# Borel Tukey morphisms and combinatorial cardinal invariants

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.