$$ is not finitely generated, and hence not finitely presented. In this note we give an explicit presentation for $${\mathbb {P}}^2$$ in terms of the unique minimal generating set; in fact, we do this more generally for $${\mathbb {P}}^K$$ , the direct product of arbitrarily many copies of $${\mathbb {P}}$$ .