Lattices of boundedly axiomatizable \(\forall\)-subclasses of \(\forall\)-classes of universal algebras (Q2300739)

Let \(\mathcal{A}\) be a \(\forall\)-class of algebras, i.e. a class axiomatizable by \(\forall\)-formulas (of a fixed signature). The \(\forall\)-class \(\mathcal{N}\subset \mathcal{A}\) is \textit{\(n\)-axiomatizable} if for each \(A\in\mathcal{A}\) the following holds: \(A\in\mathcal{N}\) if and only if any \(n\)-generated subalgebra of \(A\) belongs to \(\mathcal{N}\). And \(\mathcal{N}\) is \textit{boundedly axiomatizable} if it is \(n\)-axiomatizable for some natural number \(n\). The author investigates some properties of lattices of boundedly axiomatizable \(\forall\)-subclasses of \(\forall\)-classes of algebras. The results are then applied to a discriminator variety.