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Evgueni Vassiliev [10]Evgueni V. Vassiliev [1]
  1.  17
    On Lovely Pairs of Geometric Structures.Alexander Berenstein & Evgueni Vassiliev - 2010 - Annals of Pure and Applied Logic 161 (7):866-878.
    We study the theory of lovely pairs of geometric structures, in particular o-minimal structures. We use the pairs to isolate a class of geometric structures called weakly locally modular which generalizes the class of linear structures in the settings of SU-rank one theories and o-minimal theories. For o-minimal theories, we use the Peterzil–Starchenko trichotomy theorem to characterize for a sufficiently general point, the local geometry around it in terms of the thorn U-rank of its type inside a lovely pair.
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  2.  8
    Lovely Pairs of Models.Itay Ben-Yaacov, Anand Pillay & Evgueni Vassiliev - 2003 - Annals of Pure and Applied Logic 122 (1-3):235-261.
    We introduce the notion of a lovely pair of models of a simple theory T, generalizing Poizat's “belles paires” of models of a stable theory and the third author's “generic pairs” of models of an SU-rank 1 theory. We characterize when a saturated model of the theory TP of lovely pairs is a lovely pair , finding an analog of the nonfinite cover property for simple theories. We show that, under these hypotheses, TP is also simple, and we study forking (...)
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  3.  1
    Generic Pairs of SU-Rank 1 Structures.Evgueni Vassiliev - 2003 - Annals of Pure and Applied Logic 120 (1-3):103-149.
    For a supersimple SU-rank 1 theory T we introduce the notion of a generic elementary pair of models of T . We show that the theory T* of all generic T-pairs is complete and supersimple. In the strongly minimal case, T* coincides with the theory of infinite dimensional pairs, which was used in 1184–1194) to study the geometric properties of T. In our SU-rank 1 setting, we use T* for the same purpose. In particular, we obtain a characterization of linearity (...)
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  4.  43
    Weakly One-Based Geometric Theories.Alexander Berenstein & Evgueni Vassiliev - 2012 - Journal of Symbolic Logic 77 (2):392-422.
    We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: "weak one-basedness", absence of type definable "almost quasidesigns", and "generic linearity". Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω-categorical geometric theory interprets an infinite vector space over (...)
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  5. Generic Trivializations of Geometric Theories.Alexander Berenstein & Evgueni Vassiliev - 2014 - Mathematical Logic Quarterly 60 (4-5):289-303.
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  6.  4
    On Pseudolinearity and Generic Pairs.Evgueni Vassiliev - 2010 - Mathematical Logic Quarterly 56 (1):35-41.
    We continue the study of the connection between the “geometric” properties of SU -rank 1 structures and the properties of “generic” pairs of such structures, started in [8]. In particular, we show that the SU-rank of the theory of generic pairs of models of an SU -rank 1 theory T can only take values 1 , 2 or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to (...)
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  7.  3
    Supersimple Structures with a Dense Independent Subset.Alexander Berenstein, Juan Felipe Carmona & Evgueni Vassiliev - 2017 - Mathematical Logic Quarterly 63 (6):552-573.
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  8.  8
    On Lovely Pairs and the (∃ y ∈ P ) Quantifier.Anand Pillay & Evgueni Vassiliev - 2005 - Notre Dame Journal of Formal Logic 46 (4):491-501.
    Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.
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  9.  6
    On the Weak Non-Finite Cover Property and the N-Tuples of Simple Structures.Evgueni Vassiliev - 2005 - Journal of Symbolic Logic 70 (1):235 - 251.
    The weak non-finite cover property (wnfcp) was introduced in [1] in connection with "axiomatizability" of lovely pairs of models of a simple theory. We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the wnfcp is preserved when passing from a simple theory T to the theory TP of lovely pairs of (...)
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  10.  8
    Countably Categorical Structures with N‐Degenerate Algebraic Closure.Evgueni V. Vassiliev - 1999 - Mathematical Logic Quarterly 45 (1):85-94.
    We study the class of ω-categorical structures with n-degenerate algebraic closure for some n ε ω, which includes ω-categorical structures with distributive lattice of algebraically closed subsets , and in particular those with degenerate algebraic closure. We focus on the models of ω-categorical universal theories, absolutely ubiquitous structures, and ω-categorical structures generated by an indiscernible set. The assumption of n-degeneracy implies total categoricity for the first class, stability for the second, and ω-stability for the third.
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