Equational spectrum of Hilbert varieties
In this paper the authors investigate the sets of equations that define different subvarieties of the variety of Hilbert algebras. The authors prove that the only subvariety of the variety of Hilbert algebras that can be defined with one equation is the trivial one, i.e., the one whose algebras are one-element algebras. They also demonstrate that the varieties of Hilbert algebras, implication algebras and commutative BCK-algebras can be defined with two independent equations. Therefore, using a result from [\textit{A. Tarski}, ``Equational logic and equational theories of algebras'', in: Contrib. Math. Logic, Proc. Logic Colloq., Hannover 1966, 275--288 (1968; Zbl 0209.01402)], they conclude that the varieties of Hilbert algebras, implication algebras and commutative BCK-algebras can be defined with \(n\)-independent equations for each \(n=2,3,\ldots \). It is worth to remark that the authors make the following conjecture: ``All finitely-based equational classes emanating from Boolean algebras and having congruence distributive property are 2-based.'' Following the ideas of the paper, the conjecture can be easily checked for the following subvarieties of Hilbert algebras not considered in the paper: linear Hilbert algebras [\textit{A. Monteiro}, ``Les algèbres de Hilbert linéaires'', in: Unpublished papers of António Monteiro I. Notas de Lógica Matemática 40 (1955)] and \(n\)-valued Hilbert algebras [\textit{L. Monteiro}, ``Algèbres de Hilbert \(n\)-valentes'', Port. Math. 36, 159--173 (1977; Zbl 0488.03036)].
scientific article; zbMATH DE number 5624217