Working in an arbitrary category endowed with a fixed \((\mathcal {E}, \mathcal {M})\) -factorization system in which the preimage functor for any given morphism preserves arbitrary joins we improve the Castellini notion of hereditary interior operators. We introduce and study a general notion of hereditary interior operators in terms of dual images. It is shown that these operators behave as well as hereditary closure operators. In particular, hereditary interior operators are characterized in terms of the notion of initiality as for hereditary closure operators. Moreover, we prove that additive hereditary interior operators are Castellini’s hereditary interior operators. Some examples are included.
