On almost universal varieties of modular lattices (Q5950756)

A category \(\mathcal A\) of algebras is almost universal if it contains a class determining a full subcategory of \(\mathcal A\) whose non-constant homomorphisms are closed under composition and form a universal category (i.e. a category for which the category of graphs has a full embedding). The authors show that the variety generated by the lattice \(M_{3,3}\) is almost universal and that no more than two non-isomorphic lattices in the variety \(M_n\) (for each \(n\) finite) have isomorphic monoids of endomorphisms. The second achievement is an extension of Schein's result: any variety of lattices of which every nontrivial member has a prime ideal can have at most two non-isomorphic members with isomorphic endomorphism monoids.