Results for 'definable group'

997 found
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  1.  6
    Definable groups in dense pairs of geometric structures.Alexander Berenstein & Evgueni Vassiliev - 2022 - Archive for Mathematical Logic 61 (3):345-372.
    We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large definable subgroups of groups definable in the original language \ are also \-definable, and definably amenable \-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group (...)
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  2.  40
    Definable group extensions in semi‐bounded o‐minimal structures.Mário J. Edmundo & Pantelis E. Eleftheriou - 2009 - Mathematical Logic Quarterly 55 (6):598-604.
    In this note we show: Let R = 〈R, <, +, 0, …〉 be a semi-bounded o-minimal expansion of an ordered group, and G a group definable in R of linear dimension m . Then G is a definable extension of a bounded definable group B by 〈Rm, +〉.
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  3.  19
    Definable groups in models of Presburger Arithmetic.Alf Onshuus & Mariana Vicaría - 2020 - Annals of Pure and Applied Logic 171 (6):102795.
    This paper is devoted to understand groups definable in Presburger Arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded abelian group definable in a model of (Z, +, <) Presburger Arithmetic is definably isomorphic to (Z, +)^n mod out by a lattice.
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  4.  11
    Orthogonal Decomposition of Definable Groups.Alessandro Berarducci, Pantelis E. Eleftheriou & Marcello Mamino - forthcoming - Journal of Symbolic Logic:1-22.
    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 (...)
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  5.  37
    Topological dynamics of definable group actions.Ludomir Newelski - 2009 - Journal of Symbolic Logic 74 (1):50-72.
    We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.
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  6.  22
    Topological dynamics and definable groups.Anand Pillay - 2013 - Journal of Symbolic Logic 78 (2):657-666.
    We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group $G(M)$ on its “external type space” $S_{G,\textit{ext}}(M)$, can explain, account for, or give rise to, the quotient $G/G^{00}$, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or $fsg$) groups in NIP theories. As part of our analysis we show the existence of “externally definable (...)
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  7.  25
    Definability and definable groups in simple theories.Anand Pillay - 1998 - Journal of Symbolic Logic 63 (3):788-796.
    We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.
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  8.  24
    A note on defining groups in stable structures.Frank O. Wagner - 1994 - Journal of Symbolic Logic 59 (2):575-578.
    If * is a binary partial function which happens to be a group law on some infinite subset of some model of a stable theory, then this subset can be embedded into a definable group such that * becomes the group operation.
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  9.  33
    Connected components of definable groups, and o-minimality II.Annalisa Conversano & Anand Pillay - 2015 - Annals of Pure and Applied Logic 166 (7-8):836-849.
  10.  12
    A remark on divisibility of definable groups.Mário J. Edmundo - 2005 - Mathematical Logic Quarterly 51 (6):639-641.
    We show that if G is a definably compact, definably connected definable group defined in an arbitrary o-minimal structure, then G is divisible. Furthermore, if G is defined in an o-minimal expansion of a field, k ∈ ℕ and pk : G → G is the definable map given by pk = xk for all x ∈ G , then we have |–1| ≥ kr for all x ∈ G , where r > 0 is the maximal (...)
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  11.  21
    On the Euler characteristic of definable groups.Mário J. Edmundo - 2011 - Mathematical Logic Quarterly 57 (1):44-46.
    We show that in an arbitrary o-minimal structure the following are equivalent: conjugates of a definable subgroup of a definably connected, definably compact definable group cover the group if the o-minimal Euler characteristic of the quotient is non zero; every infinite, definably connected, definably compact definable group has a non trivial torsion point.
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  12.  9
    On non-compact p-adic definable groups.Will Johnson & Ningyuan Yao - 2022 - Journal of Symbolic Logic 87 (1):188-213.
    In [16], Peterzil and Steinhorn proved that if a group G definable in an o-minimal structure is not definably compact, then G contains a definable torsion-free subgroup of dimension 1. We prove here a p-adic analogue of the Peterzil–Steinhorn theorem, in the special case of abelian groups. Let G be an abelian group definable in a p-adically closed field M. If G is not definably compact then there is a definable subgroup H of dimension (...)
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  13.  36
    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 (...) G-flows”, and discuss issues of amenability and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal compactification and universal G-ambit in model-theoretic terms, when G is a topological group. (shrink)
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  14.  40
    On Lascar rank and Morley rank of definable groups in differentially closed fields.Anand Pillay & Wai Yan Pong - 2002 - Journal of Symbolic Logic 67 (3):1189-1196.
    Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.
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  15.  10
    The evolution of (intergroup) peace hinges on how we define groups and peace.Anne C. Pisor, Kristopher M. Smith & Jeffrey P. Deminchuk - 2024 - Behavioral and Brain Sciences 47:e22.
    Glowacki defines peace as harmonious relationships between groups maintained without the threat of violence, where groups can be anything from families to nation states. However, defining such contentious concepts like “peace” and “groups” is a difficult task, and we discuss the implications of Glowacki's definitions for understanding intergroup relationships and their evolutionary history.
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  16.  8
    Tame Expansions of $\omega$ -Stable Theories and Definable Groups.Haydar Göral - 2019 - Notre Dame Journal of Formal Logic 60 (2):161-194.
    We study groups definable in tame expansions of ω-stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair, where K is an algebraically closed field and G is a multiplicative subgroup of K× with the Mann property, we show that the pair satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in in terms of (...)
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  17.  6
    Abelian groups definable in P-adically closed fields.Will Johnson & Y. A. O. Ningyuan - forthcoming - Journal of Symbolic Logic:1-22.
    Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. (...)
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  18.  10
    On maximal stable quotients of definable groups in nip theories.Mike Haskel & Anand Pillay - 2018 - Journal of Symbolic Logic 83 (1):117-122.
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  19.  7
    On Groups with Definable F_-Generics Definable in _P-Adically Closed Fields.Anand Pillay & Y. A. O. Ningyuan - 2023 - Journal of Symbolic Logic 88 (4):1334-1353.
    The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context (...)
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  20.  12
    Definability of groups in ℵ₀-stable metric structures.Itaï Ben Yaacov - 2010 - Journal of Symbolic Logic 75 (3):817-840.
    We prove that in a continuous ℵ₀-stable theory every type-definable group is definable. The two main ingredients in the proof are: 1. Results concerning Morley ranks (i.e., Cantor-Bendixson ranks) from [Ben08], allowing us to prove the theorem in case the metric is invariant under the group action; and 2. Results concerning the existence of translation-invariant definable metrics on type-definable groups and the extension of partial definable metrics to total ones.
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  21.  18
    Corrigendum to: "On Lascar Rank and Morley Rank of Definable Groups in Differentially Closed Fields".Anand Pillay & Wai Yan Pong - 2009 - Journal of Symbolic Logic 74 (4):1436 - 1437.
  22.  49
    Type-definability, compact lie groups, and o-minimality.Anand Pillay - 2004 - Journal of Mathematical Logic 4 (02):147-162.
    We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I (...)
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  23.  30
    Splitting definably compact groups in o-minimal structures.Marcello Mamino - 2011 - Journal of Symbolic Logic 76 (3):973 - 986.
    An argument of A. Borel [Bor—61, Proposition 3.1] shows that every compact connected Lie group is homeomorphic to the Cartesian product of its derived subgroup and a torus. We prove a parallel result for definably compact definably connected groups definable in an o-minimal expansion of a real closed field. As opposed to the Lie case, however, we provide an example showing that the derived subgroup may not have a definable semidirect complement.
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  24.  46
    Groups, group actions and fields definable in first‐order topological structures.Roman Wencel - 2012 - Mathematical Logic Quarterly 58 (6):449-467.
    Given a group , G⊆Mm, definable in a first-order structure equation image equipped with a dimension function and a topology satisfying certain natural conditions, we find a large open definable subset V⊆G and define a new topology τ on G with which becomes a topological group. Moreover, τ restricted to V coincides with the topology of V inherited from Mm. Likewise we topologize transitive group actions and fields definable in equation image. These results require (...)
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  25.  14
    Definable topological dynamics and real Lie groups.Grzegorz Jagiella - 2015 - Mathematical Logic Quarterly 61 (1-2):45-55.
    We investigate definable topological dynamics of groups definable in an o‐minimal expansion of the field of reals. Assuming that a definable group G admits a model‐theoretic analogue of Iwasawa decomposition, namely the compact‐torsion‐free decomposition, we give a description of minimal subflows and the Ellis group of its universal definable flow in terms of this decomposition. In particular, the Ellis group of this flow is isomorphic to. This provides a range of counterexamples to a (...)
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  26.  30
    Interpretable groups are definable.Pantelis E. Eleftheriou, Ya'acov Peterzil & Janak Ramakrishnan - 2014 - Journal of Mathematical Logic 14 (1):1450002.
    We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals. We discuss the general open question of elimination of imaginaries in an o-minimal structure.
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  27.  10
    Definable topological dynamics for trigonalizable algebraic groups over Qp.Ningyuan Yao - 2019 - Mathematical Logic Quarterly 65 (3):376-386.
    We study the flow of trigonalizable algebraic group acting on its type space, focusing on the problem raised in [17] of whether weakly generic types coincide with almost periodic types if the group has global definable f‐generic types, equivalently whether the union of minimal subflows of a suitable type space is closed. We shall give a description of f‐generic types of trigonalizable algebraic groups, and prove that every f‐generic type is almost periodic.
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  28.  33
    Definable homomorphisms of abelian groups in o-minimal structures.Ya'acov Peterzil & Sergei Starchenko - 1999 - Annals of Pure and Applied Logic 101 (1):1-27.
    We investigate the group of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure . We prove the existence of a “large”, definable subgroup of . If contains an infinite definable set of homomorphisms then some definable subgroup of B admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure but also in any (...)
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  29.  8
    More on Galois Cohomology, Definability and Differential Algebraic Groups.Omar León Sánchez, David Meretzky & Anand Pillay - forthcoming - Journal of Symbolic Logic:1-19.
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  30.  27
    Definably compact Abelian groups.Mário J. Edmundo & Margarita Otero - 2004 - Journal of Mathematical Logic 4 (02):163-180.
    Let M be an o-minimal expansion of a real closed field. Let G be a definably compact definably connected abelian n-dimensional group definable in M. We show the following: the o-minimal fundamental group of G is isomorphic to ℤn; for each k>0, the k-torsion subgroup of G is isomorphic to n, and the o-minimal cohomology algebra over ℚ of G is isomorphic to the exterior algebra over ℚ with n generators of degree one.
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  31.  9
    Locally definable subgroups of semialgebraic groups.Elías Baro, Pantelis E. Eleftheriou & Ya’Acov Peterzil - 2019 - Journal of Mathematical Logic 20 (2):2050009.
    We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. 18 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the (...)
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  32.  20
    Definable connectedness of randomizations of groups.Alexander Berenstein & Jorge Daniel Muñoz - 2021 - Archive for Mathematical Logic 60 (7):1019-1041.
    We study randomizations of definable groups. Whenever the underlying theory is stable or NIP and the group is definably amenable, we show its randomization is definably connected.
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  33.  10
    Definable nilpotent and soluble envelopes in groups without the independence property.Ricardo de Aldama - 2013 - Mathematical Logic Quarterly 59 (3):201-205.
  34.  15
    Review: A. A. Fridman, On the Relation between the Word Problem and the Conjugacy Problem in Finitely Defined Groups. [REVIEW]J. L. Britton - 1969 - Journal of Symbolic Logic 34 (3):507-508.
  35.  29
    Groups Definable in Ordered Vector Spaces over Ordered Division Rings.Pantelis E. Eleftheriou & Sergei Starchenko - 2007 - Journal of Symbolic Logic 72 (4):1108 - 1140.
    Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex V-definable subgroup U of 〈Mⁿ, +〉 and a lattice L of rank n. As two (...)
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  36.  16
    On definability of normal subgroups of a superstable group.Akito Tsuboi - 1992 - Mathematical Logic Quarterly 38 (1):101-106.
    In this note we treat maximal and minimal normal subgroups of a superstable group and prove that these groups are definable under certain conditions. Main tool is a superstable version of Zil'ber's indecomposability theorem.
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  37.  2
    Review: A. A. Fridman, Degrees of Unsolvability of the Word Problem for Finitely Defined Groups. [REVIEW]D. J. Collins - 1969 - Journal of Symbolic Logic 34 (1):114-115.
  38.  63
    A. A. Fridman. Stépéni nérazréšimosti problémy toždéstva v konéčno oprédélénnyh gruppah. Doklady Akadémii Nauk SSSR, vol. 147 , pp. 805–808. - A. A. Fridman. Degrees of insolvability of the word problem in finitely defined groups. English translation of the preceding by Sue Ann Walker. Soviet mathematics, vol. 3 no. 6 , pp. 1733–1737. - C. R. J. Clapham. Finitely presented groups with word problems of arbitrary degrees of insolubility. Proceedings of the London Mathematical Society, ser. 3 vol. 14 , pp. 633–676. - William W. Boone. Finitely presented group whose word problem has the same degree as that of an arbitrarily given Thue system . Proceedings of the National Academy of Sciences, vol. 53 , pp. 265–269. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A first paper on Thue systems. Annals of mathematics, ser. 2 vol. 83 , pp. 520–571. - William W. Boone. Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely p. [REVIEW]J. C. Shepherdson - 1968 - Journal of Symbolic Logic 33 (2):296-297.
  39.  18
    Type-Definable and Invariant Groups in O-Minimal Structures.Jana Maříková - 2007 - Journal of Symbolic Logic 72 (1):67 - 80.
    Let M be a big o-minimal structure and G a type-definable group in Mⁿ. We show that G is a type-definable subset of a definable manifold in Mⁿ that induces on G a group topology. If M is an o-minimal expansion of a real closed field, then G with this group topology is even definably isomorphic to a type-definable group in some Mk with the topology induced by Mk. Part of this result (...)
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  40.  10
    Definability of Henselian Valuations by Conditions on the Value Group.Lothar Sebastian Krapp, Salma Kuhlmann & Moritz Link - 2023 - Journal of Symbolic Logic 88 (3):1064-1082.
    Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a (...)
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  41.  21
    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.
    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: (...)
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  42.  17
    On definable Galois groups and the strong canonical base property.Daniel Palacín & Anand Pillay - 2017 - Journal of Mathematical Logic 17 (1):1750002.
    In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. 19 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite rank context than (...)
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  43.  66
    Invariance results for definable extensions of groups.Mário J. Edmundo, Gareth O. Jones & Nicholas J. Peatfield - 2011 - Archive for Mathematical Logic 50 (1-2):19-31.
    We show that in an o-minimal expansion of an ordered group finite definable extensions of a definable group which is defined in a reduct are already defined in the reduct. A similar result is proved for finite topological extensions of definable groups defined in o-minimal expansions of the ordered set of real numbers.
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  44.  37
    Groups definable in linear o-minimal structures: the non-compact case.Pantelis E. Eleftheriou - 2010 - Journal of Symbolic Logic 75 (1):208-220.
    Let $\scr{M}=\langle M,+,<,0,S\rangle $ be a linear o-minimal expansion of an ordered group, and $G=\langle G,\oplus ,e_{G}\rangle $ an n-dimensional group definable in $\scr{M}$ . We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L, for some convex ${\ssf V}\text{-definable}$ subgroup U of $\langle M^{n},+\rangle $ and a lattice L of rank equal to the dimension of the 'compact part' of (...)
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  45. Defining Terrorism for Public Policy Purposes: The Group-Target Definition.Eric Reitan - 2010 - Journal of Moral Philosophy 7 (2):253-278.
    For the sake of developing and evaluating public policy decisions aimed at combating terrorism, we need a precise public definition of terrorism that distinguishes terrorism from other forms of violence. Ordinary usage does not provide a basis for such a definition, and so it must be stipulative. I propose essentially pragmatic criteria for developing such a stipulative public definition. After noting that definitions previously proposed in the philosophical literature are inadequate based on these criteria, I propose an alternative, which I (...)
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  46.  3
    Definable Tietze extension property in o-minimal expansions of ordered groups.Masato Fujita - 2023 - Archive for Mathematical Logic 62 (7):941-945.
    The following two assertions are equivalent for an o-minimal expansion of an ordered group $$\mathcal M=(M,<,+,0,\ldots )$$. There exists a definable bijection between a bounded interval and an unbounded interval. Any definable continuous function $$f:A \rightarrow M$$ defined on a definable closed subset of $$M^n$$ has a definable continuous extension $$F:M^n \rightarrow M$$.
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  47.  7
    Pseudofinite h-structures and groups definable in supersimple h-structures.Tingxiang Zou - 2019 - Journal of Symbolic Logic 84 (3):937-956.
    In this article we explore some properties of H-structures which are introduced in [2]. We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudofiniteness. That is, the H-structures we construct are ultraproducts of finite structures. We also prove that under the assumption that the base theory is supersimple of SU-rank one, there are no new definable groups in H-structures. This improves the corresponding result in [2].
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  48.  12
    Definably topological dynamics of p-adic algebraic groups.Jiaqi Bao & Ningyuan Yao - 2022 - Annals of Pure and Applied Logic 173 (4):103077.
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  49.  50
    Compact domination for groups definable in linear o-minimal structures.Pantelis E. Eleftheriou - 2009 - Archive for Mathematical Logic 48 (7):607-623.
    We prove the Compact Domination Conjecture for groups definable in linear o-minimal structures. Namely, we show that every definably compact group G definable in a saturated linear o-minimal expansion of an ordered group is compactly dominated by (G/G 00, m, π), where m is the Haar measure on G/G 00 and π : G → G/G 00 is the canonical group homomorphism.
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  50.  13
    Definably simple stable groups with finitary groups of automorphisms.Ulla Karhumäki - 2019 - Journal of Symbolic Logic 84 (2):704-712.
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