A growth dichotomy for o-minimal expansions of ordered groups

Abstract

Abstract. Let R be an o-minimal expansion of a divisible ordered abelian group (R, <, +, 0, 1) with a distinguished positive element 1. Then the following dichotomy holds: Either there is a 0-definable binary operation · such that (R, <, +, ·, 0, 1) is an ordered real closed field; or, for every definable function f : R → R there exists a 0-definable λ ∈ {0} ∪ Aut(R, +) withThis has some interesting consequences regarding groups definable in o-minimal structures. In particular, for an o-minimal structure M := (M, <, . . . ) there are, up to definable isomorphism, at most two continuous (with respect to the product topology induced by the order) M-definable groups with underlying set M .R. Poston showed in [8] that given an o-minimal expansion R of (R, <, +), if multiplication is not definable in R, then for every definable function f : R → R there exist r, c ∈ R such that lim x→+∞ [f (x) − rx] = c. In this paper, this fact is generalized appropriately for o-minimal expansions of arbitrary ordered groups.We say that an expansion (G, <, * , . . . ) of an ordered group (G, <, * ) is linearly bounded (with respect to * ) if for each definable function f : G → G there exists a definable λ ∈ End(G, * ) such that ultimately |f (x)| ≤ λ(x). (Here and throughout, ultimately abbreviates "for all sufficiently large positive arguments".)We now list the main results of this paper. Let R := (R, <, . . . ) be o-minimal. Theorem A (Growth Dichotomy). Suppose that R is an expansion of an ordered group (R, <, +). Then exactly one of the following holds: (a) R is linearly bounded; (b)R defines a binary operation · such that (R, <, +, ·) is an ordered real closed field. If R is linearly bounded, then for every definable f : R → R there exist c ∈ R and a definable λ ∈ {0} ∪ Aut(R, +) with Theorem B.Suppose that R is a linearly bounded expansion of an ordered group (R, <, +, 0, 1) with 1 > 0. Then every definable endomorphism of (R, +) is 0-definable. If R (with underlying set R ) is elementarily equivalent to R, then the ordered division ring of all R -definable endomorphisms of (R , +) is canonically isomorphic to the ordered division ring of all R-definable endomorphisms of (R, +).The growth dichotomy imposes some surprising constraints on continuous definable groups with underlying set R. (Here and throughout, all topological notions are taken with respect to the product topologies induced by the order topology.)

On Definable Skolem Functions in Weakly O-Minimal Nonvaluational Structures

“…The desired conclusion now follows from [8] as follows. First, by Theorem A thereof, either R is linearly bounded or there is a definable binary operation • such that (R, ≤, +, •) is a real closed field.…”

Returning to semi-bounded sets

“…Some of the results in this section, such as 4.4 and 4.8, were proved in [15] for unbounded intervals instead of long ones. Here are some facts about affine and linear functions: (1) f is affine on (a, b).…”

“…Structural results about semibounded sets can be found in [21], [17], [22], [13], [5] (in the o-minimal setting) and [1] (in arbitrary ordered abelian groups). Some results in [15] apply as well.…”

Solvable Lie Groups Definable in O-Minimal Theories

“…Proof. Theorem 4.1 in [9] asserts that if an o-minimal expansion of a real closed field is not exponential, then it is power-bounded. By [9,Theorem B], if such a field is power-bounded, then every definable endomorphisms of (R >0 , •) is 0-definable, as required.…”