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Galoitica: Journal of Mathematical Structures and Applications

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Galoitica: Journal of Mathematical Structures and Applications
Full Length Article

Volume 10Issue 1PP: 26-30 • 2023

On The Algebraic Properties of l-Congruencies in Groups

Lee Xu 1*
1University of Chinese Academy of Sciences, CAS, Mathematics Department, Beijing, China
* Corresponding Author.
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© 2023 The Author(s). Published by ASPG. This article is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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

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Xu, Lee. "On The Algebraic Properties of l-Congruencies in Groups." Galoitica: Journal of Mathematical Structures and Applications, vol. Volume 10, no. Issue 1, 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, Volume 10(Issue 1), 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 Volume 10, no. Issue 1 (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, Volume 10(Issue 1), pp. 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. 2023;Volume 10(Issue 1):26-30. DOI: https://doi.org/10.54216/GJMSA.0100103
L. Xu, "On The Algebraic Properties of l-Congruencies in Groups," Galoitica: Journal of Mathematical Structures and Applications, vol. Volume 10, no. Issue 1, pp. 26-30, 2023. DOI: https://doi.org/10.54216/GJMSA.0100103
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