## State Property Systems and Closure Spaces: a study of categorical equivalence (1999)

Citations: | 28 - 25 self |

### BibTeX

@MISC{Aerts99stateproperty,

author = {Diederik Aerts and Eva Colebunders and Ann Van der Voorde and Bart Van Steirteghem},

title = {State Property Systems and Closure Spaces: a study of categorical equivalence},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that the natural mathematical structure to describe a physical entity by means of its states and its properties within the Geneva-Brussels approach is that of a state property system. We prove that the category of state property systems (and morphisms), SP, is equivalent to the category of closure spaces (and continuous maps), Cls. We show the equivalence of the `state determination axiom' for state property systems with the `T 0 separation axiom' for closure spaces. We also prove that the category SP 0 of state determined state property systems is equivalent to the category L 0 of based complete lattices. In this sense the equivalence of SP and Cls generalizes the equivalence of Cls 0 (T 0 closure spaces) and L 0 , proven in (Erne 1984).

### Citations

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(Show Context)
Citation Context ...was proven. The category SP consists of the state property systems [2] and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties =-=[3, 4, 5, 6, 7, 8]-=-. The category Cls consists of the closure spaces and the continuous maps. In earlier work it has been shown, using the equivalence between Cls and SP, that some of the axioms of quantum axiomatics ar... |

36 | Foundations of quantum physics: a general realistic and operational approach
- Aerts
- 1999
(Show Context)
Citation Context ... Universiteit Brussel, 1160 Brussels, Belgium E-Mail: diddesen@vub.ac.be Abstract In [1] an equivalence of the categories SP and Cls was proven. The category SP consists of the state property systems =-=[2]-=- and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties [3, 4, 5, 6, 7, 8]. The category Cls consists of the closure spaces an... |

35 |
Description of many physical entities without the paradoxes encountered in quantum mechanics
- Aerts
- 1982
(Show Context)
Citation Context ...was proven. The category SP consists of the state property systems [2] and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties =-=[3, 4, 5, 6, 7, 8]-=-. The category Cls consists of the closure spaces and the continuous maps. In earlier work it has been shown, using the equivalence between Cls and SP, that some of the axioms of quantum axiomatics ar... |

31 |
Mècanique Quantique: bases et applications,, Press Polytechnique de
- Piron
- 1990
(Show Context)
Citation Context ...was proven. The category SP consists of the state property systems [2] and their morphisms, which are the mathematical structures that describe a physical entity by means of its states and properties =-=[3, 4, 5, 6, 7, 8]-=-. The category Cls consists of the closure spaces and the continuous maps. In earlier work it has been shown, using the equivalence between Cls and SP, that some of the axioms of quantum axiomatics ar... |

25 |
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Citation Context |

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der Voorde, Separation Axioms in Extension Theory for Closure Spaces and Their Relevance to State Property Systems, Doctoral Thesis
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(Show Context)
Citation Context ...and Cls1) The functors establish an equivalence of categories. F : SPa → Cls1 G : Cls1 → SPa For a more extensive study of separation axioms and their relation with state property systems we refer to =-=[12]-=-. In the present text our final aim is to use the described equivalence to translate the concept of connectedness in closure spaces into terms of state property systems. It will give us a means to dis... |

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der Voorde, A categorical approach to T1 separation and the product of state property systems
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(Show Context)
Citation Context ...ms of quantum axiomatics are equivalent with separation axioms on the corresponding closure space. More particularly it was proven that the axiom of atomicity is equivalent to the T1 separation axiom =-=[9]-=-. In the present article we analyze the intimate relation that exists between classical and nonclassical in the state property systems and disconnected and connected in the corresponding closure space... |

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3 |
der Voorde, Connectedness applied to closure spaces and state property systems
- Aerts, Deses, et al.
- 2001
(Show Context)
Citation Context ... intimate relation that exists between classical and nonclassical in the state property systems and disconnected and connected in the corresponding closure space, elaborating results that appeared in =-=[10, 11]-=-. We introduce classical properties using the concept of super selection rule, i.e. two properties are separated by a superselection rule iff there do not exist ‘superposition states’ related to these... |

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2 |
der Voorde, Classicality and connectedness for state property systems and closure spaces
- Aerts, Deses, et al.
(Show Context)
Citation Context ... intimate relation that exists between classical and nonclassical in the state property systems and disconnected and connected in the corresponding closure space, elaborating results that appeared in =-=[10, 11]-=-. We introduce classical properties using the concept of super selection rule, i.e. two properties are separated by a superselection rule iff there do not exist ‘superposition states’ related to these... |

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