How to split antichains in infinite posets

Péter L. Erdos    and    Lajos Soukup

A maximal antichain $A$ of poset $P$ splits if and only if there is a set $B\subset A$ such that for each $p\in P$ either $b\le p$ for some $b\in B$ or $p\le c$ for some $c\in A\setminus B$. The poset $P$ is cut-free if and only if there are no $x<y<z$ in $P$ such that ${[x,z]_{}}_P ={[x,y]_{}}_P\cup {[y,z]_{}}_P$. 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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