Equational bases and nonmodular lattice varieties

McKenzie, Ralph

doi:10.1090/S0002-9947-1972-0313141-1

Equational bases and nonmodular lattice varieties
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Trans. Amer. Math. Soc. 174 (1972), 1-43 Request permission

Abstract:

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.

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Sur certaines axiomes de la théorie des structures

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