Almost $ff$-universal and $q$-universal varieties of modular $0$-lattices

V. Koubek ¹   ; J. Sichler ²

¹ Department of Theoretical Computer Science and Institute of Theoretical Computer Science Faculty of Mathematics and Physics Charles University Malostranské nám. 25 118 00 Praha 1, Czech Republic
² Department of Mathematics University of Manitoba Winnipeg, Manitoba, Canada R3T 2N2

Colloquium Mathematicum, Tome 101 (2004) no. 2, pp. 161-182

Résumé

\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 $\Bbb V$.

Détail

DOI : 10.4064/cm101-2-3

Keywords: def bbb mathbb variety bbb algebras finite type almost f universal there finiteness preserving faithful functor bbb rightarrow bbb category bbb graphs their compatible maps gamma nonconstant every gamma every nonconstant homomorphism rightarrow has form gamma gamma rightarrow variety bbb q universal its lattice subquasivarieties has lattice subquasivarieties quasivariety algebras finite type quotient its sublattice variety bbb modular lattices shown bbb almost f universal only bbb q universal equivalent non distributivity bbb