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A NOTE ON THE LENGTH OF PERFECT SEQUENCES IN ORDERED SETS WITH PT-ORDER

A NOTE ON THE LENGTH OF PERFECT SEQUENCES IN ORDERED SETS WITH PT-ORDER DEMONSTRATIO MATHEMATICAVol. XXXVINo 42003Aleksander RutkowskiA NOTE ON THE LENGTH OF PERFECT SEQUENCESIN ORDERED SETS WITH PT-ORDERAbstract. For any ordinal a there exists a bipartite ordered set P containing noinfinite fences such that P has a perfect sequence of length a .1. IntroductionThe aim of this note is to answer Question 2 of [5]. Before we state thatquestion let us briefly set up notation, terminology and basic definitions, inparticular those concerning the PT-order.We shall denote by P an ordered set with < as an ordering relation.For any x,y G P, x ~ y means x is comparable to y, i.e. x < y V y < x.We assume such notions as chain completeness, retract, irreducible element,dismantlability, fence are well known to the reader. Pd is P with a dualordering. Ordinals will be donoted by Greek characters. Ord stands for theclass of all ordinals.The PT-order is a relation < p o n P which has been defined by Li in[2] and it was investigated (partially together with E. C. Milner) in thesequence of articles [3]—[5]. Define < p by the following formula:a <p b O (Vx 6 P){x ~ a =S> x ~ b)(or equivalently: each maximal chain containing a contains http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

A NOTE ON THE LENGTH OF PERFECT SEQUENCES IN ORDERED SETS WITH PT-ORDER

Rutkowski, Aleksander
Demonstratio Mathematica , Volume 36 (4): 6 – Oct 1, 2003
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References (7)

Publisher
de Gruyter
Copyright
© by Aleksander Rutkowski
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-2003-0403
Publisher site
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Abstract

DEMONSTRATIO MATHEMATICAVol. XXXVINo 42003Aleksander RutkowskiA NOTE ON THE LENGTH OF PERFECT SEQUENCESIN ORDERED SETS WITH PT-ORDERAbstract. For any ordinal a there exists a bipartite ordered set P containing noinfinite fences such that P has a perfect sequence of length a .1. IntroductionThe aim of this note is to answer Question 2 of [5]. Before we state thatquestion let us briefly set up notation, terminology and basic definitions, inparticular those concerning the PT-order.We shall denote by P an ordered set with < as an ordering relation.For any x,y G P, x ~ y means x is comparable to y, i.e. x < y V y < x.We assume such notions as chain completeness, retract, irreducible element,dismantlability, fence are well known to the reader. Pd is P with a dualordering. Ordinals will be donoted by Greek characters. Ord stands for theclass of all ordinals.The PT-order is a relation < p o n P which has been defined by Li in[2] and it was investigated (partially together with E. C. Milner) in thesequence of articles [3]—[5]. Define < p by the following formula:a <p b O (Vx 6 P){x ~ a =S> x ~ b)(or equivalently: each maximal chain containing a contains

Journal

Demonstratio Mathematica – de Gruyter

Published: Oct 1, 2003

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