imaginary element

In the category of sets, we have a bijective correspondence (up to isomorphism) between surjective functions on XX and equivalence relations on XX, where a surjective function π:X→Y\pi : X \to Y is taken to the equivalence relation E πE_{\pi} gotten by pulling back the diagonal of YY, and an equivalence relation EE is taken to the function f Ef_E which projects XX onto the set of EE-classes. If we replace Set with the category of definable sets Def(T)\mathbf{Def}(T) of a first-order theory TT, this correspondence generally fails, because when EE is a definable equivalence relation, there may not be a corresponding definable f Ef_E. That is to say, internal congruences in Def(T)\mathbf{Def}(T) are not generally effective.

We say that:

  1. TT has uniform elimination of imaginaries if the above correspondence does hold, that is, if internal congruences in Def(T)\mathbf{Def}(T) are effective.

  2. TT has elimination of imaginaries if for every definable set X={mβˆˆπ•„ | Ο†(m,b)}X = \{m \in \mathbb{M} \operatorname{ | } \varphi(m,b)\}, there exists a formula ψ(x,y)\psi(x,y) and a tuple cc with the same sort as yy such that cc uniquely satisfies X={m∈M | ψ(m,c)}.X = \{m \in M \operatorname{ | } \psi(m,c)\}.

Inside a saturated model π•„βŠ§T\mathbb{M} \models T, elimination of imaginaries is equivalent to the following statement: for every definable set XX, there exists a tuple cc such that for all ΟƒβˆˆAut(𝕄)\sigma \in \operatorname{Aut}(\mathbb{M}), Οƒ(X)=X\sigma(X) = X setwise if and only if Οƒ(c)=c\sigma(c) = c pointwise.

Poizat noticed that TT having elimination of imaginaries allows one to develop a classical Galois theory classifying definably-closed extensions of small parameter sets in terms of the closed subgroups of a profinite automorphism group in any sufficiently saturated model of TT.

The imaginary elements of a theory TT are precisely EE-classes, where EE is a 00-definable equivalence relation on TT.

  • Bruno Poizat, Une thΓ©orie de Galois imaginaire, J. Symbolic Logic 48 (1984), no.4, 1151-1170, MR85e:03083, doi
  • wikipedia imaginary element
  • Anand Pillay, Some remarks on definable equivalence relations in O-minimal structures, J. Symbolic Logic 51 (1986), 709-714, MR87h:03046, doi
  • Jan Holly, Definable operations on sets and elimination of imaginaries, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1149–1157, MR93e:03052, doi, pdf
  • Ehud Hrushovski, Groupoids, imaginaries and internal covers, arxiv/math.LO/0603413; On finite imaginaries, arxiv/0902.0842
  • D. Haskell, E. Hrushovski, H.D.Macpherson, Definable sets in algebraically closed valued fields: elimination of imaginaries, J. reine und angewandte Mathematik 597 (2006)
  • Saharon Shelah, Classification theory and the number of non-isomorphic models, Studies in Logic and the Foundations of Mathematics 92, North Holland, Amsterdam 1978
Revised on April 8, 2016 21:19:54 by Anonymous Coward (