Ratio Mathematica
Vol. 33, 2017, pp. 39-46

ISSN: 1592-7415
eISSN: 2282-8214

Contributions in Mathematics:
Hyperstructures of Professor Thomas

Vougiouklis

Violeta Leoreanu - Fotea ∗

†doi:10.23755/rm.v33i0.391

Abstract

After presenting some basic notions of hyperstructures and their applica-
tions, I shall point out on the contribution of Professor Thomas Vougiouklis
to this field of research: algebraic hyperstructures.
Keywords: weak hyperstructure
2010 AMS subject classifications: 20N20.

∗Faculty of Mathematics, Al.I. Cuza University of Iaşi, Bd. Carol I, no. 11; foteavio-
leta@gmail.com
† c©Violeta Leoreanu - Fotea. Received: 31-10-2017. Accepted: 26-12-2017. Published: 31-

12-2017.

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Violeta Leoreanu - Fotea

1 Hyperstructures and applications
Theory of hyperstructures is a field of algebra, around 80 years old and very

rich in applications, for instance in geometry, fuzzy and rough sets, automata,
cryptography, codes, probabilities, graphs and hypergraphs (see [2], [3]).

Some basic definitions:

A hyperoperation on a nonempty set H is a map ◦ : H ×H →P∗(H), where
P∗(H) denotes the set of nonempty subsets of H.

For subsets A,B of H, set A◦B =
⋃
a∈A;b∈B a◦b, and for h ∈ H write h◦A

and A◦h for {h}◦A and A◦{h}.
The pair (H,◦) is a hypergroup if for all a,b,c of H we have

(a◦ b)◦ c = a◦ (b◦ c) and a◦H = H ◦a = H.

If only the associativity is satisfied then (H,◦) is a semihypergroup. The condition
a◦H = H ◦a = H for all a of H is called the reproductive law.

A nonempty subset K of H is a subhypergroup if K ◦ K ⊆ K and for all
a ∈ K, K ◦a = K = a◦K.

A commutative hypergroup (H,◦) is a join space iff the following implication
holds: for all a,b,c,d,x of H,

a ∈ b◦x,c ∈ d◦x ⇒ a◦d∩ b◦ c 6= ∅.

A semijoin space is a commutative semihypergroup satisfying the join condition.
Hypergroups have been introduced by Marty [5] and join spaces by Prenowitz

[6]. Join spaces are an important tool in the study of graphs and hypergraphs,
binary relations, fuzzy and rough sets and in the reconstruction of several types of
noneuclidean geometries, such as the descriptive, spherical and projective geome-
tries [3], [6]. Several interesting books have been writen on hyperstructures [2],
[3], [4], [6], [8].

2 Emeritus Professor Thomas Vougiouklis and his
contribution to hyperstructures

Professor Thomas Vougiouklis is an author of more than 150 research papers
and seven text books in mathematics. He have over 3000 references. He also
wrote eight books on poetry, one CD music and lyrics.

He participated in Congresses (invited) about 60 congresses, over 20 countries.
His monograph:

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Contributions in Mathematics: Hyperstructures of Professor Thomas Vougiouklis

Hyperstructures and their representations, Hadronic Press monograph in Math-
ematics, USA (1994)

is an important book on the theory of algebraic hyperstructures.

Let us mention here some of his main contributions in Hyperstructures, es-
pecially Hv -Structures, Lie Algebras of infinite dimension, ring theory, Mathe-
matical Models.

He first introduced and studied:

• The term hope=hyperoperation (2008)

• P-hypergroups, single-power cyclicity (1981).

• Fundamental relations in hyper-rings (γ∗-relation) and Representations of
hypergroups by generalized permutations and hypermatrices (1985).

• Very Thin hyperstructures, S-construction (1988).

• Uniting elements procedure (1989), with P.Corsini.

• General hyperring, hyperfield (1990).

• The weak properties and the Hv-structures (1990).

• General Hypermodules, hypervector spaces(1990).

• Representation Theory by Hv-matrices (1990).

• Fundamental relations in hyper-modulus and hyper-vector spaces (ε∗ - rela-
tion) (1994).

• The e-hyperstructures, Hv-Lie algebras (1996).

• The h/v-structures (1998).

• ∂ - operations (2005),

• The helix hyperoperations, with S. Vougiouklis,

• n-ary hypergroups (2006), with B.Davvaz.,

• Bar instead of scale, (2008), with P. Vougioukli, etc

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Violeta Leoreanu - Fotea

Let us present here some of these notions.

HV - structures

These notions were introduced in 1990 and they satisfy the weak axioms,
where the non-empty intersection replaces the equality.

WASS means weak associativity:

∀x,y,z ∈ H, (xy)z ∩x(yz) 6= ∅.

COW means weak commutativity:

∀x,y ∈ H, xy ∩yx 6= ∅.

A hyperstructure (H, ·) is called HV -semigroup if it is WASS and it is called HV -
group if it is a reproductive HV -semigroup, i.e.

xH = Hx = H, ∀x ∈ H.

Similarly, HV -vector spaces, HV -algebras and HV -Lie algebras are defined
and their applications are mentioned in the above books.

Fundamental relations

The fundamental relations β∗, γ∗ and �∗ are defined in HV -groups, HV -rings
and HV -vector spaces being the smallest equivalences, such that the quotient
structures are a group, a ring or a vector space respectively.

The following theorem holds:

Theorem. Let (H, ·) be an HV -group and denote by U the set of all finite
products of elements of H. We define the relation β in H as follows:

xβy ⇔∃u ∈ U : {x,y}⊆ u

Then β∗ is the transitive closure of β.

In a similar way, relation γ∗ is defined in an HV -ring and relation �∗ is defined
in an HV -vector space.

An HV -ring (R,+, ·) is called an HV -field if R/γ∗ is a field.

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Contributions in Mathematics: Hyperstructures of Professor Thomas Vougiouklis

If (H, ·), (H,∗) are HV -semigroups defined on the same set H, then the hy-
peroperation (·) is smaller than (∗) (and (∗) is greater than (·) if there exists an
f ∈ Aut(H,∗), such that

x ·y ⊆ f(x∗y).
Theorem. Greater hopes than the ones which are WASS or COW are also

WASS or COW, respectively.

This theorem leads to a partial order on HV -structures and mainly to a corre-
spondence between hyperstructures and posets.

The determination of all HV -groups and HV -rings is very interesting, but dif-
ficult. There are many results of R. Bayon and N. Lygeros in this direction.

In paper [1] one can see how many HV -groups and HV -rings there exist, up to
isomorphism, for several chasses of hyperstructures of two, three or four elements.

∂- operations

The hyperoperations, called theta-operations, are motivated from the usual
property, which the derivative has on the derivation of a product of functions.

If H is a set endowed with n operations (or hyperoperations) ◦1,◦2, ...,◦n and
with one map or multivalued map f : H → H (or f : H → P(H) respectively),
then n hyperoperations ∂1,∂2, ...,∂n on H can be defined as follows:
∀x,y ∈ H,∀i ∈{1,2, ...,n},

x∂iy = {f(x)◦i y, x◦i f(y)}

or in the case ◦i is a hyperoperation or f is a multivalued map, we have
∀x,y ∈ H,∀i ∈{1,2, ...,n},

x∂iy = (f(x)◦i y)∪ (x◦i f(y)).

If ◦i is WASS, then ∂i is WASS too.

n-ary hypergroups

A mapping f : H ×···×H︸ ︷︷ ︸
n

−→ P∗(H) is called an n-ary hyperoperation,

where P∗(H) is the set of all the nonempty subsets of H. An algebraic sys-
tem (H,f), where f is an n-ary hyperoperation defined on H, is called an n-
ary hypergroupoid.

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Violeta Leoreanu - Fotea

We shall use the following abbreviated notation:
The sequence xi,xi+1, ...,xj will be denoted by x

j
i . For j < i, x

j
i is the empty

symbol. When yi+1 = · · · = yj = y the last expression will be written in the form
f(xi1,y

(j−i),znj+1).

For nonempty subsets A1, ...,An of H we define

f(An1) = f(A1, ...,An) =⋃
{f(xn1) | xi ∈ Ai, i = 1, ...,n}.

An n-ary hyperoperation f is called associative if

f(xi−11 ,f(x
n+i−1
i ),x

2n−1
n+i ) =

f(x
j−1
1 ,f(x

n+j−1
j ),x

2n−1
n+j ),

hold for every 1 ≤ i < j ≤ n and all x1,x2, ...,x2n−1 ∈ H. An n-ary hyper-
groupoid with the associative n-ary hyperoperation is called an

n-ary semihypergroup.

An n-ary hypergroupoid (H,f) in which the equation b ∈ f(ai−11 ,xi,ani+1)
has a solution xi ∈ H for every ai−11 ,ani+1,b ∈ H and 1 ≤ i ≤ n, is called an

n-ary quasihypergroup.

Moreover, if (H,f) is an n-ary semihypergroup, (H,f) is called an
n-ary hypergroup.

An n-ary hypergroupoid (H,f) is
commutative if for all σ ∈ Sn and for every an1 ∈ H we have
f(a1, ...,an) = f(aσ(1), ...,aσ(n)).

Let (H,f) be an n-ary hypergroup and B be a non-empty subset of H. B
is called an n-ary subhypergroup of (H,f), if f(xn1) ⊆ B for xn1 ∈ B, and the
equation b ∈ f(bi−11 ,xi,bni+1) has a solution xi ∈ B for every b

i−1
1 ,b

n
i+1,b ∈ B

and 1 ≤ i ≤ n.

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Contributions in Mathematics: Hyperstructures of Professor Thomas Vougiouklis

References
[1] R. Bayon and N. Lygeros, Advanced results in enumeration of hyperstruc-

tures, Journal of Algebra (Computational Algebra), 320/2 (2008),821-835

[2] P. Corsini, Prolegomena of Hypergroup Theory, Aviani Editore, Italy,
(1993)

[3] P. Corsini, P and V. Leoreanu, Applications of Hyperstructure Theory,
Kluwer Aademic Publishers, Advances in Mathematics, vol. 5, 2003.

[4] B. Davvaz and V. Leoreanu - Fotea, Hyperring theory and Applications,
Hadronic Press, 2008.

[5] F. Marty, Sur une généralisation de la notion de groupe, Actes du IV
Congrès des Mathématiciens Scandinaves, Stockholm, (1934), 45-49.

[6] W. Prenowitz and J. Jantosciak, Join geometries, Springer-Verlag, UTM,
1979.

[7] T. Vougiouklis, PhD: Cyclicity in Hypergroups, in Greek, Xanthi, 1980

[8] T. Vougiouklis, Hyperstructures and their Representations, Hadronic Press,
Inc., 1994.

[9] T. Vougiouklis, Generalization of P-hypergroups, Rendiconti del Circolo
Matematico di Palermo, Serie II, Tomo XXXVI¡ (1987), 114-121

[10] T. Vougiouklis, Representation of hypergroups by generalized permutations,
Algebra Universalis 29 (1992), 172-183.

[11] T. Vougiouklis, Representations of Hv-structures, Proceedings of the Inter-
national Conference on Group Theory, Timisoara, (1992), 159-184.

[12] T. Vougiouklis, Hv-structures:Birth and... childhood, J. of Basic Science, 4,
No.1 (2008), 119-133.

[13] T. Vougiouklis, ∂ - operations and HV -fields, Acta Math. Sinica, English
Series, vol. 24 (2008), no. 7, 1067-1078.

[14] T. Vougiouklis, On the hyperstructures called ∂ - hope, 10th Int. Congress
on Algebraic Hyperstructures and Appl., (2008), 97-112,

[15] T. Vougiouklis, Hypermatrix representations of finite Hv - groups, European
J. of Combinatorics, 44 (2015), 307-3113.

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Violeta Leoreanu - Fotea

[16] T. Vougiouklis, Hypermathematics, Hv - Structures, Hypernumbers, Hyper-
matrices and Lie-Santilli Addmissibility, American J. of Modern Physics,
(2015), 4(5), 38-46.

[17] T. Vougiouklis, Special Elements on P-hopes and ∂ - hopes, Southeast Asian
Bull. of Math. (2016), 40, 451-460.

[18] T. Vougiouklis, On the Hyperstructure Theory, Southeast Asian Bull. of
Math. (2016), 40, 603-620.

[19] T. Vougiouklis, Hypernumbers, finite hyper-fields , Algebra, Groups and
Geometries, vol. 33, no. 4 (2016), 471-490.

[20] S .Vougioukli and T. Vougiouklis, Helix-hopes on finite Hv-fields , Algebra,
Groups and Geometries, vol. 33, no. 4 (2016), 491-506.

[21] R. Majoob and T. Vougiouklis, Applications of the uniting elements method,
It. J. of Pure and Appl. Math., 36 (2016), 23-34.

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