### Abstract

Every linear extension L: [x_{1}<x_{2}<...<x_{m}] of an ordered set P on m points arises from the simple algorithm: For each i with 0≤i<m, choose x_{i+1} as a minimal element of P-{x_{j}:j≤i}. A linear extension is said to be greedy, if we also require that x_{i+1} covers x_{i} in P whenever possible. The greedy dimension of an ordered set is defined as the minimum number of greedy linear extensions of P whose intersection is P. In this paper, we develop several inequalities bounding the greedy dimension of P as a function of other parameters of P. We show that the greedy dimension of P does not exceed the width of P. If A is an antichain in P and |P-A|≥2, we show that the greedy dimension of P does not exceed |P-A|. As a consequence, the greedy dimension of P does not exceed |P|/2 when |P|≥4. If the width of P-A is n and n≥2, we show that the greedy dimension of P does not exceed n^{2}+n. If A is the set of minimal elements of P, then this inequality can be strengthened to 2 n-1. If A is the set of maximal elements, then the inequality can be further strengthened to n+1. Examples are presented to show that each of these inequalities is best possible.

Original language | English (US) |
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Pages (from-to) | 145-164 |

Number of pages | 20 |

Journal | Order |

Volume | 2 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1985 |

Externally published | Yes |

### Keywords

- AMS (MOS) subject classifications (1980): 06A05, 06A10
- Ordered sets
- greedy dimensions
- linear extension

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics

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## Cite this

*Order*,

*2*(2), 145-164. https://doi.org/10.1007/BF00334853