Vol. 82, No. 2, 1979

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ISSN: 0030-8730
Lattice varieties covering the smallest nonmodular variety

Bjarni Jónsson and Ivan Rival

Vol. 82 (1979), No. 2, 463–478

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
Received: 9 November 1976
Revised: 21 March 1978
Published: 1 June 1979
Bjarni Jónsson
Ivan Rival