On the number of join-irreducibles in a congruence representation of a finite distributive lattice (Q2496143)

Let \(L\) be a finite lattice, \(\trianglelefteq \) the reflexive and transitive closure of the join-independency relation and \(J(L)\) be the set of join-irreducibles of \(L\). It is shown that for a quasi-ordered finite set \(P\), the following are equivalent: \quad (i) there exists a finite lattice \(L\) such that \((J(L),\trianglelefteq )\) is isomorphic to \(P\); \quad (ii) the cardinality of \(\{x\in P; p\leq x\}\) differs from 2 for each \(p\in P\); For a finite distributive lattice a formula computing the join-excess function is involved. A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices. Six open problems are included in the paper.