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

Wehrung, Friedrich

HAL, hal-00004017 / Harvested from HAL

Résumé

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.

Détail

Publié le : 2002-07-05
Classification:  box product,  joinsemidistributive,  lower bounded,  primitive monoids,  strong separativity,  tensor product,  box product.,  Lattice,  monoid,  Dimension,  join-irreducible,  join dependency,  06B05, 06B10, 06B99, 06B35,  [MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM]