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Some model theory of Th ( N , · ) $\operatorname{Th}(\mathbb {N},\cdot )$
Author(s) -
Stonestrom Atticus
Publication year - 2022
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.202100049
Subject(s) - mathematics , multiplicative function , monoid , closure (psychology) , square free integer , combinatorics , characterization (materials science) , discrete mathematics , mathematical analysis , physics , law , optics , political science
Abstract ‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid( N , · ) $(\mathbb {N},\cdot )$ . We give a full characterization of the ⌀ $\varnothing$ ‐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.