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’(⊤) and ‘no information’ (⊥), 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, Jn, for n ⩾ 0, with J0 being Belnap’s seminal example. We gave a duality for the variety Vn generated by Jn, with the dual category Xn consisting of multi-sorted topological structures. Here we study the dual category in depth. We axiomatise the category Xn and show that it is isomorphic to a category Yn of single-sorted topological structures. The objects of Yn are ranked Priestley spaces endowed with a continuous retraction. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in Vn via its dual in Yn; as an application we show that the size of the free algebra FVn(1) is given by a polynomial in n of degree 6.