Ratio Mathematica
Vol. 32, 2017, pp. 37–44

ISSN: 1592-7415
eISSN: 2282-8214

Quasi-Order Hypergroups determinated by
T -Hypergroups

Šárka Hošková-Mayerová
University of Defence,

Faculty of Military Technology,

Department of Mathematics and Physics,

Kounicova 65, 612 00 Brno, Czech Republic

sarka.mayerova@unob.cz

Received on: 15-03-2017. Accepted on: 02-05-2017. Published on: 30-06-2017

doi: 10.23755/rm.v32i0.333

c©Šárka Hošková-Mayerová

Abstract

Quasi-order hypergroups were introduced by Jan Chvalina in 90s of the
twentieth century. He proved that they form a subclass of the class of all
hypergroups, i.e. structures with one associative hyperoperation fulfilling the
reproduction axiom. In this paper a theorem which allows an easy descrip-
tion of all quasi-order hypergroups is presented. Moreover, some results
concerning the relation of quasi-order and upper quasi-order hypergroups
are given. Furthermore, the transformation hypergroups acting on tolerance
spaces are defined and an example of them is mentioned.
Keywords. Quasi-order hypergroup, order hypergroup, tolerance relation,
transformation semihypergroup, transformation hypergroup.
2010 AMS subject classifications: 20F60, 20N20.

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Šárka Hošková-Mayerová

The applications of mathematics in other disciplines, for example, in informat-
ics, play a key role and they represent, in the last decades, one of the purpose, of the
study of the experts of hyperstructures theory all over the world. Hyperstructure
theory was introduced in 1934 by the French mathematician Marty [16], at the 8th
Congress of Scandinavian Mathematicians, where he defined hypergroups based
on the notion of hyperoperation, began to analyze their properties, and applied
them to groups. In the following decades and nowadays, a number of different
hyperstructures are widely studied from the theoretical point of view and for their
applications to many subjects of pure and applied mathematics by many mathe-
maticians. In a classical algebraic structure, the composition of two elements is
an element, while in an algebraic hyperstructure, the composition of two elements
is a set. Several books have been written on hyperstructure theory, see [6, 10, 17].
A recent book on hyperstructures [9] points out on their applications in rough
set theory, cryptography, codes, automata, probability, geometry, lattices, binary
relations, graphs and hypergraphs. Another book [10] is devoted especially to
the study of hyperring theory. Several kinds of hyperrings are introduced and
analyzed. The volume ends with an outline of applications in chemistry and
physics, analyzing several special kinds of hyperstructures: hyperstructures and
transposition hypergroups.

Hypergroups in the sense of Marty [16] form the largest class of multivalued
systems that satisfies group-like axioms. It should be noted that various prob-
lems in non-commutative algebra lead to the introduction of algebraic systems in
which the operations are not single-valued. The motivation for generalization of
the notion of group resulted naturally from various problems in non-commutative
algebra, another motivation for such an investigation came from geometry. Hy-
pergroups have been used in algebra, geometry, convexity, automata theory, com-
binatorial problems of coloring, lattice theory, Boolean algebras, logic etc., over
the years. Over the following decades, new and interesting results again appeared,
but it is above all that a more luxuriant flourishing of hyperstructures has been
seen in the last 20 years. It is not surprising that hypergroups as well as hyper-
groupoids, quasi-hypergroups, semihypergroups, hyperfields, hyper vector spaces,
hyperlattices etc. have been studied.

The most complete bibliography up to 2002 can be found in the monograph
of Pierguilio Corsini and Violeta Leoreanu: Applications of Hyperstructure The-
ory [9]. Another comprehensive list of literature is in monograph [17] and updated
information is included in web site: http://aha.eled.duth.gr.

In the paper [2] special types of hypergroups, so called quasi-order hypergroups
(QOHG) and order hypergroups (OHG), were introduced (cf. also [6, 9, 14, 5]).

First of all recall some basic terms and definitions. A hyperoperation “◦” on
a nonempty set H is a mapping from H ×H to P∗(H) (all nonempty subsets of

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Quasi-Order Hypergroups determinated by T -Hypergroups

H). The hypergroupoid is a pair (H,◦). The quasi-hypergroup is a hypergroupoid
if the reproduction axiom (a◦H = H = H ◦a for any a ∈ H) is fulfilled. The
quasi-hypergroup (H,◦) is called a hypergroup if moreover the hyperoperation
“◦” is associative

(
(a◦b)◦c = a◦(b◦c) for any a,b,c ∈ H

)
. Here for nonempty

A,B ⊆ H we put A ◦ B =
⋃

a∈A,b∈B
a ◦ b. We denote a ◦ B instead of {a}◦ B,

a ∈ H. See, e.g. [5, 6, 9, 7, 11].
Let (H,∗) and (H′,?) be hypergroupoids. Then a mapping f : H −→ H′ is

called inclusion homomorphism if it satisfies the condition:

f(x∗y) ⊆ f(x) ? f(y) for all pairs x,y ∈ H.

Let X be a set and τ be a tolerance relation (i.e., reflexive and symmetric
binary relation)—see [1]. Then the pair (X,τ) is a tolerance space.

Definition 1. The hypergroup (H,◦) is called a quasi-order hypergroup—cf.
[2, 4, 9]—if

(i) a ∈ a3 = a2 for any a ∈ H, (1)
(ii) a◦ b = a2 ∪ b2 for any a,b ∈ H. (2)

The hypergroup (H,◦) is called an order hypergroup if moreover

(iii) a2 = b2 implies a = b for any a,b ∈ H. (3)

Using the methods occurring in [2, 4] the following theorem characterizing all
quasi-order hypergroups can be proved. For the prove see [13]. (By a2 we mean
a◦a.)

Theorem 1. Let (H,◦) is a quasi-order hypergroup. Denote K(a) = a2 for any
a ∈ H. Then the system of sets K(a) fulfills the following conditions:

(i) a ∈ K(a) for any a ∈ H, (4)
(ii) if b ∈ K(a) then K(b) ⊆ K(a). (5)

Conversely, if any system of subsets K(a) of the set H, a ∈ H, fulfills (4) and
(5), then there exists the only hyperoperation “◦” on H such that a ◦ a = K(a)
and (H,◦) is a quasi-order hypergroup.

With respect to (3) the following corollary evidently holds:

Corollary 1. Under the assumptions of Theorem 1 the quasi-order hypergroup
(H,◦) is an order hypergroup if and only if for a 6= b there is K(a) 6= K(b).

It is easy to show that if R is a quasi-ordering on a set H, then the pair (H,◦),
where a ◦ b = R(a) ∪ R(b), a,b ∈ H, is a quasi-order hypergroup. (R(x) is

39



Šárka Hošková-Mayerová

an upper end of an element x ∈ H, i.e. the set {a ∈ H;aRx for each element
a ∈ H}). See e.g. [3, 11].

In [3] J. Chvalina introduced the concept of an upper quasi-order and upper
order hypergroup.

Definition 2. A hypergroup (H,◦) is said an upper quasi-order (upper order)
hypergroup if there exists a quasi-ordering (ordering) R such that a◦ b = R(a)∪
R(b) for a,b ∈ H.

It can be shown that the classes of all quasi-order hypergroups and upper
quasi-order hypergroups coincide. The same is true for the classes of all order
hypergroups and upper order hypergroups. See [2, Theorem 1] or [9, Proposition
2 on p. 96]. These results can be easily proved using Theorem 1.

As we will need the above mentioned result of Prof. Jan Chvalina several
times in this text we recall its formulation:

Proposition 1. [9, Proposition 2 on p. 96] A hypergroupoid (H, ·) is a (quasi)-
order hypergroup if and only if there exists a (quasi)-ordeg ρ on the set H, such
that

∀(a,b) ∈ H ×H, a · b = ρ(a)∪ρ(b),

where ρ(a) = {x ∈ H,aρx}.

Theorem 2. Every quasi-order (order) hypergroup is an upper quasi-order (upper
order) hypergroup.

Proof. Let (H,◦) be a quasi-order hypergroup. Let us define a relation R on
H as follows: aRb iff b ∈ a2 for each a,b ∈ H. Evidently aRb iff b2 ⊆ a2.
Then (4) and (5) imply that R is a quasi-ordering. Moreover, R(a) = a2. Thus
a◦ b = a2 ∪ b2 = R(a)∪R(b).

If (H,◦) is even an order hypergroup, then by Corollary 1 there is
R(a) 6= R(b) for a 6= b, a,b ∈ H. Thus R is an ordering.

In [12] a more general concept of subquasi-order hypergroup is introduced. It
is an open question whether a similar representation result as in Theorem 1 can be
found for this generalization.

Now let us recall the definition of a transformation hypergroup. It was intro-
duced in [15].

Recall first that tolerance relation is a reflective and symmetric relation on a set.
This relation yields the concept of singularity in abstract mathematical expressions.
This relation namely in connection with other structures moves corresponding
mathematical theories to useful applications. Many publications are devoted to
systematic investigation to tolerances on algebraic structures compatible with

40



Quasi-Order Hypergroups determinated by T -Hypergroups

all operations of corresponding algebras. A certain survey of important results
including valuable investment can be found in [1]. Tolerance space is a set
endowed with a tolerance relation.

Definition 3. Let X be a set, (G,•) be a hypergroup and π : X × G → X a
mapping such that π(π(x,t),s) ∈ π(x,t•s), where

π(x,t•s) = {π(x,u);u ∈ t•s)}

for each x ∈ X, s,t ∈ G.
Then the triple T = (X,G,π) is called a discrete transformation hypergroup

or an action of the hypergroup G on the phase set X. The mapping π is also
usually said to be simply an action.

More generally, it is possible to consider the situation, where the phase space
X is endowed with some additional structure. The interesting case is given in the
following definition.

Definition 4. Let (X,τ) be a tolerance space (so called phase tolerance space),
(G,•) be a semihypergroup (so called phase semihypergroup) and π : X×G → X
a mapping such that

(i) π(π(x,t),s) ∈ π(x,t•s), where π(x,t•s) = {π(x,u);u ∈ t•s)} for each
x ∈ X, s,t ∈ G;

(ii) if x,y ∈ X are such that xτ y, then π(x,g)τ π(y,g) holds for any g ∈ G.

Then T = (X,G,π) is a transformation semihypergroup with phase tolerance
space. If, moreover, the pair (G,•) is a hypergroup (so called phase hypergroup),
then the triple T = (X,G,π) is a transformation hypergroup with phase tolerance
space.

In case the tolerance τ is trivial, i.e., xτ y if and only if x = y, the preceding
definition coincides in fact with Definition 3.

Let us consider a discrete transformation hypergroup T = (X,G,π). It is
possible to assign to each transformation hypergroup a commutative, extensive
hypergroup with the support X (i.e., phase set of T ) as follows:

Let us define for arbitrary pair of elements x,y ∈ X a binary hyperoperation
�: X ×X → P∗(X) in this way:

x�y = π(x,G)∪π(y,G)∪{x,y},

where π(x,G) = {π(x,u),u ∈ G} and similarly for π(y,G).
In the following we will need the next Lemma. The proof can be found in [4].

41



Šárka Hošková-Mayerová

Lemma 1. A hypergroupoid (H, ·) such that a ∈ a3 ⊂ a2, a · b = a2 ∪ b2 for any
a,b ∈ H is a quasi-order hypergroup.

Proposition 2. The pair (X,�) is an extensive, commutative hypergroup.

The extensivity and commutativity of the hyperoperation is evident, so the pair
(X,�) is an extensive, commutative hypergroupoid. The conditions of Lemma 1
are satisfied too, so (X,�) is an extensive, commutative hypergroup.

Remark 1. Even in case when T is a transformation semihypergroup we can
assign a commutative, extensive hypergroup to this semihypergroup by the above
described way.

The considered mapping is a functorial assignment which is described in the
following way:

The above defined assignment determines a functor F from the category DTH
of all discrete transformation hypergroups into the category AH of all commutative
(abelian) hypergroups.

The functor F = (FO,Fm) (O-as objects, m-as morphisms) is defined as
follows: FO(T ) = (X,�); Fm(hX,hG) = hX. Consider Ti = (Xi,Gi,πi) ∈
DTH where (Xi,�i) are hypergroups, i = 1,2 and the morhpisms hX : X1 → X2.
Then

hX(x�1 y) = h
(
π(x,G)∪π(y,G)∪{x,y}

)
=
{
π(hX(x),hG(g),g ∈ G1)

}
∪
{
π(hX(y),hG(g),g ∈ G1)

}
∪
{
hX(x),hX(y)

}
⊆

π(hX(x),G2)∪π(hX(y),G2)∪{hX(x),hX(y)}
= hX(x)�2 hX(y)

holds for all x,y ∈ X1.

Theorem 3. The pair (X,�) is a quasi-order hypergroup determined by T ,
shortly quasi-order T -hypergroup.

Proof. Let us define on (X,�) a binary relation “ρ” as follows:

xρy ⇔∃u ∈ G such that, π(x,u) = y or x = y.

This relation is evidently reflexive. We will show, that it is transitive as well.
Let

1) x = y and y = z, then x = z and xρz,
2) x = y and π(y,v) = z, then π(x,v = z) so xρz,
3) π(x,u) = y and y = z, then π(x,u = z) so xρz,

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Quasi-Order Hypergroups determinated by T -Hypergroups

4) π(x,u) = y and π(y,v) = z, then z = π(y,v) = π(π(x,u),v). From
Definition 4 we have π(π(x,u),v) ∈ π(x,u�v), thus there exists w ∈ u�v
such that z = π(x,w). Hence we have xρz. So ρ is a quasi-order. It is well
known that ρ2 = ρ ⊃ diag(X), where diag(X) = {(x,x);x ∈ X}.

Now for any pair of elements x,y ∈ X we get x � y = ρ(x) ∪ ρ(y). So
according Proposition 1 the pair (X,�) is a quasi-order hypergroup determined
by T . Shortly it is a quasi-order T -hypergroup.

References
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Olomouc, Czech Republic (1991).

[2] J. Chvalina, Commutative hypergroups in the sence of Marthy, Proceeding
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[3] J. Chvalina, Functional Graphs, Quasi-ordered Sets and Commutative Hy-
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[4] J. Chvalina, L. Chvalinová, State hypergroup of automata, Acta Mat. et. Inf.
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Šárka Hošková-Mayerová

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