Noncommutative Pierce Duality between Steinberg Rings and Ample Ringoid Bundles

TL;DR

This paper extends classical duality results to Steinberg rings and ringoid bundles over ample groupoids, generalizing noncommutative Stone duality to a broader algebraic setting.

Contribution

It introduces a noncommutative Pierce duality between Steinberg rings and ringoid bundles, expanding the scope of duality theories in algebra and topology.

Findings

Established duality between Steinberg rings and ringoid bundles

Extended Lawson's noncommutative Stone duality to new algebraic structures

Generalized classical dualities to noncommutative and algebraic contexts

Abstract

Classic work of Pierce and Dauns-Hofmann shows that biregular rings are dual to simple ring bundles over Stone spaces. We extend this duality to Steinberg rings, a purely algebraic generalisation of Steinberg algebras, and ringoid bundles over ample groupoids. We base this largely on an even more general extension of Lawson's noncommutative Stone duality, specifically between Steinberg semigroups, a generalisation of Boolean inverse semigroups, and category bundles over ample groupoids.