Geodesic
Structure of Quasivariety Lattices. I. Independent Axiomatizability
Algebra i logika, Tome 57 (2018) no. 6, pp. 684-710.

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We find a sufficient condition for a quasivariety $\mathbf{K}$ to have continuum many subquasivarieties that have no independent quasi-equational bases relative to $\mathbf{K}$ but have $\omega$-independent quasi-equational bases relative to $\mathbf{K}$. This condition also implies that $\mathbf{K}$ is $Q$-universal.
Keywords: independent basis, quasi-identity, quasivariety, quasivariety lattice, Q-universality. 
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A. V. Kravchenko; A. M. Nurakunov; M. V. Schwidefsky. Structure of Quasivariety Lattices. I. Independent Axiomatizability. Algebra i logika, Tome 57 (2018) no. 6, pp. 684-710. http://geodesic.mathdoc.fr/item/AL_2018_57_6_a3/

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