On relative universality and \(Q\)-universality (Q1770613)

A concrete category \(\mathcal K\) is universal if there exists a full and faithfull functor \(F: {\mathcal G}\rightarrow {\mathcal K}\) from the category \(\mathcal G\) of all graphs into \(\mathcal K\). If \(F\) assigns a \(\mathcal K\)-object with finite underlying set to every finite graph, \(\mathcal K\) is said to be finite-to-finite universal (or ff-universal). A quasivariety \(\mathcal K\) of algebras of finite type is \(Q\)-universal if the inclusion-ordered lattice \(L({\mathcal K})\) of all subquasivarieties of \(\mathcal K\) has the property that for any quasivariety \(\mathcal M\) of algebras of finite type, the lattice \(L({\mathcal M})\) is a quotient of a sublattice of \(L({\mathcal K})\). \textit{M. E. Adams} and \textit{W. Dziobiak} [Algebra Univers. 46, 253--283 (2001; Zbl 1059.08002)] proved that any ff-universal quasivariety must be \(Q\)-universal, and they asked whether a somewhat weaker hypothesis could lead to the same conclusion. The authors show that the hypothesis cannot be weakened to its naturally extreme form. They introduce a form of relative universality with respect to some ideal of morphisms. As the main result of the paper they present an example of a variety of distributive double \(p\)-algebras showing that for a comparatively large ideal, the relative universality does not imply \(Q\)-universality.