The operator \(L_n\) on quasivarieties of universal algebras (Q2010196)

Let $n$ be an arbitrary natural number 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 $a/R$ $(a\in A)$ modulo the least congruence $R$ including $A\times A$ is an algebra in $M$. In this paper the following main results are proved: \begin{itemize} \item[1.] If $M$ is a hereditary class of algebras then so is $L_n(M)$. \item[2.] If $M$ is a quasivariety of algebras then so is $L_n(M)$. \item[3.] If $M$ is a hereditary and multiplicatively closed class of algebras then so is $L_n(M)$. \item[4.] If $M$ is a hereditary and ultraclosed class of algebras then so is $L_n(M)$. \item[5.] If $M$ is a universally axiomatizable class of algebras then so is $L_n(M)$. \item[6.] If $M$ is a congruence-permutable and homomorphically closed class of algebras then $L_n(M)$ is homomorphically closed too. \item[7.] If $M$ is a congruence-permutable variety of algebras then $L_n(M)$ is a variety too. \end{itemize}