With Tamás Mátrai and Juris Steprāns. Fundamenta mathematicae, 2013. (arχiv | journal)

Abstract: We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van~Douwen’s diagram. For instance, although the usual proof of the inequality $\mathfrak p\leq\mathfrak b$ does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we use our results to give an embedding from the inclusion ordering on $\mathcal P(\omega)$ into the Borel Tukey ordering on cardinal invariants.

Further notes: The main object of study in this article are cardinal invariants that are defined by relations. One may know that ideals carry cardinal invariants such as the cofinality and additivity. Think of this as a generalization not only to arbitrary partial orders, but to all relations.

Briefly, if $R\subset A\times B$ then the dominating number of $R$ is the minimum cardinality of a subset $F\subset B$ such that for any $a\in A$ there is $b\in F$ with $a\mathrel{R}b$. Intuitively, $F$ can “cover” all of $A$ in the sense of the relation $R$. For instance, the dominating number for the eventually less than* relation $\leq^*$ on $\mathbb N^\mathbb N$ is just the usual dominating number $\mathfrak{d}$. And it turns out that plenty of commonly used cardinal invariants can be expressed as a dominating number.

If $R$ and $R’$ are relations, then a morphism (or Tukey map) from $R$ to $R’$ is a pair of maps $\phi\colon A’\to A$ and $\psi\colon B\to B’$ such that $\phi(a)\mathrel{R}\implies a\mathrel{R’}\psi(b)$. Originally considered by Vojtáš, morphisms allow one to regard cardinal invariants from a categorical point of view.

What does a morphism mean? The existence of a morphism from $R$ to $R’$ implies the dominating number of $R$ is greater than or equal to the dominating number of $R’$. In most cases, the axiom of choice implies the converse holds too: if the dominating number of $R$ is $\geq$ that of $R’$, then there is a morphism from $R$ to $R’$. However, such a morphism need not be definable in any way.

In this article, we considered morphisms which are definable in the sense that they are Borel. For instance, it is fairly easy to see that the inequality $\mathfrak{p}\leq\mathfrak{b}$ is always true. But the usual argument does not show how to construct a morphism. By analyzing the two cardinals in slightly more depth, we showed that in fact this inequality is witnessed by a Borel morphism.

We also have a few (easy) results on the other side of the coin: showing that a well-known cardinal inequality does not arise from a Borel morphism justifies that the inequality is in some sense very complex.

In the last part of the paper, we consider generalizations of the splitting number $\mathfrak{s}$. Recall that if $A,B\subset\mathbb N$, then we say $B$ splits $A$ iff $A$ both meets and misses $B$ infinitely. The cardinal invariant $\mathfrak{s}$ is then the least size of a family in $\mathcal P(\mathbb N)$ that can split any given set.

It is natural to ask for stronger forms of splitting: let $\mathfrak{s}_2$ be the least size of a family such that for any two sets, some set in the family suffices to split them both. Well, it turns out that $\mathfrak{s}_2=\mathfrak{s}$, but once again, this equality is not witnessed by a Borel morphism. By playing with splitting further, we were able to find a continuum of variants that are distinct in the Borel category. Perhaps it’s not too surprising, but this implies there is a high level of richness in the Borel Tukey ordering.