Quasi-identities of congruence-distributive quasivarieties of algebras (Q5932631)

Theorem 1. Let \(R\) be a congruence-distributive quasivariety of algebras of finite signature whose class of finitely subdirect \(R\)-indecomposable algebras is finitely axiomatizable. Then there exists a finite number \(n\) (depending on \(R\)) such that, if the congruence lattice of the \(R\)-free algebra of rank \(n\) has at most countable width or satisfies the minimality or maximality condition then \(R\) has a finite quasi-identities basis. Theorem 2. Assume \(K\) is a congruence-distributive quasivariety of algebras of finite signature, \(R=K\cap V\), where \(V\) is a variety of algebras, and the class of finitely subdirect \(R\)-indecomposable algebras is finitely axiomatizable. Then \(R\) is finitely axiomatizable with respect to \(K\).