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A semiring-like representation of lattice pseudoeffect algebras

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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Correspondence to Antonio Ledda.

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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