Posets generated by irreducible elements (Q1404382)

Let \(J\) be a poset and \(L\) the Alexandroff completion of \(J\). Let be \(C_0L= \{P: J\subseteq P\subseteq L\), \(J(P)= J\}\) where \(J(P)\) is the set of all \(\vee\)-irreducible elements of \(P\). The complete lattices in \(C_0L\) form a closure system \(C_\infty L\). The posets for which \(C_0L\) is a complete Boolean atomic lattice and posets for which \(C_\infty L\) is distributive (or modular) are described. Similar results for \(C_k L\), the closure system of all posets in \(C_0 L\) that are closed under meets of less than \(k\) elements (\(k\) a cardinal number), are obtained.