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Soft Computing

, Volume 22, Issue 24, pp 8025–8040 | Cite as

States, state operators and quasi-pseudo-MV algebras

  • Wenjuan Chen
  • Wieslaw A. Dudek
Foundations
  • 78 Downloads

Abstract

Quasi-pseudo-MV algebras (quasi-pMV algebras, for short) arising from quantum computational logics are the generalizations of both quasi-MV algebras and pseudo-MV algebras. In this paper, we introduce the notions of states, state-morphisms, state operators and state-morphism operators to quasi-pMV algebras. First, we present the related properties of states on quasi-pMV algebras and show that states and Bosbach states coincide on any quasi-pMV algebra. And then we investigate the relationship between state-morphisms and the normal and maximal ideals of quasi-pMV algebras. We prove state-morphisms and extremal states are equivalent. The existence of states on quasi-pMV algebras is also discussed. Finally, state operators and state-morphism operators are introduced to quasi-pMV algebras, and the corresponding structures are called state quasi-pMV algebras and state-morphism quasi-pMV algebras, respectively. We investigate the related properties of ideals under state operators and state-morphism operators. Meanwhile, we show that there is a bijective correspondence between normal \(\sigma \)-ideals and ideal congruences on state quasi-pMV algebras.

Keywords

Ideals Quasi-pseudo-MV algebras States State operators State quasi-pMV algebras 

Notes

Acknowledgements

This study was funded by the National Natural Science Foundation of China (Grant No. 11501245), China Postdoctoral Science Foundation (No. 2017M622177) and Shandong Province Postdoctoral Innovation Projects of Special Funds (No. 201702005).

Compliance with ethical standards

Conflict of interest

Author A declares that she has no conflict of interest. Author B declares that he has no conflict of interest.

Ethical standard

This article does not contain any studies with human participants or animals 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.School of Mathematical SciencesUniversity of JinanJinanPeople’s Republic of China
  2. 2.Institute of MathematicsWrocław University of TechnologyWrocławPoland

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