### Abstract

In the power set P(E) of a set E, the sets of a fixed finite cardinality k form a cross-cut, that is, a maximal unordered set C such that if X, Y {square image of or equal to}E satisfy X{square image of or equal to}Y, X {square image of or equal to} some X′ in C, and Y{square image of or equal to} some Y′ in C, then X{square image of or equal to}Z{square image of or equal to}Y for some Z in C. For E=ω, ω_{1}, and ω_{2}, it is shown with the aid of the continuum hypothesis that P(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for ω and ω_{1}.

Original language | English |
---|---|

Pages (from-to) | 139-145 |

Number of pages | 7 |

Journal | Order |

Volume | 1 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jun 1984 |

### Fingerprint

### Keywords

- AMS (MOS) subject classifications (1980): primary 04A20, secondary 06A10, 04A30
- cross-cut
- grading
- Unordered set (antichain)

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Order*,

*1*(2), 139-145. https://doi.org/10.1007/BF00565649

**Cross-cuts in the power set of an infinite set.** / Baumgartner, J. E.; Erdős, P.; Higgs, D.

Research output: Contribution to journal › Article

*Order*, vol. 1, no. 2, pp. 139-145. https://doi.org/10.1007/BF00565649

}

TY - JOUR

T1 - Cross-cuts in the power set of an infinite set

AU - Baumgartner, J. E.

AU - Erdős, P.

AU - Higgs, D.

PY - 1984/6

Y1 - 1984/6

N2 - In the power set P(E) of a set E, the sets of a fixed finite cardinality k form a cross-cut, that is, a maximal unordered set C such that if X, Y {square image of or equal to}E satisfy X{square image of or equal to}Y, X {square image of or equal to} some X′ in C, and Y{square image of or equal to} some Y′ in C, then X{square image of or equal to}Z{square image of or equal to}Y for some Z in C. For E=ω, ω1, and ω2, it is shown with the aid of the continuum hypothesis that P(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for ω and ω1.

AB - In the power set P(E) of a set E, the sets of a fixed finite cardinality k form a cross-cut, that is, a maximal unordered set C such that if X, Y {square image of or equal to}E satisfy X{square image of or equal to}Y, X {square image of or equal to} some X′ in C, and Y{square image of or equal to} some Y′ in C, then X{square image of or equal to}Z{square image of or equal to}Y for some Z in C. For E=ω, ω1, and ω2, it is shown with the aid of the continuum hypothesis that P(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for ω and ω1.

KW - AMS (MOS) subject classifications (1980): primary 04A20, secondary 06A10, 04A30

KW - cross-cut

KW - grading

KW - Unordered set (antichain)

UR - http://www.scopus.com/inward/record.url?scp=34250135462&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250135462&partnerID=8YFLogxK

U2 - 10.1007/BF00565649

DO - 10.1007/BF00565649

M3 - Article

VL - 1

SP - 139

EP - 145

JO - Order

JF - Order

SN - 0167-8094

IS - 2

ER -