Algebraic systems with partial operations have different ways to interpret equality between two terms of the language. A strong identity is a formula which says that two terms are equal in the algebra if the existence of one of them implies the existence of the other one and in the case of existence their values are equal. A class of partial algebras defined by a set of strong identities is called a strong variety. In the characterization of strong varieties in the case of partial algebras by means of a Birkhoff-type theorem there appeared a new concept, regularity of partial homomorphisms and partial subalgebras. Here we define and study these operators from two different perspectives.Firstly, in their relation with other well known concecpts of partial homomorphisms and partial subalgebras, as well as with the po-monoid of Pigozzi for the H, S and P operators. Secondly, in regard to the preservation of the different types of formulae that represent equality in the case of partial algebras for these operators. Finally, we give a characterization of the strong varieties as classes closed under regular homomorphisms, regular subalgebras, direct products and that satisfy a closure condition.