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) |{x ∈ P | p E x }| 6= 2, for any p ∈ P . For a finite lattice L, let je(L) = | J(L)| − | J(ConL)|, where ConL is the congruence lattice of L. It is well-known that the inequality je(L) ≥ 0 holds. For a finite distributive lattice D, let us define the join-excess function: JE(D) = min(je(L) | ConL ∼= D). We provide a formula for computing the join-excess function of a finite distributive lattice D. This formula implies that JE(D) ≤ (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.