4,097 research outputs found

    Between quantum logic and concurrency

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    We start from two closure operators defined on the elements of a special kind of partially ordered sets, called causal nets. Causal nets are used to model histories of concurrent processes, recording occurrences of local states and of events. If every maximal chain (line) of such a partially ordered set meets every maximal antichain (cut), then the two closure operators coincide, and generate a complete orthomodular lattice. In this paper we recall that, for any closed set in this lattice, every line meets either it or its orthocomplement in the lattice, and show that to any line, a two-valued state on the lattice can be associated. Starting from this result, we delineate a logical language whose formulas are interpreted over closed sets of a causal net, where every line induces an assignment of truth values to formulas. The resulting logic is non-classical; we show that maximal antichains in a causal net are associated to Boolean (hence "classical") substructures of the overall quantum logic.Comment: In Proceedings QPL 2012, arXiv:1407.842

    Set optimization - a rather short introduction

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    Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

    Discrete Convex Functions on Graphs and Their Algorithmic Applications

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    The present article is an exposition of a theory of discrete convex functions on certain graph structures, developed by the author in recent years. This theory is a spin-off of discrete convex analysis by Murota, and is motivated by combinatorial dualities in multiflow problems and the complexity classification of facility location problems on graphs. We outline the theory and algorithmic applications in combinatorial optimization problems

    Representation of States on Effect-Tribes and Effect Algebras by Integrals

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    We describe σ\sigma-additive states on effect-tribes by integrals. Effect-tribes are monotone σ\sigma-complete effect algebras of functions where operations are defined by points. Then we show that every state on an effect algebra is an integral through a Borel regular probability measure. Finally, we show that every σ\sigma-convex combination of extremal states on a monotone σ\sigma-complete effect algebra is a Jauch-Piron state.Comment: 20 page

    Loomis--Sikorski Theorem and Stone Duality for Effect Algebras with Internal State

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    Recently Flaminio and Montagna, \cite{FlMo}, extended the language of MV-algebras by adding a unary operation, called a state-operator. This notion is introduced here also for effect algebras. Having it, we generalize the Loomis--Sikorski Theorem for monotone σ\sigma-complete effect algebras with internal state. In addition, we show that the category of divisible state-morphism effect algebras satisfying (RDP) and countable interpolation with an order determining system of states is dual to the category of Bauer simplices Ω\Omega such that eΩ\partial_e \Omega is an F-space
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