Some varieties and convexities generated by fractal lattices (Q1013987)

In colloquial language, a fractal is a geometric shape that is self-similar (at least approximately) to arbitrarily small parts of itself. A fractal lattice, or shortly fractal, is defined to be a lattice which is isomorphic to each of its nontrivial intervals. A variety generated by a fractal lattice is called fractal generated, with analogous terminology for other variants. In this paper, the author shows that a semifractal generated nondistributive lattice variety cannot be of residually finite length. This easily implies that there are exactly continuously many lattice varieties which are not semifractal generated. Semifractal considerations are applied to construct examples of convexities that include no minimal convexity, where a convexity is a class of lattices closed under taking homomorphic images, convex sublattices and direct products.