Length of an intersection

Abstract

A poset $\bfp$ is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\em length}, $\ell(\bfp)$ of $\bfp$. We prove that if the vertex set $X$ of $\bfp$ is infinite, of cardinality $\kappa$, and the ordering $\le$ is the intersection of finitely many partial orderings ${\le }_i$ on $X$, $1\le i\le n$, then, letting $\ell(X,\leq_i)=κ\multordby q_i+r_i$, with $r_i<\kappa$, denote the euclidian division by $\kappa$ (seen as an initial ordinal) of the length of the corresponding poset~:\[ \ell(\bfp)< κ\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n} r_i\Big|^+ \] where $|\sum r_i{|}^{+}$ denotes the least initial ordinal greater than the ordinal $\sum r_i$. This inequality is optimal (for $n\ge 2$).

arXiv comment13 pages