E.T.: How to derive finite semimodular lattices from distributive lattices

Abstract. It is proved that the class of finite semimodular lattices is the same as the class of cover-preserving join-homomorphic images of direct products of finitely many finite chains. There is a trivial “representation theorem ” for finite lattices: each of them is a join-homomorphic image of a finite distributive lattice. This follows from the fact that the finite free join semilattices (with zero) are the finite Boolean lattices. The goal of the present paper is to give two analogous but stronger representation theorems for finite semimodular (also called upper semimodular) lattices. Both theorems state that these lattices are very special join-homomorphic images of appropriate finite distributive lattices. This way we generalize the main results of G. Grätzer and E. Knapp [4] and [5]. Since even the above-mentioned trivial representation theorem was useful in [1], there is a hope that the new achievements will be of some interest in the future. To formulate our results we need the following notions. A sublattice {a1 ∧ a2, a1, a2, a1 ∨ a2} of a lattice is called a covering square if a1 ∧ a2 ≺ ai ≺ a1 ∨ a2 for i = 1, 2. A planar lattice is called slim if every covering square is an interval. Now let L and K be finite lattices. A join-homomorphism ϕ: L → K is said to be cover-preserving iff it preserves the relation . Similarly, a join-congruence Φ of L is called cover-preserving if the natural join-homomorphism L → L/Φ, x 7 → [x]Φ is cover-preserving. As usual, J(L) stands for the poset of all nonzero join-irreducible elements of L. For a poset P, H(P) denotes the lattice of all hereditary subsets (order ideals) of P. The width w(P) of a (finite) poset P is defined to be max{n: P has an n-element antichain}. First we prove: Lemma. Let Φ be a join-congruence of a finite semimodular lattice M. Then Φ is cover-preserving if and only if for any covering square S = {a ∧ b, a, b, a ∨ b} i