Th (N，·)$ \operatorname{Th}(\mathbb {N}，\cdot)$

Pub Date : 2022-04-28 DOI:10.1002/malq.202100049

Atticus Stonestrom

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引用次数: 1

摘要

“Skolem算术”是乘法单群(N，·)$ (\mathbb {N}，\cdot)$的完备理论T。给出了T的 \ \var \ \可定义稳定嵌入集的完整刻划，特别证明了在具有相同可定义闭包的关系之前，只有一个非平凡的闭包:无平方元的集合。然后我们证明了T有弱消虚数，但没有消有限虚数。

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Some model theory of Th ( N , · ) $\operatorname{Th}(\mathbb {N},\cdot )$

‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid $(N, \cdot)$ . We give a full characterization of the $⌀$ -definable stably embedded sets of T, showing in particular that, up to the relation of having the same definable closure, there is only one non-trivial one: the set of squarefree elements. We then prove that T has weak elimination of imaginaries but not elimination of finite imaginaries.

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