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Presentations for $${\mathbb {P}}^K$$ P K
Monatshefte für Mathematik ( IF 0.8 ) Pub Date : 2021-05-26 , DOI: 10.1007/s00605-021-01575-z
James East

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



中文翻译:

$$ {\ mathbb {P}} ^ K $$ PK的演示文稿

一个经典的结果是,当且仅当AB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AB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。

更新日期:2021-05-26
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