Presentations for Pk

It is a classical result that the direct product A × 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 P = {1, 2, 3,...} of positive integers. Here P is freely generated by a single element, but P² is not finitely generated, and hence not finitely presented. In this note we give an explicit presentation for P² in terms of the unique minimal generating set; in fact, we do this more generally for Pᴷ, the direct product of arbitrarily many copies of P.