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THE ADDITIVE GROUPS OF $\mathbb {Z}$ AND $\mathbb {Q}$ WITH PREDICATES FOR BEING SQUARE-FREE

Published online by Cambridge University Press:  05 October 2020

NEER BHARDWAJ
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGNURBANA, IL61801, USAE-mail: nbhard4@illinois.edu
CHIEU-MINH TRAN
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAMENOTRE DAME, IN46556, USAE-mail: mtran6@nd.edu

Abstract

We consider the structures $(\mathbb {Z}; \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Z}; <, \mathrm {SF}^{\mathbb {Z}})$ , $(\mathbb {Q}; \mathrm {SF}^{\mathbb {Q}})$ , and $(\mathbb {Q}; <, \mathrm {SF}^{\mathbb {Q}})$ where $\mathbb {Z}$ is the additive group of integers, $\mathrm {SF}^{\mathbb {Z}}$ is the set of $a \in \mathbb {Z}$ such that $v_{p}(a) < 2$ for  every prime p and corresponding p-adic valuation $v_{p}$ , $\mathbb {Q}$ and $\mathrm {SF}^{\mathbb {Q}}$ are defined likewise for rational numbers, and $<$ denotes the natural ordering on each of these domains. We prove that the second structure is model-theoretically wild while the other three structures are model-theoretically tame. Moreover, all these results can be seen as examples where number-theoretic randomness yields model-theoretic consequences.

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Article
Copyright
© Association for Symbolic Logic 2020

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