DM-GAA

ISSN 1509-9415 (print version)

ISSN 2084-0373 (electronic version)

https://doi.org/10.7151/dmgaa

Discussiones Mathematicae - General Algebra and Applications

Cite Score: 0.4

SJR: 0.203

SNIP: 0.562

MCQ: 0.12

Index Copernicus: 121.02

Discussiones Mathematicae - General Algebra and Applications

Article in volume

Authors:

Dr. Kumduang

Thodsaporn Kumduang

Department of Mathematics, Faculty of Science and Technology
Rajamangala University of Technology Rattanakosin
Nakhon Pathom 73170, Thailand

email: kumduang01@gmail.com

R. Chinram

Ronnason Chinram

Division of Computational Science
Faculty of Science, Prince of Songkla University
Hat Yai, Songkhla 90110, Thailand

email: ronnason.c@psu.ac.th

Title:

Fuzzy ideals and fuzzy congruences on menger algebras with their homomorphism properties

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Source:

Discussiones Mathematicae - General Algebra and Applications 43(2) (2023) 189-206

Received: 2020-12-16 , Revised: 2021-11-28 , Accepted: 2021-11-28 , Available online: 2023-01-11 , https://doi.org/10.7151/dmgaa.1418

Abstract:

It is well known that Menger algebras, sometime called superassociative algebras, play a major role in both mathematical sciences and related areas. The notion of fuzzy sets was initiated by L.A. Zadeh as a general mathematical machinery of classical sets. The present paper establishes a strong interaction between fuzzy sets and Menger algebras. We show that the set of all fuzzy subsets on $G$ together with one $(n+1)$-ary operations forms a Menger algebra. The concept of several kinds of fuzzy ideals in Menger algebras is introduced and some related properties are investigated. Furthermore, we provide a construction of quotient Menger algebras via fuzzy congruence relations. Finally, homomorphism theorems in terms of fuzzy congruences are studied. Our results can be considered as a generalization in the study of semigroup theory too.

Keywords:

Menger algebra, fuzzy ideal, fuzzy congruence relation, quotient Menger algebra

References:

  1. S. S. Ahn, C. Kim, Rough set theory applied to fuzzy filters in $BE$-algebras, Commun. Korean Math. Soc. 31 (2016), No. 3, 451-460. https://doi.org/10.4134/CKMS.c150168
  2. M. Demirci, J. Recasens, Fuzzy groups, fuzzy functions and fuzzy equivalence relations, Fuzzy Sets Syst. 144 (2004), 441–458. https://doi.org/10.1016/S0165-0114(03)00301-4
  3. B. Davvaz, W. A. Dudek, Fuzzy $n$-ary groups as a generalization of Rosenfeld’s fuzzy groups, J. Mult-Valued Log. S. 15 (2009), 451-469.
  4. W. A. Dudek, Fuzzication of n-ary groupoids, Quasigroups Relat. Syst. 7 (2000), 45-66.
  5. W. A. Dudek, V. S. Trokhimenko, Algebras of Multiplace Functions, De Gruyter, Berlin, 2012.
  6. W. A. Dudek, V. S. Trokhimenko, De Morgan $(2, n)$-semigroups of $n$-place functions. Commun. Algebra 44 (2016), 4430–4437. https://doi.org/10.1080/00927872.2015.1094486
  7. W. A. Dudek, V. S. Trokhimenko, Menger algebras of idempotent $n$-ary operations, Stud. Sci. Math. Hung. 55 (2019), no. 2, 260-269. https://doi.org/10.1556/012.2018.55.2.1396
  8. Y. B. Jun, Fuzzy BCK/BCI-algebras with operators, Sci. Math. 3 (2000), No. 2, 283-287.
  9. S. Kar, P. Sarkar, Fuzzy Ideals of Ternary Semigroups, Fuzzy Inf. Eng. 2 (2012), 181-193. https://doi.org/10.1007/s12543-012-0110-4
  10. J. P. Kim, D. R. Bae, Fuzzy congruences in groups, Fuzzy Sets Syst. 85 (1997), 115–120.
  11. H. S. Kim, J. Neggers, S. S. Ahn, Almost $\varphi$-fuzzy semi-ideals in groupoids, J. Intell. Fuzzy Syst. 36 (2019), 5361–5368. https://doi.org/10.3233/JIFS-181220
  12. M. Kondo, Fuzzy congruences on groups, Quasigroups Relat. Syst. 11 (2004), 59-70.
  13. S. Kuka, K. Hila, K. Naka, Application of L-fuzzy sets in $m$-ary semigroups, J. Intell. Fuzzy Syst. 34 (2018), No. 6, 4031-4040. https://doi.org/10.3233/JIFS-171365
  14. T. Kumduang, S. Leeratanavalee, Left translations and isomorphism theorems for Menger algebras of rank $n$, Kyungpook Math. J. 61 (2021), 223-237. https://doi.org/10.5666/KMJ.2021.61.2.223
  15. T. Kumduang, S. Leeratanavalee, Semigroups of terms, tree languages, Menger algebra of $n$-ary functions and their embedding theorems, Symmetry, 13 (2021), No. 4, Article 558. https://doi.org/10.3390/sym13040558
  16. N. Kuroki, Fuzzy congruences on $T^*$-pure semigroups, Inf. Sci. 84 (1995), 239–24
  17. W. J. Liu, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets Syst. 21 (1982), 133-139.
  18. X. P. Liu, D. J. Xiang, J. M. Zhan, Isomorphism Theorems for soft ring, Algebra Colloq. 19 (2012), 649-656. https://doi.org/10.1142/S100538671200051X
  19. X. P. Liu, D. J. Xiang, J. M. Zhan, Fuzzy isomorphism theorems of soft rings, Neural Comput. Appl. 21 (2012), 391-397. https://doi.org/10.1007/s00521-010-0439-8
  20. K. Menger, Superassociative systems and logical functors, Math. Ann. 157 (1964), 278–295.
  21. V. Murali, Fuzzy congruence relations, Fuzzy Sets Syst. 41 (1991), 359-369.
  22. A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512-517.
  23. J. Somjanta, N. Thuekaew, P. Kumpeangkeaw, A. Iampan, Fuzzy sets in UP-algebras, Ann. Fuzzy Math. Inform. 12 (2016), No. 6, 739-756.
  24. Y. J. Tan, Fuzzy congruences on a regular semigroup, Fuzzy Sets Syst. 117 (2001), 447–453. https://doi.org/10.1016/S0165-0114(98)00275-9
  25. A. Ullah, A. Khan, A. Ahmadian, N. Senu, F. Karaaslan, I. Ahmad, Fuzzy congruences on AG-group, AIMS Mathematics, 6 (2021), No. 2, 1754-1768. https://doi.org/10.3934/math.2021105
  26. K. Wattanatripop, T. Kumduang, T. Changphas, S. Leeratanavalee, Power Menger algebras of terms induced by order-decreasing transformations and superpositions, Int. J. Math. Comput. Sci., 16 (2021), No. 4, 1697-1707.
  27. X. Zhou, D.Xiang, J. Zhan, A novel study on fuzzy congruence on $n$-ary semigroup, U.P.B. Sci. Bull., Series A, 78 (2016), No. 3, 19-30.
  28. X. Zhou, D.Xiang, J. Zhan, Quotient rings via fuzzy congruence relations, Ital. J. Pure Appl. Math. 33 (2014), 411-424.
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