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 $(\mathbf{R},+,.,0,1)$ as a field. Then the structure $$ is first order definable in the structure $(\mathbf{R},+,.,0,1)$ as for all $x,y\in {\mathbf{R}}^{2}$ we have $x\le y$ iff $\exists z({z}^{2}=y-x)$. Thus we know that $(\mathbf{R},+,.,0,1)$ is unstable as it has a definable order on an infinite subset.
Returning to the first example, $Z$ is normal in $G$, so the set of (left) cosets of $Z$ form a factor group. The domain of the factor group is the quotient of $G$ under the equivalence relation^{} $x\equiv y$ iff $\exists z\in Z(xz=y)$. Therefore the factor group $G/Z$ will not (in general) be a definable structure, but would seem to be a “natural” structure. We therefore weaken our formalisation of “natural” from definable to interpretable. Here we require that a structure is isomorphic to some definable structure on equivalence classes^{} of definable equivalence relations. The equivalence classes of a $\mathrm{\varnothing}$-definable equivalence relation are called imaginaries.
In [2] Poizat defined the property of Elimination of Imaginaries. This is equivalent^{} to the following definition:
Definition 0.1
A structure $\mathrm{A}$ with at least two distinct $\mathrm{\varnothing}$-definable elements admits elimination of imaginaries iff for every $n\mathrm{\in}\mathrm{N}$ and $\mathrm{\varnothing}$-definable equivalence relation $\mathrm{\sim}$ on ${\mathrm{A}}^{n}$ there is a $\mathrm{\varnothing}$-definable function $f\mathrm{:}{\mathrm{A}}^{n}\mathrm{\to}{\mathrm{A}}^{p}$ (for some $p$) such that for all $x$ and $y$ from ${\mathrm{A}}^{n}$ we have
$$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 $\U0001d504$ a multi-sorted structure ${\U0001d504}^{eq}$. 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. ${\U0001d504}^{eq}$ has elimination of imaginaries. See [1] chapter 4 for a good presentation^{} of imaginaries and ${\U0001d504}^{eq}$. The idea of passing to ${\U0001d504}^{eq}$ is very useful for many purposes. Unfortunately ${\U0001d504}^{eq}$ 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.
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.
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 |