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0. Introduction. It is shown in
this paper that the equational class generated by the family of all projective planes
can be characterized by a finite set of lattice identities. The methods developed here
may provide a framework to attack similar problems and a useful medium for
studying modular lattices in general.
By a variety, or equational class, of lattices we mean the class of all lattices
satisfying a given set of lattice identities. A lattice variety is finitely based if it can be
defined by a finite set of identities. Let Λ be the lattice of all lattice varieties.
A systematic study of the lattice Λ dates back seven or eight years ago.
Most recent results in this field, including ours here, are stimulated by an
important discovery of Bjarni Jónsson in [7], Corollary 3.2. (See Baker [1], [2],
Grätzer [4], Hong [5], Jónsson [7], [S], McKenzie [9], [10], Wille [11].)
Our study here continues the works of Grätzer in [4] and of Jónsson in
[S], where the latter completed an unfinished result of the former and in
particular proved that the variety generated by all projective lines is finitely
based.
The rest of the paper is divided into four sections. In §1 we state our main
theorem and its applications but postpone the proofs until §4. In §2 we discuss the
main methods employed here: the method of strong covering, and the notions
of normality and strong normality of sequences of transposes. In case the
family of all varieties that strongly covers a given variety is finite, then the
variety is finitely based. The notions of normality and strong normality, due
to Grätzer and Jónsson respectively, are developed rather completely in
Theorem 3.1. We hope that this theorem will have some applications elsewhere.
Section 4 gives details of the proof of the main lemma stated in Seotion
1.
In the sequel, almost every theorem and lemma has its dual, even though we
rarely make explicit mention of this faot. Also, the notation L denotes a
fixed modular lattice. We wish to express our sinoere gratitude to Professor
Bjarni Jónsson for his helpful suggestions in the ideas as well as the
presentation of this paper. We also wish to thank the referee for his detailed
suggestions.
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