# nLab imaginary element

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

We say that:

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

2. $T$ has elimination of imaginaries if for every definable set $X = \{m \in \mathbb{M} \operatorname{ | } \varphi(m,b)\}$, there exists a formula $\psi(x,y)$ and a tuple $c$ with the same sort as $y$ such that $c$ uniquely satisfies $X = \{m \in M \operatorname{ | } \psi(m,c)\}.$

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

Poizat noticed that $T$ 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 $T$.

The imaginary elements of a theory $T$ are precisely $E$-classes, where $E$ is a $0$-definable equivalence relation on $T$.

• Bruno Poizat, Une théorie 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 (66.205.168.234)