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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bou F, Paoli F, Ledda A, Freytes H (2008) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\)quasi-MV algebras Part II. Soft Comput 12:341–352
Bou F, Paoli F, Ledda A, Spinks M, Giuntini R (2010) The logic of quasi-MV algebras. J Log Comput 20:619–643
Chang CC (1958) Algebraic analysis of many valued logics. Trans Am Math Soc 88:467–490
Chen WJ, Dudek WA (2014) Quantum computational algebra with a non-commutative generalization, Math Slovaca (accepted). Article ID 166
Di Nola A, Georgescu G, Iorgulescu A (2002) Pseudo BL algebras: part I. Multi Val Log 8:673–714
Freytes H, Ledda A (2009) Categories of semigroups in quantum computational structures. Math Slovaca 59:413–432
Georgescu G, Iorgulescu A (2001) Pseudo MV algebras. Mult Val Log 6:95–135
Giuntini R, Ledda A, Paoli F (2007) Expanding quasi-MV algebras by a quantum operator. Stud Log 87:99–128
Giuntini R, Paoli F, Ledda A (2010) Categorical equivalences for \(\sqrt{^{\prime }}\) quasi-MV algebras. J Log Comput 20:795–810
Jipsen P, Ledda A, Panli F (2013) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\) quasi-MV algebras. Part IV. Rep Math Log 48:3–36
Kowalski T, Paoli F (2010) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\)quasi-MV algebras. Part III. Rep Math Log 45:161–199
Kowalski T, Paoli F (2011) Joins and subdirect products of varieties. Algebra Univers 65:371–391
Kowalski T, Paoli F, Giuntini R, Ledda A (2010) The lattice of subvarieties of square root quasi-MV algebras. Stud Log 95:37–61
Kowalski T, Paoli F, Spinks M (2011) Quasi-subtractive varieties. J Symb Log 76:1261–1286
Ledda A, Konig M, Paoli F, Giuntini R (2006) MV algebras and quantum computation. Stud Log 82:245–270
Mundici D (1986) Interpretation of AF C\(^{*}\)-algebra in Łukasiewicz sentential calculus. J Funct Anal 65:15–63
Paoli F, Ledda A, Giuntini R, Freytes H (2009) On some properties of quasi-MV algebras and \(\sqrt{^{\prime }}\)quasi-MV algebras. Part I. Rep Math Log 44:31–63
Paoli F, Ledda A, Spinks M, Freytes H, Giuntini R (2011) Logics from \(\sqrt{^{\prime }}\) quasi-MV algebras. Intern J Theor Phys 50:3882–3902
Rachunek J (2002) A non-commutative generalization of MV algebras. Czechoslov Math J 52(127):255–273
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Dvurečenskij.
Rights and permissions
About this article
Cite this article
Chen, W., Dudek, W.A. The representation of square root quasi-pseudo-MV algebras. Soft Comput 19, 269–282 (2015). https://doi.org/10.1007/s00500-014-1466-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00500-014-1466-7