4 found
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  1.  15
    On Compactifications and the Topological Dynamics of Definable Groups.Jakub Gismatullin, Davide Penazzi & Anand Pillay - 2014 - Annals of Pure and Applied Logic 165 (2):552-562.
    For G a group definable in some structure M, we define notions of “definable” compactification of G and “definable” action of G on a compact space X , where the latter is under a definability of types assumption on M. We describe the universal definable compactification of G as View the MathML source and the universal definable G-ambit as the type space SG. We also point out the existence and uniqueness of “universal minimal definable G-flows”, and discuss issues of amenability (...)
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  2.  92
    G-Compactness and Groups.Jakub Gismatullin & Ludomir Newelski - 2008 - Archive for Mathematical Logic 47 (5):479-501.
    Lascar described E KP as a composition of E L and the topological closure of E L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an (...)
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  3.  27
    Model Theoretic Connected Components of Finitely Generated Nilpotent Groups.Nathan Bowler, Cong Chen & Jakub Gismatullin - 2013 - Journal of Symbolic Logic 78 (1):245-259.
    We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for (...)
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  4.  15
    On Model-Theoretic Connected Components in Some Group Extensions.Jakub Gismatullin & Krzysztof Krupiński - 2015 - Journal of Mathematical Logic 15 (2):1550009.
    We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new (...)
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