Module Structure on Effect Algebras

Simin Saidi Goraghani; Rajab Ali Borzooei

doi:10.18778/0138-0680.2020.17

Authors

Simin Saidi Goraghani Farhangian University Department of Mathematics Tehran, Iran Author https://orcid.org/0000-0002-3271-4812
Rajab Ali Borzooei Shahid Beheshti University Department of Mathematics Tehran, Iran Author https://orcid.org/0000-0001-7538-7885

DOI:

https://doi.org/10.18778/0138-0680.2020.17

Keywords:

effect algebra, product effect algebra, effect module, topology

Abstract

In this paper, by considering the notions of effect algebra and product effect algebra, we define the concept of effect module. Then we investigate some properties of effect modules, and we present some examples on them. Finally, we introduce some topologies on effect modules.

References

[1] M. K. Bennett, D. J. Foulis, Phi-symmetric effect algebras, Foundations of Physics, vol. 25 (1995), pp. 1699–1722, DOI: http://dx.doi.org/10.1007/BF02057883 DOI: https://doi.org/10.1007/BF02057883

[2] R. A. Borzooei, A. Dvurečenskij, A. H. Sharafi, Material Implications in Lattice Effect Algebras, Information Sciences, vol. 433–434 (2018), pp. 233–240, DOI: http://dx.doi.org/10.1016/j.ins.2017.12.049 DOI: https://doi.org/10.1016/j.ins.2017.12.049

[3] R. A. Borzooei, S. S. Goraghani, Free Extended BCK-Module, Iranian Journal of Mathematical Science and Informatics, vol. 10(2) (2015), pp. 29–43, DOI: http://dx.doi.org/10.7508/ijmsi.2015.02.004

[4] R. A. Borzooei, S. S. Goraghani, Free MV-modules, Journal of Intelligent and Fuzzy System, vol. 31(1) (2016), pp. 151–161, DOI: http://dx.doi.org/10.3233/IFS-162128 DOI: https://doi.org/10.3233/IFS-162128

[5] R. A. Borzooei, N. Kohestani, G. R. Rezaei, Metrizability on (Semi)topological BL-algebra, Soft Computing, vol. 16(10) (2012), pp. 1681–1690, DOI: http://dx.doi.org/10.1007/s00500-012-0852-2 DOI: https://doi.org/10.1007/s00500-012-0852-2

[6] R. A. Borzooei, G. R. Rezaei, N. Kohestani, On (semi)topological BL-algebra, Iranian Journal of Mathematical Science and Informatics, vol. 6(1) (2011), pp. 59–77, DOI: http://dx.doi.org/10.7508/ijmsi.2011.01.006

[7] R. A. Borzooei, O. Zahiri, Topology on BL-algebras, Fuzzy Sets and Systems, vol. 289 (2016), pp. 137–150, DOI: http://dx.doi.org/10.1016/j.fss.2014.11.014 DOI: https://doi.org/10.1016/j.fss.2014.11.014

[8] R. Cignoli, I. M. L. D'Ottaviano, D. Mundici, Algebraic Foundations of Many-valued Reasoning, Kluwer Academic-Dordrecht (2000). DOI: https://doi.org/10.1007/978-94-015-9480-6

[9] A. Di Nola, P. Flondor, I. Leustean, MV-modules, Journal of Algebra, vol. 267 (2003), pp. 21–40, DOI: http://dx.doi.org/10.1016/S0021-8693(03)00332-6 DOI: https://doi.org/10.1016/S0021-8693(03)00332-6

[10] A. Di Nola, C. Russo, Semiring and semimodules issues in MV-algebras, Communications in Algebra, vol. 41(3) (2013), pp. 1017–1048, DOI: http://dx.doi.org/10.1080/00927872.2011.610074 DOI: https://doi.org/10.1080/00927872.2011.610074

[11] A. Dvurečenskij, New trends in quantum structures, Kluwer Academic/Ister Science-Dordrecht/Bratislava (2000). DOI: https://doi.org/10.1007/978-94-017-2422-7

[12] A. Dvurečenskij, Product effect algebras, International Journal of Theoretical Physics, vol. 41(10) (2002), pp. 193–215, DOI: http://dx.doi.org/10.1023/A:1021017905403 DOI: https://doi.org/10.1023/A:1021017905403

[13] A. Dvurečenskij, States on Pseudo effect algebras and integrals, Foundations of Physics, vol. 41 (2011), pp. 1143–1162, DOI: http://dx.doi.org/10.1007/s10701-011-9537-4 DOI: https://doi.org/10.1007/s10701-011-9537-4

[14] A. Dvurečenskij, T. Vetterlein, Pseudo effect algebras. II. Group represetation, International Journal of Theoretical Physics, vol. 40 (2001), pp. 703–726, DOI: http://dx.doi.org/10.1023/A:1004144832348 DOI: https://doi.org/10.1023/A:1004144832348

[15] F. Forouzesh, E. Eslami, A. B. Saeid, Spectral topology on MV-modules, New Mathematics and Natural Computation, vol. 11(1) (2015), pp. 13–33, DOI: http://dx.doi.org/10.1142/S1793005715500027 DOI: https://doi.org/10.1142/S1793005715500027

[16] D. J. Foulis, M. K. Bennett, Effect algebras and unsharp quantum logics, Foundations of Physics, vol. 24(10) (1994), pp. 1331–1352, DOI: http://dx.doi.org/10.1007/BF02283036 DOI: https://doi.org/10.1007/BF02283036

[17] S. S. Goraghani, R. A. Borzooei, Results on Prime Ideals in PMV-algebras and MV-modules, Italian Journal of Pure and Applied Mathematics, vol. 37 (2017), pp. 183–196.

[18] M. R. Rakhshani, R. A. Borzooei, G. R. Rezaei, On topological effect algebras, Italian Journal of Pure and Applied Mathematics, vol. 39 (2018), pp. 312–325.