TY - JOUR
ID - 102443
TI - Expanding Belnap 2: the dual category in depth
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Craig, Andrew P. K.
AU - Davey, Brian A.
AU - Haviar, Miroslav
AD - Department of Mathematics
and Applied Mathematics
University of Johannesburg
PO Box 524, Auckland Park, 2006,
South Africa
AD - Department of Mathematics and Statistics, La Trobe University, Victoria 3086, Australia.
AD - Department of Mathematics
Faculty of Natural Sciences,
M. Bel University
Tajovskeho 40, 974~01 Banska Bystrica, Slovakia.
Y1 - 2022
PY - 2022
VL - 17
IS - 1
SP - 47
EP - 84
KW - Bilattice
KW - default bilattice
KW - natural duality
KW - multi-sorted natural duality
KW - Priestley duality
KW - piggyback duality
DO - 10.52547/cgasa.2022.102443
N2 - 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.
UR - https://cgasa.sbu.ac.ir/article_102443.html
L1 - https://cgasa.sbu.ac.ir/article_102443_eee83c99c3f1c6a0c84198810233eae9.pdf
ER -