A combinatorial representation for a special class of complete distributive lattices (Q1600817)

As is well known, representation theorems by Tarski (for complete atomic Boolean algebras) and Birkhoff (for finite distributive lattices) involve purely lattice-theoretic techniques. For distributive lattices in general, the topological tool of Priestley duality is required. Here the authors produce a representation theorem for a certain class of distributive lattices by using only lattice-theoretic techniques and avoiding topological notions. The (restricted) class concerned consists of completely distributive lattices that are what the authors call Pezzoli lattices, such a lattice being defined as a bounded distributive lattice \(D\) in which there is an order isomorphism \(\gamma\) from the set of completely join-irreducible elements to the set of completely meet-irreducible elements such that if \(b=\gamma(a)\) then \([a,1]\cap [0,b]= \emptyset\) and \([a,1] \cup [0,b]=D\).