Given an algebraic structure to investigate, mathematicians consider substructures, restrictions of the structure, quotient structures and the like. A natural question for a mathematician to ask if he is to understand is “What structures naturally live in ?” We can formalise this question in the following manner: Given some logic appropriate to the structure , we say another structure is definable in iff there is some definable subset of , a bijection and a definable function (respectively relation) on for each function (resp. relation) on so that is an isomorphism (of the relevant type for ).
For an example take some infinite group . Consider the centre of , . Then is a first order definable subset of , which forms a group with the restriction of the multiplication, so is a first order definable structure in .
As another example consider the structure as a field. Then the structure is first order definable in the structure as for all we have iff . Thus we know that is unstable as it has a definable order on an infinite subset.
Returning to the first example, is normal in , so the set of (left) cosets of form a factor group. The domain of the factor group is the quotient of under the equivalence relation iff . Therefore the factor group 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.
A structure with at least two distinct -definable elements admits elimination of imaginaries iff for every and -definable equivalence relation on there is a -definable function (for some ) such that for all and from we have
Given this property, we think of the function as coding the equivalence classes of , and we call a code for . If a structure has elimination of imaginaries then every interpretable structure is definable.
In  Shelah defined, for any structure a multi-sorted structure . 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. has elimination of imaginaries. See  chapter 4 for a good presentation of imaginaries and . The idea of passing to is very useful for many purposes. Unfortunately 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.
- 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.
|Date of creation||2013-03-22 13:25:50|
|Last modified on||2013-03-22 13:25:50|
|Last modified by||mathcam (2727)|
|Defines||elimination of imaginaries|