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Length of an intersection

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 $κ$, and the ordering $\leq$ is the intersection of finitely many partial orderings $\leq_i$ on $X$, $1\leq i\leq n$, then, letting $\ell(X,\leq_i)=κ\multordby q_i+r_i$, with $r_i<κ$, denote the euclidian division by $κ$ (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 $n\geq 2$).