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.


R. Ahlswede - P.L. Erdos - N. Graham: A splitting property of maximal antichains, Combinatorica 15 (1995), 475-480.
R. Ahlswede - L. H. Khachatrian: Splitting properties in partially ordered sets and set systems, in Numbers, Information and Complexity (Althöfer et. al. editors) Kluvier Academic Publisher, (2000), 29-44.
F. Bernstein: Zur Theorie der triginomischen Reihen, Leipz. Ber (Berichte über die Verhandlungen der Königl. Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Math.-phys. Klasse) 60 (1908), 325-338.
D. Duffus - B. Sands: Finite distributive lattices and the splitting property, Algebra Universalis 49 (2003), 13-33.
Mirna Dzamonja: Note on splitting property in strongly dense posets of size $ \aleph_0$, Radovi Matematicki 8 (1992), 321-326.
P.L. Erdos: Splitting property in infinite posets, Discrete Mathematics 163 (1997), 251-256.
P.L. Erdos: Some generalizations of property $ B$ and the splitting property, Annals of Combinatorics 3 (1999), 53-59.
T. Jech, Set Theory, Springer-Verlag, Berlin Heilderberg New York 2003.
J. Klimó, On the minimal coverint of infinite sets, Discrete Applied Mathematics, 45 (1993) 161-168.

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