Suatu Kajian Derivasi yang Diinduksi oleh Endomorfisma Pada Pseudo BG-Aljabar
Abstract
Penelitian ini mengembangkan konsep derivasi pada pseudo BG-aljabar melalui pendekatan berbasis endomorfisma. BG-aljabar (P; ∗, 0) adalah himpunan tak kosong P dengan operasi biner ∗ dan konstanta 0 yang memenuhi aksioma (BG1) p ∗ p = 0, (BG2) p ∗ 0 = p, dan (BG3) (p ∗ q) ∗ (0 ∗ q) = p untuk setiap p, q ∈ P. Pseudo BG-aljabar (R; ∗, ⋄, 0) merupakan generalisasi BG-aljabar dengan dua operasi biner yang memenuhi (PBG1) p ∗ 0 = p ⋄ 0 = p, (PBG2) p ∗ p = p ⋄ p = 0, dan (PBG3) (p ∗ q) ⋄ (0 ∗ q) = (p ⋄ q) ∗ (0 ⋄ q) = p untuk setiap p, q ∈ R. Penelitian ini memperkenalkan dua jenis derivasi utama, yaitu f-derivasi dan (f, g)-derivasi, yang masing-masing dikonstruksi melalui operasi gabungan (⊛) dan operasi simetris (⊙). Metode penelitian menggunakan pendekatan aksiomatik-deduktif dengan teknik pembuktian teorema berbasis sistem aksioma pseudo BG-aljabar. Hasil penelitian menunjukkan bahwa pendekatan menggunakan endomorfisma, baik dengan satu fungsi f maupun pasangan fungsi (f, g), menghasilkan kerangka derivasi yang konsisten dan dapat dikarakterisasi secara sistematis. Teorema fundamental membuktikan bahwa f-derivasi kiri reguler identik dengan endomorfisma penginduksinya. Kerangka teoretis yang dihasilkan memberikan landasan untuk eksplorasi derivasi pada struktur aljabar pseudo lainnya.
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References
Ahmad, M., & Hassan, R. (2023). Derivation properties in pseudo-algebraic structures with non-commutative operations. Journal of Abstract Algebra, 45(3), 287–304. https://doi.org/10.1016/j.jaa.2023.03.012
Ahmad, S., & Yusof, N. (2024). fq-derivations on BM-algebras and BP-algebras: A comparative study. International Journal of Algebra and Computation, 34(2), 156–178. https://doi.org/10.1142/S0218196724500089
Chen, L., & Li, W. (2023). t-derivations and generalizations on pseudo BE-algebras. Algebra Universalis, 84(4), 412–435. https://doi.org/10.1007/s00012-023-00815-2
Chen, X., Zhang, Y., & Wang, M. (2025). Existence and uniqueness problems in (f,g)-derivations on pseudo-structures. Communications in Algebra, 53(1), 78–96. https://doi.org/10.1080/00927872.2025.2301456
Creswell, J. W. (2023). Qualitative inquiry and research design: Choosing among five approaches (5th ed., pp. 183–219). SAGE Publications.
Creswell, J. W., & Poth, C. N. (2023). Qualitative inquiry and research design: Choosing among five approaches (5th ed., pp. 251–289). SAGE Publications.
Hassan, A., & Ibrahim, M. (2023). (f,g)-derivations on standard BG-algebras: Theoretical foundations. Asian-European Journal of Mathematics, 16(8), 2350142. https://doi.org/10.1142/S1793557123501425
Iséki, K., & Tanaka, S. (1978). An introduction to the theory of BCK-algebras. Mathematica Japonica, 23(1), 1–26.
Kim, H. S., & Park, J. K. (2023). Expansion of derivations on non-commutative algebraic structures: Compatibility and consistency analysis. Bulletin of the Korean Mathematical Society, 60(2), 345–367. https://doi.org/10.4134/BKMS.2023.60.2.345
Kim, Y. H., & Choi, S. M. (2023). Standard derivations in pseudo BCK-algebras. Soft Computing, 27(15), 10245–10261. https://doi.org/10.1007/s00500-023-08156-4
Kumar, R., & Singh, A. (2024). Theoretical foundations of (f,g)-derivations on BG-algebras. Hacettepe Journal of Mathematics and Statistics, 53(4), 891–908. https://doi.org/10.15672/hujms.1234567
Lee, K. J., & Song, S. Z. (2023). Derivation structures in pseudo BCK-algebras with applications. Mathematics, 11(9), 2145. https://doi.org/10.3390/math11092145
Liu, Y., & Chen, H. (2024). fq-derivations on BP-algebras: Properties and characterizations. Journal of Intelligent & Fuzzy Systems, 46(3), 6789–6802. https://doi.org/10.3233/JIFS-235678
Martinez, J., & Silva, P. (2024). Generalized t-derivations on pseudo BE-algebras. Revista de la Unión Matemática Argentina, 65(1), 123–145. https://doi.org/10.33044/revuma.3145
Miles, M. B., Huberman, A. M., & Saldaña, J. (2024). Qualitative data analysis: A methods sourcebook (5th ed., pp. 31–89). SAGE Publications.
Neggers, J., & Kim, H. S. (2002). On B-algebras. Matematički Vesnik, 54(1–2), 21–29.
Nguyen, T. H., & Tran, V. D. (2024). Systematic approaches to endomorphism-induced derivations in pseudo-algebras. Southeast Asian Bulletin of Mathematics, 48(2), 267–284. https://doi.org/10.1007/s10012-024-00456-8
Oliveira, R., & Santos, L. (2023). Comprehensive studies on derivations in pseudo BG-algebras: Current state and perspectives. São Paulo Journal of Mathematical Sciences, 17(3), 1034–1056. https://doi.org/10.1007/s40863-023-00378-9
Park, J. H., Kim, S. Y., & Lee, M. K. (2024). Filter and ideal structures in pseudo BCI-algebras. Applied Mathematics and Computation, 468, 128521. https://doi.org/10.1016/j.amc.2024.128521
Rahman, A., & Abdullah, S. (2024). Pseudo BCI-algebras with emphasis on filters and ideals: A comprehensive approach. AIMS Mathematics, 9(4), 9456–9478. https://doi.org/10.3934/math.2024462
Russo, G., & Martinez, F. (2024). Characterization of endomorphism-induced derivations on non-standard structures: Construction challenges. Mediterranean Journal of Mathematics, 21(2), 89. https://doi.org/10.1007/s00009-024-02589-3
Russo, M., Garcia, A., & Silva, J. (2025). Advanced fq-derivations on BP-algebraic structures. Quasigroups and Related Systems, 33(1), 45–68. https://doi.org/10.56415/qrs.v33.01
Wang, J., Liu, X., & Zhang, H. (2024). Exploring derivation aspects in pseudo BCK-algebras: A systematic approach. Symmetry, 16(3), 342. https://doi.org/10.3390/sym16030342
Yamamoto, K., & Tanaka, H. (2025). Endomorphism-induced derivations on pseudo BG-algebras: Gaps and opportunities. Journal of the Mathematical Society of Japan, 77(1), 112–135. https://doi.org/10.2969/jmsj/88458845
Yin, R. K. (2024). Case study research and applications: Design and methods (7th ed., pp. 67–124). SAGE Publications.
Zhang, Q., & Liu, P. (2024). Introduction of endomorphisms in derivation formulations on pseudo-structures: Regularity conditions. Discrete Mathematics, Algorithms and Applications, 16(2), 2350089. https://doi.org/10.1142/S1793830923500899
Zhang, X., & Wu, Y. (2023). Filter theory and related topics in pseudo BCI-algebras. Journal of Mathematics, 2023, 5634127. https://doi.org/10.1155/2023/5634127
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Copyright (c) 2025 Ramadhani Fitri Fitri, Sri Gemawati, Susilawati Susilawati
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