On almost universal varieties of modular lattices

Abstract. If a variety V of lattices contains the modular lattice $ M_{3,3} $ then every monoid S can be represented by all non-constant endomorphisms of a lattice $ L\in V $, and non-isomorphic such lattices form a proper class. On the other hand, if V is contained in the variety $ \bold M_n $ generated by a finite modular lattice $ M_n $ then the full endomorphism monoid determines lattices from V up to an isomorphism or an anti-isomorphism. It is not known whether or not the variety $ \bold M_{\infty} $ generated by all finite lattices $ M_n $ is of the first or of the second kind.