A semiring-like representation of lattice pseudoeffect algebras

Foundations
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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.

Keywords

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

Notes

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.

Compliance with ethical standards

Conflict of interest

The authors declared that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants performed by any of the authors.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Algebra and GeometryPalacký UniversityOlomoucCzech Republic
  2. 2.Università di CagliariCagliariItaly

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