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Quasi-identities of congruence-distributive quasivarieties of algebras. (English. Russian original) Zbl 0966.08010

Sib. Math. J. 42, No. 1, 108-118 (2001); translation from Sib. Mat. Zh. 42, No. 1, 123-135 (2001).
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 \(K\).

MSC:

08C15 Quasivarieties
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