FROM JOIN-IRREDUCIBLES TO DIMENSION THEORY FOR LATTICES WITH CHAIN CONDITIONS

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D_L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form:

Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations

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 Dim L is strongly separative, iff it satisfies the axiom

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: