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## Boolean algebras arising from information systems (2003)

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### Other Repositories/Bibliography

Venue: | Proceedings of the Tarski Centenary Conference, Banach |

Citations: | 6 - 4 self |

### Citations

246 | Boolean algebras with operators - J'onsson, Tarski |

208 |
Modal Correspondence Theory
- Benthem
- 1976
(Show Context)
Citation Context ... © 2 U ��© R��� is called the full modal complex algebra of K. Correspondence theory investigates the relationship between properties of the relations of a frame and properties of its complex algebra =-=[18]-=-. Examples, which we will need later – and which are easily proved –, are (3.18) (3.19) (3.20) (3.21) R is ����� R��� X��¢ reflexive � X R is weakly ����© R��� U� ¦ reflexive ¢�© R��� X��� X R is ����... |

52 | Modal environment for boolean speculations - Gargov, Passy, et al. - 1987 |

34 |
Monotone bounded distributive lattice expansions
- Gehrke, Jónsson
(Show Context)
Citation Context ...£�� ¡ n 1 : � n £���� X � ¡ ¡ m 1 : � m � n 1 : � n £�£�� � X ¡ � m 1 : � m � 1 ¡ � n 2 : � n £ § X � 12 � X£��sNow, f � � � f σ ¡ ω£�� ¥�� � ¥�� ¥ ¡ ω£ § Examples of a similar nature can be found in =-=[7]-=-. 5 Mixed algebras ¡ � f � X� f : � B� X � X£�� ω ¡�¡ � n 2 : � n £ X : � B� X � ω £�� X Now that we have the concept of a canonical extension, we can look at the algebras arising from algebras of the... |

33 |
General theory of Boolean algebras
- Koppelberg
- 1989
(Show Context)
Citation Context ...�§�§ � cn£ . For each i � n choose an ultrafilter Ui of B which contains ci and no c j� j � ¥ i. c0� Let hC : � B C be defined by hC� b� ¥ ∑ ¡ ci : b � Ui£ § Then, hC is a retraction of B onto C, see =-=[13]-=-, p. 78. For each x � B let Then, gC is an operator on C, and hC © B� g����© C� gC� : is a homomorphism. gC� 0� ¥ gC� hC� 0��� ¥ Now, g� 0��� ¥ hC� 1� ¥ 1, and, for x� y � C, hC� gC� hC� x��� ¥ hC� g�... |

32 |
Inaccessible worlds, Notre Dame
- Humberstone
- 1983
(Show Context)
Citation Context ...ary modal logics can offer. In one direction, one considers operations which are intended to express properties not expressible by the possibility operator, such as the inaccessibility of Humberstone =-=[10]-=-, or the suffiency operators of Gargov et al. [6]. Such operators lead to various classes of Boolean algebras with operators, and in earlier work we have introduced the classes of sufficiency algebras... |

26 |
On the canonicity of Sahlqvist identities
- Jónsson
- 1994
(Show Context)
Citation Context ... such that ��� Sg��� ¥ g. 3. S� � R� � ¥ R. We invite the reader to consult [5] for details. The algebra CmSUA� K� ¥ © 2 U ����� R����� is called the full sufficiency complex algebra of K. B. Jónsson =-=[11]-=- has remarked that MOA is generated by its finite members. A similar result is true for SUA. 7sProposition 3.6. SUA is generated by its finite members. Proof. Let © B� g��� SUA. It is our aim to show ... |

18 | Modal definability in enriched languages. Notre Dame - Goranko - 1990 |

16 | The preservation of Sahlqvist equations in completions of Boolean algebras with operators
- Givant, Venema
- 1999
(Show Context)
Citation Context ...A. 2. x � gPgP� x� . 3. gP� x����� x. 4. fa fa fa� x��� fa� x� .sProof. Observe that each of the axioms for the classes of algebras in Table 3 can be presented as an equation which is MOA – canonical =-=[8, 11]-=-, or it leads to a SUA canonical equation after the transformation � , defined on p. 3. For example, is canonical, since the inequalities (6.1) (6.2) x � f ∂ P fP� x� � x � x��� f � x f ∂ x� � are can... |

14 | Beyond modalities: Sufficiency and mixed algebras
- Düntsch, Orłowska
- 2000
(Show Context)
Citation Context ...ses of Boolean algebras with operators, and in earlier work we have introduced the classes of sufficiency algebras (SUA) and mixed algebras (MIA) and have started an investigation of their properties =-=[5]-=-. Other novel features of the logics that have been considered in connection with information systems are – on the level of semantics – frames with a family ¡ RQ : Q ¢ PAR£ of relative relations. Thes... |

9 | Relational attribute systems - Düntsch, Gediga, et al. |

9 |
Information algebras
- Orłowska
- 1995
(Show Context)
Citation Context ...ssigns to an illness, then one a� x� ¥ can allow /0 to express that the expert does not have (or does not want to publicise) an opinion [see 3]. Each attribute a defines various information relations =-=[14, 16, 17]-=- on the Universe OB in the following way: If T is a relation on 2 VALa , we let RT be the relation on U defined by If A ¢ OB, we let T� s xR xR T a y ��� a� x� Ta� y� § A y ��� xRT � A� T� a y for all... |

9 |
Information systems, similarity relations and modal logics
- Vakarelov
- 1998
(Show Context)
Citation Context ...ssigns to an illness, then one a� x� ¥ can allow /0 to express that the expert does not have (or does not want to publicise) an opinion [see 3]. Each attribute a defines various information relations =-=[14, 16, 17]-=- on the Universe OB in the following way: If T is a relation on 2 VALa , we let RT be the relation on U defined by If A ¢ OB, we let T� s xR xR T a y ��� a� x� Ta� y� § A y ��� xRT � A� T� a y for all... |

8 | Editor): Incomplete Information: Rough Set Analysis. Physica-Verlag - Orłowska - 1998 |

7 | S.: 2002. Incomplete information: structure, inference, complexity - Demri, Orlowska |

4 | Informational representability of models for information logics
- Demri, Orłowska
- 1999
(Show Context)
Citation Context ... relative counterparts to the relations (2.1) – (2.3) in terms of local and global properties. Representability theorems which exhibit the connections between the relations of Table 1 can be found in =-=[1]-=- and [4]. 3 Modal and sufficiency algebras Throughout, © B������������� 0� 1� we let be a Boolean algebra, which we will just call B. If A is any Boolean At� A� algebra, then is the set of all atoms o... |

4 |
Introduction: What you always wanted to know about rough sets
- Orłowska
- 1997
(Show Context)
Citation Context ...ssigns to an illness, then one a� x� ¥ can allow /0 to express that the expert does not have (or does not want to publicise) an opinion [see 3]. Each attribute a defines various information relations =-=[14, 16, 17]-=- on the Universe OB in the following way: If T is a relation on 2 VALa , we let RT be the relation on U defined by If A ¢ OB, we let T� s xR xR T a y ��� a� x� Ta� y� § A y ��� xRT � A� T� a y for all... |

1 |
A multimodal logic for disagreement and exhaustiveness
- Düntsch
- 2001
(Show Context)
Citation Context ...e counterparts to the relations (2.1) – (2.3) in terms of local and global properties. Representability theorems which exhibit the connections between the relations of Table 1 can be found in [1] and =-=[4]-=-. 3 Modal and sufficiency algebras Throughout, © B������������� 0� 1� we let be a Boolean algebra, which we will just call B. If A is any Boolean At� A� algebra, then is the set of all atoms of A, and... |

1 |
Discriminator–Algebras, vol. 6 of Studien zur Algebra und ihre Anwendungen
- Werner
- 1978
(Show Context)
Citation Context ...ed by m� x� ¥ f ∂ � x��� g��� x� satisfies If x� y� z � B, then m� x� ¥ � � � if x ¥ 1� 1� otherwise§ 0� t� x� y� z� ¥ z � m����� x � y����� x � m����� x � y��� is the ternary discriminator, see e.g. =-=[19]-=-. Here, x � y is the symmetric difference. We can use this to show a correspondence result. Suppose that � is a language for mixed logic. Let P1 be the set of all first order sentences ϕ in a language... |