An Algebra of Fuzzy (m, n)-Semihyperrings

DOI: 10.4236/ajcm.2013.31012   PDF   HTML   XML   10,254 Downloads   14,228 Views  


We propose a new class of algebraic structure named as (m, n)-semihyperring which is a generalization of usual semihyperring. We define the basic properties of (m, n)-semihyperring like identity elements, weak distributive (m, n)-semihyperring, zero sum free, additively idempotent, hyperideals, homomorphism, inclusion homomorphism, congruence relation, quotient (m, n)-semihyperring etc. We propose some lemmas and theorems on homomorphism, congruence relation, quotient (m, n)-semihyperring, etc. and prove these theorems. We further extend it to introduce the relationship between fuzzy sets and (m, n)-semihyperrings and propose fuzzy hyperideals and homomorphism theorems on fuzzy (m, n)-semihyperrings and the relationship between fuzzy (m, n)-semihyperrings and the usual (m, n)-semihyper-rings.

Share and Cite:

S. Alam, S. Aljahdali and N. Hundewale, "An Algebra of Fuzzy (m, n)-Semihyperrings," American Journal of Computational Mathematics, Vol. 3 No. 1, 2013, pp. 73-79. doi: 10.4236/ajcm.2013.31012.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] B. Davvaz, “Fuzzy Hyperideals in Ternary Semihyperrings,” Iranian Journal of Fuzzy Systems, Vol. 6, No. 4, 2009, pp. 21-36.
[2] T. Vougiouklis, “On Some Representations of Hypergroups,” Annales Scientifiques de l’Universite de Clermont, Serie Mathematique, Vol. 26, 1990, pp. 21-29.
[3] S. Chaopraknoi, S. Hobuntud and S. Pianskool, “Admitting a Semihyperring with Zero of Certain Linear Transformation Subsemigroups of Part (ii),” Journal of Mathematics, 2008, pp. 45-58.
[4] B. Davvaz and N. S. Poursalavati, “On Polygroup Hyperrings and Representations of Polygroups,” Journal of the Korean Mathematical Society, Vol. 36, No. 6, 1999, pp. 1021-1031.
[5] R. Ameri and H. Hedayati, “On k-Hyperideals of Semihyperrings,” Journal of Discrete Mathematical Sciences and Cryptography, Vol. 10, No. 1, 2007, pp. 41-54.
[6] L. A. Zadeh, “Fuzzy Sets,” Information and Control, Vol. 8, No. 3, 1965, pp. 338-353. doi:10.1016/S0019-9958(65)90241-X
[7] B. Davvaz, “Fuzzy Hv-Groups,” Fuzzy Sets and Systems, Vol. 101, No. 1, 1999, pp. 191-195. doi:10.1016/S0165-0114(97)00071-7
[8] R. Ameri and T. Nozari, “Fuzzy Hyperalgebras,” Computers and Mathematics with Applications, Vol. 61, No. 2, 2011, pp. 149-154. doi:10.1016/j.camwa.2010.08.059
[9] I. Cristea, “On the Fuzzy Subhypergroups of Some Particular Complete Hypergroups(I),” World Applied Sciences Journal, Vol. 7, 2009, pp. 57-63.
[10] B. Davvaz and W. A. Dudek, “Fuzzy n-ary Groups as a Generalization of Rosenfield’s Fuzzy Groups,” Journal of Multiple-Valued Logic and Soft Computing, Vol. 15, No. 5-6, 2009, pp. 471-488.
[11] R. Ameri and H. Hedayati, “Homomorphism and Quotient of Fuzzy k-Hyperideals,” Ratio Mathematica, Vol. 20, 2010.
[12] S. Mirvakili and B. Davvaz, “Relations on Krasner (m, n)-Hyperrings,” European Journal of Combinatorics, Vol. 31, No. 3, 2010, pp. 790-802. doi:10.1016/j.ejc.2009.07.006
[13] B. Davvaz, “Fuzzy Krasner (m, n)-Hyperrings,” Computers and Mathematics with Applications, Vol. 59, No. 12, 2010, pp. 3879-3891.
[14] B. Davvaz and T. Vougiouklis, “n-ary Hypergroups,” Iranian Journal of Science and technology, Vol. 30, 2006, pp. 165-174.
[15] B. Davvaz, P. Corsini and V. L. Fotea, “Fuzzy n-ary Subpolygroups,” Computers and Mathematics with Applications, Vol. 57, 2009, pp. 141-152.
[16] V. L. Fotea, “A New Type of Fuzzy n-ary Hyperstructures,” Information Sciences, Vol. 179, No. 15, 2009, pp. 2710-2718. doi:10.1016/j.ins.2009.03.017
[17] S. E. Alam, S. Rao and B. Davvaz, “(m, n)-Semirings and a Generalized Fault Tolerance Algebra of Systems,” General Mathematics, 2010.
[18] W. A. Dudek and V. V. Mukhin, “On Topological n-ary Semigroups,” Quasigroups and Related Systems, Vol. 3, 1996, pp. 73-88.
[19] W. A. Dudek, “Idempotents in n-ary Semigroups,” Southeast Asian Bulletin of Mathematics, Vol. 25, No. 1, 2001, pp. 97-104. doi:10.1007/s10012-001-0097-y
[20] H. Hedayati and R. Ameri, “Construction of k-Hyperideals by P-Hyperoperations,” Journal of Applied Mathematics, Vol. 15, 2005, pp. 75-89.
[21] R. Ameri and M. M. Zahedi, “Hyperalgebraic Systems,” Italian Journal of Pure and Applied Mathematics, Vol. 6, 1999, pp. 21-32.
[22] M. K. Sen and U. Dasgupta, “Some Aspects of GH-Rings,” Journal of Annals of the Alexandru Ioan Cuza University—Mathematics, 2010.
[23] S. Burris and H. P. Sankappanavar, “A Course in Universal Algebra of Graduate Texts in Mathematics,” Springer-Verlag, Berlin, 1981. doi:10.1007/978-1-4613-8130-3
[24] S. Kar and B. K. Maity, “Congruences on Ternary Semigroups,” Journal of the Chungcheong Mathematical Society, Vol. 20, No. 3, 2007.
[25] X. Ma, J. Zhan, B. Davvaz and B. Y. Jun, “Some Kinds of -Interval-Valued Fuzzy Ideals of BCI-Algebras,” Information Sciences, Vol. 178, No. 19, 2008, pp. 3738-3754. doi:10.1016/j.ins.2008.06.006
[26] H. Hedayati and R. Ameri, “Fuzzy k-Hyperideals,” International Journal of Pure and Applied Mathematical Sciences, Vol. 2, No. 2, 2005, pp. 247-256.
[27] B. Y. Jun, A. M. Ozturk and Z. S. Song, “On Fuzzy hIdeals in Hemirings,” Information Sciences, Vol. 162, No. 3-4, 2004, pp. 211-226. doi:10.1016/j.ins.2003.09.007

comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.