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.