The operator $L_n$ on quasivarieties of universal algebras

Let $n$ be an arbitrary natural and let $\mathcal{M}$ be a class of universal algebras. Denote by $L_n(\mathcal{M})$ the class of algebras $G$ such that, for every $n$-generated subalgebra $A$ of $G$, the coset $a/R$ ($a\in A$) modulo the least congruence $R$ including $A\times A$ is an algebra in $\mathcal{M}$. We investigate the classes $L_n(\mathcal{M})$. In particular, we prove that if $\mathcal{M}$ is a quasivariety then $L_n(\mathcal{M})$ is a quasivariety. The analogous result is obtained for universally axiomatizable classes of algebras. We show also that if $\mathcal{M}$ is a congruence-permutable variety of algebras then $L_n(\mathcal{M})$ is a variety. We find a variety $\mathcal{P}$ of semigroups such that $L_1(\mathcal{P})$ is not a variety.