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Weakly one-based geometric theories

Alexander Berenstein & Evgueni Vassiliev

Journal of Symbolic Logic 77 (2):392-422 (2012)

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Fields with a dense-codense linearly independent multiplicative subgroup.Alexander Berenstein & Evgueni Vassiliev - 2020 - Archive for Mathematical Logic 59 (1-2):197-228.details We study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \\), satisfying a density/codensity condition. Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and (...) Download Export citation Bookmark 3 citations
Weak One-Basedness.Gareth Boxall, David Bradley-Williams, Charlotte Kestner, Alexandra Omar Aziz & Davide Penazzi - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):435-448.details We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geometries arising from infinite-dimensional vector spaces over division rings. Download Export citation Bookmark
Supersimple structures with a dense independent subset.Alexander Berenstein, Juan Felipe Carmona & Evgueni Vassiliev - 2017 - Mathematical Logic Quarterly 63 (6):552-573.details Based on the work done in [][] in the o‐minimal and geometric settings, we study expansions of models of a supersimple theory with a new predicate distiguishing a set of forking‐independent elements that is dense inside a partial type, which we call H‐structures. We show that any two such expansions have the same theory and that under some technical conditions, the saturated models of this common theory are again H‐structures. We prove that under these assumptions the expansion is supersimple and (...) Download Export citation Bookmark
Generic trivializations of geometric theories.Alexander Berenstein & Evgueni Vassiliev - 2014 - Mathematical Logic Quarterly 60 (4-5):289-303.details We study the theory of the structure induced by parameter free formulas on a “dense” algebraically independent subset of a model of a geometric theory T. We show that while being a trivial geometric theory, inherits most of the model theoretic complexity of T related to stability, simplicity, rosiness, the NIP and the NTP2. In particular, we show that T is strongly minimal, supersimple of SU‐rank 1, has the NIP or the NTP2 exactly when has these properties. We show that (...) Download Export citation Bookmark 2 citations

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