An Algebra of Fuzzy (*m*, *n*)-Semihyperrings ()

S. E. Alam, Sultan Aljahdali, Nisar Hundewale

College of Computers and Information Technology, Taif University, Taif, KSA.

**DOI: **10.4236/ajcm.2013.31012
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College of Computers and Information Technology, Taif University, Taif, KSA.

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.

Keywords

(*m*, *n*)-Semihyperring; Hyperoperation; Hyperideal; Homomorphism; Congruence Relation; Fuzzy (*m*, *n*)-Semihyperring

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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 |

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