Equational bases and nonmodular lattice varieties
Ralph McKenzie · Transactions of the American Mathematical Society · 1972
This paper is focused on equational theories and equationally defined varieties of lattices which are not assumed to be modular. It contains both an elementary introduction to the subject and a survey of open problems and recent work. The concept of a “splitting” of the lattice of lattice theories is defined here for the first time in print. These splittings are shown to correspond bi-uniquely with certain finite lattices, called “splitting lattices". The problems of recognizing whether a given finite lattice is a splitting lattice, whether it can be embedded into a free lattice, and whether a given interval in a free lattice is atomic are shown to be closely related and algorithmically solvable. Finitely generated projective lattices are characterized as being those finitely generated lattices that can be embedded into a free lattice.