The following proposition answers the question of the relation between cardinalities of antichains of P and anti-chains of F? Proposition 1

1. Every anti-chain in P has cardinality 1 =) every anti-chain in F has cardinality 1 2. There exists an anti-chain in P of cardinality 2, but no element of P is incomparable with two different elements =) every anti-chain in F has cardinality at most 2 3. There exists an element in P which is incomparable with two different elements =) there exists anti-chains of any cardinality in F