A chain complete poset with no infinite antichain has a finite core (Q1311420)

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''.