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The Goldblatt-Thomason Theorem for Coalgebras
[chapter]

*
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. ...

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

2016
*
Lecture Notes in Computer Science
*

We show how previous work on modal extensions of Ł n -valued logics fits naturally into

doi:10.1007/978-3-662-52921-8_8
fatcat:sh3jm2dwufhixk6lpivdwhvewq
*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] . ...##
###
Distributive Substructural Logics as Coalgebraic Logics over Posets

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.) ]. ...

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

2020
*
arXiv
*
pre-print

We prove analogues of

arXiv:1904.12997v4
fatcat:we3s3ixc5bdfxngpwcuueqoswa
*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. ...##
###
Goldblatt-Thomason Theorems for Modal Intuitionistic Logics
[article]

2022
*
arXiv
*
pre-print

We prove a

arXiv:2011.10221v3
fatcat:jth72fcgr5aali722nheix5gpa
*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. ...##
###
Goldblatt-Thomason-style Theorems for Graded Modal Language

2010
*
Advances in Modal Logic
*

We prove two main

dblp:conf/aiml/SanoM10
fatcat:y63benskvvhi5gprdphbxexatu
*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. ...##
###
Duality for powerset coalgebras
[article]

2022
*
arXiv
*
pre-print

This allows us to describe an endofunctor H on CABA such that

arXiv:2008.01849v6
fatcat:ldexspld4vaftjhjwrnx6tbcz4
*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. ...##
###
First-Order Modal Logic: Frame Definability and Lindström Theorems
[article]

2016
*
arXiv
*
pre-print

This paper involves generalizing

arXiv:1602.00201v2
fatcat:6xewfnc24fetbco2ubfzjpvfby
*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*. ...##
###
How to write a coequation
[article]

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. ...

##
###
Ultrafilter Extensions for Coalgebras
[chapter]

2005
*
Lecture Notes in Computer Science
*

Our main contributions are

doi:10.1007/11548133_17
fatcat:bxsfjcbgwfbcnchuzmq3sxzqmi
*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. ...##
###
6 Algebras and coalgebras
[chapter]

2007
*
Studies in Logic and Practical Reasoning
*

In

doi:10.1016/s1570-2464(07)80009-7
fatcat:e77uebqxdferlcg2nlfe5hobpq
*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. ...##
###
Local Goldblatt–Thomason theorem

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]. ...

##
###
Equational Coalgebraic Logic

2009
*
Electronical Notes in Theoretical Computer Science
*

Finally, we argue that

doi:10.1016/j.entcs.2009.07.097
fatcat:qwz6upbojjck3lsqgdy3vjrady
*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] . ...##
###
Neighbourhood Structures: Bisimilarity and Basic Model Theory

2009
*
Logical Methods in Computer Science
*

In

doi:10.2168/lmcs-5(2:2)2009
fatcat:5tgcljiirrafpfn3lsbgway3ku
*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. ...##
###
Goldblatt-Thomason for LE-logics
[article]

2018
*
arXiv
*
pre-print

We prove a uniform version of

arXiv:1809.08225v1
fatcat:tpuhd7pf65f2lhzfhwaezxg4ym
*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. ...
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