A non-commutative Priestley duality

TL;DR

This paper establishes a duality between strongly distributive skew lattices with zero and sheaves over local Priestley spaces, extending classical dualities to a non-commutative setting.

Contribution

It introduces a non-commutative Priestley duality for skew lattices, generalizing classical dualities and providing a canonical embedding for these structures.

Findings

Duality between skew lattices and sheaves over local Priestley spaces

Skew lattices correspond to Priestley orders on Boolean spaces

Skew lattice structures are natural in sheaf contexts over Priestley spaces

Abstract

We prove that the category of left-handed strongly distributive skew lattices with zero and proper homomorphisms is dually equivalent to a category of sheaves over local Priestley spaces. Our result thus provides a non-commutative version of classical Priestley duality for distributive lattices and generalizes the recent development of Stone duality for skew Boolean algebras. From the point of view of skew lattices, Leech showed early on that any strongly distributive skew lattice can be embedded in the skew lattice of partial functions on some set with the operations being given by restriction and so-called override. Our duality shows that there is a canonical choice for this embedding. Conversely, from the point of view of sheaves over Boolean spaces, our results show that skew lattices correspond to Priestley orders on these spaces and that skew lattice structures are naturally appropriate in any setting involving sheaves over Priestley spaces.