# Elimination of imaginaries for $1$-tuples, but not $n$-tuples

Edit: Following discussion with Tomasz below, I want to clarify that I mean for definable subsets here to range over arbitrary sorts; ie, $$\mathfrak{C}$$ can be multi-sorted, but I'm not considering it as having a distinguished "home sort" or using the term "definable subsets" to refer to subsets of a distinguished sort.

Otherwise, as he points out, one can consider the theory of infinite sets as a multi-sorted theory $$T^*$$, with sorts $$(S_n)_{n\in\omega}$$ and $$n$$-ary projection maps $$\pi_n:S_1^n\to S_n$$, where $$S_1$$ is taken to be the "home sort" and each $$S_n$$ is taken to be the sort for the subsets of $$S_1$$ of size at most $$n$$. Then any finite subset $$\{x_1,\dots,x_n\}\subset S_1$$ of the home sort has a canonoical parameter given by $$\pi_{n}(x_1,\dots,x_n)$$, but (as Tomasz argues) the pair of tuples $$\{(a,b),(c,d)\}\subset S_1^2$$ for distinct elements $$a,b,c,d\in S_1$$ has no canonical parameter.

However, this is not quite what I'm looking for, since $$T^*$$ only has elimination of $$1$$-imaginaries for subsets of the home sort. Indeed, for instance, the finite subset $$\{\pi_2(a,b),\pi_2(c,d)\}\subset S_2$$ has no canonical parameter, so $$T^*$$ does not eliminate $$1$$-imaginaries for subsets of arbitrary sorts.

Let $$T$$ be a complete first-order theory and fix a monster model $$\mathfrak{C}\models T$$. Recall that, for a $$\mathfrak{C}$$-definable collection of $$n$$-tuples $$E\subseteq\mathfrak{C}^n$$, a canonical parameter for $$E$$, denoted $$\ulcorner E\urcorner$$, is a finite tuple $$\overline{e}\in\mathfrak{C}$$ such that an automorphism of $$\mathfrak{C}$$ fixes $$E$$ setwise if and only if it fixes $$\overline{e}$$; this is unique up to interdefinability. Now, $$T$$ eliminates imaginaries if every $$\mathfrak{C}$$-definable set of $$n$$-tuples has a canonical parameter, and $$T$$ eliminates finite imaginaries if every finite set of $$n$$-tuples has a canonical parameter.

Example: If $$T$$ is the theory of infinite sets, then $$T$$ does not eliminate finite imaginaries, since for any $$c_1\neq c_2\in\mathfrak{C}$$ the set $$\{c_1,c_2\}$$ has no canonical parameter.

Example: If $$T$$ is totally ordered, then $$T$$ eliminates finite imaginaries, since an automorphism fixes a finite collection of $$n$$-tuples setwise if and only if it fixes it pointwise.

Now, I'm curious about how much elimination of imaginaries can fail for tuples as opposed to elements. Say that $$T$$ eliminates $$1$$-imaginaries if every $$\mathfrak{C}$$-definable subset of $$\mathfrak{C}$$ has a canonical parameter, and say that $$T$$ eliminates finite $$1$$-imaginaries if every finite subset of $$\mathfrak{C}$$ has a canonical parameter.

Question: Is there a theory that eliminates $$1$$-imaginaries but does not eliminate imaginaries? Is there a theory that eliminates finite $$1$$-imaginaries but does not eliminate finite imaginaries?

I suspect the answer is yes, but I'm struggling to come up with any examples.

• I think that if you take a pure infintie set expanded by canonical parameters for finite subsets, then you trivially have elimination of finite 1-imaginaries, but a set of the form $\{(a,b),(c,d)\}$ has no canonical paremeter, since its real dcl will consist of $\{a,c\}$ and $\{b,d\}$, and the tuple $\{a,c\}\{b,d\}$ is not interdefinable with $\{(a,b),(c,d)\}$ (the latter has the same type as $\{(c,b),(a,d)\}$. More or less the same example should work for your other question. Commented Apr 15, 2021 at 0:19
• Actually, exactly the same example, as in this case, the only definable sets in the home sort are finite or cofinite, so eliminating finite 1-imaginaries is the same as eliminating all imaginaries. It all gets a bit more tricky if you insist on a one-sorted structure, but just throwing in some sort of random encoding should do the trick. Commented Apr 15, 2021 at 0:24
• @tomasz I'm sure I'm missing something, but – denoting by $e$ the canonical parameter for $\{a,b\}$ and $f$ the canonical parameter for $\{c,d\}$ – don't we then have the problem that the finite subset $\{e,f\}$ has no canonical parameter? Commented Apr 15, 2021 at 0:33
• (indeed, note that, for any $a'\in\{b,c,d\}$, there is an automorphism fixing $\{e,f\}$ setwise and swapping $a$ and $a'$. so the only possible candidate for a canonical parameter of $\{e,f\}$ is the canonical parameter of $\{a,b,c,d\}$. but this element is fixed by the automorphism swapping $a$ and $c$ and fixing all other real elements, and this automorphism does fix $\{e,f\}$ setwise.) Commented Apr 15, 2021 at 0:57
• Yes, that is what I am arguing (since then you can make the whole thing one-sorted). Commented Apr 15, 2021 at 19:51

Elimination of 1-imaginaries does not imply elimintaion of imaginaries in a one sorted theory. For example, every o-minimal structure eliminates 1-imaginaries, since definable subsets of the home sort are finite unions of intervals which are coded by the endpoints. However, an o-minimal structure need not eliminiate imaginaries, see e.g here - the example is $$(\mathbf{Q}, <)$$ with a 4-ary predicate for $$x + y = w + z$$
• thank you so much, this resolves the question completely!! this is a great class of examples for theories eliminating $1$-imaginaries Commented Jul 5, 2021 at 23:48