### Abstract

The index of an irreflexive binary relation R is the smallest cardinal number σ(R) such that R equals the union of σ(R) partial orders. With s(n) the largest index for an R defined on n points, it is shown that s(n)/log_{2} n→1 as n →∞. The index function is examined for symmetric R’s and almost transitive R’s, and a characterization for σ(R)≦2 is presented. It is shown also that inf{n:s(n)>3}≦13, but the exact value of inf {n:s(n)>3} is presently unknown.

Original language | English (US) |
---|---|

Pages (from-to) | 149-161 |

Number of pages | 13 |

Journal | Pacific Journal of Mathematics |

Volume | 39 |

Issue number | 1 |

State | Published - 1971 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Pacific Journal of Mathematics*,

*39*(1), 149-161.

**Directed graphs as unions of partial orders.** / Fishburn, Peter C.; Spencer, Joel.

Research output: Contribution to journal › Article

*Pacific Journal of Mathematics*, vol. 39, no. 1, pp. 149-161.

}

TY - JOUR

T1 - Directed graphs as unions of partial orders

AU - Fishburn, Peter C.

AU - Spencer, Joel

PY - 1971

Y1 - 1971

N2 - The index of an irreflexive binary relation R is the smallest cardinal number σ(R) such that R equals the union of σ(R) partial orders. With s(n) the largest index for an R defined on n points, it is shown that s(n)/log2 n→1 as n →∞. The index function is examined for symmetric R’s and almost transitive R’s, and a characterization for σ(R)≦2 is presented. It is shown also that inf{n:s(n)>3}≦13, but the exact value of inf {n:s(n)>3} is presently unknown.

AB - The index of an irreflexive binary relation R is the smallest cardinal number σ(R) such that R equals the union of σ(R) partial orders. With s(n) the largest index for an R defined on n points, it is shown that s(n)/log2 n→1 as n →∞. The index function is examined for symmetric R’s and almost transitive R’s, and a characterization for σ(R)≦2 is presented. It is shown also that inf{n:s(n)>3}≦13, but the exact value of inf {n:s(n)>3} is presently unknown.

UR - http://www.scopus.com/inward/record.url?scp=84972565031&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84972565031&partnerID=8YFLogxK

M3 - Article

VL - 39

SP - 149

EP - 161

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -