Imaginary element

Imaginary element

In mathematical model theory, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by harvtxt|Shelah|1990, and elimination of imaginaries was introduced by harvtxt|Poizat|1983

Definitions

*"M" is a model of some theory.
*x and y stand for "n"-tuples of variables, for some natural number "n".
*An equivalence formula is a formula φ(x,y) that is a symmetric and transitive relation. Its domain is the set of elements a of "M""n" such that φ(a,a); it is an equivalence relation on its domain.
*An imaginary element a/φ of "M" is an equivalence formula φ together with an equivalence class a.
*"M" has elimination of imaginaries if for every imaginary element a/φ there is a formula θ(x,y) such that there is a unique tuple b so that the equivalence class of a consists of the tuples x such that θ(x,y)
*A model has uniform elimination of imaginaries if the formula θ can be chosen independently of a.
*A theory has elimination of imaginaries if every model does (and similarly for uniform elimination).

Examples

*ZFC set theory has elimination of imaginaries.
*Peano arithmetic has uniform elimination of imaginaries.
*A vector space of dimension at least 2 over a finite field with at least 3 elements does not have elimination of imaginaries.

References

*Citation | last1=Hodges | first1=Wilfrid | author1-link=Wilfrid Hodges | title=Model theory | publisher=Cambridge University Press | isbn=978-0-521-30442-9 | year=1993
*citation|id=MR|0727805
last=Poizat|first= Bruno
title=Une théorie de Galois imaginaire. [An imaginary Galois theory]
journal=J. Symbolic Logic |volume=48 |year=1983|issue= 4|pages= 1151-1170
*Citation | last1=Shelah | first1=Saharon | author1-link=Saharon Shelah | title=Classification theory and the number of nonisomorphic models | origyear=1978 | publisher=Elsevier | edition=2nd | series=Studies in Logic and the Foundations of Mathematics | isbn=978-0-444-70260-9 | year=1990


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Element — El e*ment, n. [F. [ e]l[ e]ment, L. elementum.] 1. One of the simplest or essential parts or principles of which anything consists, or upon which the constitution or fundamental powers of anything are based. [1913 Webster] 2. One of the ultimate …   The Collaborative International Dictionary of English

  • Imaginary part — In mathematics, the imaginary part of a complex number z, is the second element of the ordered pair of real numbers representing z, i.e. if z = (x, y) , or equivalently, z = x+iy, then the imaginary part of z is y. It is denoted by Im (z) or Im{… …   Wikipedia

  • Magnetic element — Element El e*ment, n. [F. [ e]l[ e]ment, L. elementum.] 1. One of the simplest or essential parts or principles of which anything consists, or upon which the constitution or fundamental powers of anything are based. [1913 Webster] 2. One of the… …   The Collaborative International Dictionary of English

  • List of mathematics articles (I) — NOTOC Ia IA automorphism ICER Icosagon Icosahedral 120 cell Icosahedral prism Icosahedral symmetry Icosahedron Icosian Calculus Icosian game Icosidodecadodecahedron Icosidodecahedron Icositetrachoric honeycomb Icositruncated dodecadodecahedron… …   Wikipedia

  • Zero-based numbering — is numbering in which the initial element of a sequence is assigned the index 0, rather than the index 1 as is typical in everyday circumstances. Under zero based numbering, the initial element is sometimes termed the zeroth element, rather than… …   Wikipedia

  • Leibniz, Gottfried Wilhelm — ▪ German philosopher and mathematician Introduction born July 1 [June 21, old style], 1646, Leipzig died November 14, 1716, Hannover, Hanover  German philosopher, mathematician, and political adviser, important both as a metaphysician and as a… …   Universalium

  • List of mathematical logic topics — Clicking on related changes shows a list of most recent edits of articles to which this page links. This page links to itself in order that recent changes to this page will also be included in related changes. This is a list of mathematical logic …   Wikipedia

  • Governmentium — is a joke which involves the definition of an imaginary element. It is a satire of the bureaucracy of large government establishments in the form of a scientific description of a newly discovered chemical element. It reads as… …   Wikipedia

  • Versor — In mathematics, a versor is a directed great circle arc that corresponds to a quaternion of norm one. In geometry and physics, a versor is sometimes defined as a unit vector indicating the orientation of a directed axis (such as a Cartesian axis) …   Wikipedia

  • phlogiston — [flō jis′tän, flō jis′tən] n. [ModL < Gr phlogistos < phlogizein, to burn, inflame < phlegein, to burn: for IE base see BLACK] an imaginary element formerly believed to cause combustion and to be given off by anything burning; matter or… …   English World dictionary

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”