imaginaries

Given an algebraic structure $S$ 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 $S$ is “What structures naturally live in $S$?” We can formalise this question in the following manner: Given some logic appropriate to the structure $S$, we say another structure $T$ is definable in $S$ iff there is some definable subset $T^{\prime }$ of $S^n$, a bijection $\sigma :T^{\prime }\to T$ and a definable function (respectively relation) on $T^{\prime }$ for each function (resp. relation) on $T$ so that $\sigma$ is an isomorphism (of the relevant type for $T$).

For an example take some infinite group $(G,.)$. Consider the centre of $G$, $Z:=\{x\in G:\forall y\in G(xy=yx)\}$. Then $Z$ is a first order definable subset of $G$, which forms a group with the restriction of the multiplication, so $(Z,.)$ is a first order definable structure in $(G,.)$.

As another example consider the structure $(𝐑,+,.,0,1)$ as a field. Then the structure is first order definable in the structure $(𝐑,+,.,0,1)$ as for all $x,y\in {𝐑}^2$ we have $x\le y$ iff $\exists z(z^2=y-x)$. 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 $x\equiv y$ iff $\exists z\in Z(xz=y)$. 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 $\mathrm{\varnothing }$-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 $\mathrm{A}$ with at least two distinct $\mathrm{\varnothing }$-definable elements admits elimination of imaginaries iff for every $n\mathrm{\in }\mathrm{N}$ and $\mathrm{\varnothing }$-definable equivalence relation $\mathrm{\sim }$ on ${\mathrm{A}}^n$ there is a $\mathrm{\varnothing }$-definable function $f\mathrm{:}{\mathrm{A}}^n\mathrm{\to }{\mathrm{A}}^p$ (for some $p$) such that for all $x$ and $y$ from ${\mathrm{A}}^n$ we have

$$x\sim y\mathit{\text{ iff }}f(x)=f(y).$$

Given this property, we think of the function $f$ as coding the equivalence classes of $\sim$, and we call $f(x)$ a code for $x/\sim$. 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 ${𝔄}^{eq}$. This is done by adding a sort for every $\mathrm{\varnothing }$-definable equivalence relation, so that the equivalence classes are elements (and code themselves). This is a closure operator i.e. ${𝔄}^{eq}$ has elimination of imaginaries. See [1] chapter 4 for a good presentation of imaginaries and ${𝔄}^{eq}$. The idea of passing to ${𝔄}^{eq}$ is very useful for many purposes. Unfortunately ${𝔄}^{eq}$ 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.

Title	imaginaries
Canonical name	Imaginaries
Date of creation	2013-03-22 13:25:50
Last modified on	2013-03-22 13:25:50
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	7
Author	mathcam (2727)
Entry type	Definition
Classification	msc 03C95
Classification	msc 03C68
Related topic	CyclicCode
Defines	imaginaries
Defines	elimination of imaginaries
Defines	definable structure
Defines	interpretable structure
Defines	code