Heyting algebras with dual pseudocomplementation (Q1061157)

An algebra \((L;\vee,\wedge,\to,+,0,1)\) is said to be an \(H_+\)-algebra if (L;\(\vee,\wedge,\to,0,1)\) is a Heyting algebra and \(+\) is the dual pseudocomplement in L, i.e. \(x\geq a^+\) iff \(x\vee a=1\). The class of all \(H_+\)-algebras is equational and comprises double Heyting algebras as well as regular double p-algebras. The author studies \(H_+\)-algebras and generalizes a lot of results obtained earlier by R. Beazer, P. Köhler and the reviewer for double Heyting algebras and regular double p-algebras (see for example \textit{R. Beazer} [Algebra Univers. 9, 238-243 (1979; Zbl 0414.06010), ibid. 10, 220-224 (1980; Zbl 0431.06014)], \textit{P. Köhler} [ibid. 10, 189-194 (1980; Zbl 0431.06015)] and the reviewer [ibid. 3, 238-246 (1973; Zbl 0276.08005) ibid. 10, 195-219 (1980; Zbl 0431.06013)]). The main results: (1) Every congruence relation on a \(H_+\)-algebra is uniquely determined by its kernel, a normal filter; (2) The class of \(H_+\)-algebras enjoys CEP; (3) Simple, (finitely) subdirectly irreducible and directly indecomposable \(H_+\)-algebras are characterized; (4) \(H_+\)-algebras with Stonean (Boolean) congruence lattices are described.