On the number of join-irreducibles in a congruence representation of a   finite distributive lattice
  For a finite lattice L, let EL denote the reflexive and transitive closure of the join-dependency relation on L, defined on the set J(L) of all join-irreducible elements of L. We characterize the relations of the form EL, as follows: Theorem. Let E be a quasi-ordering on a finite set P. Then the following conditions are equivalent: (i) There exists a finite lattice L such that (J(L),EL) is isomorphic to the quasi-ordered set (P,E). (ii) There are not exactly two elements x in P such that p E x, for any p in P. For a finite lattice L, let je(L) = |J(L)|-|J(Con L)|, where Con L is the congruence lattice of L. It is well-known that the inequality je(L) $\ge$ 0 holds. For a finite distributive lattice D, let us define the join-excess function: JE(D) = min(je(L) | Con L isomorphic to D). We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that JE(D) $\le$ (2/3)| J(D)|, for any finite distributive lattice D; the constant 2/3 is best possible. A special case of this formula gives a characterization of congruence lattices of finite lower bounded lattices. 
George Gr\"atzer, Friedrich Wehrung (LMNO)