## 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
Full Text:

### References:

  Ginsburg, S; Isbell, J, The category of cofinal types I, Trans. amer. math. soc., 116, 386-393, (1965) · Zbl 0212.32602  A. Hajnal and N. Sauer, Complete subgraphs of infinite multipartite graphs and antichains in partially ordered sets. · Zbl 0655.05002  Isbell, J, The category of cofinal types II, Trans. amer. math. soc., 116, 394-416, (1965) · Zbl 0212.32701  Milner, E.C; Prikry, K, The cofinality of a partially ordered set, (), to appear · Zbl 0511.06002  Milner, E.C; Sauer, N, Remarks on the cofinality of a partially ordered set and a generalization of König’s lemma, Discrete math., 35, 165-171, (1981) · Zbl 0465.05041  Milner, E.C; Pouzet, M, On the cofinality of partially ordered sets, (), 279-298 · Zbl 0489.06001  Pouzet, M, Parties cofinales des ordres partiels ne contenant pas d’antichaines infinies, J. London math. soc., (1980), to appear
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.