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

Document Type : Research Paper

Authors

Department of Mathematics, Yazd University, Yazd, Iran

Abstract

$\!\!\!\!$ 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.

Keywords


\bibitem{book-corsini-1993}
P. Corsini, {\it Prolegomena of hypergroup theory},
 Aviani editore, Second edition, 1993.

\bibitem{book-corsini-2003}
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.

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

\bibitem{davvaz1}
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.
\bibitem{krasner1}
M. Krasner, {\it A class of hyperrings and hyperfields},
 Internat. J. Math. Math. Sci.,
  {\bf 6}\textbf{(2)} (1983), 307-312.

\bibitem{krasner2}
M. Krasner, {\it Approximation des corps values complets de
caracteristque $p\neq0$
  par ceux de caracteristique 0}, Actes due Colloque d' Algebre Superieure C.B.R.M, Bruxelles,
  (1965), 12-22.

\bibitem{leoreanu-davvaz}
V.~Leoreanu-Fotea and B.~Davvaz, {\em $n$-hypergroups and binary
relations}, European J. Combin., {\bf 29}\textbf{(5)} (2008),
1207-1218.

\bibitem{mirvakili-krasner}
S. Mirvakili and B. Davvaz, {\it  Relations on krasner $(m,
n)$-hyperrings}, European J. Combin., {\bf 31}\textbf{(3)} (2010),
790-802.

\bibitem{rosenfeld1}
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.

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

\bibitem{zahedi2}
M. M. Zahedi and A. Hasankhani, {\it F-polygroups {I}}, J. Fuzzy
Math., {\bf 4}\textbf{(3)} (1996), 533-548.

\bibitem{zahedi3}
M. M. Zahedi and A. Hasankhani, {\it  F-polygroups {II}}, Information
Sciences, {\bf 89}\textbf{(3-4)} (1996), 225-243.