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Title:On elementary pairs ofo-minimal structures
Author(s):Lewenberg, Adam H.
Doctoral Committee Chair(s):Henson, C. Ward
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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 + $\delta$).
Issue Date:1995
Rights Information:Copyright 1995 Lewenberg, Adam H.
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9543648
OCLC Identifier:(UMI)AAI9543648

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