A chain complete poset with no infinite antichain has a finite core
type: Item
instance of: scholarly article
title: A chain complete poset with no infinite antichain has a finite core
zbMATH Open document ID: 0789.06002
DOI: 10.1007/BF01108708
published in: Order
publication date: 1994-06-14T00:00:00Z
review text: For an ordered set \(P\) define a quasi-order \(\trianglelefteq\) (passing- through order) as follows: \(a\trianglelefteq b\) \((a,b\in P)\) if every maximal chain of \(P\) which passes through \(a\) also passes through \(b\). By means of this relation, a perfect sequence \(\Pi=\langle P_ \alpha: \alpha\leq \lambda\rangle\) for \(P\) is defined, where \(\lambda\) is an ordinal number and \(P= P_ 0\supset P_ 1\supset\cdots\supset P_ \lambda\). A stronger version of a conjecture from an earlier paper of the authors [ibid. 9, 321-331 (1992; Zbl 0782.06001)] is proved: ``If \(\langle P_ \alpha: \alpha\leq \lambda\rangle\) and \(\langle P_ \alpha': \alpha\leq \lambda\rangle\) are perfect sequences for a chain-complete poset \(P\) which has no infinite antichain, then \(\lambda= \lambda'< \omega\), \(P_ \alpha\) is order-isomorphic to \(P_ \alpha'\) for all \(\alpha\leq \lambda\), \(P_ \lambda\) is finite, and is a retract which contains no irreducible elements and reflects the fixed-point property''.
Mathematics Subject Classification ID: 06A06
zbMATH DE Number: 484785
zbMATH Keywords: core
zbMATH Keywords: passing-through order
zbMATH Keywords: quasi-order
zbMATH Keywords: perfect sequences
zbMATH Keywords: chain- complete poset
zbMATH Keywords: retract
zbMATH Keywords: irreducible elements
zbMATH Keywords: fixed-point property
author: Boyu Li
author: Eric C. Milner
reviewed by: Ladislav Skula
MaRDI profile type: Publication
cites work: Fixed points in partially ordered sets
cites work: Partially Ordered Sets
cites work: The PT-order and the fixed point property
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description: scientific article; zbMATH DE number 484785