Let $CSub\,(\text{\bf K})$ denote the variety of lattices generated by convex sublattices of lattices in $\text{\bf K}$. For any proper variety $\text{\bf V}$, the variety $CSub\,(\text{\bf V})$ is proper. There are uncountably many varieties $\text{\bf V}$ with $CSub\,(\text{\bf V})=\text{\bf V}$.
@article{118725, author = {V\'aclav Slav\'\i k}, title = {A note on convex sublattices of lattices}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {36}, year = {1995}, pages = {7-9}, zbl = {0821.06007}, mrnumber = {1334407}, language = {en}, url = {http://dml.mathdoc.fr/item/118725} }
Slavík, Václav. A note on convex sublattices of lattices. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) pp. 7-9. http://gdmltest.u-ga.fr/item/118725/
General Lattice Theory, Birkhäuser Verlag, 1978. | MR 0504338
Lattices with unique complementation, Pacif. J. Math. 92 (1981), 1-13. (1981) | MR 0618040 | Zbl 0468.06005
Equational bases for lattice theories, Math. Scand. 27 (1970), 24-38. (1970) | MR 0274353 | Zbl 0307.08001
Primitive subsets of lattices, Alg. Universalis 2 (1972), 95-98. (1972) | MR 0311524 | Zbl 0269.06001