The chain covering number of a poset with no infinite antichains
The chain covering number $\operatorname{Cov}(P)$ 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, $\operatorname{Cov}(P)\...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-10-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.511/ |
| _version_ | 1797638555385724928 |
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| author | Abraham, Uri Pouzet, Maurice |
| author_facet | Abraham, Uri Pouzet, Maurice |
| author_sort | Abraham, Uri |
| collection | DOAJ |
| description | The chain covering number $\operatorname{Cov}(P)$ 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, $\operatorname{Cov}(P)\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 $[\nu ]^2$ and its dual, and four elements if $\nu $ is a limit cardinal with $\operatorname{cf}(\nu )$ 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 $. |
| first_indexed | 2024-03-11T13:05:16Z |
| format | Article |
| id | doaj.art-1de968e8e7a34ed1b825bb3f88f65160 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T13:05:16Z |
| publishDate | 2023-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-1de968e8e7a34ed1b825bb3f88f651602023-11-03T14:36:26ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-10-01361G81383139910.5802/crmath.51110.5802/crmath.511The chain covering number of a poset with no infinite antichainsAbraham, Uri0Pouzet, Maurice1Math & CS Dept., Ben-Gurion University, Beer-Sheva, 84105 IsraelICJ, Mathématiques, Université Claude-Bernard Lyon1, 43 bd. 11 Novembre 1918, 69622 Villeurbanne Cedex, France; Mathematics & Statistics Department, University of Calgary, Calgary, Alberta, Canada T2N 1N4The chain covering number $\operatorname{Cov}(P)$ 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, $\operatorname{Cov}(P)\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 $[\nu ]^2$ and its dual, and four elements if $\nu $ is a limit cardinal with $\operatorname{cf}(\nu )$ 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 $.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.511/ |
| spellingShingle | Abraham, Uri Pouzet, Maurice The chain covering number of a poset with no infinite antichains Comptes Rendus. Mathématique |
| title | The chain covering number of a poset with no infinite antichains |
| title_full | The chain covering number of a poset with no infinite antichains |
| title_fullStr | The chain covering number of a poset with no infinite antichains |
| title_full_unstemmed | The chain covering number of a poset with no infinite antichains |
| title_short | The chain covering number of a poset with no infinite antichains |
| title_sort | chain covering number of a poset with no infinite antichains |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.511/ |
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