The aim of this article is to represent the general description of an entity by means of its states, contexts and properties. The entity that we want to describe does not necessarily have to be a physical entity, but can also be an entity of a more abstract nature, for example a concept, or a cultural artifact, or the mind of a person, etc..., which means that we aim at very general description. The effect that a context has on the state of the entity plays a fundamental role, which means that our approach is intrinsically contextual. The approach is inspired by the mathematical formalisms that have been developed in axiomatic quantum mechanics, where a specific type of quantum contextuality is modelled. However, because in general states also influence context – which is not the case in quantum mechanics – we need a more general setting than the one used there. Our focus on context as a fundamental concept makes it possible to unify ‘dynamical change ’ and ‘change under influence of measurement’, which makes our approach also more general and more powerful than the traditional quantum axiomatic approaches. For this reason an experiment (or measurement) is introduced as a specific kind of context. Mathematically we introduce a state context property system as the structure to describe an entity by means of its states, contexts and properties. We also strive from the start to a categorical setting and derive the morphisms between

this paper consists of lifting the --- categorically --- equivalent descriptions of physical systems by a (i) `state space' or a (ii) `property lattice' --- see [14,20,25,26] --- to an asymmetrical --- i.e., not anymore isomorphic --- duality on the level of: (i)

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. We formally introduce and study a notion of 'soft induction' on entities with an operationally motivated logico-algebraic description, and in particular the derived notions of 'induced state transition' and 'induced property transition'. We study the meaningful collections of these soft inductions which all have a quantale structure due to the introduction of temporal composition and arbitrary choice on the level of these state transitions and the corresponding property transitions. 1 Introduction The essential physical concepts that lie at the base of this paper are the notion of a property according to the Geneva school operational approach [2, 10, 13] and the idea that measurements on an entity provoke a real change of the state of the system [3], i.e., a change of its 'actual' properties [2, 10, 13]. Within this conceptual context, we introduce the notion of an 'induction', and in particular of 'soft inductions'. For other aspects related to these inductions we refer to [5, 7]....

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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.

: We recover the rays in the tensor product of Hilbert spaces within a larger class of so called `states of compoundness', structured as a complete lattice with the `state of separation' as its top element. At the base of the construction lies the assumption that the cause of actuality of a property of one individual entity is actuality of a property of the other. Keywords: property lattice, tensor product of Hilbert spaces, separated system, Galois duality. 1. INTRODUCTION Most approaches towards a realistic description of compound ---quantum--- systems are based on the recognition of subsystems, imposing some mathematical universal property as a structural criterion (Hellwig and Krauser, 1977; Zecca, 1977; Aerts and Daubechies, 1978; Aerts, 1982; Pulmannova, 1984; Ischi, 1999; Valckenborgh, 1999). In this paper we take a di#erent point of view, essentially focusing on a structural characterization of the interaction between the individual entities, rather than on the compound syst...

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 set-theoretic 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 so-called “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.

We enlarge the hom-sets of categories of complete lattices by introducing `state transitions' as generalized morphisms. The obtained category will then be compared with a functorial quantaloidal enrichment and a contextual quantaloidal enrichment that uses a specific concretization in the category of sets and partially defined maps (Parset). Keywords: operational quantum logic, state transition, categorical approach, complete lattice, quantaloid. 1 Introduction In this paper we present a construction that abstracts the concept of `state transition' as introduced in (Amira et al., 1998) and (Coecke and Stubbe, 1999a; 1999b), making it applicable to any subcategory A of J CLat. We compare this construction, the result of which is a quantaloid that we call Q st A, with two other quantaloidal extensions that arise naturally when considering the action of the power-functor on A. In fact, one of these natural extentions is functorial, we denote it by Q - A, and the other, calle...

We present a detailed synthetic overview of the utilisation of categorical techniques in the study of order...