The dual geometry of Boolean semirings

It is well known that the variety of Boolean semirings, which is generated by the three element semiring ${\mathbb{S}}$ , 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 S? and obtain an optimal natural duality between the quasi-variety ISP( ${\mathbb{S}}$ ) and the category IS _c P+(S?). Then we construct an optimal and very small structure S? _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.