Lattice varieties covering the smallest nonmodular variety

Bjarni Jónsson and Ivan Rival

Vol. 82 (1979), No. 2, 463–478

DOI: 10.2140/pjm.1979.82.463

Abstract

There are sixteen varieties of lattices that are known to cover N, the variety generated by the five-element nonmodular lattice N. Fifteen of these are generated by finite subdirectly irreducible lattices L₁,L₂,⋯,L₁₅, and the sixteenth is jointly generated by N and the diamond M₃. We show that every variety of lattices that properly contains N includes one of the lattices M₃,L₁,L₂,⋯,L₁₅. Of these sixteen lattices, the first six fail to be semidistributive; in fact, every variety of lattices in which the semidistributive law fails contains one of these six.

Mathematical Subject Classification 2000

Primary: 06B20

Milestones

Received: 9 November 1976

Revised: 21 March 1978

Published: 1 June 1979