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A Geometric Approach to MV-Algebras

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On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 336))

Abstract

Markov unrecognizability theorem puts an end to the classical program of equipping any combinatorial manifold M with a computable set \(\mathscr {I}_M\) of invariants such that a manifold N is homeomorphic to M iff \(\mathscr {I}_M=\mathscr {I}_N\). To make sense of the statement of the theorem manifolds are replaced by finite strings of symbols for triangulated rational polyhedra, and homeomorphisms are understood as rational PL-homeomorphisms. Thus, objects and arrows undergo a radical transformation—and yet with no essential loss of generality for the original recognition problem. A further restriction on the arrows arises if one views the recognizability problem from the viewpoint of algorithmic complexity theory: here one must take into account the amount of information needed to specify rational polyhedra. We are thus left with the category of rational polyhedra (objects) with integer PL-maps (arrows). A new geometry arises, where the affine group over the integers takes on the same role as the isometry group does in euclidean space. Differently from the category of rational polyhedra with rational PL-maps, a wealth of new geometric computable invariants emerges in this new category. We discuss in particular the rational measure of rational polyhedra. Its role and applicability is amplified by the duality between rational polyhedra and finitely presented MV-algebras.

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Acknowledgments

I am grateful to my friend Peter Klement, whose many papers and books [20, and references therein] taught me the importance of t-norms, and whose kind hospitality at Magdalena Bildungshaus allowed me to get in contact with a community of mathematicians—of which he has been for decades one of the focal points—involved in all aspects of fuzzy logic.

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  1. Department of Mathematics and Computer Science, University of Florence, Florence, Italy

    Daniele Mundici

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  1. Dept of Knowledge-Based Math. Systems, Johannes Kepler University Linz, Linz, Austria

    Susanne Saminger-Platz

  2. Dept. of Mathematics & Descriptive Geo., Slovak University of Technology, BRATISLAVA, Slovakia

    Radko Mesiar

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Mundici, D. (2016). A Geometric Approach to MV-Algebras. In: Saminger-Platz, S., Mesiar, R. (eds) On Logical, Algebraic, and Probabilistic Aspects of Fuzzy Set Theory. Studies in Fuzziness and Soft Computing, vol 336. Springer, Cham. https://doi.org/10.1007/978-3-319-28808-6_4

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