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Finite cutsets and finite antichains

โœ Scribed by Norbert Sauer; Robert E. Woodrow

Publisher
Springer Netherlands
Year
1984
Tongue
English
Weight
661 KB
Volume
1
Category
Article
ISSN
0167-8094

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โœฆ Synopsis

An ordered set (P, <) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x) u Fx meets every maximal chain of (P, <). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardiinality less than II. We specify n for m < w. Indeed, n -1= m = widthPform=1,2,n=5ifm=3andn>n, ifm>4.Withtheaddedhypothesisthatevevbounded chain has a supremum and i&mum in P, it is shown that for 4 < m < Hi, n = H,. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain. AMS (MOS) subject classifications (1980). 06~0.

๐Ÿ“œ SIMILAR VOLUMES

The finite cutset property
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โœ J.-M. Brochet ๐Ÿ“‚ Article ๐Ÿ“… 1993 ๐Ÿ› John Wiley and Sons ๐ŸŒ English โš– 388 KB

## Abstract A cutset of __H__ is a subset of โˆช __H__ which meets every element of __H.__ __H__ has the finite cutset property if every cutset of __H__ contains a finite one. We study this notion, and in particular how it is related to the compactness of __H__ for the natural topology. MSC: 04A20, 5

Maximum Antichains in Random Subsets of
Maximum Antichains in Random Subsets of a Finite Set
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We consider the random poset P(n, p) which is generated by first selecting each subset of [n]=[1, ..., n] with probability p and then ordering the selected subsets by inclusion. We give asymptotic estimates of the size of the maximum antichain for arbitrary p= p(n). In particular, we prove that if p

Extensions of ordered sets having the fi
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โœ John Ginsburg ๐Ÿ“‚ Article ๐Ÿ“… 1986 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 892 KB

Let P be an ordered set. P is said to have the finite cutset property if for every x in P there is a finite set F of elements which are noncomparable to x such that every maximal chain in P meets {x} t.J F. It is well known that this property is equivalent to the space of maximal chains of P being c