Journal of Mahani Mathematical Research

Journal of Mahani Mathematical Research

Multiplicative derivations in $\vee$-hoop algebras

Document Type : Research Paper

Authors
Ali Madanshekaf
Mohammad Mahdi Motamedi Nezhad
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran
10.22103/jmmr.2025.24470.1727
Abstract
In this paper, first, while introducing multiplicative derivations, we examine some properties of these derivations and present properties of multiplicative derivations in $\vee$-hoop algebras. Then we show that the set of multiplicative derivations on $\vee$-hoop algebras forms a distributive lattice under certain conditions. Also, while examining the relationship between the square root and the derivation on $\vee$-hoop algebras and introducing the critical point by using the composition of them, we present some characteristics of the critical point. Finally, we show that the set of critical points forms a distributive lattice.
Keywords
Hoop algebra
Derivation
Square root
fixed point
critical point
Subjects
[1] Agliano, P., Ferreirim, I. M. A., & Montagna, F., (2007). Basic hoops: an algebraic study of continuous t-norms, Studia Logica, 87(1), 73-98. https://doi.org/10.1007/s11225-007-9078-1
[2] Alshehri, N. O., (2010). Derivations of MV -algebras, Int. J. Math. Math. Sci., 312027, 7 pages. https://doi:10.1155/2010/312027.
[3] Bell, H. E., & Kappe, L. C., (1989). Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar., 53(3-4), 339-346. https://doi.org/10.1007/bf01953371
[4] Blok, W. J., & Ferririm, I. M. A., (2000). On the structure of hoops, Algebra Universalis, 43(2-3), 233-257. https://doi.org/10.1007/s000120050156
[5] Borzooei, R. A., & Aaly Kologani, M., (2014). Filter theory of hoopalgebras, Journal of advanced research in pure mathematics, 6(4), 72-86. https://doi.org/10.5373/JARPM.1895.120113
[6] Bosbach, B., (1969). Komplementare Halbgruppen. Axiomatik und Arithmetik, Fund. Math. 64, 257-287. http://eudml.org/doc/214075
[7] Bosbach, B., (1970). Komplementare Halbgruppen. Kongruenzen und Quotienten, Fund. Math. 69, 1-14. https://doi.org/10.4064/fm-69-1-1-14
[8] Buchi, J. R., & Owenz, T. M., Complemented monoids and hoops, unpublished manuscript. Not applicable
[9] Davey, B. A., & Priestley, H. A., 2002. Introduction to Lattices and Order, Cambridge University Press, Second edition.
[10] Davvaz, B., Zareyan, L. & Leoreanu-Fotea, V., (2012). (3; 3)-Ary di erential rings, Mediterr. J. Math., 9, 357-378. https://doi.org/10.1007/s00009-011-0124-x
[11] Georgescu, G., Leustean, L. , & Preoteasa, V., (2005). Pseudo-hoops, Mult.-Valued Logic Soft Comput., 11(1), 153-184.
[12] Ghorbani, Sh., Torkzadeh, L., & Motamed, S., (2013). ( ;)-derivations and (    ; )-derivations on MV -algebras, Iran. J. Math. Sci. Inform., 8(1), 75-90. https://doi.org/10.7508/ijmsi.2013.01.008
[13] Hohle, U., (1995). Commutative, residuated `-monoids. In: U. Hohle., E.P. Klement (eds) Non-Classical Logics and their Applications to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory, 32, Springer, Dordrecht, 53-106.
[14] Jun, Y. B., & Xin, X. L., (2004). On derivations of BCI-algebras, Inform. Sci., 159(3-4), 167-176. https://doi.org/10.1016/j.ins.2003.03.001
[15] Keubeng Yemene, D. L., Diekouam Fotso, L. E., Akume, D. & Lele, C., (2020). Some types of derivations in bounded commutative residuated lattices, J. Algebr. Hyperstruct. Log. Algebras., 1(4), 21-37. https://doi.org/10.52547/HATEF.JAHLA.1.4.2
[16] Kondo, M., (2011). Some types of  lters in hoops, 41st IEEE International Symposium on Multiple-Valued Logic, 50-53. https://doi.org/10.1109/ISMVL.2011.9
[17] Madanshekaf, & A., Motamedi Nezhad, M. M., (2024). Application of Hohle's square roots on hoop algebras, Submitted, https://doi.org/10.48550/arXiv.2407.12460.
[18] Namdar, A., & Borumand Saeid, A., (2024). (Generalized) Derivation on hoop algebras, New Math. Nat. Comput., 20(02), 451-471. https://doi.org/10.1142/S1793005724500248
[19] Nasr Esfahani, M. H., Khaksar Haghami, F. & Heidarian, S., Some results on ideals and  -derivations in BL-algebras, Soft Computing Journal, In press, https://doi.org/10.22052/scj.2024.253237.1166.
[20] Piciu, D., 2007. Algebras of Fuzzy Logic, Editura Universtaria, Craiova.
[21] Posner, E. C., (1957). Derivations in prime rings, Proc. Amer. Math. Soc., 8, 1093-1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0
[22] Szasz, G., (1975). Derivations of lattices, Acta Sci., Math. 37, 149-154. Not applicable
[23] Torkzadeh, L., & Abbasian, L., (2013). On ( ; _)-derivation for BL-algebras, J. Hyperstruct., 2(2), 151-162. https://doi.org/10.22098/JHS.2013.2564
[24] Wang, J. T., She, Y. H., & Qian, T., (2019). Study of MV -algebras via derivations, An. St. Univ. Ovidius Constanta, Ser. Mat., 27(3), 259-278. https://doi.org/10.2478/auom-2019-0044
[25] Xin, X. L., Li, T. Y., & Lu, J. H., (2008). On derivations of lattices, Inform. Sci., 178(2), 307-316. https://doi.org/10.1016/j.ins.2007.08.018
[26] Yazarli, H., (2013). A note on derivations in MV -algebras, Miskolc Math Notes., 14(1), 345-354. https://doi.org/10.18514/MMN.2013.420
[27] Zhang, L. J., & Xin, X. L., (2019). Derivations and di erential  lters on semihoops, Ital. J. Pure Appl. Math., 42, 916-933. Not applicable
[28] Zhu, K., Wang, J. & Yang, Y., (2020). On generalized derivations in residuated lattices, Int. J. Appl. Math., 50(2), 1-6. Not applicable 
Journal of Mahani Mathematical Research
Volume 15, Issue 1 - Serial Number 33
January 2026
Pages 139-165

  • Receive Date 04 December 2024
  • Revise Date 07 May 2025
  • Accept Date 05 August 2025