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Interpretable groups are definable

Pantelis E. Eleftheriou, Ya'acov Peterzil & Janak Ramakrishnan

Journal of Mathematical Logic 14 (1):1450002 (2014)

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Fundamental group in o-minimal structures with definable Skolem functions.Bruno Dinis, Mário J. Edmundo & Marcello Mamino - 2021 - Annals of Pure and Applied Logic 172 (8):102975.details In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allow us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally (...) Download Export citation Bookmark
Orthogonal Decomposition of Definable Groups.Alessandro Berarducci, Pantelis E. Eleftheriou & Marcello Mamino - forthcoming - Journal of Symbolic Logic:1-22.details Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal (...) Download Export citation Bookmark
Topologizing Interpretable Groups in p-Adically Closed Fields.Will Johnson - 2023 - Notre Dame Journal of Formal Logic 64 (4):571-609.details We consider interpretable topological spaces and topological groups in a p-adically closed field K. We identify a special class of “admissible topologies” with topological tameness properties like generic continuity, similar to the topology on definable subsets of Kn. We show that every interpretable set has at least one admissible topology, and that every interpretable group has a unique admissible group topology. We then consider definable compactness (in the sense of Fornasiero) on interpretable groups. We show that an interpretable group is (...) Download Export citation Bookmark 2 citations
Euler characteristic of imaginaries in o-minimal structures.Sofya Kamenkovich & Ya'acov Peterzil - 2017 - Mathematical Logic Quarterly 63 (5):376-383.details We define the notion of Euler characteristic for definable quotients in an arbitrary o-minimal structure and prove some fundamental properties. Download Export citation Bookmark 1 citation
Interpretable Sets in Dense o-Minimal Structures.Will Johnson - 2018 - Journal of Symbolic Logic 83 (4):1477-1500.details We give an example of a dense o-minimal structure in which there is a definable quotient that cannot be eliminated, even after naming parameters. Equivalently, there is an interpretable set which cannot be put in parametrically definable bijection with any definable set. This gives a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Additionally, we show that interpretable sets in dense o-minimal structures admit definable topologies which are “tame” in several ways: (a) they are Hausdorff, (b) every point (...) Download Export citation Bookmark 4 citations

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