Almost ff-universality Implies Q-universality

Abstract

A concrete category $\mathbb {Q}$ is finite-to-finite (algebraically) almost universal if the category of graphs and graph homomorphisms can be embedded into $\mathbb {Q}$ in such a way that finite $\mathbb {Q}$-objects are assigned to finite graphs and non-constant $\mathbb {Q}$-morphisms between any $\mathbb {Q}$-objects assigned to graphs are exactly those arising from graph homomorphisms. A quasivariety $\mathbb {Q}$ of algebraic systems of a finite similarity type is Q-universal if the lattice of all subquasivarieties of any quasivariety $\mathbb {R}$ of algebraic systems of a finite similarity type is isomorphic to a quotient lattice of a sublattice of the subquasivariety lattice of $\mathbb {Q}$. This paper shows that any finite-to-finite (algebraically) almost universal quasivariety $\mathbb {Q}$ of a finite type is Q-universal.