Expanding Belnap 2: the dual category in depth

Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled 'How a computer should think'. Prioritised default bilattices include not only Belnap's four values, for `true' ($t$), `false'($f$), `contradiction' ($\top$) and `no information' ($\bot$), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence. In our companion paper, we introduced a new family of prioritised default bilattices, $\mathbf J_n$, for $n \in \omega$, with $\mathbf J_0$ being Belnap's seminal example. We gave a duality for the variety $\mathcal V_n$ generated by $\mathbf J_n$, with the objects of the dual category $\mathcal X_n$ being multi-sorted topological structures. Here we study the dual category in depth. We give an axiomatisation of the category $\mathcal X_n$ and show that it is isomorphic to a category $\mathcal Y_n$ of single-sorted topological structures. The objects of $\mathcal Y_n$ are Priestley spaces endowed with a continuous retraction in which the order has a natural ranking. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in $\mathcal V_n$ via its dual in $\mathcal Y_n$; as an application we show that the size of the free algebra $\mathbf F_{\mathcal V_n}(1)$ is given by a polynomial in $n$ of degree $6$.