How To Split Antichains In Infinite Posets*

A maximal antichain A of poset P splits if and only if there is a set BA such that for each pP either bp for some bB or pc for some cA\B. The poset P is cut-free if and only if there are no x < y < z in P such that [x,z] P = [x,y] P ∪ [y,z] P . By [1] every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see [2])) we prove here that if a maximal antichain in a cut-free poset “resembles” to a finite set then it splits. We also show that a version of this theorem is just equivalent to Axiom of Choice.

We also investigate possible strengthening of the statements that “A does not split” and we could find a maximal strengthening.

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Correspondence to Péter L. Erdős.

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* This work was supported, in part, by Hungarian NSF, under contract Nos. T37846, T34702, T37758, AT 048 826, NK 62321. The second author was also supported by Bolyai Grant.

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Erdős, P.L., Soukup, L. How To Split Antichains In Infinite Posets*. Combinatorica 27, 147–161 (2007).

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Mathematics Subject Classification (2000):

  • 06A07
  • 03E05