Presentations for P^K

TL;DR

The paper explores presentations of direct products of the semigroup of positive integers, providing explicit descriptions for P^K, highlighting differences from group and monoid cases.

Contribution

It offers an explicit presentation for P^K, the direct product of positive integer semigroups, extending known results and clarifying the structure of these semigroups.

Findings

Explicit presentation for P^K provided

Demonstrates differences from group and monoid direct products

Generalizes to arbitrary K for the semigroup P

Abstract

It is a classical result that the direct product AxB 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 P^2 is not finitely generated, and hence not finitely presented. In this note we give an explicit presentation for P^2 in terms of the unique minimal generating set; in fact, we do this more generally for P^K, the direct product of arbitrarily many copies of P.