## 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

### MSC:

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

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