A semiring-like representation of lattice pseudoeffect algebras

  • Ivan Chajda
  • Davide Fazio
  • Antonio Ledda


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.


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



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.


  1. Belluce LP, Di Nola A, Ferraioli AR (2013) MV-semirings and their sheaf representations. Order 30(1):165–179MathSciNetCrossRefzbMATHGoogle Scholar
  2. Beran L (2011) Orthomodular lattices: an algebraic approach. Mathematics and its applications. Springer, NetherlandsGoogle Scholar
  3. Bonzio S, Chajda I, Ledda A (2016) Representing quantum structures as near semirings. Logic J IGPL 24(5):719–742MathSciNetCrossRefGoogle Scholar
  4. Chajda I (2012) Basic algebras and their applications. An overview. In: Czermak J et al (eds) Proceedings of 81st workshop on general algebra, Salzburg, Austria, 2011. Johannes Heyn, Klagenfurt, pp 1–10Google Scholar
  5. Chajda I, Länger H (2015) Commutative basic algebras and coupled near semirings. Soft Comput 19:1129–1134CrossRefzbMATHGoogle Scholar
  6. Chajda I, Länger H (2017) A representation of lattice effect algebras by means of right near semirings with involution. Int J Theor Phys 56:3719–3726MathSciNetCrossRefzbMATHGoogle Scholar
  7. Dalla Chiara ML, Giuntini R, Greechie R (2004) Reasoning in quantum theory: sharp and unsharp quantum logic. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  8. Dvurečenskij A (2001) Pseudo MV algebras are intervals in \(\ell \)-groups. J Aust Math Soc 72:427–445MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dvurečenskij A (2015) Lexicographic pseudo MV-algebras. J Appl Logic 13:825–841MathSciNetCrossRefzbMATHGoogle Scholar
  10. Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Mathematics and its applications. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  11. Dvurečenskij A, Vetterlein T (2001a) Pseudoeffect algebras I, basic properties. Int J Theor Phys 40:685–701MathSciNetCrossRefzbMATHGoogle Scholar
  12. Dvurečenskij A, Vetterlein T (2001b) Pseudoeffect algebras II, group representations. Int J Theor Phys 40:703–726MathSciNetCrossRefzbMATHGoogle Scholar
  13. Dvurečenskij A, Vetterlein T (2004) Non-commutative algebras and quantum structures. Int J Theor Phys 43(7/8):15–63MathSciNetzbMATHGoogle Scholar
  14. Foulis DJ, Bennett MK (1994) Effect algebras and unsharp quantum logics. Found Phys 24:719–742MathSciNetCrossRefzbMATHGoogle Scholar
  15. Foulis DJ, Pulmannová S, Vincenková E (2011) Lattice pseudoeffect algebras as double residuated structures. Soft Comput 12:2479–2488CrossRefzbMATHGoogle Scholar
  16. Georgescu G, Iorgulescu A (2001) Pseudo MV-algebras. Mult Valued Logic 6:95–135MathSciNetzbMATHGoogle Scholar
  17. Giuntini R, Greuling H (1989) Toward a formal language for unsharp properties. Found Phys 19:931–945MathSciNetCrossRefGoogle Scholar
  18. Głazek K (2002) A guide to the literature on semirings and their applications in mathematics and information sciences. Springer, BerlinzbMATHGoogle Scholar
  19. Kadji A, Lele C, Nganou JB (2016) A non-commutative generalization of Łukasiewicz rings. J Appl Logic 16:1–13MathSciNetCrossRefzbMATHGoogle Scholar
  20. Kalmbach G (1983) Orthomodular lattices, volume 18 of London mathematical society monographs. Academic Press, LondonGoogle Scholar
  21. Mundici D (1986) Interpretation of AFC\(^*\)-algebras in Łukasiewicz sentential calculus. J Funct Anal 65:15–63MathSciNetCrossRefzbMATHGoogle Scholar
  22. Rachůnek J (2001) A non-commutative generalization of MV-algebras. Czechoslov Math J 52:255–273MathSciNetCrossRefzbMATHGoogle Scholar
  23. Vitolo P (2010) Compatibility and central elements in pseudo-effect algebras. Kybernetica 46(6):996–1008MathSciNetzbMATHGoogle Scholar

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

Personalised recommendations