On sublattice lattice varieties
In this note the author gives solutions to Problem I.6 and Problem I.7 in \textit{G. Grätzer}'s book: General lattice theory (1978; Zbl 0436.06001). The solutions are as follows: (1) For any non-trivial lattice variety \({\mathcal K}\), Sub(\({\mathcal K})\) is the variety of all lattices. (2) The only lattice varieties \({\mathcal K}\) for which Sub(\({\mathcal K})={\mathcal K}\) are the variety of all lattices and the trivial variety of the one- element lattice. Here Sub(\({\mathcal K})\) denotes the variety generated by the class of all sublattice lattices of members of \({\mathcal K}\). The key to the problems is the following theorem: Any finite lattice can be embedded into the sublattice of a finite Boolean lattice.
scientific article; zbMATH DE number 4191756