The commutator in equivalential algebras and Fregean varieties
type: Item
instance of: scholarly article
title: The commutator in equivalential algebras and Fregean varieties
zbMATH Open document ID: 1233.08003
author: Paweł M. Idziak
author: Katarzyna Słomczyńska
author: Andrzej Wroński
published in: Algebra Universalis
publication date: 2011-08-16T00:00:00Z
review text: A variety \(\mathcal {V}\) with a distinguished constant \(\mathbf{0}\) is called Fregean if every algebra \(\mathbf{A}\in \mathcal {V}\) is congruence \(\mathbf{0}\)-regular (i.e., \(\mathbf{0}/\alpha = \mathbf{0}/\beta\) implies \(\alpha = \beta\) for all congruences \(\alpha, \beta \in \mathbf{Con(A)}\)) and congruence-orderable (i.e., \(\Theta_{\mathbf{A}}(\mathbf{0},a)=\Theta_{\mathbf{A}}(\mathbf{0},b)\) implies \(a=b\) for all \(a,b\in \mathbf{A}\). Among the Fregean varieties there are Boolean algebras, Boolean groups, Brouwerian semilattices, Hilbert algebras, Heyting algebras, and many other algebras that arise as algebratizations of classical, intuitionistic and intermediade logics. Equivalential algebras, introduced by \textit{J. K. Kabzińki} and \textit{A. Wroński} [``On equivalential algebras'', in: Proceedings of the 1975 international symposium on multiple-valued logic. Fifth annual symposium held at Indiana University, Bloomington, Ind., 1975. Long Beach, Calif.: IEEE Computer Society. 419--428 (1975)] as an algebraic counterpart of the purely equivalential fragment of intuitionistic logic, constitute a natural example of a Fregean variety of special importance. Namely, every congruence-permutable Fregean variety has a binary term that turns each of its members into an equivalential algebra. In the paper mentioned above, equivalential algebras were defined as being of the form \(\mathbf{A}=(A,\leftrightarrow )\) that satisfy all identities \(\mathbf{t}=\mathbf{s}\), where \(\mathbf{t}\leftrightarrow \mathbf{s}\) is a tautology of intuitionistic logic. In this paper, the authors give a full characterization of the commutator for equivalential algebras (Theorem 3.2) and solvable Fregean varieties (Theorem 4.2). It is shown that in a solvable algebra from a Fregean variety, the commutator coincides with the commutator of its purely equivalential reduct. Moreover, an intrinsic characterization of the commutator in this setting is given.
reviewed by: Florentina Chirteş
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 03B50
Mathematics Subject Classification ID: 03B55
Mathematics Subject Classification ID: 03G25
Mathematics Subject Classification ID: 08B99
zbMATH DE Number: 5939541
zbMATH Keywords: commutator theory
zbMATH Keywords: Fregean varieties
zbMATH Keywords: equivalential algebras
MaRDI profile type: Publication
full work available at URL: s00012-011-0133-4
OpenAlex ID: W2160372569
cites work: On the structure of varieties with equationally definable principal congruences. II
cites work: Q3775604
cites work: Ideals in universal algebras
cites work: On \(n\)-permutable congruences
cites work: Polynomially rich algebras
cites work: FREGEAN VARIETIES
cites work: Q4011728
cites work: Equivalential algebras. I: Representation
cites work: Free equivalential algebras
cites work: Equational logic and theories in sentential languages
DOI: 10.1007/S00012-011-0133-4
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description: scientific article; zbMATH DE number 5939541