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\begin {center}
\vskip 1cm{\LARGE\bf The Hyperrings of Order 3}
\vskip 1cm
\large
R. Bayon \\
14 av. du bosquet \\
06160 Juan les Pins \\
FRANCE\\
\href{mailto:roman.bayon@gmail.com}{\tt roman.bayon@gmail.com} \\
\ \\
N. Lygeros\\
14 av. Condorcet\\
69100 Villeurbanne\\
FRANCE\\
\end {center} 

\vskip .2in

\begin {abstract}
We first explain the historical and logical relations of hyperstructures
introduced by M. Krasner and R. Rota, and generalized by T.
Vougiouklis. Then, with our new algorithm based on our previous results
on hypergroups and $H_v$-groups of order 2, 3 and 4, we enumerate
hyperrings and $H_v$-rings. More precisely, we found 63 hyperrings of
order 2, 875 $H_v$-rings of order 2 and 33,277,642 hyperrings of order
3. Finally, in this new context, we study a new connection between 
groups and hypergroups via the notion of duality.
\end {abstract}



\section {Introduction and Definitions}

More than seventy years have gone by since the creation of the concept of hypergroup \cite{Mar34}.
M. Krasner and H. S. Wall \cite{Rib83,Wal37} introduced similar definitions, but only F. Marty's concept is accepted \cite{Cor93,Cor03,Dra95,Dra95b,Kun37,Vou91,Vou94,Vou95}.

\begin{definition}
A hypergroupoid $\langle H,. \rangle$ is a set $H$ equipped with an hyperoperation $(.): H \times H \longrightarrow {\cal P}(H)$. \end{definition}
\begin{definition} A quasigroup is an hypergroupoid verifying the axiom of reproduction: $\forall x \in H \: xH=Hx=H$.\end{definition}
\begin{definition} A semigroup is an hypergroupoid verifying associativity: $\forall x,y,z\in H \: x(yz)=(xy)z$.\end{definition}
\begin{definition}[F. Marty \cite{Mar34,Mar35,Mar36}] A hypergroup
$\langle H,. \rangle$ is a quasigroup verifying associativity: $\forall x,y,z \in H \: x(yz)=(xy)z$.\end{definition}
In 1991 T. Vougiouklis generalized the definition of F. Marty by weakening associativity \cite{Vou91}.
\begin{definition} A hyperoperation is weakly associative if for any $x,y,z \in H$, $x(yz) \cap (xy)z \neq \emptyset$.\end{definition}

\begin{definition}[T. Vougiouklis \cite{Vou91}] $\langle H,.\rangle$ is a  $H_v$-group if $(.): H \times H \rightarrow p(H)$ is a weakly associative hyperoperation for which the reproduction axiom $hH=Hh=H$ is valid for any $h$ of $H$.\end{definition}

The essential idea which governs the existence of these $H_v$-groups is a weaken associativity.
This weakening simply consists in considering the two terms of the associative law as sets,
since this is possible, and in requiring that their intersection shall not be empty.

The $H_v$-groups have a property of which the hypergroups are deprived.
This one is built from the definition of the following partial order.
\begin{definition}[T. Vougiouklis \cite{Vou94}] Let  $\langle H,.\rangle$
and $\langle H,* \rangle$ two $H_v$-groups.
 We say that $(.)$ is less or equal than $(*)$, and note $\leq$,
if and only if there exists $f \in Aut(H,*)$ such that $xy \subseteq f(x*y)$ for any $x,y$ of $H$.\end{definition}
From this definition we can deduce the following theorem:
\begin{theorem}[T. Vougiouklis \cite{Vou94}] If a hyperoperation is weakly associative,
then any hyperoperation superior to it and defined on the same set is weakly associative too.\end{theorem}
From this property, we can show the concept of minimality in a natural way.

\begin{definition}[T. Vougiouklis \cite{Vou91}]  $(R,+,.)$ A hyperstructure is called a hyperring if $(R,+)$ is a hypergroup, $(R,.)$ is a semigroup and $(.)$ is distributive in respect to $(+)$.\end{definition}

\begin{definition}[S. Spartalis, A. Dramalides and T. Vougiouklis \cite{Vou88c}]  $(R,+,.)$ A hyperstructure is called an $H_v$-ring if $(R,+)$ is an $H_v$-group, $(R,.)$ is a weak semigroup and $(.)$ is weakly distributive in respect to $(+)$.\end{definition}
A. Dramalidis enumerated a restricted class 

of $H_v$-rings, the dual $H_v$-rings.
\begin{definition}[A. Dramalidis \cite{Dra95}]
An $H_v$-ring $(R,+,.)$ is dual if $(R,.,+)$ is an $H_v$-ring.\end{definition}
He classified all $H_v$-ring such that $R=\{ 0,1,a \}$ where $0$ is the scalar unit of $H_v$-group $(R,+)$ and absorbing element of semi-hypergroup $(H,.)$ and $1$ is the scalar unit of semi-hypergroup $(H,.)$.
In the same way, he classified all hyperannoids, where $(.)$ is not distributive in respect to $(+)$.
He needed to avoid a maximum of computations because they were done case-by-case. So he tried to minimize the role of associativity because of its high computational cost and use the symmetry of duality. 

We enumerate hyperrings and $H_v$-rings of small orders and this will probably improve our understanding of the hyperannoids \cite{Bay0y}. Indeed certain categories of hypergroups were studied because of their low computational cost, but they were useless for the understanding of hyperstructures. Our research \cite{Bay05b,Bay08} already showed the greater importance of cyclic and single-power hypergroups than the canonical hypergroups \cite{Mit72,Mit86}. 


From an historical point of view, M. Krasner introduced the notion of
hyperring in 1966, ten years after the notion of hyperfield.  So the
hyperring in M. Krasner's sense generalizes his notion of hyperfield.
This one was considered as the natural extension of F. Martys's
hypergroups. But this extension is not as natural as it seems. In order
to avoid technical problems, Krasner used {\it ad hoc} properties
which were studied by his disciple J. Mittas. Mittas introduced
canonical hypergroups which are, in short, a restriction of hyperring
and consequently of hyperfield in Krasner's sense.  This global schema
seemed complete and closed, but in fact was not. The radically different
approach of T. Vougiouklis showed this critical point. T. Vougiouklis
started his work by weakening associativity in the hypergroup of 
Marty. It was then easy to extend this notion to hyperring and to
hyperfield in a natural way. Moreover, this approach generalizes
Krasner's and Rota's approaches.  Vougiouklis does not work in a
specific case as canonical hypergroups. His approach is based on the
hypergroup in Marty's sense and moreover, he introduces $H_v$-groups.
He avoids the pitfall of representativity in the world of hypergroups.
Indeed in our research, we show the low importance of canonical
hypergroups in the set of hypergroups. From this observation, we easily
deduce that M. Krasner's generalization of hyperrings and hyperfields are
analogous in the corresponding world. So the generalization of 
Vougiouklis embraces the whole set of hyperstructures.

\section{Enumeration}

In enumeration theory we have already obtained some results in different
fields (see \cite{BLS03,BLS05,ChLy92,ChLy94,FrLy91,Lyg05} and
\seqnum{A108089}, \seqnum{A132590}.

In our previous work we enumerate and classify the hypergroups of order 3 \cite{Bay05,Bay05b} and abelian hypergroups of order 4
\cite{Bay05c} with Birkhoff's point of view \cite{Bir46}. We then study the $H_v$-groups of order 3 \cite{Bay0x}, and abelian  $H_v$-groups of order 4 with Marty-Moufang hypergroups \cite{CAI05}.

Thanks to these enumerative results we can characterize some
hyperstructures, as shown with rigid hyperstructures or with
hypocomplete hypergroups \cite{Bay05b}.  The obtained results
contribute consequently to validate our algorithm. We could also
confirm the results of R. Migliorati \cite{Mig85}, some results of S-C.
Chung and B-M Choi \cite{Chu01} and results of T. Vougiouklis
\cite{Vou96,Vou97} too.  Now we present the best computational results
in these fields \seqnum{A132591}.

\begin{theorem}[G. Nordo \cite{Nor95}] There are 3,999 isomorphism classes of hypergroups of order 3.\end{theorem}
We give a more precise presentation with the classification of hypergroups by projectivity and cyclicity (see table \ref{hg3}).

\begin{table} [H]
\begin{center}
\begin{tabular}{|c |c |c|c c|c|c c|}
\hline
\multicolumn{2}{|c}{}&\multicolumn{6}{|c|}{Classes of isomorphism} \\
 \cline{3-8}
\multicolumn{2}{|c}{}&\multicolumn{3}{|c|}{Abelians}&\multicolumn{3}{|c|}{non Abelians} \\
 \cline{3-8}
\multicolumn{2}{|c|}{}&\multirow{2}*{Cyclics}&\multicolumn{2}{|c|}{non Cyclics}&\multirow{2}*{Cyclics}&\multicolumn{2}{|c|}{non Cyclics} \\
\multicolumn{2}{|c|}{}& &Proj.&non Proj.& &Proj.&non Proj. \\
\hline
\multirow{4}*{$|Aut(H)|$} &1&4&2&-&-&-&-\\
 &2&3&-&-&6&1&- \\
 &3&70&3&5&154&8&4 \\
 &6&360&2&17&3279&20&61 \\
 \hline
\end{tabular}
\par
\caption{Classification of Hypergroups of Order 3}
\par
\label{hg3}
\end{center}
\end{table}

\begin{theorem}[R. Bayon \& N. Lygeros \cite{Bay0x}] There are 20 isomorphism classes of $H_v$-groups of order 2 (see table \ref{hvg2}), $*$ indicates hypergroups).\end{theorem}
\begin{table}[H]

\begin{center}
%\vspace{0.2cm}
\begin{tabular}{|c|c||c|c|}
\hline
$H_v$-group & $|Aut(H_v)|$ & $H_v$-group & $|Aut(H_v)|$ \\
\hline
$(a;b;b;a)^*$&2&$(H;a;H;b)^*$&2\\
\hline
$(H;b;b;a)$&2&$(a;H;H;b)^*$&1\\
\hline
$(a;H;b;a)$&2&$(H;a;a;H)$  &2\\
\hline
$(a;b;H;a)$&2&$(H;b;a;H)$  &1\\
\hline
$(H;a;a;b)^*$&2&$(H;a;b;H)$  &1\\
\hline
$(H;H;b;a)$  &2&$(H;H;H;a)^*$&2\\
\hline
$(H;b;H;a)$  &2&$(H;H;H;b)^*$&2\\
\hline
$(a;H;H;a)$  &2&$(H;H;a;H)$&2\\
\hline
$(b;H;H;a)$  &1&$(H;H;b;H)$&2\\
\hline
$(H;H;a;b)^*$&2&$(H;H;H;H)^*$&1\\
\hline
\end{tabular}
\end{center}
\par
\caption{$H_v$-groups of Order 2 ($H=\{a,b \}$)}
\label{hvg2}
\par
\end{table}
%\FloatBarrier
\begin{theorem}[R. Bayon \& N. Lygeros \cite{Bay0x}] There are 1,026,462 isomorphism classes of $H_v$-groups of order 3 (see table \ref{hvg3}).\end{theorem}
\begin{table} [H]

\begin{center}
\begin{tabular}{|c |c |c|c c|c|c c|}
\hline
\multicolumn{2}{|c}{}&\multicolumn{6}{|c|}{Classes of isomorphism} \\
 \cline{3-8}
\multicolumn{2}{|c}{}&\multicolumn{3}{|c|}{Abelian}&\multicolumn{3}{|c|}{non
Abelians} \\
 \cline{3-8}
\multicolumn{2}{|c|}{}&\multirow{2}*{Cyclics}&\multicolumn{2}{|c|}{non
Cyclics}&\multirow{2}*{Cyclics}&\multicolumn{2}{|c|}{non Cyclics} \\
\multicolumn{2}{|c|}{}& &Proj.&non Proj.& &Proj.&non Proj. \\
\hline
\multirow{4}*{$|Aut(H_v)|$} &1&5&2&-&4&2&-\\
 &2&8&1&1&47&5&7 \\
 &3&243&8&14&2034&66&76 \\
 &6&7439&10&195&1003818&1083&11394 \\
 \hline
\end{tabular}
\par
\caption{Classification of $H_v$-groups of Order 3}
\label{hvg3}
\par
\end{center}
\end{table}




\section{Hyperrings}

A natural approach to hyperrings is to construct them from their
underlying hyperstructures. With this manner, we can easily check
intermediate results. Consequently, we use the enumeration of
hypergroups, semi-hypergroups, $H_v$-groups and $S_v$-groups (which are
analogue of $H_v$-groups for semi-hypergroups).

\begin{fact} [R. Bayon \& N. Lygeros] Let $(R,+,.)$ be a hyperring then $Aut(R) = Aut(+) \cup Aut(.)$.\end{fact}

\begin{corollary} [R. Bayon \& N. Lygeros] Let $(R,+,.)$ be a hyperring then $|Aut(R)| \geq max(|Aut(+)|,|Aut(.)|)$.\end{corollary}

\begin{theorem}[R. Bayon \& N. Lygeros] There are 63 isomorphism classes of hyperrings of order 2 (see table \ref{hr2}).\end{theorem}
\begin{table} [H]
\begin{center}
\begin{tabular}{|c |c | c|}
\hline
\multicolumn{2}{|c|}{}&Classes \\
\hline
\multirow{2}*{$|Aut(R)|$} &1&6\\
 &2&114 \\
 \hline
\end{tabular}
\par
\caption{Classification of Hyperrings of Order 2}
\par
\label{hr2}
\end{center}
\end{table}
{\theorem[R. Bayon \& N. Lygeros] There are 875 isomorphism classes of ${H_v}$-rings of order 2 (see table \ref{hvr2}).}
\begin{table} [H]
\begin{center}
\begin{tabular}{|c |c | c|}
\hline
\multicolumn{2}{|c|}{}&Classes \\
\hline
\multirow{2}*{$|Aut(R)|$} &1&33\\
 &2&1684 \\
 \hline
\end{tabular}
\par
\caption{Classification of $H_v$-rings of Order 2}
\label{hvr2}
\par
\end{center}
\end{table}
\begin{theorem}[R. Bayon \& N. Lygeros] There are 33,277,642 isomorphism classes of hyperrings of order 3 (see Table \ref{hr3}).\end{theorem}
\begin{table} [H]
\begin{center}
\begin{tabular}{|c |c | c|}
\hline
\multicolumn{2}{|c|}{}&Classes \\
\hline
\multirow{4}*{$|Aut(R)|$} &1&31\\
 &2&506 \\
 &3&67,857\\
 &6&199,528,434 \\
 \hline
\end{tabular}
\par
\caption{Classification of Hyperrings of Order 3}
\label{hr3}
\par
\end{center}
\end{table}

This global approach generalize the partial results obtained by T.
Vougiouklis and A. Dramalidis \cite{Dra95,Vou94}.

\subsection{Algorithm}
We generate all the simple hyperstructures (hypergroups, semigroups,
$H_v$-groups,... )  and for each of them we compute the order of its
automorphisms group.  The number of hyperstructures, up to isomorphism,
$p$ is:  $$p=\sum_{i=1}^{n!} \frac{s_i}{i}$$ where $n$ is the order of
the hyperstructures, and $s_i$ is the number of hyperstructures having
an automorphisms group of order $i$.

The algorithm for hyperrings is a natural extension of this one. We
generate all simple hyperstructures and check distributivity for each
valid pair of hyperstructures. If the hyperringoid verifies
distributivity, we compute and we store the order of their
automorphisms group. As all paires have been checked, we determine the
number of hyperrings, up to isomorphism.

\section{Remarks on the notion of the dual of a group}

Our research on the notion of the dual of a group is leaded by the
program of M. Mizony. We want to establish a structural link between
groups and hypergroups.  The dual of a group $G$, denoted $\hat{G}$ is
the set of characters which are morphism from the group to the
multiplicative group $\mathbb{C^*}$. So $\hat{G}$ is a group for
product of morphism. If $G$ is a finite group of order $n$  then the
elements of $\hat{G}$ are the morphisms from $G$ to the group of the
$n^{th}$ root of unity.

For the cyclic group $\mathbb{Z}/p\mathbb{Z}$ if we denote $\omega=e^{\frac{2\nu x}{p}}$, with $i \in \{0,\ldots ,n-1\}$, all the elements of $\hat{G}$ are written:
$$ \chi_i : G \to \mathbb{C}^*  $$
$$ x \mapsto (\omega^i)^x = e^{\frac{2\nu x i}{p}}$$
We have $G \cong \hat{G}$ and $\hat{G}$ is an orthogonal basis of $E$
the set of functions from $G$ to $\mathbb{C}$, vector space with an
Hermitian product:  $\langle f,g \rangle = \sum_{x \in G}
\bar{f(x)}.g(x)$ ($\bar{f(x)}$ is the conjugate of $f(x)$), and we know
that this product definition is important for classical hypergroups. It
is a way to catch hyperrings in the sense of F. Marty.  It is possible
resolve an more generic case, the one of abelian finite group. In this
case, the dual of $G$ is an orthogonal basis of $E$ too. We can observe
it by considering the decomposition of $\mathbb{Z}$-modules:
$$G \cong \mathbb{Z}/p_1\mathbb{Z}\times \ldots \times \mathbb{Z}/p_r \mathbb{Z}$$
But this kind of approach is not possible in the non-abelian case. To
see this, we can study the symmetric group of order $n$. Indeed, by
choosing two specific transpositions and combining them to have a
permutation, we obtain only two characters. This problem in classical
group theory is an opening to the theory of hypergroups.

\section*{Acknowledgements}
We would thank Nikos Antampoyfis who
provided us some useful references.


\begin{thebibliography}{10}

\bibitem{CAI05}
R.~Bayon and N.~Lygeros.
\newblock The ${H_v}$-groups and {Marty}-{Moufang} {hypergroups}.
\newblock In {\em Proceedings of {First} {International} {Conference} on
  {Algebraic} {Informatics}}, pp.\ 285--294. {Aristotle} {University} of
  {Thessaloniki}, 2005.

\bibitem{Bay08}
R.~Bayon and N.~Lygeros.
\newblock Advanced results in enumeration of hyperstructures.
\newblock {\em Journal of Algebra}, {\bf 320} (2008), 821--835.

\bibitem{Bay05}
R.~Bayon and N.~Lygeros.
\newblock Les hypergroupes d'ordre 3.
\newblock {\em Italian J. Pure Appl. Math.}, {\bf 20} (2006), 223--236.

\bibitem{Bay05b}
R.~Bayon and N.~Lygeros.
\newblock Cat\'egories sp\'ecifiques d'hypergroupes d'ordre 3.
\newblock In {\em El\'ements structurels de la th\'eorie des hyperstructures:
  Colloque de l'Universit\'e de Thrace}, March 2005.

\bibitem{Bay05c}
R.~Bayon and N.~Lygeros.
\newblock Les hypergroupes ab\'eliens d'ordre 4.
\newblock In {\em El\'ements structurels de la th\'eorie des hyperstructures:
  Colloque de l'Universit\'e de Thrace}, march 2005.

\bibitem{Bay0y}
R.~Bayon and N.~Lygeros.
\newblock Hyperstructures and automorphism groups.
\newblock Submitted.

\bibitem{Bay0x}
R.~Bayon and N.~Lygeros.
\newblock Les hypergroupes et ${H_v}$-groupes d'ordre 3.
\newblock {\em Bull. Greek Math. Soc.}, to appear, 2008.

\bibitem{BLS03}
R.~Bayon, N.~Lygeros, and J.-S. Sereni.
\newblock Nouveaux progr\`es dans l'\'enum\'eration des mod\`eles mixtes.
\newblock In {\em Knowledge discovery and discrete mathematics: JIM'2003},
  pages 243--246, Universit\'e de Metz, France, 2003. INRIA.

\bibitem{BLS05}
R.~Bayon, N.~Lygeros, and J.-S. Sereni.
\newblock New progress in enumeration of mixed models.
\newblock {\em Appl. Math. E-Notes}, {\bf 5} (2005), 60--65.

\bibitem{Bir46}
G.~Birkhoff.
\newblock Sobre los grupos de automorphimos.
\newblock {\em Revista Union Math. Arg}, {\bf 11} (1946), 155--157.

\bibitem{ChLy92}
C.~Chaunier and N.~Lygeros.
\newblock The number of orders with thirteen elements.
\newblock {\em Order}, {\bf 9} (3) (1992), 203--204.

\bibitem{ChLy94}
C.~Chaunier and N.~Lygeros.
\newblock Le nombre de posets \`a isomorphie pr\`es ayant {$12$} \'el\'ements.
\newblock {\em Theoret. Comput. Sci.}, {\bf 123} (1994), 89--94.

\bibitem{Chu01}
S-C. Chung and B-M Choi.
\newblock ${H_v}$-groups on the set $\{e,a,b\}$.
\newblock {\em Italian J. Pure Appl. Math.}, {\bf 10} (2001),  133--140.

\bibitem{Cor93}
P.~Corsini.
\newblock {\em Prolegomena of Hypergroup Theory}.
\newblock Aviani Editore, 1993.

\bibitem{Cor03}
P.~Corsini and V.~Leoreanu.
\newblock {\em Applications of Hyperstructure Theory}.
\newblock Kluwer Academic Publishers, 2003.

\bibitem{Dra95}
A.~Dramalidis.
\newblock Dual ${H_v}$-rings.
\newblock {\em Riv. Math. Pura Appl.}, {\bf 17} (1995), 55--62.

\bibitem{Dra95b}
A.~Dramalidis.
\newblock {\em Geometrical - Algebraic Hyperstructures and Their Instructional
  Approach}.
\newblock PhD thesis, in Greek, 1995.

\bibitem{FrLy91}
R.~Fra{\"{\i}}ss{\'e} and N.~Lygeros.
\newblock Petits posets: d\'enombrement, repr\'esentabilit\'e par cercles et
  ``compenseurs''.
\newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, {\bf 313} (7) (1991),
417--420.

\bibitem{Kun37}
M.~Kuntzmann.
\newblock Op\'erations multiformes. hypergroupes.
\newblock {\em C. R. Acad. Sci. Paris Math.}, {\bf 204} (1937),
1787--1788.

\bibitem{Lyg05}
N.~Lygeros, P.V. Marchand, and M.~Massot.
\newblock Enumeration and 3d-representation of the stereo-isomers of paraffinic
  molecules.
\newblock {\em J. Symbolic Computation}, {\bf 40} (2005), 1225--1241.

\bibitem{Mar34}
F.~Marty.
\newblock Sur une g\'en\'eralisation de la notion de groupe.
\newblock In {\em 8\`eme congr\`es des Math\'ematiciens Scandinaves,
  Stockholm}, pp.\ 45--49, 1934.

\bibitem{Mar35}
F.~Marty.
\newblock R\^ole de la notion d'hypergroupe dans l'\'etude des groupes non
  ab\'eliens.
\newblock {\em C. R. Acad. Sci. Paris Math.}, {\bf 201} (1935), 636-638.

\bibitem{Mar36}
F.~Marty.
\newblock Sur les groupes et hypergroupes attach\'es \`a une fraction
  rationnelle.
\newblock {\em Annales scientifiques de l'E.N.S.}, {\bf 53} (1936),
83--123.

\bibitem{Mig85}
R.~Migliorato.
\newblock Ipergruppi di cardinalit\`a 3 e isomorfismi di ipergruppoidi
  commutativi totalmente regolari.
\newblock {\em Atti Convegno su Ipergruppi, altre Strutture Multivoche e loro
  applicazioni, Udine}, 1985.

\bibitem{Mit72}
J.~Mittas.
\newblock Hypergroupes canoniques.
\newblock {\em Math. Balkanica}, {\bf 2} (1972), 165--179.

\bibitem{Mit86}
J.~Mittas.
\newblock Hypergroupes renaissants.
\newblock {\em Atti Sem. Mat. Fis. Un. Modena}, {\bf 33} (1984),
319--328.

\bibitem{Nor95}
G.~Nordo.
\newblock An algorithm on number of isomorphism classes of hypergroups of order
  3.
\newblock {\em Italian J. Pure Appl. Math.}, No.\ 2 (1997), 37--42.

\bibitem{Rib83}
P.~Ribenboim.
\newblock {\em Il mondo Krasneriano}.
\newblock Queen's Mathematical Preprint, No.\ 1983-12, 158 pp., 1983.

\bibitem{Vou88c}
S.~Spartalis, A.~Dramalides, and T. Vougiouklis.
\newblock On ${H_v}$-group rings.
\newblock {\em Algebras, Groups, and Geometries}, {\bf 15} (1988),
47--54.

\bibitem{Vou91}
T. Vougiouklis.
\newblock The fundamental relation in hyperrings. The general hyperfield.
\newblock In {\it Algebraic Hyperstructures and Applications 
(X\'anthi, 1990)}, World Scientific Press, 1991, p.\ 203--211.

\bibitem{Vou94}
T. Vougiouklis.
\newblock {\em Hyperstructures and their Representations}.
\newblock Hadronic Press, 1994.

\bibitem{Vou95}
T. Vougiouklis.
\newblock A new class of hyperstructures.
\newblock {\em J. Combin. Inform. System Sci.} {\bf 20} (1995),
229--235.

\bibitem{Vou96}
T. Vougiouklis.
\newblock ${H_v}$-groups defined on the same set.
\newblock {\em Discrete Math.}, {\bf 155} (1996), 259--265.

\bibitem{Vou97}
T. Vougiouklis, S.~Spartalis, and M.~Kessoglides.
\newblock Weak hyperstructures on small sets.
\newblock {\em Ratio Math.}, {\bf 12} (1997), 90--96.

\bibitem{Wal37}
H. S. Wall.
\newblock Hypergroups.
\newblock {\em American J. Math.}, {\bf 59} (1937), 78--98.

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\noindent 2000 {\it Mathematics Subject Classification}: 
Primary 20N20.

\noindent \emph {Keywords:} Automorphism groups, Hypergroups, Hyperrings, Hyperstructures, $H_v$-groups.

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\noindent (Concerned with sequences
\seqnum{A108089},
\seqnum{A132590},
and
\seqnum{A132591}.)

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\vspace*{+.1in}
\noindent
Received September 9 2007;
revised July 25 2008. 
Published in {\it Journal of Integer Sequences}, August 1 2008.

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\noindent
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