The dual geometry of Boolean semirings

TL;DR

This paper explores the duality between Boolean semirings and partially Stone spaces, introducing optimal natural dualities and a novel geometric structure called 'hairy cubes' with polynomial representations.

Contribution

It develops new optimal and strong dualities for Boolean semirings using topological structures and introduces 'hairy cubes' as a geometric model with polynomial element representations.

Findings

Established an optimal natural duality between ISP(S) and IS_cP+(4A0S)

Constructed a small structure 4A0S_os that yields a strong duality

Introduced 'hairy cubes' as a geometric representation with polynomial descriptions

Abstract

It is well known that the variety of Boolean semirings, which is generated by the three element semiring S, is dual to the category of partially Stone spaces. We place this duality in the context of natural dualities. We begin by introducing a topological structure \uS and obtain an optimal natural duality between the quasi-variety ISP(S) and the category IS_cP+(\uS). Then we construct an optimal and very small structure \uS_os that yields a strong duality. The geometry of some of the partially Stone spaces that take part in these dualities is presented, and we call them "hairy cubes", as they are n-dimensional cubes with unique incomparable covers for each element of the cube. We also obtain a polynomial representation for the elements of the hairy cube.