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The Goldblatt-Thomason Theorem for Coalgebras [chapter]

Alexander Kurz, Jiří Rosický
Algebra and Coalgebra in Computer Science  
Goldblatt and Thomason's theorem on modally definable classes of Kripke frames and Venema's theorem on modally definable classes of Kripke models are generalised to coalgebras.  ...  Acknowledgements Our greatest debts are to David Gabelaia whose sharp eye allowed us to correct a serious mistake in the statement of our main theorem.  ...  We are also grateful to the referees for their suggestions.  ... 
doi:10.1007/978-3-540-73859-6_23 dblp:conf/calco/KurzR07 fatcat:bi6y6xzcezhzhiwfovz775mble

Expressivity of Many-Valued Modal Logics, Coalgebraically [chapter]

Marta Bílková, Matěj Dostál
2016 Lecture Notes in Computer Science  
We show how previous work on modal extensions of Ł n -valued logics fits naturally into the coalgebraic framework and indicate some of the ensuing generalisations.  ...  We denote by FR the class of frames. *  ...  This shows that (3) falls into the framework of [7] and allows us to obtain the Goldblatt-Thomason theorems of [9] from the coalgebraic Goldblatt-Thomason theorem of [6] .  ... 
doi:10.1007/978-3-662-52921-8_8 fatcat:sh3jm2dwufhixk6lpivdwhvewq

Distributive Substructural Logics as Coalgebraic Logics over Posets

Marta Bílková, Rostislav Horcík, Jiri Velebil
2012 Advances in Modal Logic  
Goldblatt-Thomason theorem for classes of resulting coalgebras for instance shows that frames for axiomatic extensions of distributive Full Lambek logic are modally definable classes of certain coalgebras  ...  As an application of this approach we prove a general version of Goldblatt-Thomason theorem that characterizes definability of classes of frames for logics extending the distributive Full Lambek logic,  ...  Our version of the theorem is an analogue of coalgebraic Goldblatt-Thomason theorem for Set coalgebras [26, Theorem 3.15(2.) ].  ... 
dblp:conf/aiml/BilkovaHV12 fatcat:x7bstrkv7vghtdiahlm3fzxobe

Correspondence, Canonicity, and Model Theory for Monotonic Modal Logics [article]

Kentarô Yamamoto
2020 arXiv   pre-print
We prove analogues of the Goldblatt-Thomason Theorem and Fine's Canonicity Theorem for classes of monotonic neighborhood frames closed under elementary equivalence in coalgebraic predicate logic.  ...  We investigate the role of coalgebraic predicate logic, a logic for neighborhood frames first proposed by Chang, in the study of monotonic modal logics.  ...  Finally, I gratefully acknowledge financial support from the Takenaka Scholarship Foundation.  ... 
arXiv:1904.12997v4 fatcat:we3s3ixc5bdfxngpwcuueqoswa

Goldblatt-Thomason Theorems for Modal Intuitionistic Logics [article]

Jim de Groot
2022 arXiv   pre-print
We prove a Goldblatt-Thomason theorem for dialgebraic intuitionistic logics, and instantiate it to Goldblatt-Thomason theorems for a wide variety of modal intuitionistic logics from the literature.  ...  I am grateful to the anonymous reviewers for many constructive and helpful comments.  ...  A general Goldblatt-Thomason theorem for coalgebraic logics for Set-coalgebras was given in [22] . In the present paper we prove Goldblatt-Thomason theorems for modal intuitionistic logics.  ... 
arXiv:2011.10221v3 fatcat:jth72fcgr5aali722nheix5gpa

Goldblatt-Thomason-style Theorems for Graded Modal Language

Katsuhiko Sano, Minghui Ma
2010 Advances in Modal Logic  
We prove two main Goldblatt-Thomason-style Theorems for graded modal language in Kripke semantics: full Goldblatt-Thomason Theorem for elementary classes and relative Goldblatt-Thomason Theorem within  ...  By neighborhood semantic view, we can define a natural generalization of Jankov-Fine formula for GML and establish relative Goldblatt-Thomason Theorem.  ...  In particular, we owe Definition 3.4, Proposition 3.5 and the remark just after Theorem 6.3 to them.  ... 
dblp:conf/aiml/SanoM10 fatcat:y63benskvvhi5gprdphbxexatu

Duality for powerset coalgebras [article]

Guram Bezhanishvili, Luca Carai, Patrick Morandi
2022 arXiv   pre-print
This allows us to describe an endofunctor H on CABA such that the category Alg(H) of algebras for H is dually equivalent to the category Coalg(P) of coalgebras for the powerset endofunctor P on Set.  ...  As a consequence, we derive Thomason duality from Tarski duality, thus paralleling how J\'onsson-Tarski duality is derived from Stone duality.  ...  Acknowledgements We would like to thank the referees for their comments. One of the referees suggested to us an alternative description of L given in Theorem 3.9.  ... 
arXiv:2008.01849v6 fatcat:ldexspld4vaftjhjwrnx6tbcz4

First-Order Modal Logic: Frame Definability and Lindström Theorems [article]

Reihane Zoghifard, Massoud Pourmahdian
2016 arXiv   pre-print
This paper involves generalizing the Goldblatt-Thomason and the Lindstr\"om characterization theorems to first-order modal logic.  ...  The authors would like to thank Mohsen Khani for carefully reading the paper and giving some useful remarks which helped us to improve the presentation of the paper.  ...  Since the Goldblatt-Thomason and Lindström theorems both have coalgebraic forms (see [12] and [13, 7] ), it is natural to look for the question of studying coalgebraic adaptation of these theorems.  ... 
arXiv:1602.00201v2 fatcat:6xewfnc24fetbco2ubfzjpvfby

How to write a coequation [article]

Fredrik Dahlqvist, Todd Schmid
2021 arXiv   pre-print
for.  ...  In this review paper, we argue that this is partly due to the multitude of syntaxes for writing down coequations, which seems to have led to some confusion about what coequations are and what they are  ...  Acknowledgements The authors are most grateful to Alexander Kurz for his services as history consultant. The responsibility for any mistake or mischaracterisation lie solely with the authors.  ... 
arXiv:2109.11967v1 fatcat:gjiohhvtfvhrhnbgwzpqiatvoa

Ultrafilter Extensions for Coalgebras [chapter]

C. Kupke, A. Kurz, D. Pattinson
2005 Lecture Notes in Computer Science  
Our main contributions are the generalisations of two classical theorems in modal logic to coalgebras, namely the Jónsson-Tarski theorem giving a settheoretic representation for each modal algebra and  ...  Stone-coalgebras (coalgebras over the category of Stone spaces), on the other hand, do provide an adequate semantics for finitary modal logics.  ...  Formulate a finitary definability result for classes of coalgebras in the style of Goldblatt-Thomason [5] , based on ultrafilter extensions.  ... 
doi:10.1007/11548133_17 fatcat:bxsfjcbgwfbcnchuzmq3sxzqmi

6 Algebras and coalgebras [chapter]

Yde Venema
2007 Studies in Logic and Practical Reasoning  
In the second and last part of the chapter we describe how modal logic, and its model theory, provides many natural manifestations of the more general theory of universal coalgebra. 1 This text has been  ...  Also, at some specific points, the final version of this text will contain references to (specific parts of) other chapters of the book; for the time being, these references are represented by the letters  ...  For a definition of Birkhoff's theorem from a coalgebraic perspective, the reader is referred to section 14. Theorem 5.40 (Goldblatt-Thomason Theorem) Let C be a class of τ -frames.  ... 
doi:10.1016/s1570-2464(07)80009-7 fatcat:e77uebqxdferlcg2nlfe5hobpq

Local Goldblatt–Thomason theorem

Evgeny Zolin
2015 Logic Journal of the IGPL  
The celebrated theorem proved by Goldblatt and Thomason in 1974 gives necessary and sufficient conditions for an elementary (i.e., first-order definable) class of Kripke frames to be modally definable.  ...  We also discuss the relationship between modal expressions and hybrid logic and leave open questions.  ...  Local Goldblatt-Thomason Theorem 17 (b') Let M be a model over a signature of cardinality κ, and U a κ-good countably incomplete ultrafilter. Then the ultrapower M U is κ-saturated [5, Th. 6.1.6].  ... 
doi:10.1093/jigpal/jzv036 fatcat:edlqi4knmzdnpor76a5br7s3ma

Equational Coalgebraic Logic

Alexander Kurz, Raul Leal
2009 Electronical Notes in Theoretical Computer Science  
Finally, we argue that the quest for a generic logic for T -coalgebras is still open in the general case.  ...  A fundamental question in this area is how to obtain, for an arbitrary functor T , a logic for T -coalgebras.  ...  It is also possible, without any consideration of syntax, to generalize from Kripke frames to all KPFs (and beyond) the Jónsson-Tarski theorem [20] and the Goldblatt-Thomason theorem [19] .  ... 
doi:10.1016/j.entcs.2009.07.097 fatcat:qwz6upbojjck3lsqgdy3vjrady

Neighbourhood Structures: Bisimilarity and Basic Model Theory

Helle Hansen, Clemens Kupke, Eric Pacuit, Till Mossakowski
2009 Logical Methods in Computer Science  
In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2^2.  ...  We prove a Hennessy-Milner theorem for modally saturated and for image-finite neighbourhood models.  ...  Peter Gumm for many fruitful discussions, and Yde Venema for initiating our cooperation on this subject. Special thanks also goes to the anonymous referees for useful comments and corrections.  ... 
doi:10.2168/lmcs-5(2:2)2009 fatcat:5tgcljiirrafpfn3lsbgway3ku

Goldblatt-Thomason for LE-logics [article]

Willem Conradie, Alessandra Palmigiano, Apostolos Tzimoulis
2018 arXiv   pre-print
We prove a uniform version of the Goldblatt-Thomason theorem for logics algebraically captured by normal lattice expansions (normal LE-logics).  ...  The original Goldblatt-Thomason theorem [17] has been extended to various classical and distributive-based logical settings which include Positive Modal Logic [3] , coalgebraic logic [22] , graded  ...  The Goldblatt-Thomason theorem for LE-logics The following proposition is an immediate consequence of Proposition 15 and Birkoff's Theorem. Proposition 49.  ... 
arXiv:1809.08225v1 fatcat:tpuhd7pf65f2lhzfhwaezxg4ym
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