Krasner $F^{(m, n)}$-Hyperrings

Document Type : Research Paper


Department of Mathematics, Yazd University, Yazd, Iran


$\!\!\!\!$ In this paper, the notion of fuzzy $\!$ Krasner $\!(m, n)$-hyperrings
($\!F^{(m, n)}\!$-hyperrings) by using the notion of
$F^m$-hyperoperations and $F^n$-operations is introduced and some
related properties are investigated. In this regards,
relationships between Krasner $F^{(m, n)}$-hyperrings and Krasner
$(m, n)$-hyperrings are considered. We shall prove that every
Krasner $F^{(m, n)}$-hyperring is extended by a Krasner $F^{(2,
n)}$-hyperring. The concepts of normal $F$-hyperideals and
homomorphisms of Krasner $F^{(m, n)}$-hyperrings are adopted.
Also, the quotient of Krasner $F^{(m, n)}$-hyperrings by defining
regular relations are studied. Finally, the classical isomorphism
theorems of groups are generalized to Krasner $F^{(m,
n)}$-hyperrings provided the $F$-hyperideals considered in them
are normal.


P. Corsini, {it Prolegomena of hypergroup theory},
 Aviani editore, Second edition, 1993.

P. Corsini and V. Leoreanu,
 {it Applications of hyperstructures theory},
 Advances in Mathematics, Kluwer Academic Publisher, 2003.

bibitem{Polygroup}  B. Davvaz, {em Polygroup theory and related systems}, World Scientific, 2013.

bibitem{k} B. Davvaz, {it Fuzzy Krasner $(m,n)$-hyperrings}, Comput. Math. Appl., {bf 59}textbf{(12)} (2010), 3879-3891.

B. Davvaz and T. Vougiouklis, {it  $n$-ary hypergroups}, Iran. J.
Sci. Technol., Trans. A Sci.,
  {bf 30}textbf{(2)} (2006), 165-174.

B. Davvaz, {it Isomorphism theorems of hyperrings},
 Indian J. Pure Appl. Math., {bf 35}textbf{(3)} (2004), 321-331.

bibitem{dc} B. Davvaz and P. Corsini, {it Fuzzy $n$-ary hypergroups},  J. Intell. Fuzzy Systems,
    {bf 18}textbf{(4)} (2007), 377-382.

bibitem{DDV}  B. Davvaz, W. A. Dudek and T. Vougiouklis, {it A
generalization of  $n$-ary algebraic systems}, Comm. Algebra, {bf
37}textbf{(4)} (2009), 1248-1263.
M. Krasner, {it A class of hyperrings and hyperfields},
 Internat. J. Math. Math. Sci.,
  {bf 6}textbf{(2)} (1983), 307-312.

M. Krasner, {it Approximation des corps values complets de
caracteristque $pneq0$
  par ceux de caracteristique 0}, Actes due Colloque d' Algebre Superieure C.B.R.M, Bruxelles,
  (1965), 12-22.

V.~Leoreanu-Fotea and B.~Davvaz, {em $n$-hypergroups and binary
relations}, European J. Combin., {bf 29}textbf{(5)} (2008),

S. Mirvakili and B. Davvaz, {it  Relations on krasner $(m,
n)$-hyperrings}, European J. Combin., {bf 31}textbf{(3)} (2010),

A.~Rosenfeld, {it Fuzzy groups},
 J. Math. Anal. Appl., {bf 35} (1971), 512-517.

bibitem{V} T. Vougiouklis, {it Hyperstructures and their representations},
        Hadronic Press, Inc, 115, Palm Harber, USA, 1994.

L.~A. Zadeh, {it Fuzzy sets},  Information and Control, {bf 8}
(1965), 338-353.

M. M. Zahedi and A. Hasankhani, {it F-polygroups {I}}, J. Fuzzy
Math., {bf 4}textbf{(3)} (1996), 533-548.

M. M. Zahedi and A. Hasankhani, {it  F-polygroups {II}}, Information
Sciences, {bf 89}textbf{(3-4)} (1996), 225-243.