S-spectral topology in MV -algebras

Forouzesh, Fereshteh; Bedrood, Mahta; Lenzi, Giacomo

doi:10.22111/ijfs.2026.52469.9254

Document Type : Research Paper

Authors

¹ Higher Education Complex of Bam

² Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132. 84084 Fisciano, SA, Italy.

https://doi.org/10.22111/ijfs.2026.52469.9254

Abstract

In this paper, we introduce new classes of ideals, called S-prime ideals and S-maximal ideals,
based on a ∧-closed system S. The connection between these ideals and classical prime ideals
is examined, and it is shown that every S-prime ideal constitutes a specific type of prime ideal.
Moreover, it is proven that any proper ideal disjoint from S is contained in an S-prime ideal.
The behavior of S-prime ideals is further analyzed in the setting of quotient MV -algebras, and
their properties under isomorphisms are investigated. In the final part of the study, the complete
∧-closed system S is introduced, and a new topology, called the S-spectral topology, is defined using
S and the family of S-ideals. Several topological properties of this space, including the Hausdorff,
T0, and T1 separation axioms, are discussed.

Keywords

Main Subjects

Fuzzy Math

References

[1] M. Bedrood, F. Sajadian, G. Lenzi, A. Borumand Saeid, Z◦-ideals and Z-ideals in MV-algebras, Bulletin of the
Belgian Mathematical Society-Simon Stevin, 30 (2023), 51-65. https://doi.org/10.36045/j.bbms.211109

[2] M. Bedrood, F. Sajadian, G. Lenzi, A. Borumand Saeid, A generalization of prime ideals in MV-algebras, Journal
of Algebra and Its Applications, 25(05) (2026). https://doi.org/10.1142/S0219498826500180

[3] L. P. Belluce, A. Di Nola, S. Sessa, The prime spectrum of an MV-algebra, Mathematical Logic Quarterly, 40(3)
(1994), 331-346. https://doi.org/10.1002/malq.19940400304

[4] C. C. Chang, Algebraic analysis of many-valued logic, Transactions of the American Mathematical Society, 88(2)
(1958), 467-490. https://doi.org/10.2307/1993227

[5] C. C. Chang, A new proof of the completeness of the Lukasiewicz axioms, Transactions of the American Mathematical
Society, 93(1) (1959), 74-80. https://doi.org/10.1090/S0002-9947-1959-0106489-0

[6] R. Cignoli, I. M. L. D’Ottaviano, D. Mundici, Algebraic foundations of many-valued reasoning, Trends in Logic,
Vol. 7, Kluwer Academic Publishers, Dordrecht, 2000. Reprinted by Springer Science & Business Media, 2013.
https://doi.org/10.1007/978-94-015-9480-6

[7] A. Di Nola, F. Liguori, S. Sessa, Using maximal ideals in the classification of MV-algebras, Portugaliae Mathematica,
50(1) (1993), 87-102.

[8] A. Filipoiu, G. Georgescu, A. Lettieri, Maximal MV-algebras, Mathware Soft Computing, 4(1) (1997), 53-62.

[9] F. Forouzesh, E. Eslami, A. Borumand Saeid, Spectral topology on MV-modules, New Mathematics and Natural
Computation, 11(01) (2015), 13-33. https://doi.org/10.1142/S1793005715500027

[10] F. Forouzesh, F. Sajadian, M. Bedrood, Inverse topology in MV-algebras, Mathematica Bohemica, 144(3) (2019),
273-285. https://doi.org/10.21136/MB.2018.0117-17

[11] F. Forouzesh, F. Sajadian, M. Bedrood, Some results in types of extensions of MV-algebras, Publications de
l’Institut Math´ematique, 105(119) (2019), 161-177. https://doi.org/10.2298/PIM1919161F

[12] C. S. Hoo, MV-algebras, ideals and semisimplicity, Mathematica Japonica, 34(4) (1989), 563-583.

[13] C. S. Hoo, Molecules and linearly ordered ideals of MV-algebras, Publicacions Matem´atiques, 41(2) (1997), 455-465.
https://www.jstor.org/stable/43737254

[14] A. Iorgulescu, J. B. Iorgulescu, Algebras of logic as BCK-algebras, Bucharest University of Economics, Bucharest,
Romania, 2008.

[15] M. Kolaˇr´ık, Independence of the axiomatic system for MV-algebras, Mathematica Slovaca, 63(1) (2013), 1-4.
https://doi.org/10.2478/s12175-012-0076-z

[16] A. H. Movahed, M. Bedrood, A. Borumand Saeid, Extensions of Z-ideals in MV-algebras, Iranian Journal of Fuzzy
Systems, 21(4) (2024), 23-34. https://doi.org/10.22111/ijfs.2024.48312.8498

[17] A. H. Movahed, M. Bedrood, A. Borumand Saeid, On 2-absorbing ideals in MV-algebras, Filomat, 39(9) (2025),
2941-2951. https://doi.org/10.2298/FIL2509941M

[18] A. H. Movahed, M. Bedrood, A. Borumand Saeid, Study of MV-algebras in view of R-ideals and 2R-ideals, Fuzzy
Sets and Systems, 508 (2025), 109313. https://doi.org/10.1016/j.fss.2025.109313

[19] D. Mundici, Advanced Lukasiewicz calculus and MV-algebras, Trends in Logic, Vol. 35, Springer, Berlin, 2011.
https://doi.org/10.1007/978-94-007-0840-2

[20] J. R. Munkres, Topology, Dorling Kindersley, India, 2000.

[21] D. Piciu, Algebras of fuzzy logic, Ed. Universitaria Craiova, 2007.

S-spectral topology in MV -algebras

References

References

Volume 23, Issue 2March and April 2026Pages 65-80

Volume 23, Issue 2
March and April 2026
Pages 65-80