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If $I$ is a set, $U$ a nonprincipal ultrafilter on $I$ and $E=(E_i)_{i\in I}$ a family of sets indexed by $I$, then the ultraproduct $E^*$ of $E$ is generally defined as the quotient of $\prod_{i\in I}E_i$ by the equivalence relation "equality on a subset of $I$ which belongs to $U$".

However, this definition is "wrong": this $E^*$ is nonempty if and only if all the $E_i$'s are nonempty, while the expected condition is "if and only if $\{i\in I\vert E_i\neq\emptyset\}\in U$". In fact, Łoś' theorem is false with this definition since "nonempty" can be defined by the formula $(\exists x)(x=x)$.

So I guess the right definition is $E^*=\varinjlim_{J\in U}\prod_{i\in J}E_i$ where $U$ is ordered by reverse inclusion and the transition maps are the projections. If each $E_i$ is nonempty this is equivalent to the standard form, which explains why the latter is used since in most applications (at least in algebra) the $E_i$'s carry some algebraic structure which excludes emptiness.

Of course there is a sheaf-theoretic version of this: let $I^\vee$ be the Stone-Čech compactification of the discrete space $I$, $j:I\to I^\vee$ the canonical inclusion. Then $(E_i)$ defines a sheaf of sets $\mathcal{E}$ on $I$. Put $\mathcal{E}^\vee:=j_*\mathcal{E}$. Then $E^*$ is the stalk of $\mathcal{E}^\vee$ at hte point corresponding to $U$. (In the case of ultraproducts of rings, this is explicitly stated in Schoutens' LNM 1999 book: see 2.6.2 there).

My question: are there accessible references where this issue is correctly addressed?

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I'd note that this is rarely an issue because ultraproducts are almost always applied to objects which are known to be non-empty. –  Henry Towsner Feb 19 '11 at 17:13
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In model theory, models of a given a theory are usually asumed to be nonempty sets (with extra structure). –  Martin Brandenburg Feb 19 '11 at 20:05

1 Answer 1

If you allow, as one way of correctly addressing the issue, simply assuming that the factors of the ultraproduct are nonempty, then the issue is correctly addressed in, for example, Chang and Keisler's "Model Theory", Bell and Slomson's "Models and Ultraproducts", and Comfort and Negrepontis's "Theory of Ultrafilters". The same can undoubtedly be said for almost all other treatments of model theory; the three I listed were just the first three I happened to pull off my bookshelf.

If, on the other hand, you require that empty factors be permitted, then you may need to go to the category-theoretic literature, where empty structures are not so cavalierly excluded from consideration.

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Thanks for your answer. You are saying more or less the same thing as Henry and Martin in their comments to the question. Perhaps I should explain why I think this "cavalier" attitude is also dangerous and short-sighted. First, it is dangerous because it means that whenever you use ultraproducts, you must make sure that you are dealing with nonempty objects. I am not quite sure everybody does that. (Continued in next comment) –  Laurent Moret-Bailly Feb 20 '11 at 17:16
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About short-sightedness, I think it is a bad idea to put restrictions on objects just to make definitions "easier". Take the following claim (notations as in question): Assume the $E_i$'s are $L$-structures, and $\phi$ is an $L$-formula with $n$ free variables. For an $L$-structure $E$ let $S(E,\phi)$ be the set of tuples in $E^n$ satisfying $\phi$. Then $S(E^*,\phi)$ is the ultraproduct of the $S(E_i,\phi)$'s. IOW, "taking definable sets given by $\phi$" commutes with ultraproducts. This is a very nice way of stating Łoś' thm, but it's true only with the right definition of ultraproducts. –  Laurent Moret-Bailly Feb 20 '11 at 17:38

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