The commutator in equivalential algebras and Fregean varieties

A class \({\mathcal {K}}\) of algebras with a distinguished constant term 0 is called Fregean if congruences of algebras in \({\mathcal {K}}\) are uniquely determined by their 0-cosets and Θ _A (0, a) = Θ _A (0, b) implies a = b for all \({a, b \in {\bf A} \in \mathcal {K}}\) . The structure of Fregean varieties was investigated in a paper by P. Idziak, K. S?omczyńska, and A. Wroński. In particular, it was shown there that every congruence permutable Fregean variety consists of algebras that are expansions of equivalential algebras, i.e., algebras that form an algebraization of the purely equivalential fragment of the intuitionistic propositional logic. In this paper we give a full characterization of the commutator for equivalential algebras and solvable Fregean varieties. In particular, we show 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.