1,337 research outputs found

    An analysis of the logic of Riesz Spaces with strong unit

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    We study \L ukasiewicz logic enriched with a scalar multiplication with scalars taken in [0,1][0,1]. Its algebraic models, called {\em Riesz MV-algebras}, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of {\em DMV-algebras} and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective

    Lukasiewicz logic and Riesz spaces

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    We initiate a deep study of {\em Riesz MV-algebras} which are MV-algebras endowed with a scalar multiplication with scalars from [0,1][0,1]. Extending Mundici's equivalence between MV-algebras and ℓ\ell-groups, we prove that Riesz MV-algebras are categorically equivalent with unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent with the class of commutative unital C∗^*-algebras. The propositional calculus RL{\mathbb R}{\cal L} that has Riesz MV-algebras as models is a conservative extension of \L ukasiewicz ∞\infty-valued propositional calculus and it is complete with respect to evaluations in the standard model [0,1][0,1]. We prove a normal form theorem for this logic, extending McNaughton theorem for \L ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in RL{\mathbb R}{\cal L} and we relate them with the analogue of de Finetti's coherence criterion for RL{\mathbb R}{\cal L}.Comment: To appear in Soft Computin

    Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory

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    In Riesz space theory it is good practice to avoid representation theorems which depend on the axiom of choice. Here we present a general methodology to do this using pointfree topology. To illustrate the technique we show that almost f-algebras are commutative. The proof is obtained relatively straightforward from the proof by Buskes and van Rooij by using the pointfree Stone-Yosida representation theorem by Coquand and Spitters

    Integrals and Valuations

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    We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and tightly connected to the Riesz space structure.Comment: Submitted for publication 15/05/0

    Notes on divisible MV-algebras

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    In these notes we study the class of divisible MV-algebras inside the algebraic hierarchy of MV-algebras with product. We connect divisible MV-algebras with Q\mathbb Q-vector lattices, we present the divisible hull as a categorical adjunction and we prove a duality between finitely presented algebras and rational polyhedra
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