Graded Lawson-Stone Duality

TL;DR

The paper generalizes Stone duality to the graded setting by introducing Gamma-graded Boolean inverse semigroups and graded Hausdorff ample groupoids. It constructs duality functors and natural isomorphisms to establish a contravariant equivalence between the graded inverse semigroup and graded groupoid categories. It provides concrete examples and shows that graded and ungraded enveloping rings are isomorphic, ensuring compatibility with graded algebras. This framework supports graded noncommutative algebras such as Steinberg and Leavitt path algebras and has implications for graded K-theory and ideal structure.

Abstract

The classical Stone duality associates to each Boolean algebra a topological space consisting of ultrafilters. Lawson's generalisation constructs a dual equivalence of categories of Boolean inverse $\land$-semigroups and Hausdorff ample topological groupoids. We further generalise the duality to the graded analogues of those categories, and provide various illustrations.

Theorems & Definitions (42)

• Definition 2.1
• Lemma 3.1
• proof
• Lemma 3.2: cf. Lawson2, Lemma 2.2(2,3)
• proof
• Lemma 3.3
• proof
• Lemma 3.4: cf. Lawson2, Lemma 2.14
• proof
• Proposition 3.5