On The Algebraic Properties of l-Congruencies in Groups

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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 :

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MLA	Lee Xu. "On The Algebraic Properties of l-Congruencies in Groups." Galoitica: Journal of Mathematical Structures and Applications, Vol. 10, No. 1, 2023 ,PP. 26-30 (Doi : https://doi.org/10.54216/GJMSA.0100103)
APA	Lee Xu. (2023). On The Algebraic Properties of l-Congruencies in Groups. Journal of Galoitica: Journal of Mathematical Structures and Applications, 10 ( 1 ), 26-30 (Doi : https://doi.org/10.54216/GJMSA.0100103)
Chicago	Lee Xu. "On The Algebraic Properties of l-Congruencies in Groups." Journal of Galoitica: Journal of Mathematical Structures and Applications, 10 no. 1 (2023): 26-30 (Doi : https://doi.org/10.54216/GJMSA.0100103)
Harvard	Lee Xu. (2023). On The Algebraic Properties of l-Congruencies in Groups. Journal of Galoitica: Journal of Mathematical Structures and Applications, 10 ( 1 ), 26-30 (Doi : https://doi.org/10.54216/GJMSA.0100103)
Vancouver	Lee Xu. On The Algebraic Properties of l-Congruencies in Groups. Journal of Galoitica: Journal of Mathematical Structures and Applications, (2023); 10 ( 1 ): 26-30 (Doi : https://doi.org/10.54216/GJMSA.0100103)
IEEE	Lee Xu, On The Algebraic Properties of l-Congruencies in Groups, Journal of Galoitica: Journal of Mathematical Structures and Applications, Vol. 10 , No. 1 , (2023) : 26-30 (Doi : https://doi.org/10.54216/GJMSA.0100103)