  In this paper, we generalize the notion of rational singularities for any
reflexive sheaf of rank $1$, link our notion of rational singularities with the
notion of rational singularities in [Kov11], and prove generalizations of
standard facts about rational singularities. Moreover, by using a definition of
non-rational locus, we introduce the notion of $(B_{q+1})$ as a dual notion of
well-known Serre's notion of $(S_{q+1})$, and prove a theorem about
$q$-birational morphisms.
