The commutator in equivalential algebras and Fregean varieties
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.
scientific article; zbMATH DE number 5939541
scientific article