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.