The Operator Ln on Quasivarieties of Universal Algebras
A. I. Budkin · Siberian Mathematical Journal · 2019
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 ∈ A ) modulo the least congruence R including A × 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 ℘ of semigroups such that L 1 ( ℘ ) is not a variety.