39 found
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Frank O. Wagner [29]Frank Wagner [11]Frank John Wagner [1]
  1. Frank O. Wagner (1993). Quasi-Endomorphisms in Small Stable Groups. Journal of Symbolic Logic 58 (3):1044-1051.
    We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.
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  2. Bruno Poizat & Frank O. Wagner (2000). Liftez Les Sylows! Une Suite À "Sous-Groupes Périodiques d'Un Groupe Stable". Journal of Symbolic Logic 65 (2):703-704.
    If G is an omega-stable group with a normal definable subgroup H, then the Sylow-2-subgroups of G/H are the images of the Sylow-2-subgroups of G. /// Sei G eine omega-stabile Gruppe und H ein definierbarer Normalteiler von G. Dann sind die Sylow-2-Untergruppen von G/H Bilder der Sylow-2-Untergruppen von G.
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  3. Frank O. Wagner (2004). Some Remarks on One-Basedness. Journal of Symbolic Logic 69 (1):34-38.
    A type analysable in one-based types in a simple theory is itself one-based.
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  4. Oleg Belegradek, Ya'Acov Peterzil & Frank Wagner (2000). Quasi-o-Minimal Structures. Journal of Symbolic Logic 65 (3):1115-1132.
    A structure (M, $ ,...) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is (...); any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U-rank 1. (shrink)
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  5.  7
    Itay Ben-Yaacov, Ivan Tomašić & Frank O. Wagner (2004). Constructing an Almost Hyperdefinable Group. Journal of Mathematical Logic 4 (02):181-212.
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  6.  5
    David M. Evans & Frank O. Wagner (2000). Supersimple Ω-Categorical Groups and Theories. Journal of Symbolic Logic 65 (2):767-776.
    An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl( $\emptyset$ )-definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU-rank.
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  7.  16
    Frank O. Wagner (2000). Minimal Fields. Journal of Symbolic Logic 65 (4):1833-1835.
    A minimal field of non-zero characteristic is algebraically closed.
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  8.  3
    Frank O. Wagner (2015). Plus Ultra. Journal of Mathematical Logic 15 (2):1550008.
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  9.  14
    Barbara Secker, Maya J. Goldenberg, Barbara Gibson, Frank Wagner, Bob Parke, Jonathan Breslin, Alison Thompson, Jonathan Lear & Peter Singer (2006). Just Regionalisation: Rehabilitating Care for People with Disabilities and Chronic Illnesses. [REVIEW] BMC Medical Ethics 7 (1):1-13.
    Background Regionalised models of health care delivery have important implications for people with disabilities and chronic illnesses yet the ethical issues surrounding disability and regionalisation have not yet been explored. Although there is ethics-related research into disability and chronic illness, studies of regionalisation experiences, and research directed at improving health systems for these patient populations, to our knowledge these streams of research have not been brought together. Using the Canadian province of Ontario as a case study, we address this gap (...)
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  10.  2
    Frank O. Wagner (1998). CM-Triviality and Stable Groups. Journal of Symbolic Logic 63 (4):1473-1495.
    We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.
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  11.  6
    Enrique Casanovas & Frank O. Wagner (2002). Local Supersimplicity and Related Concepts. Journal of Symbolic Logic 67 (2):744-758.
    We study local strengthenings of the simplicity condition. In particular, we define and study a local Lascar rank, as well as short, low, supershort and superlow theories. An example of a low, non supershort theory is given.
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  12.  18
    Frank O. Wagner (1991). Small Stable Groups and Generics. Journal of Symbolic Logic 56 (3):1026-1037.
    We define an R-group to be a stable group with the property that a generic element (for any definable transitive group action) can only be algebraic over a generic. We then derive some corollaries for R-groups and fields, and prove a decomposition theorem and a field theorem. As a nonsuperstable example, we prove that small stable groups are R-groups.
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  13.  12
    Itay Ben-Yaacov, Ivan Tomasic & Frank O. Wagner (2002). The Group Configuration in Simple Theories and its Applications. Bulletin of Symbolic Logic 8 (2):283-298.
    In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or in the $\omega$-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity. The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and (...)
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  14.  4
    Frank Wagner (1990). Subgroups of Stable Groups. Journal of Symbolic Logic 55 (1):151-156.
    We define the notion of generic for an arbitrary subgroup H of a stable group, and show that H has a definable hull with the same generic properties. We then apply this to the theory of stable fields.
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  15.  3
    Itay Ben-Yaacov & Frank O. Wagner (2004). On Almost Orthogonality in Simple Theories. Journal of Symbolic Logic 69 (2):398 - 408.
    1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which multi-acts (...)
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  16.  1
    Ivan Tomašić & Frank O. Wagner (2003). Applications of the Group Configuration Theorem in Simple Theories. Journal of Mathematical Logic 3 (02):239-255.
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  17.  7
    Frank O. Wagner (1994). Nilpotent Complements and Carter Subgroups in Stable ℜ-Groups. Archive for Mathematical Logic 33 (1):23-34.
    The following theorems are proved about the Frattini-free componentG Φ of a soluble stable ℜ-group: a) If it has a normal subgroupN with nilpotent quotientG Φ/N, then there is a nilpotent subgroupH ofG Φ withG Φ=NH. b) It has Carter subgroups; if the group is small, they are all conjugate. c) Nilpotency modulo a suitable Frattini-subgroup (to be defined) implies nilpotency. The last result makes use of a new structure theorem for the centre of the derivative of the Frattini-free component (...)
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  18.  5
    Bruno Poizat & Frank Wagner (1993). Sous-Groupes Periodiques d'Un Groupe Stable. Journal of Symbolic Logic 58 (2):385-400.
    We develop a Sylow theory for stable groups satisfying certain additional conditions (2-finiteness, solvability or smallness) and show that their maximal p-subgroups are locally finite and conjugate. Furthermore, we generalize a theorem of Baer-Suzuki on subgroups generated by a conjugacy class of p-elements.
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  19.  1
    Françoise Point & Frank O. Wagner (2000). Essentially Periodic Ordered Groups. Annals of Pure and Applied Logic 105 (1-3):261-291.
    A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals. We study definable sets and functions in such groups, and relate them to the quasi-o-minimal groups introduced in Belegradek (...)
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  20.  8
    Frank O. Wagner (1993). Commutator Conditions and Splitting Automorphisms for Stable Groups. Archive for Mathematical Logic 32 (3):223-228.
    We show that a stable groupG satisfying certain commutator conditions is nilpotent. Furthermore, a soluble stable group with generically splitting automorphism of prime order is nilpotent-by-finite. In particular, a soluble stable group with a generic element of prime order is nilpotent-by-finite.
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  21.  5
    Frank Wagner (2001). Fields of Finite Morley Rank. Journal of Symbolic Logic 66 (2):703-706.
    If K is a field of finite Morley rank, then for any parameter set $A \subseteq K^{eq}$ the prime model over A is equal to the model-theoretic algebraic closure of A. A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl( $\emptyset$ ).
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  22.  5
    Frank O. Wagner (2013). Quelques Réflexions Inévitables. Archive for Mathematical Logic 52 (1-2):159-171.
    We generalize Frécon’s construction of the inevitable radical to groups in stable and even simple theories.
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  23.  5
    Frank O. Wagner (1994). A Note on Defining Groups in Stable Structures. 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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  24.  3
    Frank Wagner (2001). Hyperdefinable Groups in Simple Theories. Journal of Mathematical Logic 1 (01):125-172.
  25.  4
    Frank Wagner (1992). Hrushovski Ehud. Unidimensional Theories Are Superstable. Annals of Pure and Applied Logic, Vol. 50 (1990), Pp. 117–138. Hrushovski Ehud. Almost Orthogonal Regular Types. Annals of Pure and Applied Logic, Vol. 45 (1989), Pp. 139–155. [REVIEW] Journal of Symbolic Logic 57 (2):762-763.
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  26.  7
    Frank O. Wagner (1998). Small Fields. Journal of Symbolic Logic 63 (3):995-1002.
    An infinite field with only countably many pure types is algebraically closed.
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  27.  7
    Frank O. Wagner (1993). Stable Groups, Mostly of Finite Exponent. Notre Dame Journal of Formal Logic 34 (2):183-192.
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  28.  3
    Frank O. Wagner (1992). A Propos E'equations Generiques. Journal of Symbolic Logic 57 (2):548-554.
    We prove that a stable solvable group $G$ which satisfies $x^n = 1$ generically is of finite exponent dividing some power of $n$. Furthermore, $G$ is nilpotent-by-finite. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).
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  29.  1
    Frank O. Wagner (1997). On the Structure of Stable Groups. Annals of Pure and Applied Logic 89 (1):85-92.
    In this paper, we shall survey results about the group-theoretic properties of stable groups. These can be classified into three main categories, according to the strength of the assumptions needed: chain conditions, generic types, and some form of rank. Each category has its typical application: Chain conditions often allow us to deduce global properties from local ones, generic properties are used to get definable groups from undefinable ones, and rank is necessary to interpret fields in certain group actions. While originally (...)
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  30. Frank Wagner (1992). Review: Ehud Hrushovski, Unidimensional Theories Are Superstable; Ehud Hrushovski, Almost Orthogonal Regular Types. [REVIEW] Journal of Symbolic Logic 57 (2):762-763.
     
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  31.  9
    Ziv Shami & Frank O. Wagner (2002). On the Binding Group in Simple Theories. Journal of Symbolic Logic 67 (3):1016-1024.
    We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p' interalgebraic with a finite tuple of realizations of p, which is generated over φ. Moreover, the group of elementary permutations of p' over all realizations of φ is type-definable.
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  32.  2
    Frank O. Wagner (1992). More on ${\Germ R}$. Notre Dame Journal of Formal Logic 33 (2):159-174.
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  33.  1
    Frank O. Wagner (1992). À Propos d'Équations Génériques. Journal of Symbolic Logic 57 (2):548-554.
    We prove that a stable solvable group G which satisfies xn = 1 generically is of finite exponent dividing some power of n. Furthermore, G is nilpotent-by-finite. A second result is that in a stable group of finite exponent, involutions either have big centralisers, or invert a subgroup of finite index (which hence has to be abelian).
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  34.  3
    Frank Wagner (2005). Subsimple Groups. Journal of Symbolic Logic 70 (4):1365 - 1370.
    We define a notion of genericity for genericity subgroups of groups interpretable in a simple theory. and show that a type generic for such a group is generic for the minimal hyperdefinable supergroup (the definable hull). In particular, at least one generic type of the definable hull is finitely satisfiable in the original subgroup. If the subgroup is a subfield, then the additive and the multiplicative definable hull both have bounded index in the smallest hyperdefinable superfield.
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  35. Oleg Belegradek, Viktor Verbovskiy & Frank O. Wagner (2003). Coset-Minimal Groups. Annals of Pure and Applied Logic 121 (2-3):113-143.
    A totally ordered group G is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G{±∞}. Continuing work in Belegradek et al. 1115) and Point and Wagner 261), we study coset-minimality, as well as two weak versions of the notion: eventual and ultimate coset-minimality. These groups are abelian; an eventually coset-minimal group, as a pure ordered group, is an ordered abelian group of finite regular rank. (...)
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  36. Daniel Palacín & Frank O. Wagner (2013). Ample Thoughts. Journal of Symbolic Logic 78 (2):489-510.
    Non-$n$-ampleness as defined by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of $\Sigma$-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.
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  37. Frank Wagner (2011). Propriétés résiduelLes dans Les groupes supersimpLes. Journal of Symbolic Logic 76 (2):361 - 367.
    Si C est une pseudo-variété, alors un groupe supersimple résiduellement C est nilpotent-par-poly-C. If C is a pseudo-variety, then a supersimple residually C group is nilpotent-by-poly-C.
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  38. Frank Wagner (2011). Residual Properties in Supersimple Groups. Journal of Symbolic Logic 76 (2):361-367.
     
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  39. Frank O. Wagner (2013). Some Inevitable Reflections. Archive for Mathematical Logic 52 (1-2):159 - 171.
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