Structure of quasivariety lattices. I: Independent axiomatizability
A quasivariety \(K\) has an \(\omega \)-independent quasi-equational basis in a quasivariety \(M\) if there are a basis \(\Phi \) of \(K\) in \(M\) and a partition \(\Phi =\cup_ {n<\omega} \Phi_n\) such that for every \(i\) \(\Phi\setminus\Phi_i\) is not a basis of \(K\) in \(M\). A quasivariety \(K\) is a \(Q\)-universal if its lattice of subquasivarieties has, as a homomorphic image of a sublattice, the lattice of subquasivarieties of any quasivariety of algebras of finite type. The authors find a sufficient condition for a quasivariety \(K\) to have continuum many subquasivarieties that have no independent quasi-equational bases in \(K\) but have \(\omega\)-independent quasi-equational bases in \(K\). This condition also implies that \(K\) is \(Q\)-universal.
scientific article; zbMATH DE number 7140958