## Ultrafilter extensions for coalgebras (2005)

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Venue: | In Algebra and Coalgebra in Computer Science, volume 3629 of LNCS |

Citations: | 13 - 6 self |

### BibTeX

@INPROCEEDINGS{Kupke05ultrafilterextensions,

author = {C. Kupke and A. Kurz and D. Pattinson},

title = {Ultrafilter extensions for coalgebras},

booktitle = {In Algebra and Coalgebra in Computer Science, volume 3629 of LNCS},

year = {2005},

pages = {263--277},

publisher = {Springer}

}

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### Abstract

Abstract. This paper studies finitary modal logics as specification languages for Set-coalgebras (coalgebras on the category of sets) using Stone duality. It is wellknown that Set-coalgebras are not semantically adequate for finitary modal logics in the sense that bisimilarity does not in general coincide with logical equivalence.

### Citations

479 | Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...lter extension is a morphism of coalgebras. 2 Preliminaries and Notation Stone Duality. Unfortunately we have space only to indicate the most important notions. For a general introduction we refer to =-=[7,2]-=-. We write��� for the category of sets and functions,�� for the category of Boolean algebras and their morphisms and����� for the category of Stone-spaces and continuous maps. The contravariant functo... |

321 | Universal coalgebra: a theory of systems
- Rutten
(Show Context)
Citation Context ...d a number of techniques for coalgebras (e.g. final semantics, isomorphism theorems, final sequence, co-Birkhoff theorems) can be obtained by dualising the corresponding concepts for algebras (Rutten =-=[18]-=-). Unfortunately, when it comes to specification languages for coalgebras, it is more difficult to achieve results parametric in the functor� . The idea that (variants of) modal logics are the natural... |

123 |
A domain equation for bisimulation
- Abramsky
- 1991
(Show Context)
Citation Context ...and Setcoalgebras. This is a special instance of a more general phenomenon in computer science where topology-based structures and set-based structures interact. This was observed already in Abramsky =-=[1]-=- where powerdomain-coalgebras and powerset-coalgebras 14 (11)swere compared. We believe that the methods used here will generalise to other such situations. First, we can treat other logics than class... |

92 | Coalgebraic logic
- Moss
- 1999
(Show Context)
Citation Context ...ges for coalgebras, it is more difficult to achieve results parametric in the functor� . The idea that (variants of) modal logics are the natural logics for coalgebras goes back to Moss seminal paper =-=[14]-=-. Applying to modal logic dualised algebraic methods, leads to the insight that modal logic for coalgebras is dual to equational logic for algebras [11,13]. But the methods derived from this approach ... |

61 |
Specifying coalgebras with modal logic. Theoret
- Kurz
- 2001
(Show Context)
Citation Context ... towards (more realistic) finitary logics for coalgebras has been somewhat ad hoc. It essentially consisted in giving up parametricity in � and restricting attention to particular classes of functors =-=[12,17,6]-=-. More recently, Pattinson [15,16] has shown how these logic arise uniformly as logics given by predicate liftings. It is one of the aims of this paper to further develop this approach towards a theor... |

54 |
Metamathematics of modal logic
- Goldblatt
- 1976
(Show Context)
Citation Context ...an algebras (see e.g. [20, Section 7]). There are also a number of more immediate open questions. Formulate a finitary definability result for classes of coalgebras in the style of Goldblatt-Thomason =-=[5]-=-, based on ultrafilter extensions. If� preserves finite sets than it has a canonical lifting to from ��� to�����; show that then this lifting agrees with �� . Find nice conditions guaranteeing that ul... |

54 | Many-sorted coalgebraic modal logic: a model-theoretic study
- Jacobs
(Show Context)
Citation Context ... towards (more realistic) finitary logics for coalgebras has been somewhat ad hoc. It essentially consisted in giving up parametricity in � and restricting attention to particular classes of functors =-=[12,17,6]-=-. More recently, Pattinson [15,16] has shown how these logic arise uniformly as logics given by predicate liftings. It is one of the aims of this paper to further develop this approach towards a theor... |

53 | Coalgebraic modal logic: soundness, completeness and decidability
- Pattinson
(Show Context)
Citation Context ...ogics for coalgebras has been somewhat ad hoc. It essentially consisted in giving up parametricity in � and restricting attention to particular classes of functors [12,17,6]. More recently, Pattinson =-=[15,16]-=- has shown how these logic arise uniformly as logics given by predicate liftings. It is one of the aims of this paper to further develop this approach towards a theory of logics for coalgebras that is... |

33 |
Duality for logics of transition systems
- Bonsangue, Kurz
- 2005
(Show Context)
Citation Context ... can treat other logics than classical ones by replacing the duality between �� and����� by one for, e.g. Heyting algebras or distributive lattices. Infinitary logics can be treated as well, see e.g. =-=[4]-=-. Second, we can replace��� by other categories of interest in semantics. Third, we can make algebraic tools available by upgrading the triangle of Diagram (1) to a square where��� is now accompanied ... |

29 | Expressive logics for coalgebras via terminal sequence induction
- Pattinson
- 2004
(Show Context)
Citation Context ...ogics for coalgebras has been somewhat ad hoc. It essentially consisted in giving up parametricity in � and restricting attention to particular classes of functors [12,17,6]. More recently, Pattinson =-=[15,16]-=- has shown how these logic arise uniformly as logics given by predicate liftings. It is one of the aims of this paper to further develop this approach towards a theory of logics for coalgebras that is... |

24 | A co-variety-theorem for modal logic
- Kurz
- 1998
(Show Context)
Citation Context ... for coalgebras goes back to Moss seminal paper [14]. Applying to modal logic dualised algebraic methods, leads to the insight that modal logic for coalgebras is dual to equational logic for algebras =-=[11,13]-=-. But the methods derived from this approach are adequate only for infinitary logics. This can be seen as a consequence of the fact that����� is equivalent to the category of complete atomic Boolean a... |

18 |
Boolean algebras with operators, part 1
- Jónsson, Tarski
- 1951
(Show Context)
Citation Context ...topological representation4 �� � �� �� � of�. Finally,� is the forgetful functor that maps a space to its carrier set. Note that the one traversal of this diagram, starting at��, produces the perfect =-=[8]-=- or canonical extension������ for any Boolean algebra�. The traversal starting at ��� produces the set of ultrafilters ������ over a set � (see e.g. [3] for more information). One of our aims is to li... |

18 |
Algebraic semantics for coalgebraic logics
- Kupke, Kurz, et al.
- 2004
(Show Context)
Citation Context ...isimilar iff they have the same theory. The deeper reason for the adequateness of finitary modal logics and Stone-coalgebras is the duality of Boolean algebras and Stone spaces, see Johnstone [7]. In =-=[9]-=-, we have shown that every sound logic�given by predicate liftings induces a functor�on the category�� of Boolean algebras. Using the dual equivalence of�� and the category����� of Stone spaces, it fo... |

15 |
From modal logic to terminal coalgebras
- Rößiger
(Show Context)
Citation Context ... towards (more realistic) finitary logics for coalgebras has been somewhat ad hoc. It essentially consisted in giving up parametricity in � and restricting attention to particular classes of functors =-=[12,17,6]-=-. More recently, Pattinson [15,16] has shown how these logic arise uniformly as logics given by predicate liftings. It is one of the aims of this paper to further develop this approach towards a theor... |

5 |
Operations and equations for coalgebras
- Kurz, Rosický
- 2005
(Show Context)
Citation Context ... for coalgebras goes back to Moss seminal paper [14]. Applying to modal logic dualised algebraic methods, leads to the insight that modal logic for coalgebras is dual to equational logic for algebras =-=[11,13]-=-. But the methods derived from this approach are adequate only for infinitary logics. This can be seen as a consequence of the fact that����� is equivalent to the category of complete atomic Boolean a... |

3 | Handbook of Modal Logic, chapter Algebras and Coalgebras - Venema |

2 |
Expressivity of Coalgebraic Modal Logic: The Limits and Beyond
- Schroeder
- 2005
(Show Context)
Citation Context ... to ������� or, equivalently, �� ������������. The previous definition pre-supposes a set£of predicate liftings to define the lifted functor �� ������������. The next proposition, which was stated in =-=[19]-=- and which is an instance of the Yoneda lemma, shows that there is a canonical choice for the set of liftings. Proposition 9. There is a 1-1 correspondence ��-ary predicate liftings�� ������ ����� �� ... |