Structure theorems for o-minimal expansions of groups
type: Item
instance of: scholarly article
title: Structure theorems for o-minimal expansions of groups
zbMATH Open document ID: 0944.03027
DOI: 10.1016/S0168-0072(99)00043-3
author: Mário J. Edmundo
published in: Annals of Pure and Applied Logic
publication date: 2000-09-20T00:00:00Z
review text: Let \(R\) be an o-minimal expansion of an ordered group \((R;0,1,+,<)\) with distinguished positive element \(1\). The following statements are equivalent: (1) \(R\) is semi-bounded; (2) \(R\) has no poles; (3) \(R\) cannot define a real closed field with domain \(R\) and order \(<\); (4) \(R\) is eventually linear; (5) every \(R\)-definable set is a finite union of cones. As a corollary, \(\text{Th}(R)\) has quantifier elimination and universal axiomatization in the language with symbols for ordered group operations, bounded \(R\)-definable sets and a symbol for each definable endomorphism of the group \((R;0,+)\).
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 06F15
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 1398857
zbMATH Keywords: o-minimal expansion
zbMATH Keywords: ordered group
MaRDI profile type: Publication
cites work: Q4392286
cites work: Definable Sets in Ordered Structures. II
cites work: Linear o-minimal structures
cites work: Additive reducts of real closed fields
cites work: A growth dichotomy for o-minimal expansions of ordered groups
cites work: A structure theorem for semibounded sets in the reals
cites work: Reducts of some structures over the reals
cites work: Definable Sets in Ordered Structures. I
cites work: Between groups and rings
cites work: Defining multiplication in o-minimal expansions of the additive reals
cites work: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function
description: scientific article; zbMATH DE number 1398857