Monadic $$k\times j$$ k × j -rough Heyting algebras

In this paper, we introduce the variety of algebras, which we call monadic $$k\times j$$ -rough Heyting algebras. These algebras constitute an extension of monadic Heyting algebras and in $$3\times 2$$ case they coincide with monadic 3-valued Łukasiewicz–Moisil algebras. Our main interest is the characterization of simple and subdirectly irreducible monadic $$k\times j$$ -rough Heyting algebras. In order to this, an Esakia-style duality for these algebras is developed.