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/
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