**Péter L. Erdos and Lajos Soukup**

A maximal antichain of poset *splits* if and only if
there is a set
such that for each either for some or for some
. The
poset is *cut-free * if and only if there are no
in such that
.
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 `` does not split'' and we could find a maximal strengthening.

- 1
- R. Ahlswede - P.L. Erdos - N. Graham: A
splitting property of maximal antichains,
*Combinatorica***15**(1995), 475-480. - 2
- 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. - 3
- 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. - 4
- D. Duffus - B. Sands: Finite distributive lattices and the
splitting property,
*Algebra Universalis***49**(2003), 13-33. - 5
- Mirna Dzamonja: Note on splitting property in
strongly dense posets of size ,
*Radovi Matematicki***8**(1992), 321-326. - 6
- P.L. Erdos: Splitting property in infinite
posets,
*Discrete Mathematics***163**(1997), 251-256. - 7
- P.L. Erdos: Some generalizations of property
and the splitting property,
*Annals of Combinatorics***3**(1999), 53-59. - 8
- T. Jech,
*Set Theory*, Springer-Verlag, Berlin Heilderberg New York 2003. - 9
- J. Klimó, On the minimal coverint of infinite sets,
*Discrete Applied Mathematics*,**45**(1993) 161-168.

**Key words and phrases**:

**2000 Mathematics Subject Classification**: