  We revisit the problem of Stone duality for lattices with various
quasioperators, first studied in [14], presenting a fresh duality result. The
new result is an improvement over that of [14] in two important respects.
First, the axiomatization of frames in [14] was rather cumbersome and it is now
simplified, partly by incorporating Gehrke's proposal [8] of section stability
for relations. Second, morphisms are redefined so as to preserve Galois stable
(and co-stable) sets and we rely for this, partly again, on Goldblatt's [11]
recently proposed definition of bounded morphisms for polarities, though we
need to strengthen the definition in order to get a Stone duality result.
  In studying the dual algebraic structures associated to polarities with
relations we demonstrate that stable/co-stable set operators result as the
Galois closure of the restriction of classical (though sorted) image operators
generated by the frame relations to Galois stable/co-stable sets. This provides
a proof, at the representation level, that non-distributive logics can be
viewed as fragments of sorted, residuated (poly)modal logics, a research
direction initiated in [16,17].
