Power, principal, and isotone $ (\odot, \vee) $-higher derivations on MV-algebras

Xueting Zhao; Yichuan Yang; Xueting Zhao; Yichuan Yang

doi:10.3934/math.20251141

AIMS Mathematics

2025, Volume 10, Issue 11: 25791-25810. doi: 10.3934/math.20251141

Previous Article Next Article

Research article

Power, principal, and isotone $ (\odot, \vee) $-higher derivations on MV-algebras

Xueting Zhao ¹,
Yichuan Yang ^{2
,
,}

1.
College of Science, North China University of Technology, Beijing 100144, China
2.
School of Mathematical Sciences, Shahe Campus, Beihang University, Beijing 102206, China

Received: 23 August 2025 Revised: 02 October 2025 Accepted: 15 October 2025 Published: 10 November 2025
MSC : 03G20, 06B10, 06D35, 08B26

This paper studies three types of $ (\odot, \vee) $-higher derivations on MV-algebras: power, principal, and isotone $ (\odot, \vee) $-higher derivations. We show that every principal $ (\odot, \vee) $-higher derivation is both power and isotone. However, counterexamples demonstrate that no pairwise implications hold beyond the established one. Furthermore, explicit constructions of power $ (\odot, \vee) $-higher derivations are provided. Additionally, we characterize isotone $ (\odot, \vee) $-higher derivations and show that the fixed point set of any principal $ (\odot, \vee) $-higher derivation forms a lattice ideal.
- MV-algebra,
- higher derivation,
- MV-chain,
- semigroup,
- fixed point,
- lattice ideal
Citation: Xueting Zhao, Yichuan Yang. Power, principal, and isotone $ (\odot, \vee) $-higher derivations on MV-algebras[J]. AIMS Mathematics, 2025, 10(11): 25791-25810. doi: 10.3934/math.20251141

Related Papers:

Abstract

This paper studies three types of $ (\odot, \vee) $-higher derivations on MV-algebras: power, principal, and isotone $ (\odot, \vee) $-higher derivations. We show that every principal $ (\odot, \vee) $-higher derivation is both power and isotone. However, counterexamples demonstrate that no pairwise implications hold beyond the established one. Furthermore, explicit constructions of power $ (\odot, \vee) $-higher derivations are provided. Additionally, we characterize isotone $ (\odot, \vee) $-higher derivations and show that the fixed point set of any principal $ (\odot, \vee) $-higher derivation forms a lattice ideal.

References

[1]	N. O. Alshehri, Derivations of MV-algebras, International Journal of Mathematics and Mathematical Sciences, 2010 (2010), 312027. https://doi.org/10.1155/2010/312027
[2]	L. P. Belluce, Semisimple algebras of infinite valued logic and Bold fuzzy set theory, Can. J. Math., 38 (1986), 1356–1379. https://doi.org/10.4153/CJM-1986-069-0 doi: 10.4153/CJM-1986-069-0
[3]	Y. Çeven, On higher derivations of lattices, Mathematical Theory and Modeling, 7 (2017), 116–121.
[4]	C. C. Chang, Algebraic analysis of many-valued logic, Trans. Amer. Math. Soc., 88 (1958), 467–490. https://doi.org/10.1090/S0002-9947-1958-0094302-9 doi: 10.1090/S0002-9947-1958-0094302-9
[5]	R. L. O. Cignoli, I. M. L. D'Ottaviano, D. Mundici, Algebraic foundations of Many-Valued reasoning, Dordrecht: Springer, 2000. https://doi.org/10.1007/978-94-015-9480-6
[6]	L. C. Ciungu, Derivation operators on generalized algebras of BCK logic, Fuzzy Set. Syst., 407 (2021), 175–191. https://doi.org/10.1016/j.fss.2020.04.020 doi: 10.1016/j.fss.2020.04.020
[7]	M. Fernandez-Lebron, L. Narvaez-Macarro, Hasse-Schmidt derivations and coefficient fields in positive characteristics, J. Algebra, 265 (2003), 200–210. https://doi.org/10.1016/S0021-8693(03)00238-2 doi: 10.1016/S0021-8693(03)00238-2
[8]	M. Ferrero, C. Haetinger, Higher derivations of semiprime rings, Commun. Algebra, 30 (2002), 2321–2333. https://doi.org/10.1081/AGB-120003471 doi: 10.1081/AGB-120003471
[9]	F. K. Schmidt, H. Hasse, Noch eine Begr ündung der theorie der höheren differentialquotienten in einem algebraischen funktio nenkörper einer unbestimmten, J. Reine Angew. Math., 177 (1937), 215–237. https://doi.org/10.1515/crll.1937.177.215 doi: 10.1515/crll.1937.177.215
[10]	P. F. He, X. L. Xin, J. M. Zhan, On derivations and their fixed point sets in residuated lattices, Fuzzy Set. Syst., 303 (2016), 97–113. https://doi.org/10.1016/j.fss.2016.01.006 doi: 10.1016/j.fss.2016.01.006
[11]	N. Heerema, Convergent higher derivations on local rings, Trans. Amer. Math. Soc., 132 (1968), 31–44. https://doi.org/10.2307/1994879 doi: 10.2307/1994879
[12]	N. Heerema, Derivations and automorphisms of complete regular local rings, Am. J. Math., 88 (1966), 33–42. https://doi.org/10.2307/2373045 doi: 10.2307/2373045
[13]	J. M. Howie, An introduction to semigroup theory, New York: Academic Press, 1976.
[14]	I. Kaygorodov, M. Khrypchenko, F. Wei, Higher derivations of finitary incidence algebras, Algebr. Represent. Theor., 22 (2019), 1331–1341. https://doi.org/10.1007/s10468-018-9822-4 doi: 10.1007/s10468-018-9822-4
[15]	M. Koppinen, Automorphisms and higher derivations of incidence algebras, J. Algebra, 174 (1995), 698–723. https://doi.org/10.1006/jabr.1995.1147 doi: 10.1006/jabr.1995.1147
[16]	J. Krňávek, J. Kühr, A note on derivations on basic algebras, Soft Comput., 19 (2015), 1765–1771. https://doi.org/10.1007/s00500-014-1586-0 doi: 10.1007/s00500-014-1586-0
[17]	Y. K. Ma, M. M. Raja, V. Vijayakumar, A. Shukla, W. Albalawi, K. S. Nisar, Existence and continuous dependence results for fractional evolution integrodifferential equations of order $r\in (1, 2)$, Alex. Eng. J., 61 (2022), 9929–9939. https://doi.org/10.1016/j.aej.2022.03.010 doi: 10.1016/j.aej.2022.03.010
[18]	M. M. Raja, V. Vijayakumar, K. C. Veluvolu, An analysis on approximate controllability results for impulsive fractional differential equations of order $1 < r < 2$ with infinite delay using sequence method, Math. Method. Appl. Sci., 47 (2024), 336–351. https://doi.org/10.1002/mma.9657 doi: 10.1002/mma.9657
[19]	S. Motamed, S. Ehterami, New types of derivations in BL-algebras, New Math. Nat. Comput., 16 (2020), 627–643. https://doi.org/10.1142/S1793005720500386 doi: 10.1142/S1793005720500386
[20]	E. C. Posner, Derivations in prime rings, Proc. Amer. Math Soc., 8 (1957), 1093–1100. https://doi.org/10.2307/2032686
[21]	J. Rachůnek, D. Šalounová, Derivations on algebras of a non-commutative generalization of the Łukasiewicz logic, Fuzzy Set. Syst., 333 (2018), 11–16. https://doi.org/10.1016/j.fss.2017.01.013 doi: 10.1016/j.fss.2017.01.013
[22]	M. M. Raja, V. Vijayakumar, K. C. Veluvolu, Higher-order caputo fractional integrodifferential inclusions of Volterra–Fredholm type with impulses and infinite delay: existence results, J. Appl. Math. Comput., 71 (2025), 4849–4874. https://doi.org/10.1007/s12190-025-02412-4
[23]	P. Ribenboim, Higher derivations of rings I, Rev. Roum. Math. Pures, 16 (1971), 77–110.
[24]	F. Wei, Z. K. Xiao, Higher derivations of triangular algebras and its generalizations, Linear Algebra Appl., 435 (2011), 1034–1054. https://doi.org/10.1016/j.laa.2011.02.027 doi: 10.1016/j.laa.2011.02.027
[25]	Z. K. Xiao, F. Wei, Jordan higher derivations on triangular algebras, Linear Algebra Appl., 432 (2010), 2615–2622. https://doi.org/10.1016/j.laa.2009.12.006 doi: 10.1016/j.laa.2009.12.006
[26]	X. L. Xin, T. Y. Li, J. H. Lu, On derivations of lattices, Inform. Sciences, 178 (2008), 307–316. https://doi.org/10.1016/j.ins.2007.08.018
[27]	X. L. Xin, The fixed set of a derivation in lattices, Fixed Point Theory Appl., 2012 (2012), 218. https://doi.org/10.1186/1687-1812-2012-218 doi: 10.1186/1687-1812-2012-218
[28]	X. T. Zhao, A. P. Gan, Y. C. Yang, $ (\odot, \vee) $-Derivations on MV-algebras, Soft Comput., 28 (2024), 1833–1849. https://doi.org/10.1007/s00500-023-09384-2. doi: 10.1007/s00500-023-09384-2

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)