Bilattices are algebraic structures, stemming from the research on knowledge representation and non-monotonic reasoning; they comprise a set equipped with two lattice orders, one modelling degree of truth and one modelling amount of information. Galois connections are very useful throughout mathematics, providing a unifying abstraction for various correspondences between ordered sets, and being in close correspondence with closure operators. We introduce notions of Galois biconnections, intended to be the bilattice analogue of classical Galois connections between lattices.
The first distinction we make is between bidirectional and unidirectional Galois biconnections. A bidirectional Galois biconnection is a (compatible) pair of Galois connections between the truth orderings and the knowledge orderings of bilattices, while a unidirectional Galois biconnection is actually a Galois connection equipped with extra properties that seek to capture the bilattice structure. A further distinction is between regular Galois biconnections, which induce order-isomorphic images of the maps, strong Galois biconnections, which furnish bilattice-isomorphic images.
We investigate all four species of Galois biconnections on pre-bilattices and on bi- lattices with negation and conflation. We examine both the survival of elegant properties of Galois connections (composability, invertibility, preservation of joins and meets, etc.) and the preservation of interesting bilattice properties (distributivity, boundedness, interlacing) for the images of the bilattices under the Galois biconnection. Finally, we discuss the naturally emerging biclosure operators on bilattices and hint on the generalisation of these concepts to sets equipped with more than two lattices.