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# 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 $\sigma:T^{{\prime}}\rightarrow T$ 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 $(\mathbf{R},<)$ is first order definable in the structure $(\mathbf{R},+,.,0,1)$ as for all $x,y\in\mathbf{R}^{{2}}$ we have $x\leq y$ 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 $\emptyset$-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 $\mathfrak{A}$ with at least two distinct $\emptyset$-definable elements admits elimination of imaginaries iff for every $n\in\mathbf{N}$ and $\emptyset$-definable equivalence relation $\sim$ on $\mathfrak{A}^{{n}}$ there is a $\emptyset$-definable function $f:\mathfrak{A}^{{n}}\rightarrow\mathfrak{A}^{{p}}$ (for some $p$) such that for all $x$ and $y$ from $\mathfrak{A}^{{n}}$ we have

$x\sim y\textrm{ iff }f(x)=f(y).$ |

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 $\mathfrak{A}$ a multi-sorted structure $\mathfrak{A}^{{eq}}$. This is done by adding a sort for every $\emptyset$-definable equivalence relation, so that the equivalence classes are elements (and code themselves). This is a closure operator i.e. $\mathfrak{A}^{{eq}}$ has elimination of imaginaries. See [1] chapter 4 for a good presentation of imaginaries and $\mathfrak{A}^{{eq}}$. The idea of passing to $\mathfrak{A}^{{eq}}$ is very useful for many purposes. Unfortunately $\mathfrak{A}^{{eq}}$ 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.

# 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.

## Mathematics Subject Classification

03C95*no label found*03C68

*no label found*

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