  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.
