## Abstract

The depth of a partially ordered set (P, >) is the smallest ordinal γ such that (P, >) does not embed γ^{*}. The width of (P, γ) is the smallest cardinal number μ such that there is no antichain of size μ+1 in P. We show that if γ > ω and γ is not an infinite successor cardinal, then any partially ordered set of depth γ can be decomposed into cf(γ) parts so that the depth of each part is strictly less than γ. If γ = ω or if γ is an infinite successor cardinal, then for any infinite cardinal ⋋ there is a linearly ordered set of depth γ such that for any ⋋-decomposition one of the parts has the same depth γ. These results are used to solve an analogous problem about width. It is well known that, for any cardinal ⋋, there is a partial order of width ω which cannot be split into ⋋ parts of finite width. We prove that, for any cardinal ⋋ and any infinite cardinal ν there is a partial order of width ν^{+} which cannot be split into ⋋ parts of smaller width.

Original language | English (US) |
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Pages (from-to) | 773-785 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 290 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1985 |

## Keywords

- Category
- Depth
- Extendable ordinal
- Graph
- Partial order
- Partition relation
- Width