\(Q\)-universal varieties of bounded lattices. (Q1771953)

For a quasivariety \(K\), let \(L(K)\) denote the lattice of all quasivarieties contained in \(K\). A quasivariety \(K\) of algebraic systems of finite type is \(Q\)-universal if, for any quasivariety \(M\) of finite type, \(L(M)\) is a homomorphic image of a sublattice of \(L(K)\). It is known that, for every variety \(K\) of \((0,1)\)-lattices, if \(K\) contains a finite nondistributive simple \((0,1)\)-lattice, then \(K\) is \(Q\)-universal. The converse is true within varieties of modular \((0,1)\)-lattices but not in the general case: There exists a family \((A_i;\;i<2^\omega )\) of locally finite varieties of \((0,1)\)-lattices each of which is \(Q\)-universal, but none of which contains a non-distributive simple \((0,1)\)-lattice. The construction of \(A_i\) is included in the paper.