(f, g)-DERIVATIONS IN RESIDUATED LATTICE

Document Type : Original Manuscript

Authors

1 National school of Applied Sciences ENSAK, Ibn Tofail University, BP 241, 14000 KenitraMorocco

2 National school of Applied Sciences ENSAK, Ibn Tofail University, BP 241, 14000 KenitraMorocco.

3 Superior School of Technology, Ibn Tofail University, BP 241, 14000 Kenitra-Morocco.

Abstract

In this paper, we present and examine the characteristics of (f, g)-derivations for a residuated lattice. Some relationships between (f, g)-derivationand isotone, contractive and ideal (f, g)-derivations are given. The set of fixedpoint of an (f, g)-derivation is introduced and its structure is studied. More precisely, we show that the set of fixed points is also a residuated lattice.

Keywords


 1. A. Ali and M. H. Rahaman, On pair of generalized derivations in rings, Khayyam J. Math., 6(1) (2020), 87–94.
2. Y. Çeven and M. A. Özturk, On f-derivations of lattices, Bull. Korean Math. Soc., 45(4) (2008), 701–707.
3. C. C. Chang, Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490.
4. D. Chaudhuri, (σ, τ)-derivations of group rings, Comm. Algebra, 47(9) (2019), 3800–3807.
5. L. C. Ciungu, Pseudo BCI-algebras with derivations, (2019), arXiv: 1902.09895.
6. K. K. Dey, S. K. Saha and A. C. Paul, On orthogonal generalized derivations of semiprime-rings, GANIT: J. Bangladesh Math. Soc., 39 (2019), 63–70.
7. F. Esteva and L. Godo, Monoidal t-norm-based logic: towards a logic forleftcontinuous t-norm, Fuzzy Sets Syst., 124 (2001), 271–288.
8. E. Guven, One sided generalized (σ, τ)-derivations on rings, Bol. Soc. Paran. Mat., 38(2) (2020), 41–50.
9. P. He, X. Xin and J. Zhan, On derivations and their fixed point sets in residuated lattices, Fuzzy Sets Syst., 303 (2016), 97–113.
10. A. Hosseini and A. Fosner, The image of Jordan left derivations on algebras, Bol. Soc. Paran. Mat., 38(6) (2020), 53–61.
11. Y. B. Jun and X. L. Xin, On derivations of BCI-algebras, Inform. Sci., 159 (2004), 167–176.
12. T. Kowalski and H. Ono, Residuated Lattices: An Algebraic Glimpse at Logic Without Contraction, 2001.
13. J. Liang, X. L. Xin and J. T. Wang, On derivations of EQ-algebras, Journal of Intelligent and Fuzzy Systems, 35 (2018), 5573–5583.
14. D. Ling and K. Zhu, On a new class of derivations on residuated lattices, Italian Journal of Pure and Applied Mathematics, 45 (2021), 843–857.
15. E. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093– 1100.
16. J. Rachunek and D. Salounova, Derivations on algebras of a non-commutative generalization of the Lukasiewicz logic, Fuzzy Sets Syst., 333 (2018), 11–16.
17. M. K. Rasheed and A. H. Majeed, Some results of (α, β)-derivations on prime semirings, Iraqi J. Sci., 60(5) (2019), 1154–1160.
18. E. Turumen, Mathematics behind fuzzy logic, Physica-Verlag, 1999.
19. X. L. Xin, T. Y. Li and J. H. Lu, On derivations of lattices, Inform. Sci., 1778 (2008), 307–316.
20. J. T. Wang, A. B. Saeid and M. Wang, On derivations of commutative multiplicative semilattices, Journal of Intelligent and Fuzzy Systems, 35 (2018), 957– 966.
21. J. T. Wang, Y. H. She and T. Qian, Study of MV-algebras via derivations, An. St. Univ. Ovidius Constantta, 27(3) (2019), 259–278. 
22. P. Ward and R. M. Dilworth, Residuatd lattice, Trans. Amer. Math. Soc., 45 (1939), 335–354.
23. J. Zhan and Y. L. Liu, On f-derivations of BCI-algebras, Int. J. Math. Math. Sci., 25 (2005), 1675–1684.
24. K. Zhu, J. Wang and Y. Yang, On Generalized Derivations in Residuated Lattices, IAENG International Journal of Applied Mathematics, 50(2) (2020).