(Quasi-)Birkhoff classes and complete homomorphic images of lattices of (quasi-) varieties (Q1374997)

We suggest in this paper a unified approach to the study of lattices of varieties and lattices of quasivarieties, as well as a method of constructing their homomorphic images. This approach is based on (quasi-)Birkhoff classes. We apply it, e.g., to find sufficient conditions of embeddability of a free lattice of denumerable rank in a given lattice of quasivarieties and show that these conditions are at the same time sufficient conditions for a quasivariety to be \(Q\)-universal. We prove that a lattice of varieties of modular lattices has a complete homomorphism onto the Boolean lattice of subsets of a denumerable set. An acquaintance with the earlier paper [\textit{V. A. Gorbunov} and \textit{V. I. Tumanov}, Dokl. Akad. Nauk SSSR 254, 272-275 (1980; Zbl 0466.08004)] would be useful for readers of this paper.