Expansions of Heyting algebras

It is well-known that congruences on a Heyting algebra are in one-to-one correspondence with filters of the underlying lattice. If an algebra A has a Heyting algebra reduct, it is of natural interest to characterise which filters correspond to congruences on A. Such a characterisation was given by Hasimoto [1]. When the filters can be sufficiently described by a single unary term, many useful properties are uncovered. The traditional example arises from boolean algebras with operators. In this setting, an algebra B = 〈B;∨,∧,¬, {fi | i ∈ I}, 0, 1〉 is a boolean algebra with (dual) operators (BAO for short) if 〈B;∨,∧,¬, 0, 1〉 is a boolean algebra, and for each i ∈ I, the operation fi is a unary map satisfying fi1 = 1 and fi(x ∧ y) = fix ∧ fiy. If B is of finite type, then congruences on B are determined by filters closed under the map d, defined by