Changing the Depth of an Ordered Set by Decomposition

Abstract

The depth of a partially ordered set $\langle P, < \rangle$ is the smallest ordinal $\gamma$ such that $\langle P, < \rangle$ does not embed ${\gamma ^\ast }$. The width of $\langle P, < \rangle$ is the smallest cardinal number $\mu$ such that there is no antichain of size $\mu + 1$ in $P$. We show that if $\gamma > \omega$ and $\gamma$ is not an infinite successor cardinal, then any partially ordered set of depth $\gamma$ can be decomposed into $\operatorname {cf}(|\gamma |)$ parts so that the depth of each part is strictly less than $\gamma$. If $\gamma = \omega$ or if $\gamma$ is an infinite successor cardinal, then for any infinite cardinal $\lambda$ there is a linearly ordered set of depth $\gamma$ such that for any $\lambda$-decomposition one of the parts has the same depth $\gamma$. These results are used to solve an analogous problem about width. It is well known that, for any cardinal $\lambda$, there is a partial order of width $\omega$ which cannot be split into $\lambda$ parts of finite width. We prove that, for any cardinal $\lambda$ and any infinite cardinal $\nu$, there is a partial order of width ${\nu ^ + }$ which cannot be split into $\lambda$ parts of smaller width.