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

Simple item page
dc.creatorWehrung, Friedrich
dc.date2005-01-24
dc.date.accessioned2026-07-07T05:16:19Z
dc.date.available2026-07-07T05:16:19Z
dc.descriptionFor 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.
dc.identifierhttps://arxiv.org/abs/math/0501404
dc.identifierhttp://arxiv.org/abs/math/0501404
dc.identifierJournal of Algebra and Its Applications Vol. 1, no. 2 (2002) 1--28
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/73946
dc.subjectGeneral Mathematics
dc.subject06B05, 06B10, 06B99, 06B35
dc.titleFrom join-irreducibles to dimension theory for lattices with chain conditions
dc.typetext

Files

Collections

arXiv