# 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 $\leq$ 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 $n\geq 2$).
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Preprints, Working Papers, ...
Domain :

http://hal.univ-reunion.fr/hal-01477258
Contributor : Réunion Univ <>
Submitted on : Monday, February 27, 2017 - 11:31:59 AM
Last modification on : Thursday, March 28, 2019 - 11:24:10 AM

### Identifiers

• HAL Id : hal-01477258, version 1
• ARXIV : 1510.00596

### Citation

Christian Delhommé, Maurice Pouzet. Length of an intersection. 2015. ⟨hal-01477258⟩

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