\(L\)-Modules

Simin Saidi Goraghani; Rajab Ali Borzooei

doi:10.18778/0138-0680.2023.27

Authors

Simin Saidi Goraghani Farhangian University, Department of Mathematics, Tehran, Iran
Rajab Ali Borzooei Shahid Beheshti University, Department of Mathematics, Faculty of Mathematical Sciences, Tehran, Iran

DOI:

https://doi.org/10.18778/0138-0680.2023.27

Keywords:

\(L\)-algebra, \(L\)-module, \(L\)-submodule, prime \(L\)-submodule

Abstract

In this paper, considering \(L\)-algebras, which include a significant number of other algebraic structures, we present a definition of modules on \(L\)-algebras (\(L\)-modules). Then we provide some examples and obtain some results on \(L\)-modules. Also, we present definitions of prime ideals of \(L\)-algebras and \(L\)-submodules (prime \(L\)-submodules) of \(L\)-modules, and investigate the relationship between them. Finally, by proving a number of theorems, we provide some conditions for having prime \(L\)-submodules.

References

P. F. A. Di Nola, I. Leustean, MV-modules, Journal of Algebra, vol. 267 (2003), pp. 21–40, DOI: https://doi.org/10.1016/S0021-8693(03)00332-6
Google Scholar DOI: https://doi.org/10.1016/S0021-8693(03)00332-6

H. A. S. Abojabal, M. Aslam, A. B. Thaheem, On actions of BCK-algebras on groups, Pan American Mathematical Journal, vol. 4 (1994), pp. 727–735.
Google Scholar DOI: https://doi.org/10.21136/CMJ.1995.128559

R. A. Borzooei, S. S. Goraghani, Prime submodules in extended BCK-module, Italian Journal of Pure and Applied Mathematics, vol. 33 (2014), pp. 433–44.
Google Scholar

R. A. Borzooei, S. S. Goraghani, Free MV-modules, Journal of Intelligent and Fuzzy System, vol. 3(1) (2016), pp. 151–161, DOI: https://doi.org/10.3233/IFS-162128
Google Scholar DOI: https://doi.org/10.3233/IFS-162128

L. C. Ciungu, Results in L-algebras, Algebra Universalis, vol. 82(7) (2021), pp. 1–18, DOI: https://doi.org/10.1007/s00012-020-00695-1
Google Scholar DOI: https://doi.org/10.1007/s00012-020-00695-1

P. Dehornoy, Set-theoretic solutions of the Yang-Baxter equation, RC-calculus, and Garside germs, Advances in Mathematics, vol. 282 (2015), pp. 93–127, DOI: https://doi.org/10.1016/j.aim.2015.05.008
Google Scholar DOI: https://doi.org/10.1016/j.aim.2015.05.008

V. G. Drinfeld, On some unsolved problems in quantum group theory, [in:] P. P. Kulish (ed.), Proceedings of Workshops held in the Euler International Mathematical Institute, Leningrad, Fall 1990, vol. 1510 of Lecture Notes in Mathematics, Springer, Berlin–Heidelberg (1992), pp. 1–8, DOI: https://doi.org/10.1007/BFB0101175
Google Scholar DOI: https://doi.org/10.1007/BFb0101175

A. Facchini, M. Gran, M. Pompini, Ideals and congruences in L-algebra and PRE-L-algebras (2023), arXiv:2305.19042 [math.CT].
Google Scholar

S. S. Goraghani, R. A. Borzooei, Most results on A-ideals in MV-moduls, Journal of Algebraic Systems, vol. 5(1) (2017), pp. 1–13, DOI: https://doi.org/10.22044/jas.2017.994
Google Scholar

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.
Google Scholar

S. S. Goraghani, R. A. Borzooei, Decomposition of A-ideals in MV-modules, Annals of the University of Craiova-mathematics and Computer Science Series, vol. 45(1) (2018), pp. 66–77, DOI: https://doi.org/10.52846/ami.v45i1.872
Google Scholar

S. S. Goraghani, R. A. Borzooei, On injective MV-modules, Bulletin of the Section of Logic, vol. 47(4) (2018), pp. 283–298, DOI: https://doi.org/10.18778/0138-0680.47.4.04
Google Scholar DOI: https://doi.org/10.18778/0138-0680.47.4.04

S. S. Goraghani, R. A. Borzooei, S. S. Ann, New definition of MV-semimoduls, Journal of Intelligent and Fuzzy Systems, vol. 41(2) (2021), pp. 3715–3726, DOI: https://doi.org/10.3233/JIFS-211130
Google Scholar DOI: https://doi.org/10.3233/JIFS-211130

A. D. Nola, C. Russo, Semiring and semimodules issues in MV-algebras, Communications in Algebra, vol. 41(3) (2013), pp. 1017–1048, DOI: https://doi.org/10.1080/00927872.2011.610074
Google Scholar DOI: https://doi.org/10.1080/00927872.2011.610074

W. Rump, A decomposition theorem for square-free unitary solutions of the quantum YangBaxter equation, Advances in Mathematics, vol. 193(1) (2005), pp. 40–55, DOI: https://doi.org/10.1016/j.aim.2004.03.019
Google Scholar DOI: https://doi.org/10.1016/j.aim.2004.03.019

W. Rump, L-algebra, self-similarity and l-groups, Journal of Algebra, vol. 320 (2008), pp. 2328–2348, DOI: https://doi.org/10.1016/j.jalgebra.2008.05.033
Google Scholar DOI: https://doi.org/10.1016/j.jalgebra.2008.05.033

W. Rump, A general Glivenko theorem, Algebra Universal, vol. 61 (2009), pp. 455–473, DOI: https://doi.org/10.1007/s00012-009-0018-y
Google Scholar DOI: https://doi.org/10.1007/s00012-009-0018-y

W. Rump, Y. Yang, Intervals in l-groups as L-algebras, Algebra Universal, vol. 6(7) (2012), pp. 121–130, DOI: https://doi.org/10.1007/s00012-012-0172-5
Google Scholar DOI: https://doi.org/10.1007/s00012-012-0172-5

Y. Wu, Y. C. Y. J. Wang, Lattice-ordered effect algebras and L-algebras, Fuzzy Sets and Systems, vol. 369 (2019), pp. 103–113, DOI: https://doi.org/10.1016/j.fss.2018.08.013
Google Scholar DOI: https://doi.org/10.1016/j.fss.2018.08.013

Y. Wu, Y. Yang, Orthomodular lattices as L-algebras, Soft Computing, vol. 24 (2020), pp. 14391–14400, DOI: https://doi.org/10.1007/s00500-020-05242-7
Google Scholar DOI: https://doi.org/10.1007/s00500-020-05242-7

Y. Yang, W. Rump, Pseudo-MV algebras as L-algebras, Journal of Multiple-Valued Logic and Soft Computing, vol. 19 (2012), pp. 621–632.
Google Scholar