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Computing on Lattice-Ordered Abelian Groups

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Book cover Fields of Logic and Computation III

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 12180))

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Abstract

Starting from a classical theorem of Gurevich and Kokorin we survey recent diverging developments of the theories of lattice-ordered abelian groups and their counterparts equipped with a distinguished order unit. We will focus on decision and recognition problems. As an application of Elliott’s classification, we will touch on word problems of AF C*-algebras.

Honoring Yuri Gurevich on his 80th birthday.

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Notes

  1. 1.

    Like its equivalent reformulation known as Farkas’ lemma, this is one of the many “Theorems of the Alternative” in Linear Programming and Convex Analysis. See [8, Theorem 2.2.1, Lemma 2.2.7, Exercise 4, p. 25], [19, 16.10 (i), (ii)], [51, §7.3, and §7.8 (31)].

  2. 2.

    This result goes back to Weinberg [54] who proved that every free \(\ell \)-group is a subdirect product of copies of the integers.

  3. 3.

    P need not be convex, nor connected. The simplexes \(S_i\) need not have the same dimension. This is the terminology of [53]. Polyhedra are called “compact polyhedra” in [50].

  4. 4.

    \(\mathcal M_{\mathbb R}(Q)\) is denoted \(\mathcal M(Q)\) in [10].

  5. 5.

    See [10, Theorem 2.2] for details.

  6. 6.

    “Unimodular” in [22] and [42].

  7. 7.

    Note that our present \(\mathcal M_{\mathbb R}(Q)\) agrees with (5) above.

  8. 8.

    Also see Baker’s analysis of finitely generated projective vector lattices in [2, Theorem 5.1].

  9. 9.

    In this way one can describe, for instance, phase transitions as singularities in the thermodynamic potentials.

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Acknowledgement

The author is grateful to the reviewer for her/his valuable remarks and for providing a short proof of Proposition 4.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, “Ulisse Dini” University of Florence, Viale Morgagni 67/A, 50134, Florence, Italy

    Daniele Mundici

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  1. Daniele Mundici
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Editors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Andreas Blass

  2. Université Paris Est Créteil, Fontainebleau, France

    Patrick Cégielski

  3. School of Computer Science, Tel Aviv University, Tel Aviv, Israel

    Nachum Dershowitz

  4. Fakultät für Mathematik und Informatik, Universität Leipzig, Leipzig, Germany

    Manfred Droste

  5. Reactive Systems Group, Universitat des Saarlandes, Saarbrücken, Germany

    Bernd Finkbeiner

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Mundici, D. (2020). Computing on Lattice-Ordered Abelian Groups. In: Blass, A., Cégielski, P., Dershowitz, N., Droste, M., Finkbeiner, B. (eds) Fields of Logic and Computation III. Lecture Notes in Computer Science(), vol 12180. Springer, Cham. https://doi.org/10.1007/978-3-030-48006-6_15

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