The Operator Ln on Quasivarieties of Universal Algebras

Resumo

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 L₁(℘) is not a variety.