imaginaries (original) (raw)

As another example consider the structure (𝐑,+,.,0,1) as a field. Then the structure (𝐑,<) is first order definable in the structure (𝐑,+,.,0,1) as for all x,y∈𝐑2 we have x≤y iff ∃z(z2=y-x). Thus we know that (𝐑,+,.,0,1) is unstable as it has a definable order on an infinite subset.

Returning to the first example, Z is normal in G, so the set of (left) cosets of Z form a factor group. The domain of the factor group is the quotient of G under the equivalence relation x≡y iff ∃z∈Z(xz=y). Therefore the factor group G/Z will not (in general) be a definable structure, but would seem to be a “natural” structure. We therefore weaken our formalisation of “natural” from definable to interpretable. Here we require that a structure is isomorphic to some definable structure on equivalence classes of definable equivalence relations. The equivalence classes of a ∅-definable equivalence relation are called imaginaries.

In [2] Poizat defined the property of Elimination of Imaginaries. This is equivalent to the following definition:

Definition 0.1

A structure A with at least two distinct ∅-definable elements admits elimination of imaginaries iff for every n∈N and ∅-definable equivalence relation ∼ on An there is a ∅-definable function f:An→Ap (for some p) such that for all x and y from An we have

Given this property, we think of the function f as coding the equivalence classes of ∼, and we call f⁢(x) a code for x/∼. If a structure has elimination of imaginaries then every interpretable structure is definable.

In [3] Shelah defined, for any structure 𝔄 a multi-sorted structure 𝔄e⁢q. This is done by adding a sort for every ∅-definable equivalence relation, so that the equivalence classes are elements (and code themselves). This is a closure operator i.e. 𝔄e⁢q has elimination of imaginaries. See [1] chapter 4 for a good presentation of imaginaries and 𝔄e⁢q. The idea of passing to 𝔄e⁢q is very useful for many purposes. Unfortunately 𝔄e⁢q has an unwieldy language and theory. Also this approach does not answer the question above. We would like to show that our structure has elimination of imaginaries with just a small selection of sorts added, and perhaps in a simple language. This would allow us to describe the definable structures more easily, and as we have elimination of imaginaries this would also describe the interpretable structures.

References

• 1 Wilfrid Hodges, A shorter model theory Cambridge University Press, 1997.
• 2 Bruno Poizat, Une théorie de Galois imaginaire, Journal of Symbolic Logic, 48 (1983), pp. 1151-1170.
• 3 Saharon Shelah, Classification Theory and the Number of Non-isomorphic Models, North Hollans, Amsterdam, 1978.