Finite distributive lattices and doubly irreducible elements
For a finite distributive lattice \(L\), let \(J(L)\) be the poset of its join-irreducible elements. The MacNeille completion of \(J(L)\) is not always distributive: a theorem of Monjardet and Wille says that this is the case if and only if \(L\) is generated by the set \(\text{Irr} (L)\) of its both join and meet irreducible elements. This leads the authors to study the following structure. If \(G\) is a finite poset, let \({\mathcal D}(G)\) be the class of all finite distributive lattices \(L\) such that \(L\) is generated by \(\text{Irr} (L)=G\), quasi-ordered by \(L\leq L'\) if there is an onto homomorphism \(L'\to L\) which is the identity on \(G\). The main result of the paper is that the partially ordered quotient of \({\mathcal D}(G)\) is a Boolean lattice \({\mathcal D}'\) whose greatest element is the free bounded distributive lattice generated by \(G\). A description of the least element of \({\mathcal D}'\) is also obtained as well as a characterization of membership in \({\mathcal D}(G)\).
scientific article; zbMATH DE number 1110083