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OPERATIONAL QUANTUM LOGIC: AN OVERVIEW
, 2000
"... The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform presentation of what we call operational quantum logic, highlighting bo ..."
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Cited by 13 (5 self)
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The term quantum logic has different connotations for different people, having been considered as everything from a metaphysical attack on classical reasoning to an exercise in abstract algebra. Our aim here is to give a uniform presentation of what we call operational quantum logic, highlighting both its concrete physical origins and its purely mathematical structure. To orient readers new to this subject, we shall recount some of the historical development of quantum logic, attempting to show how the physical and mathematical sides of the subject have influenced and enriched one another.
Tropological systems are points of quantales
 J. Pure Appl. Algebra
, 2002
"... We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C*algebras. The other sees quantales as ..."
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Cited by 12 (7 self)
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We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C*algebras. The other sees quantales as algebras of abstract experiments on physical or computational systems, and has been applied to the study of the semantics of concurrent systems. We investigate connections between the two areas, in particular showing that concurrent systems, in the form of either settheoretic or localic tropological systems, can be identified with points of quantales by means of a suitable adjunction, which indeed holds for a much larger class of socalled “tropological models”. We show that in the case of tropological models in factor quantales, which still generalize tropological systems, the identification of models and (generalized) points preserves all the information needed for describing the observable behaviour of systems. We also define a notion of morphism of models that generalizes previous definitions of morphism of systems, and show that morphisms, too, can be defined in terms of either side of the adjunction, in fact giving us isomorphisms of categories. The relation between completeness notions for tropological systems and spatiality for quantales is also addressed, and a preliminary partial preservation result is obtained.
A Noncommutative Theory of Penrose Tilings
 International Journal of Theoretical Physics 44: 655689, 2005. [3] A. Palmigiano and
"... Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible ..."
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Cited by 11 (6 self)
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Considering quantales as generalised noncommutative spaces, we address as an example a quantale Pen based on the Penrose tilings of the plane. We study in general the representations of involutive quantales on those of binary relations, and show that in the case of Pen the algebraically irreducible representations provide a complete classification of the set of Penrose tilings from which its representation as a quotient of Cantor space is recovered.
QUANTALES AND OBSERVATIONAL SEMANTICS PEDRO RESENDE Departamento
"... We illustrate the idea that quantales can be regarded as algebras of experimental observations on physical systems, and we give a survey of some research in computer science where this idea has been used. We extend the mathematical framework hitherto available so that it can be applied to more gener ..."
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We illustrate the idea that quantales can be regarded as algebras of experimental observations on physical systems, and we give a survey of some research in computer science where this idea has been used. We extend the mathematical framework hitherto available so that it can be applied to more general systems than before, in particular to quantum systems and systems whose behaviour is partially unobservable. 1.