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 \({\mathbb {P}}\).

一个经典的结果是，当且仅当A和B都被有限生成（有限表示）时，两个组的直接乘积\（A \乘以B \）才是有限生成的（有限表示）。对于mono半词的直接乘积也是如此，但对于半群则不是。典型的（反）示例是A和B都是正整数的加法半群\（{\ mathbb {P}} = \ {1,2,3，\ ldots \} \）。在这里，\（{\ mathbb {P}} \）由单个元素自由生成，但是\（{\ mathbb {P}} ^ 2 \）不是有限生成的，因此也不是有限表示的。在本说明中，我们对\（{\ mathbb {P}} ^ 2 \）进行了明确的介绍。 就独特的最小发电机组而言；实际上，我们更普遍地针对 \（{\ mathbb {P}} ^ K \）（任意多个\（{\ mathbb {P}} \）的直接乘积执行此操作。