Graded Lawson–Stone duality

The classical Stone duality associates to each Boolean algebra a topological space consisting of ultrafilters. Lawson's generalization constructs a dual equivalence between the categories of Boolean inverse (Formula presented.) -semigroups and Hausdorff ample topological groupoids. We further generalize the duality to graded versions of those categories, while allowing larger classes of morphisms, and provide various illustrations.