Lattice varieties covering the smallest nonmodular variety
Bjàrni Jónsson, Ivan Rival · Pacific Journal of Mathematics · 1979
There are sixteen varieties of lattices that are known to cover N 9 the variety generated by the five-element nonmodular lattice N. Fifteen of these are generated by finite subdirectly irreducible lattices L ly L 2 y "t L lΰ9 and the sixteenth is jointly generated by N and the diamond M z .We show that every variety of lattices that properly contains N includes one of the lattices M Zf L u L 2t -,L lδ .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.9. I. Rival, Varieties of nonmodular lattices, Notices Amer.