Orbits of Antichains in Ranked Posets

We consider the permutation f of antichains of a ranked poset P , moving the set of lower units of any monotone boolean function on P to the set of its upper zeros. A duality relation on orbits of this permutation is found, which is used for proving a conjecture by M. Deza and K. Fukuda. For P a direct product of two chains, possible lengths of orbits are completely determined.