Compactness and subsets of ordered sets that meet all maximal chains
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LetP be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains ofP is compact. The symbolsP1 andP2 refer to two particular ordered sets considered below. It is shown that the space of maximal chains ℳ (P) is compact ifP satisfies any of the following conditions: (i)P contains no copy ofP1 or its dual and all antichains inP are finite. (ii)P contains no properN and every element ofP belongs to a finite maximal antichain ofP. (iii)P contains no copy ofP1 orP2 and for everyx inP there is a finite subset ofP which is coinitial abovex. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain.
AMS (MOS) subject classifications (1980)06A10 54D30
Key wordsMaximal chain cutset chain complete antichain cofinal subset
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- 1.M. Bell and J. Ginsburg: Compact spaces and spaces of maximal complete subgraphs, to appear inTrans. Amer. Math. Soc. Google Scholar
- 2.J. Ginsburg, I. Rival, and B. Sands: Antichains and finite sets that meet all maximal chains, to appear.Google Scholar
- 3.P. A. Grillet (1969) Maximal chains and antichains,Fund. Math. 65, 157–167.Google Scholar
- 4.K. Kunen (1977) Combinatorics, inHandbook of Mathematical Logic (ed. Jon Barwise), North-Holland Amsterdam, pp. 371–401.Google Scholar
- 5.I. Rival and N. Zaguia: Crosscuts and cutsets, to appear.Google Scholar
- 6.N. Sauer and R. Woodrow (1984) Finite cutsets and Finite antichains,Order 1, 35–46.Google Scholar