Fuzzy Sub-Equality Algebras Based on Fuzzy Points

Rajab Ali Borzooei; Mona Aaly Kologani; Mohammad Mohseni Takallo; Young Bae Jun

doi:10.18778/0138-0680.2023.31

Authors

Rajab Ali Borzooei Shahid Beheshti University https://orcid.org/0000-0001-7538-7885
Mona Aaly Kologani https://orcid.org/0000-0002-5234-2876
Mohammad Mohseni Takallo
Young Bae Jun https://orcid.org/0000-0002-0181-8969

DOI:

https://doi.org/10.18778/0138-0680.2023.31

Keywords:

equality algebra, fuzzy set, fuzzy point, fuzzy ideal, sub-equality algebras, \((\in, \in)\)-fuzzy sub-equality algebras, \((\in, \in\! \vee \, {q})\)-fuzzy sub-equality algebras, \((q, \in\! \vee \, {q})\)-fuzzy sub-equality algebras

Abstract

In this paper, by using the notion of fuzzy points and equality algebras, the notions of fuzzy point equality algebra, equality-subalgebra, and ideal were established. Some characterizations of fuzzy subalgebras were provided by using such concepts. We defined the concepts of \((\in, \in)\) and \((\in, \in\! \vee \, {q})\)-fuzzy ideals of equality algebras, discussed some properties, and found some equivalent definitions of them. In addition, we investigated the relation between different kinds of \((\alpha,\beta)\)-fuzzy subalgebras and \((\alpha,\beta)\)-fuzzy ideals on equality algebras. Also, by using the notion of \((\in, \in)\)-fuzzy ideal, we defined two equivalence relations on equality algebras and we introduced an order on classes of \(X\), and we proved that the set of all classes of \(X\) by these order is a poset.

References

M. Aaly Kologani, M. Mohseni Takallo, R. Borzooei, Y. Jun, Implicative equality algebras and annihilators in equality algebras, Journal of Discrete Mathematical Sciences and Cryptography, (2021), pp. 1–18, DOI: https://doi.org/10.1080/09720529.2021.1876293.
Google Scholar DOI: https://doi.org/10.1080/09720529.2021.1876293

M. Aaly Kologani, M. Mosheni Takallo, R. A. Borzooei, Y. B. Jun, Right and left mappings in equality algebras, Kragujevac Journal of Mathematics, vol. 46(5) (2022), pp. 815–832, DOI: https://doi.org/10.46793/KgJMat2205.815K.
Google Scholar DOI: https://doi.org/10.46793/KgJMat2205.815K

M. Aaly Kologani, X. Xin, Y. Jun, M. Mohseni Takallo, Positive implicative equality algebras and equality algebras with some types, Journal of Algebraic Hyperstructures and Logical Algebras, vol. 3(2) (2022), pp. 69–86, DOI: https://doi.org/10.52547/HATEF.JAHLA.3.2.6.
Google Scholar DOI: https://doi.org/10.52547/HATEF.JAHLA.3.2.6

P. Allen, H. S. Kim, J. Neggers, Smarandache disjoint in BCK/d-algebras, Scientiae Mathematicae Japonicae, vol. 61(3) (2005), pp. 447–450.
Google Scholar

P. J. Allen, H. S. Kim, J. Neggers, Companion d-algebras, Mathematica Slovaca, vol. 57(2) (2007), pp. 93–106, DOI: https://doi.org/10.2478/s12175-007-0001-z.
Google Scholar DOI: https://doi.org/10.2478/s12175-007-0001-z

P. J. Allen, H.-S. Kim, J. Neggers, Deformations of d/BCK-algebras, Bulletin of the Korean Mathematical Society, vol. 48(2) (2011), pp. 315–324, DOI: https://doi.org/10.4134/BKMS.2011.48.2.315.
Google Scholar DOI: https://doi.org/10.4134/BKMS.2011.48.2.315

R. Borzooei, F. Zebardast, M. Aaly Kologani, Some types of filters in equality algebras, Categories and General Algebraic Structures with Applications, vol. 7 (Special Issue on the Occasion of Banaschewski’s 90th Birthday (II)) (2017), pp. 33–55.
Google Scholar

R. A. Borzooei, M. Mohseni Takallo, M. Aaly Kologani, Y. B. Jun, Quotient Structures in Equality Algebras, Journal of Algebraic Systems, vol. 11(2) (2024), pp. 65–82, DOI: https://doi.org/10.22044/JAS.2022.11919.1608.
Google Scholar

L. C. Ciungu, On pseudo-equality algebras, Archive for Mathematical Logic, vol. 53(5–6) (2014), pp. 561–570, DOI: https://doi.org/10.1007/s00153-014-0380-0.
Google Scholar DOI: https://doi.org/10.1007/s00153-014-0380-0

A. Dvurečenskij, O. Zahiri, Pseudo equality algebras: revision, Soft Computing, vol. 20 (2016), pp. 2091–2101, DOI: https://doi.org/10.1007/s00500-015-1888-x.
Google Scholar DOI: https://doi.org/10.1007/s00500-015-1888-x

B. Ganji Saffar, Fuzzy n-fold obstinate and maximal (pre) filters of EQ-algebras, Journal of Algebraic Hyperstructures and Logical Algebras, vol. 2(1) (2021), pp. 83–98, DOI: https://doi.org/10.52547/HATEF.JAHLA.2.1.6.
Google Scholar DOI: https://doi.org/10.52547/HATEF.JAHLA.2.1.6

S. Jenei, Equality algebras, Studia Logica, vol. 100(6) (2012), pp. 1201–1209, DOI: https://doi.org/10.1007/s11225-012-9457-0.
Google Scholar DOI: https://doi.org/10.1007/s11225-012-9457-0

S. Jenei, L. Kóródi, On the variety of equality algebras, [in:] Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology, Atlantis Press (2011), pp. 153–155, DOI: https://doi.org/10.2991/eusflat.2011.1.
Google Scholar DOI: https://doi.org/10.2991/eusflat.2011.1

Y. B. Jun, On (α, β)-fuzzy subalgebras of BCK/BCI-algebras, Bulletin of the Korean Mathematical Society, vol. 42(4) (2005), pp. 703–711, DOI: https://doi.org/10.4134/BKMS.2005.42.4.703.
Google Scholar DOI: https://doi.org/10.4134/BKMS.2005.42.4.703

V. Novák, B. De Baets, EQ-algebras, Fuzzy sets and systems, vol. 160(20) (2009), pp. 2956–2978, DOI: https://doi.org/10.1016/j.fss.2009.04.010.
Google Scholar DOI: https://doi.org/10.1016/j.fss.2009.04.010

A. Paad, Ideals in bounded equality algebras, Filomat, vol. 33(7) (2019), pp. 2113–2123, DOI: https://doi.org/10.2298/FIL1907113P.
Google Scholar DOI: https://doi.org/10.2298/FIL1907113P

M. M. Takallo, M. A. Kologani, MBJ-neutrosophic filters of equality algebras, Journal of Algebraic Hyperstructures and Logical Algebras, vol. 1(2) (2020), pp. 57–75, DOI: https://doi.org/10.29252/HATEF.JAHLA.1.2.6.
Google Scholar DOI: https://doi.org/10.29252/hatef.jahla.1.2.6

L. A. Zadeh, Fuzzy sets, Information and control, vol. 8(3) (1965), pp. 338–353, DOI: https://doi.org/10.1016/S0019-9958(65)90241-X.
Google Scholar DOI: https://doi.org/10.1016/S0019-9958(65)90241-X

F. Zebardast, R. A. Borzooei, M. A. Kologani, Results on equality algebras, Information Sciences, vol. 381 (2017), pp. 270–282, DOI: https://doi.org/10.1016/j.ins.2016.11.027.
Google Scholar DOI: https://doi.org/10.1016/j.ins.2016.11.027