Abstract
We investigate expansions of Heyting algebras (EHAs) in possession of a unary term describing the filters that correspond to congruences. Hasimoto proved that Heyting algebras equipped with finitely many (dual) normal operators have such a term, generalising a standard construction on finite-type boolean algebras with operators (BAOs). We utilise Hasimoto’s technique, extending the existence condition to a larger class of EHAs and some classes of double-Heyting algebras. Such a term allows us to characterise varieties with equationally definable principal congruences using a single equation. Moreover, in the presence of a dual pseudocomplement operation, discriminator varieties are characterised by a pair of equations. We also prove that a variety of dually pseudocomplemented EHAs with a normal filter term is semisimple if and only if it is a discriminator variety. This generalises two known results, one by Kowalski and Kracht for finite-type varieties of BAOs, and the other by the present author for dually pseudocomplemented Heyting algebras without additional operations.
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