A note on blockers in posets (Q1881051)

For finite bounded posets \(P\), blockers \(A^*\) of antichains \(A\) are determined as \(A^*=\min\{x\in P\mid \wedge(x)\cap\wedge(a)\neq\emptyset\) for every \(a\in A\}\) where \(\wedge(x)\) is the set of atoms below \(x\) and where \(\min E\) denotes the set of minimal elements of \(E\subseteq P\). Otherwise blockers may also be defined. In general \(A^*=A^{***}\), \(A^{**}\subseteq A\), \(A\subset B\) implies \(B^*\subset A^*\). Thus, it becomes a question of interest to consider the question of identifying posets \(P\) such that \(A=A^{**}\) for all antichains in \(P\) in interesting ways. This the authors do successfully by following theorem 2.7 (general) with corollaries (special consequences for special cases). After this they consider special antichains (e.g., symmetric antichains in \(\Pi_n)\) in special posets derived from special situations (e.g., subspace arrangements of some type, the Turan property) to add further useful details to the results obtained in their main theorem where \(P\) is seen to be isomorphic to a well-complemented subposet of a Boolean lattice which does not itself have to be a lattice.