2026-09-22T23:52:38Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/20612 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Henson, C. Ward Lewenberg, Adam H. 2011-05-07T12:44:16Z 2011-05-07T12:44:16Z 10000-01-01 1995 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 + $\delta$). Made available in DSpace on 2011-05-07T12:44:16Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9543648.pdf: 3635561 bytes, checksum: 961c23ef8cc1684a9b678bbc9156ea3b (MD5) Previous issue date: 1995 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:45:04Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:19:55-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9543648 (UMI)AAI9543648 http://hdl.handle.net/2142/20612 eng Copyright 1995 Lewenberg, Adam H. Mathematics On elementary pairs ofo-minimal structures text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.