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http://dx.doi.org/10.5831/HMJ.2020.42.2.219

COMMUTATIVE RINGS DERIVED FROM FUZZY HYPERRINGS  

Davvaz, Bijan (Department of Mathematics, Yazd University)
Firouzkouhi, Narjes (Department of Mathematics, Yazd University)
Publication Information
Honam Mathematical Journal / v.42, no.2, 2020 , pp. 219-234 More about this Journal
Abstract
The fundamental relation on a fuzzy hyperring is defined as the smallest equivalence relation, such that the quotient would be the ring, that is not commutative necessarily. In this paper, we introduce a new fuzzy strongly regular equivalence on fuzzy hyperrings, where the ring is commutative with respect to both sum and product. With considering this relation on fuzzy hyperring, the set of the quotient is a commutative ring. Also, we introduce fundamental functor between the category of fuzzy hyperrings and category of commutative rings and some related properties. Eventually, we introduce α-part in fuzzy hyperring and determine some necessary and sufficient conditions so that the relation α is transitive.
Keywords
Fuzzy hyperring; Fuzzy strongly regular equivalance; Commutative ring; Fundamental functor;
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1 Vougiouklis, T., (1990). The fundamental relation in hyperrings. The general hyperfield. Proc. of the 4th Int. In Congress on Algebraic Hyperstructures and Appl.(AHA 1990), Xanthi (pp. 203-211).
2 Vougiouklis, T., (1994). Hyperstructures and Their Representations, Hadronic Press Inc., Florida.
3 Zahedi, M.M., Bolurian, M., Hasankhani, A. (1995). On polygroups and fuzzy subpolygroups. J. Fuzzy Math. 3, 115.
4 Ameri, A., Nozari, T. (2010). Complete parts and fundamental relation of fuzzy hypersemigroup, J. of Mult-Valued Logic and Soft Computing, 19, 451-460.
5 Corsini, P., (1991) Prolegomena of Hypergroup Theory. Aviani Editore, Italy.
6 Corsini, P. (1993). Join spaces, power sets, fuzzy sets. Proc. Fifth Internat.Congress of Algebraic Hyperstructures and Application, Iasi, Romania, Hadronic Press, Palm Harbor,USA, pp.4552.
7 Corsini, P. (2000). Fuzzy sets, join spaces and factor spaces. Pure Math. Appl., 11(3), 439-446.
8 Corsini, P., Leoreanu, V. (2013). Applications of Hyperstructure Theory. Adv. Math., Kluwer Academic Publishers, Dordrecht, Hardbound.
9 Corsini, P., Leoreanu, V. (1996). About the heart of a hypergroup. Acta Universitatis Carolinae. Mathematica et Physica, 37(1), 17-29.
10 Corsini, P., Leoreanu, V. (2002). Fuzzy sets and join spaces associated with rough sets. Rendiconti di Circolo Matematico di Palermo, 51, 527536.
11 Corsini, P., Leoreanu, V. (1995). Join spaces associated with fuzzy sets. J. Combinatorics Inf. Syst. Sci., 20(14), 293303.
12 Corsini, P., Tofan, I. (1997). On fuzzy hypergroups. Pure Math. Appl., 8, 2937.
13 Davvaz, B. (1999). Fuzzy $H_v$-groups. Fuzzy Sets and Systems, 101, 191195.   DOI
14 Davvaz, B. (2001). Fuzzy $H_v$-submodules. Fuzzy Sets and Systems, 117, 477484.   DOI
15 Davvaz, B. (2013). Polygroup Theory and Related Systems. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
16 Corsini, P. (1980). Contributo alla teoria degli ipergruppi. Atti Soc. Pelor. Sc. Mat. Fis. Nat. Messina, Messina, Italy, 1-22.
17 Davvaz, B., Cristea, I. (2015). Fuzzy Algebraic Hyperstructures-An Introduction. Studies in Fuzziness and Soft Computing, 321. Springer, Cham.
18 Davvaz, B., Leoreanu-Fotea, V. (2007) Hyperring Theory and Applications. International Academic Press, Palm Harbor, USA.
19 Davvaz, B., Vougiouklis, T. (2007). Commutative rings obtained from hyperrings ($H_v$-rings) with ${\alpha}^{\ast}$-relations. Comm. Algebra, 35(11), 3307-3320.   DOI
20 Kehagias, A. (2002). L-fuzzy join and meet hyperoperations and the associated L-fuzzy hyperalgebras. Rendiconti di Circolo Matematico di Palermo, 51, 503526.
21 Koskas, M. (1970). Groupoids, demi-hypergroupes et hypergroupes. J. Math. Pures Appl., 49(9), 155-192.
22 Leoreanu-Fotea, V. (2009). Fuzzy hypermodules. Comput. Math. Appl. 57(3), 466-475.   DOI
23 Leoreanu-Fotea, V., Davvaz, B. (2009). Fuzzy hyperrings. Fuzzy Sets and Systems, 160(16), 2366-2378.   DOI
24 Leoreanu-Fotea, V., Zhan, J., Leoreanu, L. (2013). Fuzzy ${\Gamma}$-hyperrings and fuzzy ${\Gamma}$-hypermodules. J. Intell. Fuzzy Systems, 24(3), 647-655.   DOI
25 Mirvakili, S., Davvaz, B. (2013). Relationship between rings and hyperrings by using the notion of fundamental relations. Comm. Algebra, 41(1), 70-82.   DOI
26 Leoreanu, V. (2000). Direct limit and inverse limit of join spaces associated with fuzzy sets. Pure Math. Appl., 11, 509512.
27 Leoreanu, V. (2005). About hyperstructures associated with fuzzy sets of type 2. Ital. J. Pure Appl. Math., 17, 127136.
28 Mirvakili, S., Anvariyeh, S. M., Davvaz, B. (2008). Transitivity of ${\Gamma}$-relation on hyperfields. Bull. Math. Soc. Sci. Math. Roumanie, 51(99), 233-243.
29 Kehagias, A. (2003). An example of L-fuzzy join space. Rendiconti di Circolo Matematico di Palermo, 52, 322350.
30 Mirvakili, S., Davvaz, B. (2012). Strongly transitive geometric spaces: applications to hyperrings. Rev. Un. Mat. Argentina, 53(1), 43-53.
31 Mirvakili, S., Davvaz, B. (2013). Relationship between rings and hyperrings by using the notion of fundamental relations. Communications in Algebra, 41(1), 70-82.   DOI
32 Mirvakili, S., Davvaz, B. (2012). Strongly transitive geometric spaces: Applications to hyperrings. Rev. Un. Mat. Argentina, 53(1), 43-53.
33 Sureau, Y. (1980). Contribution a la thorie des hypergroupes et hypergroupes operant transivement sur un ensemble. Doctoral Thesis.
34 Mirvakili, S., Anvariyeh, S. M., Davvaz, B. (2008). On ${\alpha}$-relation and transitivity conditions of ${\alpha}$. Communications in Algebra, 36(5), 1695-1703.   DOI
35 Sen, M. K., Ameri, R., Chowdhury, G., (2008). Fuzzy hypersemigroups. Soft Computing, 12(9), 891-900.   DOI