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Let $a$ and $b$ are filters. The product $a\times b$ is defined as the filter (on the set of pairs) induced by the base $\{ A\times B | A\in a, B\in b \}$.

It is simple to show that product of a non-trivial ultrafilter with itself is not an ultrafilter (as it is not finer than the principal filter corresponding to the identity relation).

My question: Is product of every two (different) non-trivial ultrafilters always not an ultrafilter?

Note: non-trivial ultrafilter is the same as non-principal ultrafilter.

edited Aug 19 2011 at 19:17

asked Aug 19 2011 at 19:05

porton
82●8

	For the benefit of anyone who (like me) happens accross this page as an ultrafilter novice: Although the filter mentioned in this question is not an ultrafilter, nevertheless there IS an ultrafilter on $A \times B$ canonically induced by ultrafilters $\mu$ on $A$, $\nu$ on $B$. Call it $\mu \otimes \nu$. It's the "outer measure" generated by $\mu,\nu$. To wit, if $W \subset A \times B$, then $W \in \mu \otimes \nu$ iff for every finite cover $W \subset \bigcup_{i=1}^n U_i \times V_i$ of $W$ by "rectangles", there exists an $i$ with $U_i \in \mu$ and $V_i \in \nu$. – Tim Campion Mar 20 at 17:09
	Note that $\mu \otimes \nu$ refines the filter the OP describes. Also, although this definition of $\mu\otimes \nu$ is obviously symmetric in $A$ and $B$, an alternative description can be given which is not so obviously symmetric. Namely, $W \in \mu \otimes \nu$ iff $\{a \in A \mid W\|_{a, -} \in \nu\} \in \mu$. Here $W\|_{a,-} \subset B$ is meant to denote the set $\{b \in B \mid (a,b) \in W\}$. – Tim Campion Mar 20 at 17:13
	And just for the heck of it, I'll point out that the asymmetric definition of $\mu \otimes \nu$ comes from the strength of the ultrafilter monad. Every monad on Set is strong, so this construction is very formal. The symmetric definition certifies that the ultrafilter monad is commutative, i.e. that the two "Fubini" maps are equal. This is [an important part of] what makes ultrafilters behave like measures, and sets them apart from weaker order-theoretic structures. For example, the filter monad and the double-powerset monad are not commutative. – Tim Campion Apr 3 at 21:46

Andreas Blass · Accepted Answer · 2011-08-19 20:22:47Z UTC

The product $a\times b$ of two ultrafilters is an ultrafilter if and only if, for every function $f$ from the underlying set of $a$ into $b$ (that's not a typo for "into the underlying set of $b$"), there is a set $A\in a$ such that $\bigcap_{x\in A}f(x)\in b$ . One way for this to happen is for the underlying set of $a$ to be small enough and $b$ to be complete enough, as in Joel's answer. Notice, though, that the condition is, despite its appearance, symmetrical between $a$ and $b$. In particular, if $b$ lives on $\omega$ while $a$ is countably complete, then the condition is satisfied because $f$ will be constant on some set in $a$ (because countably complete ultrafilters are closed under intersection of continuum many sets). [Archaeologists may be interested to know that this characterization of ultrafilters whose product is ultra occurs on page 22 of my 1970 Ph.D. thesis, which is, thanks to patient scanning, available from my web page.]

Joel David Hamkins · Answer 2 · 2011-08-19 19:22:55Z UTC

No. If $a$ and $b$ are principal ultrafilters, then so is the product filter as you have defined it. If $a$ and $b$ contain $\{x\}$ and $\{y\}$, respectively, then the base of your product includes the singleton $\{(x,y)\}$, and hence it is the principal ultrafilter.

But perhaps by "nontrivial" you meant nonprincipal. In this case, here is another example. Let $\mu$ be any ultrafilter on $\omega$ and let $\nu$ be a $\kappa$-complete ultrafilter on a measurable cardinal $\kappa$. If we consider the product filter $\mu\times\nu$ on $\omega\times\kappa$, as you have defined it, then it is an ultrafilter, since for any $X\subset \omega\times\kappa$, there are fewer than $\kappa$ many possible horizontal slices $X_\alpha=\{n\mid (n,\alpha)\in X\}$, and so there is some $A\subset \omega$ such that $\{\alpha\lt\kappa\mid X_\alpha=A\}\in \nu$. If $A\in\mu$, then $X$ is in the product filter, and if $A\notin\mu$, then the complement of $X$ is in the product filter. So $\mu\times\nu$ as you have defined it is an ultrafilter.

Product of ultrafilters, is it an ultrafilter?

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