Two problems about perfect distributive lattices (Q1085189)

In this note we solve two problems left open by \textit{E. Fried}, \textit{G. Hansoul}, \textit{E. T. Schmidt} and \textit{J. Varlet} [in Contributions to general algebra 3, Proc. Conf., Vienna 1984, 125-142 (1985; Zbl 0572.06009)]. First, any perfect bounded distributive lattice is isomorphic to a weak Boolean product of directly indecomposable perfect bounded lattices. Moreover, a bounded distributive lattice is perfect if and only if it admits a (essentially unique) sheaf representation by bounded chains. Second, a poset is representable over the class of perfect distributive lattices if and only if it is a cardinal sum of chains \(C_ i\) such that \(0\oplus C_ i\oplus 1\) is algebraic for every i.