Konstruksi dan Analisis r-Ideal di BG-aljabar

Meivy Andhika Beauty; Sri Gemawati; Leli Deswita

doi:10.37905/jjom.v7i1.30097

Konstruksi dan Analisis r-Ideal di BG-aljabar

Meivy Andhika Beauty, Sri Gemawati, Leli Deswita

Abstract

A non-empty set G with a binary operation ∗ and a constant 0 that satisfies the following axioms: , , and for all is called a BG-algebra. A non-empty subset I of G is said to be an ideal in G if it satisfies: (i) and (ii) and implies for all . This article introduces the new concept of r-ideal in BG-algebra, which is an extension of the ideal in BN-algebra. Unlike the definition of an ideal in BN-algebra, an r-ideal only requires a non-empty subset I of G without the need to satisfy the full ideal conditions. This study examines the properties of r-ideals and their relationships with subalgebras, normal, and ideals in BG-algebra. In the final part, it is concluded that every subalgebra is an r-ideal in BG-algebra, and every normal ideal is also an r-ideal.

Keywords

BG-algebra; r-ideal BG-algebra; normal; subalgebra

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References

C. B. Kim and H. S. Kim, “On BG-algebras,” Demonstratio Mathematica, vol. 41, no. 3, pp. 497–505, 2008.

Kamaludin, S. Gemawati, and Kartini, “Derivations in BG-algebras,” International Journal of Algebra, vol. 13, no. 5, pp. 249–257, 2019, doi: 10.12988/ija.2019.9620.

S. Widianto, S. Gemawati, and Kartini, “Direct product in BG-algebras,” International Journal of Algebra, vol. 13, no. 5, pp. 239–247, 2019, doi: 10.12988/ija.2019.9619.

D. K. Basnet and L. B. Singh, “(∈, ∈∨q)-Fuzzy Ideals of BG-Algebra,” 2011.

E. Yattaqi, S. Gemawati, and I. Hasbiyati, “fq-derivasi di BM-aljabar,” Jambura Journal of Mathematics, vol. 3, no. 2, pp. 155–166, Jun. 2021, doi: 10.34312/jjom.v3i2.10379.

S. Adnan Bajalan, R. Raheem Mohammed Amin, and A. K. Bajalan, “Structures of Pseudo-BG Algebra and Sime pseudo-BG-Algebra.” [Online]. Available: http://tjps.tu.edu.iq/index.php/j

B. Elavarasan, G. Muhiuddin, K. Porselvi, and Y. B. Jun, “Hybrid structures applied to ideals in near-rings,” Complex and Intelligent Systems, vol. 7, no. 3, pp. 1489–1498, Jun. 2021, doi: 10.1007/s40747-021-00271-7.

C. M. Tiwari and S. Swati, “AN INTRODUCTION OF GRAPH THEORY IN APPLIED MATHEMATICS,” 2023. [Online]. Available: https://www.researchgate.net/publication/371138959

R. Muthuraj and S. Devi, “Multi-Fuzzy BG-ideals in BG-algebra,” Annals of Pure and Applied Mathematics, vol. 15, no. 2, pp. 193–200, Dec. 2017, doi: 10.22457/apam.v15n2a5.

S. Gemawati, Mashadi, Kartini, Musraini, and E. Fitria, “Sheffer Stroke BN-algebras and Connected Topics,” European Journal of Pure and Applied Mathematics, vol. 18, no. 1, p. 5783, Jan. 2025, doi: 10.29020/nybg.ejpam.v18i1.5783.

X. Zhang, R. A. Borzooei, and Y. B. Jun, “Q-filters of quantum B-algebras and basic implication algebras,” Symmetry (Basel), vol. 10, no. 11, Nov. 2018, doi: 10.3390/sym10110573.

H. H. Abbass and A. A. Hamza, “On U-BG-filter of a U-BG-BH-algebra,” Applied Mathematical Sciences, vol. 11, pp. 1297–1305, 2017, doi: 10.12988/ams.2017.73114.

Q. Mohsin Luhaib and H. Hadi Abbass, “On a Smarandache Closed and Completely Filter of a Smarandache BH-Algebra,” in IOP Conference Series: Materials Science and Engineering, IOP Publishing Ltd, Nov. 2020. doi: 10.1088/1757-899X/928/4/042017.

C. B. Kim and H. S. Kim, “On BN-algebras,” Kyungpook Mathematical Journal, vol. 53, no. 2, pp. 175–184, 2013, doi: 10.5666/KMJ.2013.53.2.175.

E. Fitria, S. Gemawati, and Kartini, “Prime ideals in B-algebras,” International Journal of Algebra, vol. 11, pp. 301–309, 2017, doi: 10.12988/ija.2017.7838.

G. Dymek and A. Walendziak, “(Fuzzy) Ideals of BN-Algebras,” Scientific World Journal, vol. 2015, 2015, doi: 10.1155/2015/925040.

G. #1, E. Fitria, A. Hadi, and M. M. #4, “Complete Ideal and n-Ideal of BN-algebras,” International Journal of Mathematics Trends and Technology, vol. 66, pp. 52–59, 2020, doi: 10.14445/22315373/IJMTT-V66I11P503.

S. Gemawati, M. M, A. Putri, R. Marjulisa, and E. Fitria, “T-IDEAL AND α-IDEAL OF BP-ALGEBRAS,” BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 18, no. 2, pp. 1129–1134, May 2024, doi: 10.30598/barekengvol18iss2pp1129-1134.

M. Murali Krishna Rao, “r-Ideals and m-k-ideals in inclines,” Discussiones Mathematicae - General Algebra and Applications, vol. 40, no. 2, pp. 297–309, Dec. 2020, doi: 10.7151/dmgaa.1340.

S. Gemawati, M. Musraini, A. Hadi, L. Zakaria, and E. Fitria, “On r-Ideals and m-k-Ideals in BN-Algebras,” Axioms, vol. 11, no. 6, Jun. 2022, doi: 10.3390/axioms11060268.

DOI: https://doi.org/10.37905/jjom.v7i1.30097