Stable Reducts of Elementary Extensions of Presburger Arithmetic
Eran Alouf, Antongiulio Fornasiero, Itay Kaplan
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Source: Crossref
Published: Oct 10, 2026
DOI: 10.1142/s0219061326500182
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Suppose [Formula: see text] is elementarily equivalent to an archimedean ordered abelian group [Formula: see text] with small quotients (for all [Formula: see text], [Formula: see text] is finite). Then every stable reduct of [Formula: see text] which expands [Formula: see text] (equivalently every reduct that does not add new unary definable sets) is interdefinable with [Formula: see text]. This extends previous results on stable reducts of [Formula: see text] to (stable) reducts of elementary extensions of [Formula: see text]. In particular this holds for [Formula: see text] and [Formula: see text]. As a result we answer a question of Conant from 2018. This result is a corollary of a more general statement about expansions of weakly-minimal 1-based expansions of abelian groups with small quotients preserving the acl operator.
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