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Finite cutsets and finite antichains

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Abstract

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} ∪ 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 cardinality less than n. We specify n for m < w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n⩾ℵ1 if m ≥4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4⩽m⩽ℵ0, n=ℵ0. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.

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References

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Additional information

This work was supported by the NSERC of Canada.

Communicated by I. Rival

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Sauer, N., Woodrow, R.E. Finite cutsets and finite antichains. Order 1, 35–46 (1984). https://doi.org/10.1007/BF00396272

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AMS (MOS) subject classifications (1980)

  • 06A10

Key words

  • Ordered set
  • chain
  • antichains
  • width
  • cutset