Nabla algebras and chu spaces

Alessandra Palmigiano*, Yde Venema

*Corresponding author for this work

Research output: Chapter in Book / Report / Conference proceedingConference contributionAcademicpeer-review

Abstract

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.

Original languageEnglish
Title of host publicationAlgebra and Coalgebra in Computer Science - Second International Conference, CALCO 2007, Proceedings
Pages394-408
Number of pages15
Publication statusPublished - 1 Dec 2007
Externally publishedYes
Event2nd International Conference on Algebra and Coalgebra in Computer Science, CALCO 2007 - Bergen, Norway
Duration: 20 Aug 200724 Aug 2007

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume4624 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference2nd International Conference on Algebra and Coalgebra in Computer Science, CALCO 2007
CountryNorway
CityBergen
Period20/08/0724/08/07

Keywords

  • Chu space
  • Coalgebra
  • Modal algebra
  • Relation lifting
  • Vietoris hyper-space

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