Equational spectrum of Hilbert varieties
type: Item
instance of: scholarly article
title: Equational spectrum of Hilbert varieties
zbMATH Open document ID: 1209.03056
author: R. Padmanabhan
author: Sergiu Rudeanu
published in: Central European Journal of Mathematics
publication date: 2009-11-02T00:00:00Z
review text: 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not considered in the paper: linear Hilbert algebras [\textit{A. Monteiro}, ``Les algèbres de Hilbert linéaires'', in: Unpublished papers of António Monteiro I. Notas de Lógica Matemática 40 (1955)] and \(n\)-valued Hilbert algebras [\textit{L. Monteiro}, ``Algèbres de Hilbert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)].
reviewed by: Leonardo Manuel Cabrer
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 06F35
Mathematics Subject Classification ID: 08B15
zbMATH DE Number: 5624217
zbMATH Keywords: Hilbert algebra
zbMATH Keywords: implication algebra
zbMATH Keywords: Tarski algebra
zbMATH Keywords: Boolean algebra
zbMATH Keywords: BCK-algebra
zbMATH Keywords: equational spectrum
zbMATH Keywords: equational class
zbMATH Keywords: subvariety
MaRDI profile type: Publication
full work available at URL: s11533-008-0060-6
OpenAlex ID: W2002498893
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DOI: 10.2478/S11533-008-0060-6
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description: scientific article; zbMATH DE number 5624217