Imaginary element

In model theory, a branch of mathematics, an imaginary element of a structure is roughly a definable equivalence class. These were introduced by Shelah (1990), and elimination of imaginaries was introduced by 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 Mⁿ 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, b).
• 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 of that theory 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

• ISBN 978-0-521-30442-9
• Poizat, Bruno (1983), "Une théorie de Galois imaginaire. [An imaginary Galois theory]", Journal of Symbolic Logic, 48 (4): 1151–1170,
  MR 0727805
• ISBN 978-0-444-70260-9