University of Illinois Urbana-Champaign

Definable equivalence relations and disc spaces of algebraically closed valued fields

Holly, Jan Elise

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https://hdl.handle.net/2142/21612

Description

Title
Definable equivalence relations and disc spaces of algebraically closed valued fields
Author(s)
Holly, Jan Elise
Issue Date
1992
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
  • 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.
Type of Resource
text
Permalink
http://hdl.handle.net/2142/21612
Copyright and License Information
Copyright 1992 Holly, Jan Elise

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