Almost $ff$-universal and $q$-universal varieties of modular $0$-lattices
Volume 101 / 2004
Abstract
\def\Bbb#1{{\mathbb#1}}A variety $\Bbb V$ of algebras of a finite type is almost $f\!f$-universal if there is a finiteness-preserving faithful functor $F:\Bbb G\rightarrow \Bbb V$ from the category $\Bbb G$ of all graphs and their compatible maps such that $F\gamma$ is nonconstant for every $\gamma$ and every nonconstant homomorphism $h:FG\rightarrow FG'$ has the form $h=F\gamma$ for some $\gamma :G\rightarrow G'$. A variety $\Bbb V$ 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 $\Bbb V$ of modular $0$-lattices it is shown that $\Bbb V$ is almost $f\!f$-universal if and only if $\Bbb V$ is $Q$-universal, and that this is also equivalent to the non-distributivity of $\Bbb V$.
