A note on blockers in posets

The blocker $A^{\ast }$ of an antichain $A$ in a finite poset $P$ is the set of elements minimal with the property of having with each member of $A$ a common predecessor. The following is done:

1. The posets $P$ for which $A^{\ast \ast }=A$ for all antichains are characterized.
2. The blocker $A^{\ast }$ of a symmetric antichain in the partition lattice is characterized.
3. Connections with the question of finding minimal size blocking sets for certain set families are discussed.