Equational Bases and Nonmodular Lattice Varieties ( ! )

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. Introduction. What we do in this paper is fairly described by the phrase "equational model theory of lattices." We take this in a broad sense, as the study of lattices and their equational theories, with particular emphasis on finding connections relating algebraic properties of lattices to various properties of their theories. Since many interesting properties of lattice theories can be defined abstractly, purely by reference to the lattice of all such theories, we are naturally led to investigate the structure of S the lattice of all equational theories of lattices. Recent studies in this largely unexplored field have been greatly stimulated by a paper of Bjarni Jónsson [6]. The central result of that paper (L6, Corollary 3.2], stated below as Lemma 1.4) has already found a number of applications, in the original paper as well as in [l], [7], L9J and the present paper. Broadly speaking, Jónsson's result tells us that the cords binding lattices to their theories are much more tightly drawn than one would expect on the basis of experience with other equationally defined classes of algebras, such as groups and rings. The happy consequences of this fact, both for the algebraic and the equational study of lattices, will hardly be exhausted by the present paper; and we expect other researchers to take up the challenge. To that end, we have given Received by the editors March 5, 1969. AMS (MOS) subject classifications (1970). Primary 08A15, 06A20, 02G05, 02G15.