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D-Effect Algebra can Be Made into a D-Total Algebra

Received: 11 April 2017     Accepted: 20 May 2017     Published: 28 November 2017
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Abstract

In this paper we prove that every D-effect algebra (E, ∆, 0, 1) can be made into a D-total algebra (E, ⍍, ¬, 1) in such a way that two elements are compatible in (E, ∆, 0, 1) if and only if they commute in(E, ⍍, ¬, 1) where x ∆ y =(x' + y')'.

Published in Mathematics Letters (Volume 3, Issue 6)
DOI 10.11648/j.ml.20170306.13
Page(s) 71-76
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

D-Basic Algebra, Weak D-Basic Algebra, Antitone Involution, D-Effect Algebra, D-Total Algebra

References
[1] Allam, A. A. E. M., Mikhaeel, N. N., & Merdach, H. H. (2016). Commutative groupoid algebra. Journal of Mathematical and Computational Science, 6(2), 262.
[2] Chajda, I., and Länger, H.: "States on basic algebras." Mathematica Bohemica 142.2 (2017): 197-210.
[3] Chajda, I., Halas, R., Kuhr, J.: Every effect algebra can be made into a total algebra. Algebra universalis. 61(2): 139-150 (2009).
[4] Chajda, I., Lattices and semilattices having an antitone involution in every upper interval. Comment. Math. Univ. Carolin. 44(4): 577-585 (2003).
[5] Chajda, I., Halaš, R., Kuhr, J.: Many-valued quantum algebras. Algebra Universalis 60, 63–90 (2009).
[6] Chajda, I., Halaš, R., Kuhr, J.: Semilattice Structures. Heldermann, Lemgo (2007).
[7] Dvurečenskij, A., and Hyčko, M.: "Hyper effect algebras." Fuzzy Sets and Systems (2017).
[8] Foulis, D. J., Bennett, M. K.: Effect algebras and unsharp quantum logic. Found. Phys. 24, 1325–1346 (1994).
[9] Jezek, J., Quackenbush, R.: Directoids: algebraic models of up-directed sets. Algebra Universalis 27, 49–69 (1990).
[10] Kopka, F., Chovanec, F.: D-posets. Math. Slovaca 44, 21–34 (1994).
[11] Kühr, J., Chajda, I., and Halaš, R.: "The join of the variety of MV-algebras and the variety of orthomodular lattices." International Journal of Theoretical Physics 54. 12: 2244 -2244(2015).
[12] Searle, SR., Khuri, AI.: Matrix algebra useful for statistics. John Wiley & Sons; (2017).
[13] Stefan, F., Ronco, M., and Showers, P.: "Polytopes and algebras of grafted trees: Stellohedra." arXiv preprint arXiv: 1608.08546 (2016).
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    Ahmed Allam, Nabila Mikhaeel, Huda Merdach. (2017). D-Effect Algebra can Be Made into a D-Total Algebra. Mathematics Letters, 3(6), 71-76. https://doi.org/10.11648/j.ml.20170306.13

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    ACS Style

    Ahmed Allam; Nabila Mikhaeel; Huda Merdach. D-Effect Algebra can Be Made into a D-Total Algebra. Math. Lett. 2017, 3(6), 71-76. doi: 10.11648/j.ml.20170306.13

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    AMA Style

    Ahmed Allam, Nabila Mikhaeel, Huda Merdach. D-Effect Algebra can Be Made into a D-Total Algebra. Math Lett. 2017;3(6):71-76. doi: 10.11648/j.ml.20170306.13

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  • @article{10.11648/j.ml.20170306.13,
      author = {Ahmed Allam and Nabila Mikhaeel and Huda Merdach},
      title = {D-Effect Algebra can Be Made into a D-Total Algebra},
      journal = {Mathematics Letters},
      volume = {3},
      number = {6},
      pages = {71-76},
      doi = {10.11648/j.ml.20170306.13},
      url = {https://doi.org/10.11648/j.ml.20170306.13},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ml.20170306.13},
      abstract = {In this paper we prove that every D-effect algebra (E, ∆, 0, 1) can be made into a D-total algebra (E, ⍍, ¬, 1) in such a way that two elements are compatible in (E, ∆, 0, 1) if and only if they commute in(E, ⍍, ¬, 1) where x ∆ y =(x' + y')'.},
     year = {2017}
    }
    

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    T1  - D-Effect Algebra can Be Made into a D-Total Algebra
    AU  - Ahmed Allam
    AU  - Nabila Mikhaeel
    AU  - Huda Merdach
    Y1  - 2017/11/28
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    N1  - https://doi.org/10.11648/j.ml.20170306.13
    DO  - 10.11648/j.ml.20170306.13
    T2  - Mathematics Letters
    JF  - Mathematics Letters
    JO  - Mathematics Letters
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    PB  - Science Publishing Group
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    UR  - https://doi.org/10.11648/j.ml.20170306.13
    AB  - In this paper we prove that every D-effect algebra (E, ∆, 0, 1) can be made into a D-total algebra (E, ⍍, ¬, 1) in such a way that two elements are compatible in (E, ∆, 0, 1) if and only if they commute in(E, ⍍, ¬, 1) where x ∆ y =(x' + y')'.
    VL  - 3
    IS  - 6
    ER  - 

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Author Information
  • Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt

  • Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt

  • Department of Mathematics, Faculty of Science, Damietta University, Damietta, Egypt

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