The chain covering number of a poset with no infinite antichains

Abraham, Uri; Pouzet, Maurice

Comptes Rendus. Mathématique, Tome 361 (2023) no. G8, pp. 1383-1399

The chain covering number $Cov\left(P\right)$ of a poset $P$ is the least number of chains needed to cover $P$. For an uncountable cardinal $\nu$, we give a list of posets of cardinality and covering number $\nu$ such that for every poset $P$ with no infinite antichain, $Cov\left(P\right)\ge \nu$ if and only if $P$ embeds a member of the list. This list has two elements if $\nu$ is a successor cardinal, namely ${\left[\nu \right]}^2$ and its dual, and four elements if $\nu$ is a limit cardinal with $cf\left(\nu \right)$ weakly compact. For $\nu ={\aleph }_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $\nu$.

Le nombre de recouvrement par chaînes d’un ensemble ordonné $P$ (poset), noté $Cov\left(P\right)$, est le plus petit nombre de chaînes nécessaires pour recouvrir $P$. Pour un cardinal donné $\nu$, on donne une liste de posets $Q$ de nombre de recouvrement par chaînes $\nu$ telle que pour tout poset $P$ sans antichaîne infinie, $Cov\left(P\right)\ge \nu$ si et seulement si $P$ contient une copie d’un membre de la liste. Cette liste est constituée de posets de cardinal $\nu$, elle a deux éléments si $\nu$ est un cardinal successeur, à savoir ${\left[\nu \right]}^2$ et son dual, et quatre éléments si $\nu$ est un cardinal limite avec $cf\left(\nu \right)$ faiblement compact. Pour $\nu ={\aleph }_1$, une liste a été donnée par le premier auteur ; sa construction a été étendue par F. Dorais à tout cardinal successeur infini $\nu$.

Classification : 03E05, 06A07

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