Generalizing Galvin and Jónsson’s classification to N5

The problem of determining (up to lattice isomorphism) the lattices that are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and Jónsson when they classified (up to lattice isomorphism)...

Full description

Saved in:
Bibliographic Details
Published in:Algebra universalis 2020, Vol.81 (3)
Main Author: Chan, Brian T.
Format: Article
Language:English
Subjects:
Algebra
Isomorphism
Lattices (mathematics)
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:The problem of determining (up to lattice isomorphism) the lattices that are sublattices of free lattices is in general an extremely difficult and an unsolved problem. A notable result towards solving this problem was established by Galvin and Jónsson when they classified (up to lattice isomorphism) all of the distributive sublattices of free lattices in 1959. In this paper, we weaken the requirement that a sublattice of a free lattice be distributive to requiring that a such a lattice belongs in the variety of lattices generated by the pentagon N 5 . Specifically, we use McKenzie’s list of join-irreducible covers of the variety generated by N 5 to extend Galvin and Jónsson’s results by proving that all sublattices of a free lattice that belong to the variety generated by N 5 satisfy three structural properties. Afterwards, we explain how the results in this paper can be partially extended to lattices from seven known infinite sequences of semidistributive lattice varieties.
ISSN:0002-5240
1420-8911
DOI:10.1007/s00012-020-00674-6