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A semiring-like representation of lattice pseudoeffect algebras

Soft Computing volume 23, pages 1465–1475 (2019)Cite this article

Abstract

In order to represent lattice pseudoeffect algebras, a non-commutative generalization of lattice effect algebras, in terms of a particular subclass of near semirings, we introduce in this article the notion of near pseudoeffect semiring. Taking advantage of this characterization, in the second part of the present work, we present, as an application, an alternative, rather straight as well as simple, explanation of the relationship between lattice pseudoeffect algebras and pseudo-MV algebras by means of a simplified axiomatization of generalized Łukasiewicz semirings, a variety of non-commutative semirings equipped with two antitone unary operations.

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Notes

  1. In this article the notion of semiring will be required only in Sect. 3. It will be introduced explicitly in Definition 6.

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Acknowledgements

The research of I. Chajda is supported by IGA, Project PřF 2018 012. D. Fazio and A. Ledda gratefully acknowledge the support of the Horizon 2020 program of the European Commission: SYSMICS Project, Number: 689176, MSCA-RISE-2015. A. Ledda expresses his gratitude for the support of Fondazione di Sardegna within the project “Science and its Logics: The Representation’s Dilemma”, Cagliari, Number: F72F16003220002, and for the support of Regione Autonoma della Sardegna within the project “Order-theoretical properties in mathematics and in physics”, CUP: F72F16002920002.

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Authors and Affiliations

  1. Department of Algebra and Geometry, Palacký University, 17. listopadu 12, 771 46, Olomouc, Czech Republic

    Ivan Chajda

  2. Università di Cagliari, via Is Mirrionis 1, 09123, Cagliari, Italy

    Davide Fazio & Antonio Ledda

Authors
  1. Ivan Chajda
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  2. Davide Fazio
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  3. Antonio Ledda
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Correspondence to Antonio Ledda.

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The authors declared that they have no conflict of interest.

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Communicated by A. Di Nola.

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Chajda, I., Fazio, D. & Ledda, A. A semiring-like representation of lattice pseudoeffect algebras. Soft Comput 23, 1465–1475 (2019). https://doi.org/10.1007/s00500-018-3157-2

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  • DOI: https://doi.org/10.1007/s00500-018-3157-2

Keywords

  • Pseudoeffect algebra
  • Pseudo-MV algebra
  • Semiring
  • Near semiring
  • Near-p semiring
  • Gl-semirings