Characterizable classes of lattices
\(\mathbf{N}(M)\) denotes the class of systems of a fixed signature \(\sigma\) which do not contain subsystems isomorphic to the systems in \(M\), a given set of finite systems of signature \(\sigma\). A subclass \(\mathcal{C}'\) of a class \(\mathcal{C}\) of algebraic systems is said to be characterizable in \(\mathcal{C}\) by a set \(M\) of finite systems if \(\mathcal{C}'=\mathcal{C}\cap \mathbf{N}(M)\). It is known that the variety \(V_l\) of \(l\)-modular lattices is characterizable in the class of all latices by \(L_{13}\). Also, the variety of \(p\)-modular lattices is characterizable by \(L_{15}\). It is shown in the paper under review that \(V_l+V_p\), the join of \(V_l\) and \(V_p\) in the lattice of all lattices, is not a characterizable variety. The author answers in the affirmative the following question (raised by V. A. Gorbunov): Do there exist two strict quasivarieties whose meet is a variety? Concretely, the author proves that \(\mathbf{N}(L_1, L_{12})\cap \mathbf{N}(L_2, L_{11})= \mathbf{N}(L_1, L_2, L_{11}, L_{12})\) is a variety and \(\mathbf{N}(L_1, L_{12})\) and \(\mathbf{N}(L_2, L_{11})\) are quasivarieties but not varieties, where \(L_2\) is the dual of \(L_1\) (\(L_1\) placed up-side down) and \(L_{12}\) is the dual lattice of \(L_{11}\). Finally, the author exhibits an example of a characterizable locally finite lattice variety.
scientific article; zbMATH DE number 6121533