FINITELY BASED LATTICE VARIETIES (I)

In R. McKenzie[12], it is shown that the cardinality of the lattice variety is $2^\aleph_0$ and K. Baker[1] contains the stronger result that M, the variety of all modular lattices, has $2^\aleph_0$ subvarieties. It follows that there exists a variety of modular lattices that is not finitely based. In fact, K. Baker[2] gave an example of such a variety. And it was proved by K. Baker [2] and B. Jonsson [8] that join of two finitely based lattice varieties is not always finitely based. K. Baker[2] gave an explicit example of case of nonmodular lattice variety. Then it is proposed whether the join of two finitely based varieties if finitely based under certain conditions. The answer to the question is not affirmative.