Decomposable ordered groups

Authors

Eliana Barriga, Alf Onshuus, Charles Steinhorn

Type: Preprint

Publication Date: 2014-01-01

Citations: 2

DOI: https://doi.org/10.48550/arxiv.1402.6520

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Abstract

Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.

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arXiv (Cornell University)
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O-stable ordered groups

2012-01-01

V. V. Verbovskiĭ

Ordered Permutation Groups

1982-01-07

A. M. W. Glass

As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets. In the special case that the underlying set is … As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets. In the special case that the underlying set is linearly ordered, there is a natural subgroup to study, namely the set of permutations that preserves that order. In some senses. these are universal for automorphisms of models of theories. The purpose of this book is to make a thorough, comprehensive examination of these groups of permutations. After providing the initial background Professor Glass develops the general structure theory, emphasizing throughout the geometric and intuitive aspects of the subject. He includes many applications to infinite simple groups, ordered permutation groups and lattice-ordered groups. The streamlined approach will enable the beginning graduate student to reach the frontiers of the subject smoothly and quickly. Indeed much of the material included has never been available in book form before, so this account should also be useful as a reference work for professionals.

ON LINEARLY ORDERED STRUCTURES OF FINITE RANK

2009-12-01

Alf Onshuus , Charles Steinhorn

O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable … O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable theories. This is the first in an anticipated series of papers whose aim is the development of model theory for ordered structures of rank greater than one. A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here. These include all ordered structures definable (as subsets of n-tuples of the universe) in o-minimal structures. The principal result in this paper, Theorem 5.1, asserts roughly that a decomposable structure [Formula: see text] can be partitioned into finitely many definable subsets such that on each set the restriction of < is a "twisted lexicographic" order. As a consequence (Corollary 5.1), for all n and linear orders ≺ definable on a subset X ⊆ M n in an o-minimal structure [Formula: see text], there is a definable partition of X such that the restriction of ≺ to each set in the partition is "lexicographic".

Fully Ordered Groups

1974-01-01

Ali Ivanovich Kokorin , V. M. Kopytov , D. Louvish

Fully ordered groups

1977-09-01

Gian‐Carlo Rota

Definable linear orders definably embed into lexicographic orders in o-minimal structures

2012-10-10

Janak Ramakrishnan

We classify definable linear orders in o-minimal structures expanding groups. For example, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma precedes right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mo>≺</mml:mo> … We classify definable linear orders in o-minimal structures expanding groups. For example, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma precedes right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mo>≺</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(P,\prec )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a linear order definable in the real field. Then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma precedes right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mo>≺</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(P,\prec )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> embeds definably in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper R Superscript n plus 1 Baseline comma greater-than Subscript lex Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>lex</mml:mtext> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {R}^{n+1},>_{\text {lex}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than Subscript lex Baseline"> <mml:semantics> <mml:msub> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>lex</mml:mtext> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">>_{\text {lex}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the lexicographic order and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the o-minimal dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This improves a result of Onshuus and Steinhorn in the o-minimal group context.

Definable linear orders definably embed into lexicographic orders in o-minimal structures

2010-01-01

Janak Ramakrishnan

We completely characterize definable linear orders in o-minimal structures expanding groups. For example, let (P,<_p) be a linear order definable in the real field R. Then (P,<_p) embeds definably in … We completely characterize definable linear orders in o-minimal structures expanding groups. For example, let (P,<_p) be a linear order definable in the real field R. Then (P,<_p) embeds definably in (R^{n+1},<_l), where <_l is the lexicographic order and n is the o-minimal dimension of P. This improves a result of Onshuus and Steinhorn in the o-minimal group context.

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Definable linear orders definably embed into lexicographic orders in o-minimal structures

2010-03-28

Janak Ramakrishnan

On reducible and decomposable representations of orders.

1978-01-01

Wilhelm Plesken

Partially ordered groups

1992-01-01

Gerard J. Murphy

Ordered groups

1983-01-01

V. M. Kopytov

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One-dimensional groups definable in o-minimal structures

1994-10-01

Adam Strzeboński

Groups definable in ordered vector spaces over ordered division rings

2007-12-01

Pantelis E. Eleftheriou , Sergei Starchenko

Abstract Let M = 〈 M , +, <, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 … Abstract Let M = 〈 M , +, <, 0, {λ} λЄ D 〉 be an ordered vector space over an ordered division ring D , and G = 〈 G , ⊕, e G 〉 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 〈 M n , +〉 and a lattice L of rank n . As two consequences, we derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L .

Ordering of partially ordered groups [Lecture]

1958-01-01

František Šik

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L-ordered and L-lattice ordered groups

2015-04-05

R. A. Borzooei , Anatolij Dvurečenskij , Omid Zahiri

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Extensions of Ordered Groups

1955-08-01

Paul Conrad

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Defining new linear functions in tame expansions of the real ordered additive group

2021-01-01

Alex Savatovsky

We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an … We explore \emph{semibounded} expansions of arbitrary ordered groups; namely, expansions that do not define a field on the whole universe. We introduce the notion of a \emph{semibounded} expansion of an arbitrary ordered group, extending the usual notion from the o-minimal setting. For $\mathcal{R}=( \mathbb{R}, <, +, \ldots)$, a semibounded o-minimal structure and $P\subseteq \mathbb{R}$ a set satisfying certain tameness conditions, we discuss under which conditions $(\mathcal R,P)$ defines total linear functions that are not definable in \mathcal{R}. Examples of such structures that does define new total linear functions include the cases when $\mathcal{R}$ is a reduct of $(\mathbb{R},<,+,\cdot_{\upharpoonright (0,1)^2},(x\mapsto \lambda x)_{\lambda\in I\subseteq \mathbb{R}})$, and $P= 2^\mathbb{Z}$, or $P$ is an iteration sequence (for any $I$) or $P=\mathbb{Z}$, for $I=\mathbb{Q}$.

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Defining new linear functions in tame expansions of the real ordered additive group

2021-10-24

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Orderable groups

2010-01-01

Cristóbal Rivas

In Chapter 1 we give the basic background and notations. We also give a new characterization of the Conrad property for orderings. In Chapter 2, we use the new characterization … In Chapter 1 we give the basic background and notations. We also give a new characterization of the Conrad property for orderings. In Chapter 2, we use the new characterization of the Conradian property to give a classification of groups admitting (only) finitely many Conradian orderings \S 2.1. Using this classification we deduce a structure theorem for the space of Conradian orderings \S 2.2. In addition, we are able to give a structure theorem for the space of left-orderings on a group by studying the possibility of approximating a given ordering by its conjugates \S 2.3. In Chapter 3 we show that, for groups having finitely many Conradian orderings, having an isolated left-ordering is equivalent to having only finitely many left-orderings. In Chapter 4, we prove that the space of left-orderings of the free group on $n\geq2$ generators have a dense orbit under the natural action of the free group on it. This gives a new proof of the fact that the space of left-orderings of the free group in at least two generators have no isolated point. In Chapter 5, we describe the space of bi-orderings of the Thompson's group $\efe$. We show that this space contains eight isolated points together with four canonical copies of the Cantor set.

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2010-12-28

Cristóbal Rivas

Classifying torsion free groups in o-minimal expansions of real closed fields

2016-07-15

Eliana Barriga , Alf Onshuus

Definable Group Extensions and o-Minimal Group Cohomology via Spectral Sequences

2013-01-12

Eliana Barriga

We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable … We provide the theoretical foundation for the Lyndon-Hochschild- Serre spectral sequence as a tool to study the group cohomology and with this the group extensions in the category of denable groups. We also present various results on denable modules and actions, denable extensions and

On torsion-free groups in o-minimal structures

2005-10-01

Ya’acov Peterzil , Sergei Starchenko

We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably … We consider groups definable in the structure Ran and certain o-minimal expansions of it.We prove: If G = G, * is a definable abelian torsion-free group, then G is definably isomorphic to a direct sum of R, + k and R >0 , • m , for some k, m 0. Futhermore, this isomorphism is definable in the structure R, +, •, G .In particular, if G is semialgebraic, then the isomorphism is semialgebraic.We show how to use the above result to give an "o-minimal proof" to the classical Chevalley theorem for abelian algebraic groups over algebraically closed fields of characteristic zero.We also prove: Let M be an arbitrary o-minimal expansion of a real closed field R and G a definable group of dimension n.The group G is torsion-free if and only if G, as a definable group-manifold, is definably diffeomorphic to R n .

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Lie Groups Beyond an Introduction

1996-01-01

Anthony W. Knapp

Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie … Preface to the Second Edition Preface to the First Edition List of Figures Prerequisites by Chapter Standard Notation Introduction: Closed Linear Groups Lie Algebras and Lie Groups Complex Semisimple Lie Algebras Universal Enveloping Algebra Compact Lie Groups Finite-Dimensional Representations Structure Theory of Semisimple Groups Advanced Structure Theory Integration Induced Representations and Branching Theorems Prehomogeneous Vector Spaces Appendices Hints for Solutions of Problems Historical Notes References Index of Notation Index

Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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