Some varieties and convexities generated by fractal lattices

Let L be a lattice. If for each a < b ∈ L there is a lattice embedding of φ : L → [a, b] then L is called a semifractal. If, in addition, 0,1 ∈ L and φ can always be chosen such that φ(0) = a and φ(1) = b then L is said to be a 0–1-semifractal. Now let L be a bounded lattice. If for each a1 < b1 ∈ L and a2 < b2 ∈ L there is a lattice embedding ψ : [a1, b1] → [a2, b2] with ψ(a1) = a2 and ψ(b1) = b2 then we say that L is a quasifractal. If ψ can always be chosen an isomorphism or, equivalently, if L is isomorphic to each of its nontrivial intervals then L will be called a fractal lattice or, shortly, a fractal. Although there is an obvious hierarchy of these notions and we construct 0–1-semifractals which are not quasifractals it remains only a conjecture that the above notions are distinct. A variety generated by a fractal lattice is called fractal generated, and analogous terminology applies for the rest of our new notions. We show that semifractal generated nondistributive lattice varieties cannot be of residually finite length. This will easily imply 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 D of all distributive lattices are the only known fractal generated lattice varieties at present. Four distinct countable distributive fractal lattices will be given such that each of them generates D. After showing that each lattice can be embedded in a quasifractal, continuously many quasifractals will be given such that each of them has the power א0 and generates the variety of all lattices. The last section of the paper is devoted to an application. A class of lattices is called a convexity if it is closed under taking homomorphic images, convex sublattices and direct products. This notion is due to Ervin Fried. Each nontrivial lattice variety includes the variety generated by the two element lattice, which is a minimal variety. The question if the same is true for convexities goes back to Jakub́ık [15]. Using appropriate semifractals we give many convexities which include no minimal convexity.