There are no intermediate structures between the group of integers and Presburger arithmetic
classification
🧮 math.LO
keywords
mathbbmathcalarithmeticexpansionfirst-ordergroupintegersinterdefinable
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We show that if a first-order structure $\mathcal{M}$, with universe $\mathbb{Z}$, is an expansion of $(\mathbb{Z},+,0)$ and a reduct of $(\mathbb{Z},+,<,0)$, then $\mathcal{M}$ must be interdefinable with $(\mathbb{Z},+,0)$ or $(\mathbb{Z},+,<,0)$.
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