From Join-irreducibles to Dimension Theory for Lattices with Chain Conditions

For a finite lattice L, the congruence lattice ConL of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation DL on J(L). We establish a similar version of this result for the dimension monoid DimL of L, a natural precursor of ConL. For L join-semidistributive, this result takes the following form: Theorem 1. Let L be a finite join-semidistributive lattice. Then DimL is isomorphic to the commutative monoid defined by generators ∆(p), for p ∈ J(L), and relations ∆(p) + ∆(q) = ∆(q), for all p, q ∈ J(L) such that p DL q. As a consequence of this, we obtain the following results: Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff DimL is strongly separative, iff it satisfies the axiom (∀x)(2x = x ⇒ x = 0). Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A B of A and B is join-semidistributive, and the following isomorphism holds: Dim(A B) ∼= DimA ⊗DimB.