Please use this identifier to cite or link to this item: http://hdl.handle.net/2381/36575
Title: Positive Fragments Of Coalgebraic Logics
Authors: Balan, A.
Kurz, Alexander Herbert
Velebil, J.
First Published: 22-Sep-2015
Publisher: IfCoLog (International Federation of Computational Logic)
Citation: Logical Methods In Computer Science, 2015, 11 (3), 18 (51)
Abstract: 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.
DOI Link: 10.2168/LMCS-11(3:18)2015
ISSN: 1860-5974
Links: http://www.lmcs-online.org/ojs/viewarticle.php?id=1564&layout=abstract
http://hdl.handle.net/2381/36575
Version: Publisher Version
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © 2015, the authors. This is an Open Access article distributed under the terms of the Creative Commons Attribution No Derivatives Licence ( http://creativecommons.org/licenses/by-nd/2.0/ ) which permits use and distribution in any medium, provided the original work is properly cited and no modifications or adaptations are made.
Appears in Collections:Published Articles, Dept. of Computer Science

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