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Center for Nonlinear Studies |
A maximal antichain A of poset P splits if and only if there is a set B ⊂ A such that for each p ∈ P either b ≤ p for some b ∈ B or p ≤ c for some c ∈ A \ B. The poset P is cut-free if and only if there are no x < y < z in P such that [xz] P = [xy]P ∪ [yz]P. By Ahlswede, Erdos and Graham (1995) every maximal antichain in a finite cut-free poset splits. Although this statement for infinite posets fails (see Ahlswede, Khachatrian (2000)) it is true that if a maximal antichain in a cut-free poset “resembles” to a finite set then it splits. We also investigate possible strengthening of the statements that “A does not split” and we could find a maximal strengthening. We will discuss some applications as well. It is interesting to remark that this property is not a stand-alone phenomenon: a version of it is just equivalent to Axiom of Choice.