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


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