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Title:Definable equivalence relations and disc spaces of algebraically closed valued fields
Author(s):Holly, Jan Elise
Doctoral Committee Chair(s):Henson, C. Ward
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:A theory T admits elimination of imaginaries (EI) if every definable equivalence relation $\sim$ is the kernel of a definable map f. (I.e., $\vec{x}\sim\vec{y}\Longleftrightarrow f(\vec{x})=f(\vec{y}).)$ This term was introduced by Poizat, and some theories that admit EI are those of ($\rm I\!N, +, \cdot$), algebraically closed fields, and real closed fields. (Note: theories here are first-order, with equality, and to be consistent with other formulations of EI, we require at least two distinct constants to be definable.)
Let $ACF\sb{val}$ be the theory of algebraically closed fields with nontrivial valuation, in the language $\{0, 1, +, -, \vert\}$ ($x\vert y\Longleftrightarrow v(x) \le v(y),$ where v is the valuation). The theory $ACF\sb{val}$ fails to admit EI, even for 1-variable definable equivalence relations. However, by considering fields of equi-characteristic zero, and adding new sorts for the space of "closed discs" and the space of "open discs", along with the canonical maps to these spaces, we obtain a theory $ACF\sp\prime\sb{val}$ such that:
Theorem. The theory $ACF\sp\prime\sb{val}$ admits EI for 1-variable definable equivalence relations on the field.
To prove this, we introduce the concepts of definable property and definable operation on sets, as well as prototypes for a theory. The following result is also necessary:
Theorem. Each K-definable set $S\subseteq K\models ACF\sb{val}$ has a unique decomposition into v-connected components. Each v-connected component is of the form $D\\(B\sb1\dot\cup\...\dot\cup B\sb{n}),$ where D is a disc or D = K, and $B\sb1,\...,B\sb{n}$ are proper subdiscs of D.
We prove this by formally developing the tree-structure of valued fields, using valued trees and valued sets, with valued fields and disc spaces of valued fields being examples of valued sets.
In addition to a detailed coding of finite sets of discs (necessary for the first theorem above), we give a full axiomatization of open disc spaces in the language $\{+, \cdot, \subseteq\},$ where + and $\cdot$ are interpreted setwise on discs.
Finally, we display the form of definable functions in algebraically closed valued fields, as well as algebraic closures and definable closures in the framework that includes disc spaces.
Issue Date:1992
Rights Information:Copyright 1992 Holly, Jan Elise
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9305554
OCLC Identifier:(UMI)AAI9305554

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