A variety 𝕍 of algebras of a finite type is almost ff-universal if there is a finiteness-preserving faithful functor F: 𝔾 → 𝕍 from the category 𝔾 of all graphs and their compatible maps such that Fγ is nonconstant for every γ and every nonconstant homomorphism h: FG → FG' has the form h = Fγ for some γ: G → G'. A variety 𝕍 is Q-universal if its lattice of subquasivarieties has the lattice of subquasivarieties of any quasivariety of algebras of a finite type as the quotient of its sublattice. For a variety 𝕍 of modular 0-lattices it is shown that 𝕍 is almost ff-universal if and only if 𝕍 is Q-universal, and that this is also equivalent to the non-distributivity of 𝕍.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm101-2-3,
author = {V. Koubek and J. Sichler},
title = {Almost ff-universal and q-universal varieties of modular 0-lattices},
journal = {Colloquium Mathematicae},
volume = {100},
year = {2004},
pages = {161-182},
zbl = {1066.06004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm101-2-3}
}
V. Koubek; J. Sichler. Almost ff-universal and q-universal varieties of modular 0-lattices. Colloquium Mathematicae, Tome 100 (2004) pp. 161-182. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm101-2-3/