The Operator Ln on Quasivarieties of Universal Algebras

Let n be an arbitrary natural and let M be a class of universal algebras. Denote by L-n(M) the class of algebras G such that, for every n-generated subalgebra A of G, the coset a/R (a is an element of A) modulo the least congruence R including A x A is an algebra in M. We investigate the classes L-n(M). In particular, we prove that if M is a quasivariety then L-n(M) is a quasivariety. The analogous result is obtained for universally axiomatizable classes of algebras. We show also that if M is a congruence-permutable variety of algebras then L-n(M) is a variety. We find a variety P of semigroups such that L-1(P) is not a variety.