Browsing by Autor "Alf Onshuus"

Now showing 1 - 15 of 15

Additivity of the dp-rank
(American Mathematical Society, 2013) Itay Kaplan; Alf Onshuus; Alexander Usvyatsov
The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank cannot be reduced to the study of its dp-minimal types, and we discuss the possible relations between dp-rank and VC-density.
Characterizing rosy theories
(Cambridge University Press, 2007) Clifton Ealy; Alf Onshuus
Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.
Definable one dimensional structures in o-minimal theories
(Hebrew University of Jerusalem, 2010) Assaf Hasson; Alf Onshuus; Ya’acov Peterzil
Definable structures in o-minimal theories: One dimensional types
(Hebrew University of Jerusalem, 2010) Assaf Hasson; Alf Onshuus; Ya’acov Peterzil
Generic stability, forking, and þ-forking
(American Mathematical Society, 2012) Darío García; Alf Onshuus; Alexander Usvyatsov
Abstract notions of “smallness” are among the most important tools that model theory offers for the analysis of arbitrary structures. The two most useful notions of this kind are forking (which is closely related to certain measure zero ideals) and thorn-forking (which generalizes the usual topological dimension). Under certain mild assumptions, forking is the finest notion of smallness, whereas thorn-forking is the coarsest. In this paper we study forking and thorn-forking, restricting ourselves to the class of generically stable types. Our main conclusion is that in this context these two notions coincide. We explore some applications of this equivalence.
Groups in NTP2
(2015) Nadja Hempel; Alf Onshuus
We prove the existence of abelian, solvable and nilpotent definable envelopes for groups definable in models of an NTP2 theory.
Properties and consequences of Thorn-independence
(Cambridge University Press, 2006) Alf Onshuus
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.
SOLVABLE LIE GROUPS DEFINABLE IN O-MINIMAL THEORIES
(Cambridge University Press, 2016) Annalisa Conversano; Alf Onshuus; Sergei Starchenko
In this paper, we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
STABILIZERS, GROUPS WITH -GENERICS, AND PRC FIELDS
(Cambridge University Press, 2018) Samaria Montenegro; Alf Onshuus; Pierre Simon
In this paper, we develop three different subjects. We study and prove alternative versions of Hrushovski’s ‘stabilizer theorem’, we generalize part of the basic theory of definably amenable NIP groups to $\text{NTP}_{2}$ theories, and finally, we use all this machinery to study groups with f-generic types definable in bounded pseudo real closed fields.
Stable types in rosy theories
(Cambridge University Press, 2010) Assaf Hasson; Alf Onshuus
Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of U þ -rank 1 must arise from definable stable sets.
þ-Forking and Stable Forking
(2016) Clifton Ealy; Alf Onshuus
Usamos una contrucción particular de una relación de independencia para demostrar que en cualquier teoría þ-bifurcación es equivalente a bifurcación con una fórmula estable (en el sentido específico de st-bifurcación dada en la Definición 1.3). También demostramos que si tenemos þ-división podemos lograr división fuerte sobre una base que pertenece a la clausura algebraica del conjunto parámetro.
The algebraic numbers definable in various exponential fields
(Cambridge University Press, 2012) Jonathan Kirby; Angus Macintyre; Alf Onshuus
Abstract We prove the following theorems. Theorem 1: for any E-field with cyclic kernel, in particular ℂ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: for the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.
The independence property in generalized dense pairs of structures
(Cambridge University Press, 2011) Alexander Berenstein; Alf Dolich; Alf Onshuus
Abstract We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.
Unstable structures definable in o-minimal theories
(Cornell University, 2007) Assaf Hasson; Alf Onshuus
Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly ordered. As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional.
Unstable structures definable in o-minimal theories
(Birkhäuser, 2010) Assaf Hasson; Alf Onshuus