Fuzzy congruences on AG-group

Aman Ullah; Akram Khan; Ali Ahmadian; Norazak Senu; Faruk Karaaslan; Imtiaz Ahmad; Aman Ullah; Akram Khan; Ali Ahmadian; Norazak Senu; Faruk Karaaslan; Imtiaz Ahmad

doi:10.3934/math.2021105

AIMS Mathematics

2021, Volume 6, Issue 2: 1754-1768. doi: 10.3934/math.2021105

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Research article

Fuzzy congruences on AG-group

1.
Department of Mathematics, University of Malakand, Chakadar, Dir (L), Pakistan
2.
Institute of IR 4.0, The National University of Malaysia, 43600 UKM, Bangi, Selangor, Malaysia
3.
Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang Selangor, Malaysia
4.
Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang Selangor, Malaysia
5.
Department of Mathematics, Faculty of Sciences, Çankırı Karatekin, University, 18100, Çankırı, Turkey

Received: 25 August 2020 Accepted: 16 November 2020 Published: 30 November 2020
MSC : 14A22, 16S38

In this paper, we establish the idea of fuzzy congruences on Abel-Grassmann's group (AG-group). We investigate different outcomes of fuzzy-congruences on AG-groups in detail and give some examples to illustrate the newly established results. We develop the relation between fuzzy congruence and fuzzy normal subgroup. In the end, we also provide some results of fuzzy homomorphism on AG-groups.
- fuzzy equivalence relation,
- compatible,
- congruence relation,
- AG-group,
- fuzzy normal AG-group
Citation: Aman Ullah, Akram Khan, Ali Ahmadian, Norazak Senu, Faruk Karaaslan, Imtiaz Ahmad. Fuzzy congruences on AG-group[J]. AIMS Mathematics, 2021, 6(2): 1754-1768. doi: 10.3934/math.2021105

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Abstract

In this paper, we establish the idea of fuzzy congruences on Abel-Grassmann's group (AG-group). We investigate different outcomes of fuzzy-congruences on AG-groups in detail and give some examples to illustrate the newly established results. We develop the relation between fuzzy congruence and fuzzy normal subgroup. In the end, we also provide some results of fuzzy homomorphism on AG-groups.

References

[1]	L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353. doi: 10.1016/S0019-9958(65)90241-X
[2]	A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517. doi: 10.1016/0022-247X(71)90199-5
[3]	E. Sanchez, Resolution of composite fuzzy relation equations, Information and Control, 30 (1976), 38-48. doi: 10.1016/S0019-9958(76)90446-0
[4]	P. Bhattacharya, N. P. Mukherjee, Fuzzy relations and fuzzy groups, Information and Control, 36 (1985), 267-282.
[5]	N. P. Mukherjee, P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Information and Control, 34 (1984), 225-239.
[6]	W. C. Nemitz, Fuzzy relations and fuzzy functions, Fuzzy Set. Syst., 19 (1986), 177-191. doi: 10.1016/0165-0114(86)90036-9
[7]	H. Fan, J. Feng, M. Meng, B. Wanga, General decmposition of fuzzy relations: semi-tensor product appraoch, Fuzzy Set. Syst., 384 (2020), 75-90. doi: 10.1016/j.fss.2018.12.012
[8]	D. Cheng, J. Feng, H. Lv, Solving fuzzy relational equations via semitensor product, IEEE T. Fuzzy Syst., 20 (2012), 390-396. doi: 10.1109/TFUZZ.2011.2174243
[9]	V. Murali, Fuzzy equivalence relation, Fuzzy Set. Syst., 30 (1989), 155-163. doi: 10.1016/0165-0114(89)90077-8
[10]	N. Kuroki, Fuzzy congruence and fuzzy normal subgroups, Information and Control, 60 (1992), 247-259.
[11]	X. Zhou, D. Xiang, J. Zhan, A noval study fuzzy congruences on n-ray semigroups, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 78 (2016), 19-30.
[12]	S. K. Shoar, Fuzzy normal congruences and fuzzy cosets relations on groups, International Journal of Pure and Applied Mathematics, 115 (2017), 211-224.
[13]	M. Shah, C. Gretton, V. Sorge, Enumerating AG-groups with a study of smarandache AG-groups, International Mathematical Forum, 6 (2011), 3079-3086.
[14]	M. S. Kamran, Conditions for LA-semigroups to resemble associative structures, Ph. D. Thesis, Quaid-i-Azam University, Islamabad, 1993.
[15]	Amanullah, A study of fuzzy AG-subgroups, Ph. D. Thesis, University of Malakand, Chakdara Dir L, Pakistan, 2016.
[16]	I. Ahmad, Amanullah, M. Shah, Fuzzy AG-Subgroups, Life Sci. J., 9 (2012), 3931-3936.
[17]	Amanullah, I. Ahmad, M. Shah, On the equal-height elements of fuzzy AG-subgroups, Life Science Journal, 10 (2013), 3143-3146.
[18]	Amanullah, I. Ahmad, F. Karaaslan, Cubic Abel-Grassmann's subgroups, J. Comput. Theor. Nanos., 13 (2016), 628-635. doi: 10.1166/jctn.2016.4852
[19]	F. Karaaslan, I. Ahmad, Amanullah, Bipolar soft groups, J. Intell. Fuzzy Syst., 31 (2016), 651-662. doi: 10.3233/IFS-162178
[20]	W. Khan, K. Hila, G. Chen, Sandwich sets and congruences in completely inverse AG-groupoid, Italian Journal of Pure and Applied Mathematics*, 39 (2018), 822-838.
[21]	W. Khan, G. Y. Chena, B. Davvaz, Fuzzy congruences on non-associative semigroups, J. Intell. Fuzzy Syst., 35 (2018), 3783-3796. doi: 10.3233/JIFS-18663
[22]	G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, 5th edition, Oxford University Press, New York, 1979.
[23]	F. Guterl, Suddenly, Number theory makes sense to industry, Math Horizons, 2 (1994), 6-8. doi: 10.1080/10724117.1994.11974900
[24]	C. Ding, Chinese remainder theorem: applications in computing, coding, cryptography, World Scientific Publishing Company, 1996.
[25]	S. Y. Yan, Number theory for computing, Springer, 2002.
[26]	M. Schroeder, Number theory in science and communication: with applications in cryptography, physics, digital information, computing, and self-similarity, Springer, 2008.
[27]	M. K. Chakraborty, M. Das, Reduction of fuzzy strict order relations, Fuzzy Set. Syst., 15 (1985), 33-44. doi: 10.1016/0165-0114(85)90014-4
[28]	V. Murali, Fuzzy equivalence relations, Fuzzy Set. Syst., 30 (1989), 155-163. doi: 10.1016/0165-0114(89)90077-8
[29]	M. Samhan, Fuzzy congruence on semigroups, Information Sciences, 74 (1993), 165-175. doi: 10.1016/0020-0255(93)90132-6
[30]	J. M. Howie, An introduction to semigroup theory, Academic Press, London, 1976.
[31]	J. N. Mordeson, D. S. Malik, N. Kuroki, Fuzzy Semigroups, Springer-Verlag Berlin Heidelberg, 2003.

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