TY - GEN

T1 - Nabla algebras and chu spaces

AU - Palmigiano, Alessandra

AU - Venema, Yde

PY - 2007/12/1

Y1 - 2007/12/1

N2 - This paper is a study into some properties and applications of Moss' coalgebraic or 'cover' modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. More precisely, we introduce the notions of a modal ∇-algebra and of a positive modal ∇-algebra. We establish a categorical isomorphism between the category of modal ∇-algebra and that of modal algebras, and similarly for positive modal ∇-algebras and positive modal algebras. We then turn to a presentation, in terms of relation lifting, of the Vietoris hyperspace in topology. The key ingredient is an F-lifting construction, for an arbitrary set functor F, on the category Chu of two-valued Chu spaces and Chu transforms, based on relation lifting. As a case study, we show how to realize the Vietoris construction on Stone spaces as a special instance of this Chu construction for the (finite) power set functor. Finally, we establish a tight connection with the axiomatization of the modal ∇-algebras.

AB - This paper is a study into some properties and applications of Moss' coalgebraic or 'cover' modality ∇. First we present two axiomatizations of this operator, and we prove these axiomatizations to be sound and complete with respect to basic modal and positive modal logic, respectively. More precisely, we introduce the notions of a modal ∇-algebra and of a positive modal ∇-algebra. We establish a categorical isomorphism between the category of modal ∇-algebra and that of modal algebras, and similarly for positive modal ∇-algebras and positive modal algebras. We then turn to a presentation, in terms of relation lifting, of the Vietoris hyperspace in topology. The key ingredient is an F-lifting construction, for an arbitrary set functor F, on the category Chu of two-valued Chu spaces and Chu transforms, based on relation lifting. As a case study, we show how to realize the Vietoris construction on Stone spaces as a special instance of this Chu construction for the (finite) power set functor. Finally, we establish a tight connection with the axiomatization of the modal ∇-algebras.

KW - Chu space

KW - Coalgebra

KW - Modal algebra

KW - Relation lifting

KW - Vietoris hyper-space

UR - http://www.scopus.com/inward/record.url?scp=38049088184&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38049088184&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:38049088184

SN - 9783540738572

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 394

EP - 408

BT - Algebra and Coalgebra in Computer Science - Second International Conference, CALCO 2007, Proceedings

T2 - 2nd International Conference on Algebra and Coalgebra in Computer Science, CALCO 2007

Y2 - 20 August 2007 through 24 August 2007

ER -