Andrew Craig – Dualities for prioritised default bilattices I

University of Johannesburg
Bilattices are algebras with two lattice structures and a unary negation operation that preserves one lattice order and reverses the other. The lattice order that is reversed by the negation is known as the ‘truth’ order while the order that is preserved by the negation is known as the ‘knowledge’ order. Bilattices have applications in a number of areas including artificial intelligence. In this work we introduce a new class of default bilattices, $J_n$, for $n \in \omega$. Here $J_0$ is Belnap's famous example of four-valued logic coming from his 1977 paper entitled How a computer should think. We demonstrate a dual categorical equivalence between varieties of bilattices and classes of multi-sorted structured topological spaces. Our main tool is the theory of natural dualities, and our approach relies on the identification of the meet-irreducible elements of certain subalgebra lattices.

This is joint work with Brian A. Davey amd Miroslav Haviar.