On elementary pairs ofo-minimal structures

Lewenberg, Adam H.

Permalink

https://hdl.handle.net/2142/20612 Copy

Description

Title

On elementary pairs ofo-minimal structures

Author(s)

Lewenberg, Adam H.

Issue Date

1995

Doctoral Committee Chair(s)

Henson, C. Ward

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Language

eng

Abstract

• In the first part of this thesis I prove some results on elementary pairs of o-minimal structures.
• Let ${\cal R}$ = (R, $<$, 0, 1, ...) be an o-minimal expansion of a dense ordered group.
• Let $T\sb{\rm tame}$ be the theory of dedekind complete elementary pairs (${\cal R}, {\cal N}$, st); here ${\cal N}$ is a proper elementary substructure dedekind complete in ${\cal R}$ and st is the standard part map induced by ${\cal N}$ on ${\cal R}$.
• Among other results, I prove that if T has quantifier elimination and is universally axiomatizable then $T\sb{\rm tame}$ has quantifier elimination. Furthermore, $T\sb{\rm tame}$ is complete, and if T is model complete $T\sb{\rm tame}$ is model complete (these last two results do not need the assumption that T has quantifier elimination or is universally axiomatizable). I also prove that if ${\cal R}$ is an o-minimal expansion of the real additive group, and if every definable map $f:{\cal R}\to{\cal R}$ is everywhere locally bounded, then any function $g:{\bf R}\sp{m}\to{\bf R}\sp{m}$ definable in ${\cal R}$ which is locally injective and continuous is in fact a homeomorphism.
• In the second part of the thesis I prove a result about embedding partial groups in groups. Let (X, p, i) be a triple where X is a set, $p : U\to X$ and $i : V\to X$ are functions, where $U\subseteq X\times X$ and $V\subseteq X$, such that, writing ab for $p(a, b)$ and $a\sp{-1}$ for $i(a)$, the following equations are satisfied for all $a,b$ and c in X when both sides of the equation are defined:$$(ab)b\sp{-1}=a, a\sp{-1}(ab)=b, a(bc)=(ab)c.$$Such a triple I call a partial group. Let (X, p, i) be a finite partial group. For each $x\in X$ let $\lambda\sb{x}$ denote the partial function on X given by $a\mapsto xa$, and let $\rho\sb{x}$ denote the partial function given by $a\mapsto ax$. Fix $0<\varepsilon, \eta<1/2$. I call ($X, p, i$) an ($\varepsilon, \eta$)-partial group if for all $x\in X: (1) \vert {\rm dom}\ i\vert\ge (1-\varepsilon)\vert X\vert, (2) \vert {\rm dom}\lambda\sb{x}\vert\ge (1-\varepsilon)\vert X\vert, (3) \vert {\rm dom}\rho\sb{x}\vert\ge (1-\varepsilon)\vert X\vert$, and (4) $\vert\{z\in X : (xz)z\sp{-1}\ {\rm is\ defined}\}\vert\ge\eta\vert X\vert$.
• The main result is that for every $\delta\in$ (0, 1) and every $\eta\in$ (0, 1/2) there is an $\varepsilon\in$ (0, 1/2) such that every ($\varepsilon, \eta$)-partial group can be embedded in some finite group such that the ratio of the cardinality of the finite group to the cardinality of the embedded partial group is less than (1 + $\delta$).

Type of Resource

text

Permalink

http://hdl.handle.net/2142/20612

Copyright and License Information

Copyright 1995 Lewenberg, Adam H.

Owning Collections