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# Positive Fragments Of Coalgebraic Logics

journal contribution
posted on 04.02.2016, 12:46 by A. Balan, Alexander Herbert Kurz, J. Velebil
Positive modal logic was introduced in an influential 1995 paper of Dunn as the positive fragment of standard modal logic. His completeness result consists of an axiomatization that derives all modal formulas that are valid on all Kripke frames and are built only from atomic propositions, conjunction, disjunction, box and diamond. In this paper, we provide a coalgebraic analysis of this theorem, which not only gives a conceptual proof based on duality theory, but also generalizes Dunn's result from Kripke frames to coalgebras for weak-pullback preserving functors. To facilitate this analysis we prove a number of category theoretic results on functors on the categories \$mathsf{Set}\$ of sets and \$mathsf{Pos}\$ of posets: Every functor \$mathsf{Set} to mathsf{Pos}\$ has a \$mathsf{Pos}\$-enriched left Kan extension \$mathsf{Pos} to mathsf{Pos}\$. Functors arising in this way are said to have a presentation in discrete arities. In the case that \$mathsf{Set} to mathsf{Pos}\$ is actually \$mathsf{Set}\$-valued, we call the corresponding left Kan extension \$mathsf{Pos} to mathsf{Pos}\$ its posetification. A \$mathsf{Set}\$-functor preserves weak pullbacks if and only if its posetification preserves exact squares. A \$mathsf{Pos}\$-functor with a presentation in discrete arities preserves surjections. The inclusion \$mathsf{Set} to mathsf{Pos}\$ is dense. A functor \$mathsf{Pos} to mathsf{Pos}\$ has a presentation in discrete arities if and only if it preserves coinserters of `truncated nerves of posets'. A functor \$mathsf{Pos} to mathsf{Pos}\$ is a posetification if and only if it preserves coinserters of truncated nerves of posets and discrete posets. A locally monotone endofunctor of an ordered variety has a presentation by monotone operations and equations if and only if it preserves \$mathsf{Pos}\$-enriched sifted colimits.

## Citation

Logical Methods In Computer Science, 2015, 11 (3), 18 (51)

## Author affiliation

/Organisation/COLLEGE OF SCIENCE AND ENGINEERING/Department of Computer Science

## Version

VoR (Version of Record)

## Published in

Logical Methods In Computer Science

## Publisher

IfCoLog (International Federation of Computational Logic)

1860-5974

2015

04/02/2016

## Publisher version

http://www.lmcs-online.org/ojs/viewarticle.php?id=1564&layout=abstract

en