Galoitica: Journal of Mathematical Structures and Applications

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Volume 10 , Issue 1 , PP: 26-30, 2023 | Cite this article as | XML | Html | PDF | Full Length Article

On The Algebraic Properties of l-Congruencies in Groups

Lee Xu 1 *

  • 1 University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China - (Leexu1244@yahoo.com)
  • Doi: https://doi.org/10.54216/GJMSA.0100103

    Received: July 10, 2023 Revised: September 25, 2023 Accepted: December 21, 2023
    Abstract

    This paper is dedicated to defining and studying the concept of congruencies in l-groups, where we prove the following main results:

    1)       If θ is a congruence relation on l-group G, then  is l-group.

    2)       If  is l-group, then for  holds:

     if and only if  are equivalent.

    Also, we illustrate many examples to clarify the validity of our work.

    Keywords :

    Group , l-Congruence , order relation , Lattice-ordered group.

    References

    [1]    Dedekind, R., (1897). Ojber Zerlegungen Von Zahlen durclihre gr 68sten gemein schaftlichen Teiler, Ges. Werke, Bd 2, XXVIII6.

    [2]    Levi, F; (1913). Arithmetische Gesetze in Gebiete diskerter Gruppen, Rend, Palermo, 35, pp.225-236.

    [3]    Birkhoff, G., (1942), lattice-ordered groups, Annals of Mathematics.

    [4]    Birkhoff, G., (1937), Rings of sets, Duke Mathematical Journal, pp.443-454.

    [5]    Jakubik, J., (2001), On the Congruence Lattice of an Abelian Lahice ordered Group. Mathematica Bohemica.

    [6]    Ploscica, M; (2021). Congruence Lattices of Abelian l-Groups, Vol 33, pp.1651-1658.

    [7]    Waaijers, L; (1968), on the structure of lttice ordered Groups. Doctoral thesis, Waltman-Delft, Holland.

    [8]    Gerberding, J; (2020), Groups of Divisibility, Honors thesis, South Dakota, pp.15-17.

    [9]    Gratzer, G; (2011), Lttice theory Foundation, Springer Science and business Media, p 30.

    [10] Gratzer, G; (2006), the Congruence of a finite lattice, Boston, pp. 14-16.

    Cite This Article As :
    Xu, Lee. On The Algebraic Properties of l-Congruencies in Groups. Galoitica: Journal of Mathematical Structures and Applications, vol. , no. , 2023, pp. 26-30. DOI: https://doi.org/10.54216/GJMSA.0100103
    Xu, L. (2023). On The Algebraic Properties of l-Congruencies in Groups. Galoitica: Journal of Mathematical Structures and Applications, (), 26-30. DOI: https://doi.org/10.54216/GJMSA.0100103
    Xu, Lee. On The Algebraic Properties of l-Congruencies in Groups. Galoitica: Journal of Mathematical Structures and Applications , no. (2023): 26-30. DOI: https://doi.org/10.54216/GJMSA.0100103
    Xu, L. (2023) . On The Algebraic Properties of l-Congruencies in Groups. Galoitica: Journal of Mathematical Structures and Applications , () , 26-30 . DOI: https://doi.org/10.54216/GJMSA.0100103
    Xu L. [2023]. On The Algebraic Properties of l-Congruencies in Groups. Galoitica: Journal of Mathematical Structures and Applications. (): 26-30. DOI: https://doi.org/10.54216/GJMSA.0100103
    Xu, L. "On The Algebraic Properties of l-Congruencies in Groups," Galoitica: Journal of Mathematical Structures and Applications, vol. , no. , pp. 26-30, 2023. DOI: https://doi.org/10.54216/GJMSA.0100103