Almost \textit{ff}-universality implies \(Q\)-universality (Q1037407)

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 \(\mathbb Q\). The main result of the paper states that any finite-to-finite (algebraically) almost universal quasivariety \(\mathbb Q\) of a finite type is \(Q\)-universal.