Ordinal sum of equality algebras

Document Type : Research Paper

Authors

1 Faculty of Medicine, Tehran Medical Sciences, Islamic Azad University, Tehran, Iran

2 Hatef Higher Education Institute, Zahedan, Iran

3 Shahid Behesti University

Abstract

In this paper, by using the notion of ordinal sum of equality algebras which is defined in [4], we
investigate some related properties and different kinds of filters of it such as prime and maximal. In
addition, by focusing on chain equality algebra, we define the concept of a cut of an equality algebra
and use it to introduce a new ordinal sum of them. By defining a relation relative to a cut, we show
that the family of classes is made a cone algebra. Finally, we prove that every chain equality algebra
can be represented as an ordinal sum of chain cone algebras.

Keywords

References

[1] M. Aaly Kologani, R. A. Borzooei, G. R. Rezaei, Y. B. Jun, Commutative equality algebras and &-equality algebras,
Annals of the University of Craiova - Mathematics and Computer Science Series, 47(2) (2020), 331-345. https:
//doi.org/10.52846/ami.v47i2.1260
[2] M. Aaly Kologani, M. Mohseni Takallo, R. A. Borzooei, Y. B. Jun, Implicative equality algebras and annihilators
in equality algebras, Journal of Discrete Mathematical Sciences and Cryptography, (2021), 1-18. https://doi.org/
10.1080/09720529.2021.1876293
[3] M. Aaly Kologani, M. Mohseni Takallo, R. A. Borzooei, Y. B. Jun, Right and left mappings in equality algebras,
Kragujevac Journal of Mathematics, 45(5) (2022), 815-832. https://doi.org/10.46793/KgJMat2205.815K
[4] M. Aaly Kologani, S. Niazian, Some different representations of equality algebras, Soft Computing, 28 (2024),
13071-13082. https://doi.org/10.1007/s00500-024-10381-2
[5] R. A. Borzooei, M. Aaly Kologani, G. R. Rezaei, Radical of filters on hoops, Iranian Journal of Fuzzy Systems,
20(7) (2023), 127-143. https://doi.org/10.22111/IJFS.2023.7661
[6] R. A. Borzooei, M. Zarean, O. Zahiri, Involutive equality algebras, Soft Computing, 22 (2018), 7505-7517. https:
//doi.org/10.1007/s00500-018-3032-1
[7] R. A. Borzooei, F. Zebardast, M. Aaly Kologani, Some type of filters in equality algebras, Categories and General
Algebraic Structures with Applications, 7 (2017), 33-55. https://doi.org/10.1137/050641867
[8] B. Bosbach, Komplement¨are halbgruppen. Axiomatik und arithmetik, Fundamenta Mathematicae, 64 (1969), 257-
287.
[9] B. Bosbach, Komplement¨are halbgruppen. Kongruenzen and quotienten, Fundamenta Mathematicae, 69 (1970),
1-14.
[10] S. Burris, H. P. Sankappanavar, A course in universal algebra, Springer Verlag, New York, 1981. https://doi.
org/10.1007/978-1-4614-6946-9
[11] L. C. Ciungu, On pseudo-equality algebras, Archive for Mathematical Logic, 53 (2014), 561-570. https://doi.
org/10.1007/s00153-014-0380-0
[12] L. C. Ciungu, Internal states on equality algebras, Soft Computing, 19(4) (2015), 939-953. https://doi.org/10.
1007/s00500-014-1494-3
[13] A. Dvureˇcenskij, J. K¨uhr, On the structure of linearly ordered pseudo-BCK-algebras, Archive for Mathematical
Logic, 48 (2009), 771-791. https://doi.org/10.1007/s00153-009-0151-5
[14] S. Jenei, Equality algebras, Studia Logica, 100(6) (2012), 1201-1209. https://doi.org/10.1109/CINTI.2010.
5672249
[15] S. Jenei, L. K´or´odi, On the variety of equality algebras, Fuzzy Logic and Technology, (2011), 153-155. https:
//doi.org/10.2991/eusflat.2011.1
[16] M. A. Hashemi, R. A. Borzooei, Tense like equality algebras, Categories and General Algebraic Structures with
Applications (CGASA), 13(1) (2020), 143-166. https://doi.org/10.29252/cgasa.13.1.143
[17] P. H´ajek, Metamathematics of fuzzy logic, In Trends in Logic-Studia Logica Library 4; Kluwer Academic Publishers:
Dordrecht, The Netherlands, 1998.
[18] J. Medina, Characterizing when an ordinal sum of t-norms is a t-norm on bounded lattices, Fuzzy Sets and Systems,
202 (2012), 75-88. https://doi.org/10.1016/j.fss.2012.03.002
[19] S. Niazian, Prime filters and Zariski topology on equality algebras, International Journal of Industrial Mathematics,
15(2) (2023), 137-150. https://doi.org/10.30495/ijim.2023.22548
[20] S. Niazian, M. Aaly Kologani, R. A. Borzooei, On co-annihilators on equality algebras, Miskolc Mathematical
Notes, 24(2) (2023), 933-951. https://doi.org/10.18514/MMN.2023.3816
[21] S. Niazian, M. Aaly Kologani, R. A. Borzooei, Relative co-annihilators in lattice equality algebras, Mathematica
Bohemica, 149(4), (2024), 585-602. https://doi.org/10.21136/MB.2024.0120-23
[22] V. Nov´ak, EQ-algebras: Primary concepts and properties, In Proc. Czech-Japan Seminar, Ninth Meeting. Kitakyushu
& Nagasaki, (2006), 219-223.
[23] V. Nov´ak, B. De Baets, EQ-algebras, Fuzzy Sets and Systems, 160(20) (2009), 2956-2978. https://doi.org/10.
1016/j.fss.2009.04.010
[24] S. M. Paul, L. S. Allen, On the structure of semigroups on a compact manifold with boundary, Annals of Mathematics,
65 (1957), 117-143. https://doi.org/10.2307/1969668
[25] G. R. Rezaei, M. Aaly Kologani, R. A. Borzooei, M. Mohseni Takallo, Derivations of equality algebras, Soft
Computing, 26 (2022), 5057-5067. https://doi.org/10.1007s00500-022-06972-6
[26] S. Saminger, On ordinal sums of triangular norms on bounded lattices, Fuzzy Sets and Systems, 157 (2006),
1403-1416. https://doi.org/10.1016/j.fss.2005.12.021
[27] S. Saminger-Platz, E. P. Klement, R. Mesiar, On extensions of triangular norms on bounded lattices, Indagationes
Mathematicae, 19 (2008), 135-150. https://doi.org/10.1016/S0019-3577(08)80019-5
[28] O. Zahiri, On a new sum of equality algebras, Journal of Algebraic Hyperstructures and Logical Algebras, 5(1)
(2024), 87-94. https://doi.org/10.61838/kman.jahla.5.1.8
[29] F. Zebardast, R. A. Borzooei, M. Aaly Kologani, Results on equality algebras, Information Sciences, 381 (2017),
270-282. https://doi.org/10.1016/j.ins.2016.11.027
Iranian Journal of Fuzzy Systems
Volume 22, Issue 2
March and April 2025
Pages 41-57

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