The chain covering number of a poset is the least number of chains needed to cover . For an uncountable cardinal , we give a list of posets of cardinality and covering number such that for every poset with no infinite antichain, if and only if embeds a member of the list. This list has two elements if is a successor cardinal, namely and its dual, and four elements if is a limit cardinal with weakly compact. For , a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal .
Le nombre de recouvrement par chaînes d’un ensemble ordonné (poset), noté , est le plus petit nombre de chaînes nécessaires pour recouvrir . Pour un cardinal donné , on donne une liste de posets de nombre de recouvrement par chaînes telle que pour tout poset sans antichaîne infinie, si et seulement si contient une copie d’un membre de la liste. Cette liste est constituée de posets de cardinal , elle a deux éléments si est un cardinal successeur, à savoir et son dual, et quatre éléments si est un cardinal limite avec faiblement compact. Pour , une liste a été donnée par le premier auteur ; sa construction a été étendue par F. Dorais à tout cardinal successeur infini .
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Accepted:
Published online:
Uri Abraham 1; Maurice Pouzet 2, 3
@article{CRMATH_2023__361_G8_1383_0, author = {Uri Abraham and Maurice Pouzet}, title = {The chain covering number of a poset with no infinite antichains}, journal = {Comptes Rendus. Math\'ematique}, pages = {1383--1399}, publisher = {Acad\'emie des sciences, Paris}, volume = {361}, year = {2023}, doi = {10.5802/crmath.511}, language = {en}, }
Uri Abraham; Maurice Pouzet. The chain covering number of a poset with no infinite antichains. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1383-1399. doi : 10.5802/crmath.511. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.511/
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