The closure-interior Galois connection and its applications to relational equations and inclusions

By using the closure–interior Galois connection and the box product of relations, we show that the duals of some of our former theorems on the relational equation R = G−1◦ S ◦ F and inclusion R ⊆ G−1◦ S ◦ F can be easily derived from some simple basic facts on Galois connections. To clarify the importance of this subject, we recall that a relation F on a set X is called regular (normal) if F = F ◦ S ◦ F F = F ◦ S ◦ (F c )−1 for some relation S on X. Note that here S may, for instance, be the identity relation of X or the inverse relation of F . Thus, in particular, all idempotent and non-mingled-valued relations are regular. Moreover, a pair ( F , G) of relations is called properly (uniformly) mildly continuous, with respect to the relations R and S , if R = G−1 ◦ S ◦ F R ⊆ G−1◦S ◦ F . Thus, in particular the relation F is regular if and only if the pair F , F −1 is properly mildly continuous with respect to F and some relation S . It is noteworthy that the largest such S is F −1 ◦ F c ◦ F −1