TY - JOUR
T1 - Product representation for default bilattices
T2 - An application of natural duality theory
AU - Cabrer, L. M.
AU - Craig, A. P.K.
AU - Priestley, H. A.
N1 - Publisher Copyright:
© 2014 Elsevier B.V.
PY - 2015/7/1
Y1 - 2015/7/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84925110853&partnerID=8YFLogxK
U2 - 10.1016/j.jpaa.2014.09.034
DO - 10.1016/j.jpaa.2014.09.034
M3 - Article
AN - SCOPUS:84925110853
SN - 0022-4049
VL - 219
SP - 2962
EP - 2988
JO - Journal of Pure and Applied Algebra
JF - Journal of Pure and Applied Algebra
IS - 7
ER -