On the minimal cofinal subsets of a directed quasi-ordered set. (English) Zbl 0537.06002

This paper is concerned with the size and structure of the cofinal subsets of an arbitrary directed quasi-ordered set. For \(<Q,\leq>\) an (upward) directed quasi-order, write \({\mathcal D}(Q)\) for the set of all the non-extendable chains of order type the minimal cardinal number of any non-extendable chain in Q. The set \({\mathcal D}(Q)\) is itself a directed set under the relation \(A\leq B\) if and only if for all \(a\in A\) there is \(b\in B\) with \(a\leq b\), so one can form \({\mathcal D}^ 2(Q)={\mathcal D}({\mathcal D}(Q)).\) With each \({\mathcal A}\in {\mathcal D}^ 2(Q)\), the union of the sets in \({\mathcal A}\) is a subset of Q, called here the original set of \({\mathcal A}\). The authors extend the iteration of the operation \({\mathcal D}\) into the transfinite, and so they have \({\mathcal D}^{\lambda}(Q)\) for arbitrary ordinals \(\lambda\), and for each \({\mathcal A}\) in \({\mathcal D}^{\lambda}(Q)\) they define its original set, a subset of Q. Their main result is that for any directed quasi-order \((Q,\leq)\) there is a minimal ordinal number \(\lambda\) such that every cofinal subset of Q contains a cofinal subset which is the original set of a special type of element in \({\mathcal D}^{\lambda}(Q)\). A special case of the result gives necessary and sufficient conditions for a directed set to contain a cofinal chain.
Reviewer: N.H.Williams


06A06 Partial orders, general
03E05 Other combinatorial set theory
03E10 Ordinal and cardinal numbers
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