Abstract
The novelty of this paper is to construct several explicit formulas for the number of distinct fuzzy matrices of a finite order which leads us to invent new integer sequences and helps to develop fuzzy subgroups of some finite groups of matrices. In order to achieve the sequences, we analyze the behavioral study of a natural equivalence relation on the set of all fuzzy matrices of a given order. In addition, this paper derives some important relevant results by enumerating non-equivalent class of fuzzy matrices. We achieve these results by incorporating the notion of \(k\)-level fuzzy matrices, \(\alpha\)-cuts and chains.
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Abdulhakeem O, Babangida I (2018) On the number of distinct fuzzy subgroups of dihedral group of order 60. J Qual Measur Anal 14:67–79
Abdulhakeem O, Garba S (2018) Counting distinct fuzzy subgroups of dihedral groups of order 2pnq where p and q are distinct primes. Dutse J Pure Appl Sci 4:444–453
Bellman R, Zadeh LA (1970) Decision making in a fuzzy environment. Manage Sci 17:141–164
Bezdek JC, Harris JD (1978) Fuzzy Partition relation: an axiomatic basis for clustering. Fuzzy Sets Syst 1:111–127
Darabi H, Imanparast M (2013) Counting number of fuzzy subgroups of some of dihedral groups. Int J Pure Appl Math 85:563–575
Degang C, Jiashang J, Congxin W, Tsang ECC (2005) Some notes on equivalent fuzzy sets and fuzzy subgroups. Fuzzy Sets Syst 152:403–409. https://doi.org/10.1016/j.fss.2004.11.003
Dubois D, Prade H (1980) Fuzzy sets and systems: theory and applications, 1st edn. Academic Press, New York
Guleria A, Bajaj RK (2019) On Pythagorean fuzzy soft matrices, operations and their applications in decision making and medical diagnosis. Soft Comput 23:7889–7900. https://doi.org/10.1007/s00500-018-3419-z
Han L, Guo X (2020) The number of subgroup chains of finite nilpotent groups. Symmetry 12:1537. https://doi.org/10.3390/sym12091537
Herrera F, Herrera-Viedma E, Martínez L, Wang PP (2006) Recent advancements of fuzzy sets: theory and practice. Inf Sci 176:349–351
Jain A (2006) Fuzzy subgroups and certain equivalence relations. Iran J Fuzzy Syst 3:75–91. https://doi.org/10.22111/ijfs.2006.469
Kahraman C (2007) Fuzzy set applications in industrial engineering. Inf Sci 177:1531–1532. https://doi.org/10.1016/j.ins.2006.09.010
Kamali Ardekani L, Davvaz B (2017) Classifying fuzzy (normal) subgroups of the group D2p × Zq and finite groups of order n ≤ 20. J Intell Fuzzy Syst 33:3615–3627. https://doi.org/10.3233/JIFS-17301
Kamali Ardekani L, Davvaz B (2020a) On the number of fuzzy subgroups of dicyclic groups. Soft Comput 24:6183–6191. https://doi.org/10.1007/s00500-020-04761-7
Kamali Ardekani L, Davvaz B (2020b) Classifying and counting fuzzy normal subgroups by a new equivalence relation. Fuzzy Sets Syst 382:148–157. https://doi.org/10.1016/j.fss.2019.05.002
Kim KH, Roush FW (1980) Generalized fuzzy matrices. Fuzzy Sets Syst 4:293–315. https://doi.org/10.1016/0165-0114(80)90016-0
Murali V (2004) Fuzzy points of equivalent fuzzy subsets. Inf Sci 158:277–288. https://doi.org/10.1016/j.ins.2003.07.008
Murali V (2005) Equivalent finite fuzzy sets and Stirling numbers. Inf Sci 174:251–263. https://doi.org/10.1016/j.ins.2004.08.008
Murali V (2006) Combinatorics of counting finite fuzzy subsets. Fuzzy Sets Syst 157:2403–2411. https://doi.org/10.1016/j.fss.2006.03.005
Murali V, Makamba BB (2001) On an equivalence of fuzzy subgroups I. Fuzzy Sets Syst 123:259–264
Murali V, Makamba BB (2003a) On an equivalence of fuzzy subgroups II. Fuzzy Sets Syst 136:93–104
Murali V, Makamba BB (2003b) On an equivalence of fuzzy subgroups III. Int J Math Math Sci 36:2303–2313. https://doi.org/10.1155/S0161171203205238
Murali V, Makamba BB (2004a) Counting the number of fuzzy subgroups of an abelian group of order pnqm. Fuzzy Sets Syst 144:459–470. https://doi.org/10.1016/S0165-0114(03)00224-0
Murali V, Makamba BB (2004b) Fuzzy subgroups of finite abelian groups. Far East J Math Sci 14:113–125
Murali V, Makamba BB (2005) Finite fuzzy sets. Int J Gen Syst 34:61–75. https://doi.org/10.1080/03081070512331318356
OEIS Foundation Inc. (2020) The online encyclopedia of integer sequences. https://oeis.org/
Ogiugo ME, Enioluwafe M (2017) Classifying a class of the fuzzy subgroups of the alternating groups An. African J Pure Appl Math 4:27–33
Rosenfeld A (1971) Fuzzy groups. J Math Anal Appl 35:512–517. https://doi.org/10.1016/0022-247X(71)90199-5
Šešelja B, Tepavčević A (2004) A note on a natural equivalence relation on fuzzy power set. Fuzzy Sets Syst 148:201–210. https://doi.org/10.1016/j.fss.2003.10.025
Shyamal AK, Pal M (2004) Two new operators on fuzzy matrices. J Appl Math Comput 15:91–107. https://doi.org/10.1007/BF02935748
Tǎrnǎuceanu M (2013) On the number of fuzzy subgroups of finite symmetric groups. J Mult-Valued Log Soft Comput 21:201–213
Tărnăuceanu M (2009) Distributivity in lattices of fuzzy subgroups. Inf Sci 179:1163–1168. https://doi.org/10.1016/j.ins.2008.12.003
Tărnăuceanu M (2012) Classifying fuzzy subgroups of finite nonabelian groups. Iran J Fuzzy Syst 9:31–41
Tărnăuceanu M (2015) The number of chains of subgroups of a finite elementary abelian p-group. Politehn Univ Bucharest Sci Bull Ser A, Appl Math Phys 77:65–68
Tărnăuceanu M (2016) A new equivalence relation to classify the fuzzy subgroups of finite groups. Fuzzy Sets Syst 289:113–121. https://doi.org/10.1016/j.fss.2015.08.024
Tǎrnǎuceanu M, Bentea L (2008) On the number of fuzzy subgroups of finite abelian groups. Fuzzy Sets Syst 159:1084–1096. https://doi.org/10.1016/j.fss.2007.11.014
Thomason MG (1977) Convergence of powers of a fuzzy matrix. J Math Anal Appl 57:476–480. https://doi.org/10.1016/0022-247X(77)90274-8
Tomescu I (1975) Introduction to combinatorics. Collet’s Publishers Ltd., London
Van Lint JH, Wilson RM (2001) A course in combinatorics. Cambridge University Press, Cambridge
Volf AC (2004) Counting fuzzy subgroups and chains of subgroups. Fuzzy Syst Artif Intell 10:191–200
Zadeh LA (1965) Fuzzy sets. Inf. Control 8:338–353
Zadeh LA (1971) Similarity relations and fuzzy orderings. Inf Sci 3:177–200
Zimmermann HJ (2001) Fuzzy set theory and its applications, 4th edn. Springer, Berlin
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Kannan, S.R., Mohapatra, R.K. & Hong, TP. The invention of new sequences through classifying and counting fuzzy matrices. Soft Comput 25, 9663–9676 (2021). https://doi.org/10.1007/s00500-020-05320-w
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DOI: https://doi.org/10.1007/s00500-020-05320-w