# Unboundedness number and domination number of a poset $(P,\leq)$

Suppose $(P,\leq)$ is a poset without maximal elements. For $X\subseteq P$ we set $X^u = \{p\in P: p \geq x \text{ for all } x\in X\}$ and call this the set of upper bounds of $X$. We say that $B\subseteq P$ is unbounded if $B^u = \emptyset$. Moreover we say $D\subseteq P$ is dominating if for all $p\in P$ there is $d\in D$ such that $p\leq d$. We set

• ${\frak b}(P) = \min\{|B|: B\subseteq P\text{ is unbounded}\}$, and
• ${\frak d}(P) = \min\{|D|: B\subseteq P\text{ is dominating}\}$.

It is easy to see that for all posets $P$ without maximal elements we have ${\frak b}(P) \leq {\frak d}(P)$. What is an example of a poset $P$ in which we can prove in $\textsf{ZFC}$ that ${\frak b}(P) < {\frak d}(P)$?

$\omega\times \omega_1$ with the product (pointwise) order.