From join-irreducibles to dimension theory for lattices with chain conditions

Abstract

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 defined by generators D(p), for p in J(L), and relations D(p) +D(q) = D(q), for all p, q in J(L) such that p D\_L q . As a consequence, 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 quasi-identity 2x=x implies x=0. Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A $\bp$ B f A and B is join-semidistributive, and Dim(A $\bp$ B) is isomorphic to $Dim A \otimes Dim B$, where $\otimes$ denotes the tensor product of commutative monoids.