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Soft Computing

, Volume 19, Issue 2, pp 269–282 | Cite as

The representation of square root quasi-pseudo-MV algebras

Foundations

Abstract

\(\sqrt{'}\) quasi-MV algebras arising from quantum computation are term expansions of quasi-MV algebras. In this paper, we introduce a generalization of \(\sqrt{'}\) quasi-MV algebras, called square root quasi-pseudo-MV algebras (\(\sqrt{\hbox {quasi-pMV}}\) algebras, for short). First, we investigate the related properties of \(\sqrt{\hbox {quasi-pMV}}\) algebras and characterize two special types: Cartesian and flat \(\sqrt{\hbox {quasi-pMV}}\) algebras. Second, we present two representations of \(\sqrt{\hbox {quasi-pMV}}\) algebras. Furthermore, we generalize the concepts of PR-groups to non-commutative case and prove that the interval of a non-commutative PR-group with strong order unit is a Cartesian \(\sqrt{\hbox {quasi-pMV}}\) algebra. Finally, we introduce non-commutative PR-groupoids which extend abelian PR-groupoids and show that the category of negation groupoids with operators and the category of non-commutative PR-groupoids are equivalent.

Keywords

Quasi-MV algebras Quasi-pMV algebras \(\sqrt{\hbox {quasi-pMV}}\) algebras Cartesian \(\sqrt{\hbox {quasi-pMV}}\) algebras PR-groups Representations 

Notes

Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant No.11126301), Promotive Research Fund for Young and Middle-aged Scientists of Shandong Province (No. BS2011SF002).

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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of JinanJinanP.R. China
  2. 2.Institute of Mathematics and Computer ScienceWrocław University of TechnologyWrocławPoland

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