Resumen de Some varieties and convexities generated by fractal lattices

Gábor Czédli

• We introduce definitions of semifractal, 0�1-fractal, quasifractal and fractal lattices. A variety generated by a fractal lattice is called fractal generated, with analogous terminology for the other variants.

  We show 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. On the other hand, for each prime field F, the variety generated by all subspace lattices of vector spaces over F is shown to be fractal generated. These countably many varieties and the class $${\mathcal{D}}$$ of all distributive lattices are the only known fractal generated lattice varieties at present. Four distinct countable distributive fractal lattices are given each of which generates $${\mathcal{D}}$$. After showing that each lattice can be embedded in a quasifractal, continuously many quasifractals are given each of which has cardinality $${\aleph_{0}}$$ and generates the variety of all lattices.

  Semifractal considerations are applied to construct examples of convexities that include no minimal convexity, thus answering a question of Jakubík. (A convexity is a class of lattices closed under taking homomorphic images, convex sublattices and direct products, a notion due to Ervin Fried.)