Length of an intersection

TL;DR

This paper investigates the length of a poset formed by the intersection of finitely many partial orders on an infinite set, providing an optimal upper bound expressed through ordinal arithmetic.

Contribution

It establishes a precise upper bound for the length of intersected partial orders on infinite sets, extending understanding of well-partially ordered structures.

Findings

Derived an explicit upper bound for the length of intersected posets.

Proved the bound is optimal for cases with two or more partial orders.

Connected the length of the intersection to ordinal division and sum.

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 $\leq$ is the intersection of finitely many partial orderings $\leq_i$ on $X$, $1\leq i\leq n$, then, letting $\ell(X,\leq_i)=\kappa\multordby q_i+r_i$, with $r_i<\kappa$, denote the euclidian division by $\kappa$ (seen as an initial ordinal) of the length of the corresponding poset~:\[ \ell(\bfp)< \kappa\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n} r_i\Big|^+ \] where $|\sum r_i|^+$ denotes the least initial ordinal greater than the ordinal $\sum r_i$. This inequality is optimal (for $n\geq 2$).