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Essentially periodic ordered groups

Françoise Point & Frank O. Wagner

Annals of Pure and Applied Logic 105 (1-3):261-291 (2000)

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Presburger sets and p-minimal fields.Raf Cluckers - 2003 - Journal of Symbolic Logic 68 (1):153-162.details We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger (...) language. (shrink) Download Export citation Bookmark 11 citations
Coset-minimal groups.Oleg Belegradek, Viktor Verbovskiy & Frank O. Wagner - 2003 - Annals of Pure and Applied Logic 121 (2-3):113-143.details A totally ordered group G is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G{±∞}. Continuing work in Belegradek et al. 1115) and Point and Wagner 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular rank. Any (...) Download Export citation Bookmark 3 citations
Expansions of Presburger arithmetic with the exchange property.Nathanaël Mariaule - 2021 - Mathematical Logic Quarterly 67 (4):409-419.details Let G be a model of Presburger arithmetic. Let be an expansion of the language of Presburger. In this paper, we prove that the ‐theory of G is ‐minimal iff it has the exchange property and is definably complete (i.e., any bounded definable set has a maximum). If the ‐theory of G has the exchange property but is not definably complete, there is a proper definable convex subgroup H. Assuming that the induced theories on H and are definable complete and (...) Download Export citation Bookmark
Ordered abelian groups that do not have elimination of imaginaries.Martina Liccardo - forthcoming - Archive for Mathematical Logic:1-17.details We investigate the property of elimination of imaginaries for some special cases of ordered abelian groups. We show that certain Hahn products of ordered abelian groups do not eliminate imaginaries in the pure language of ordered groups. Moreover, we prove that, adding finitely many constants to the language of ordered abelian groups, the theories of the finite lexicographic products $$\mathbb {Z}^n$$ and $$\mathbb {Z}^n \times \mathbb {Q}$$ have definable Skolem functions. Download Export citation Bookmark
Integration and cell decomposition in p-minimal structures.Pablo Cubides Kovacsics & Eva Leenknegt - 2016 - Journal of Symbolic Logic 81 (3):1124-1141.details Download Export citation Bookmark 4 citations

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