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

Ahmed Allam; Nabila Mikhaeel; Huda Merdach

doi:doi:10.11648/j.ml.20170306.13

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

Ahmed Allam, Nabila Mikhaeel, Huda Merdach

Published in Mathematics Letters (Volume 3, Issue 6)

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), 2024. 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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APA Style

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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TY  - JOUR
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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T2  - Mathematics Letters
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JO  - Mathematics Letters
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PB  - Science Publishing Group
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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
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Author Information

Ahmed Allam

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
Nabila Mikhaeel

Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt
Huda Merdach

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

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APA Style

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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TY  - JOUR
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
PY  - 2017
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
SP  - 71
EP  - 76
PB  - Science Publishing Group
SN  - 2575-5056
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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