  Bilattices (that is, sets with two lattice structures) provide an algebraic
tool to model simultaneously the validity of, and knowledge about, sentences in
an appropriate language. In particular, certain bilattices have been used to
model situations in which information is prioritised and so can be viewed
hierarchically. These default bilattices are not interlaced: the lattice
operations of one lattice structure do not preserve the order of the other one.
The well-known product representation theorem for interlaced bilattices does
not extend to bilattices which fail to be interlaced and the lack of a product
representation has been a handicap to understanding the structure of default
bilattices. In this paper we study, from an algebraic perspective, a hierarchy
of varieties of default bilattices, allowing for different levels of default.
We develop natural dualities for these varieties and thereby obtain a concrete
representation for the algebras in each variety. This leads on to a form of
product representation that generalises the product representation as this
applies to distributive bilattices.
