# Dilworth’s theorem

###### Theorem.

If $P$ is a poset with width $$, then $w$ is also the smallest integer such that $P$ can be written as the union of $w$ chains.

Remark. The smallest cardinal $c$ such that $P$ can be written as the union of $c$ chains is called the *chain covering number* of $P$. So Dilworth’s theorem^{} says that if the width of $P$ is finite, then it is equal to the chain covering number of $P$. If $w$ is infinite^{}, then statement is not true. The proof of Dilworth’s theorem and its counterexample in the infinite case can be found in the reference below.

## References

- 1 J.B. Nation, “Lattice Theory”, http://www.math.hawaii.edu/ jb/lat1-6.pdfhttp://www.math.hawaii.edu/ jb/lat1-6.pdf

Title | Dilworth’s theorem |
---|---|

Canonical name | DilworthsTheorem |

Date of creation | 2013-03-22 15:49:37 |

Last modified on | 2013-03-22 15:49:37 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 14 |

Author | CWoo (3771) |

Entry type | Theorem |

Classification | msc 06A06 |

Classification | msc 06A07 |

Synonym | Dilworth chain decomposition theorem |

Related topic | DualOfDilworthsTheorem |

Defines | chain covering number |