Supersimple theories

We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined by a possibly infinite conjunction of first order formulas, then it is the intersection of definable equivalence relations.

     

An approximate Herbrand’s theorem and definable functions in metric structures

We develop a version of Herbrand's theorem for continuous logic and use it to prove that definable functions in infinite‐dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand's theorem can be used to characterize definable functions in absolutely ubiquitous structures from classical logic.     

Quasi-decompositions for Self-Small Abelian Groups

Abstract

We will prove a Krull-Schmidt type theorem for the quasi decompositions of a self-small abelian group of finite torsion free rank, which extends the classical result proved for finite rank torsion free abelian groups.     

On Turner’s theorem and first-order theory

A theorem of E.C. Turner states that if F is a finitely generated free group, then the test words are precisely the elements not contained in any proper retract. In this paper, we examine some ideas in model theory and logic related to Turner’s characterization of test words and introduce Turner groups, a class of groups containing all finite groups and all stably hyperbolic groups satisfying this characterization. We show that Turner’s theorem is not first-order expressible. However, we prove that every finitely generated elementary free group is a Turner group.     

Perturbations of Smooth Actions with Nontrivial Cohomology

We prove a general perturbation result for smooth Lie group actions with nontrivial finite‐dimensional cohomology. It describes sufficient conditions on cohomology over an action which imply that the action lies in a finite‐dimensional family of actions such that any small perturbation of the family intersects the smooth conjugacy class of the given action. We cast the classical KAM result on perturbations of Diophantine vector fields on tori into this general setup, and we address a few applications and potential applications of this result to homogeneous Lie group actions with finite‐dimensional first cohomology. © 2014 Wiley Periodicals, Inc.     

On definability of types of finite Cantor‐Bendixson rank

We prove that every type of finite Cantor‐Bendixson rank over a model of a first‐order theory without the strict order property is definable and has a unique nonforking extension to a global type. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On pseudolinearity and generic pairs

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 (complete) theory of generic pairs of models of an SU ‐rank 1 theory T can only take values 1 (if and only if T is trivial), 2 (if and only if T is linear) or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to linearity for ω ‐categorical SU ‐rank 1 structures, established in [7], from the conjecture that an ω ‐categorical supersimple theory has finite SU ‐rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Purity and Quasiequality of Abelian Groups

We study the Cohn purity in an abelian group regarded as a left module over its endomorphism ring. We prove that if a finite rank torsion-free abelian group G is quasiequal to a direct sum in which all summands are purely simple modules over their endomorphism rings then the module _E(G) G is purely semisimple. This theorem makes it possible to construct abelian groups of any finite rank which are purely semisimple over their endomorphism rings and it reduces the problem of endopure semisimplicity of abelian groups to the same problem in the class of strongly indecomposable abelian groups.     

A group in a group

1. In an Abelian group, a module, or more generally a one-based group H, the only definable groups are the obvious ones: if G is interpretable in H, then it has a definable subgroup of finite index which is definably isomorphic to a quotient A/B, where A and B are definable subgroups of a Cartesian power of H. 2. In such a group the introduction of those quotient groups weakly eliminates imaginary elements. More generally, for a stable theory the existence of canonical real bases for complete types implies the elimination of imaginary elements. 3. A group which is interpretable in a one-based structure is one-based. The property of being one-based is preserved by interpretation for theories of finite rank but not in general.Translated from French.Translated from Algebra i Logika, No. 3, pp. 368–378, May–June, 1990.     

A constructive proof of the Peter‐Weyl theorem

We present a new and constructive proof of the Peter‐Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*‐algebras to give a proof which may be seen as a direct generalization of Burnside's algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of Bishop. In fact, the present theory of compact groups may be seen as a natural continuation in the line of Bishop's work on locally compact, but Abelian, groups [2]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The automorphism group of the Gaussian measure cannot act pointwise

Classical ergodic theory deals with measure (or measure class) preserving actions of locally compact groups on Lebesgue spaces. An important tool in this setting is a theorem of Mackey which provides spatial models for BooleanG-actions. We show that in full generality this theorem does not hold for actions of Polish groups. In particular there is no Borel model for the Polish automorphism group of a Gaussian measure. In fact, we show that this group as well as many other Polish groups do not admit any nontrivial Borel measure preserving actions.     

On the Euler characteristic of definable groups

We show that in an arbitrary o‐minimal structure the following are equivalent: (i) 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; (ii) every infinite, definably connected, definably compact definable group has a non trivial torsion point (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Ideals without ccc and without property

We prove a strong version of a theorem of Balcerzak-Roslanowski-Shelah by showing, in ZFC, that there exists a simply definable Borel -ideal for which both the ccc and property (M) fail. The proof involves Polish group actions.

     

The question of the definability of Abelian groups by the centers of their endomorphism rings

We obtain necessary and sufficient conditions for an Abelian group in the class of completely decomposable torsion-free Abelian groups of a chosen finite rank to be definable in this class by the center of the endomorphism ring of the group.     

On recognition of finite simple groups with connected prime graph

The spectrum of a finite group is the set of its element orders. We prove a theorem on the structure of a finite group whose spectrum is equal to the spectrum of a finite nonabelian simple group. The theorem can be applied to solving the problem of recognizability of finite simple groups by spectrum.     

Definability in the structure of words with the inclusion relation

We develop a theory of (first-order) definability in the subword partial order in parallel with similar theories for the h-quasiorder of finite k-labeled forests and for the infix order. In particular, any element is definable (provided that the words of length 1 or 2 are taken as parameters), the first-order theory of the structure is atomic and computably isomorphic to the first-order arithmetic. We also characterize the automorphism group of the structure and show that every predicate invariant under the automorphisms of the structure is definable in the structure.     

On the canonical base property

We give an example of a finite rank, in fact $\aleph _{1}$ -categorical, theory where the canonical base property (CBP) fails. In fact, we give a “group-like” example in a sense that we will describe below. We also prove, in a finite Morley rank context, that if all definable Galois groups are “rigid,” then $T$ has the CBP.     

Groups and C*‐Algebras from a 4‐Dimensional Anzai Flow

The rotation flow on the circle T gives a concrete representation of the irrational rotation algebra, which is an in finite dimensional simple quotient of the group C*‐algebra of the discrete Heisenberg group H₃ analogously certain 2‐ and 3‐dimensional Anzai flows on T ² and T ³are known to give concrete representations of the corresponding quotients of the group C*‐algebras of the groups H₄ and H_5,5. Considered here is the (minimal, effective) 4‐dimensional Anzai flow F = (ℤ, T ⁴) generated by the homeomorphism (y, x, w, v) ↦ (λy, yx, xw, wv); a group H_6,10 is determined by F the faithful in finite dimensional simple quotients of whose group C*‐algebra C*‐(H_6,10 have concrete representations given by F. Furthermore, the rest of the infinite dimensional simple quotients of C*‐(H_6,10 are identified and displayed as C*‐crossed products generated by minimal effective actions and also as matrix algebras over simple C*‐algebras from groups of lower dimension; these lower dimensional groups are H₃ and subgroups of H₄ and H_5,5.     

On Schreier graphs of gyrogroup actions

The notion of a group action can be extended to the case of gyrogroups. In this article, we examine a digraph and graph associated with a gyrogroup action on a finite nonempty set, called a Schreier digraph and graph. We show that algebraic properties of gyrogroups and gyrogroup actions such as being gyrocommutative, being transitive, and being fixed-point-free are reflected in their Schreier digraphs and graphs. We also prove graph-theoretic versions of the three fundamental theorems involving actions: the Cauchy–Frobenius lemma (also known as the Burnside lemma), the orbit-stabilizer theorem, and the orbit decomposition theorem. Finally, we make a connection between gyrogroup actions and actions of symmetric groups by evaluation via Schreier digraphs and graphs.     

Minimality conditions on circularly ordered structures

We explore analogues of o‐minimality and weak o‐minimality for circularly ordered sets. Much of the theory goes through almost unchanged, since over a parameter the circular order yields a definable linear order. Working over ?? there are differences. Our main result is a structure theory (with infinitely many doubly transitive examples related to Jordan permutation groups) for ?₀‐categorical weakly circularly minimal structures. There is a 5‐homogeneous (or ‘5‐indiscernible’) example which is not 6‐homogeneous, but any example which is k‐homogeneous for some k ≥ 6 is k‐homogeneous for all k. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)