Cross-cuts in the power set of an infinite set

J. E. Baumgartner, P. Erdős, D. Higgs

Research output: Contribution to journalArticle

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, 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.

Original languageEnglish
Pages (from-to)139-145
Number of pages7
JournalOrder
Volume1
Issue number2
DOIs
Publication statusPublished - Jun 1984

Fingerprint

Power set
Continuum Hypothesis
Unordered
Cardinality
Complement

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

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

In: Order, Vol. 1, No. 2, 06.1984, p. 139-145.

Research output: Contribution to journalArticle

Baumgartner, JE, Erdős, P & Higgs, D 1984, 'Cross-cuts in the power set of an infinite set', Order, vol. 1, no. 2, pp. 139-145. https://doi.org/10.1007/BF00565649
Baumgartner, J. E. ; Erdős, P. ; Higgs, D. / Cross-cuts in the power set of an infinite set. In: Order. 1984 ; Vol. 1, No. 2. pp. 139-145.
@article{ec6252362b1f4cc489dfd739c5e6706c,
title = "Cross-cuts in the power set of an infinite set",
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, 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.",
keywords = "AMS (MOS) subject classifications (1980): primary 04A20, secondary 06A10, 04A30, cross-cut, grading, Unordered set (antichain)",
author = "Baumgartner, {J. E.} and P. Erdős and D. Higgs",
year = "1984",
month = "6",
doi = "10.1007/BF00565649",
language = "English",
volume = "1",
pages = "139--145",
journal = "Order",
issn = "0167-8094",
publisher = "Springer Netherlands",
number = "2",

}

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

AN - SCOPUS:34250135462

VL - 1

SP - 139

EP - 145

JO - Order

JF - Order

SN - 0167-8094

IS - 2

ER -