RATIO MATHEMATICA 23 (2012), 81–86 ISSN: 1592-7415

Closed, Reflexive, Invertible, and
Normal Subhypergroups of

Special Hypergroups

Pavĺına Račková
University of Defence, Brno, Czech Republic

pavlina.rackova@unob.cz

Abstract

In [5] J. Jantosciak introduced several special types of subhyper-
groups (invertible, closed, normal, reflexive) of a general hypergroup
and studied their relationship. In this article, the full description of
such subhypergroups in hypergroups induced by quasiordered groups
is given. Further, it is shown that there are no such non-trivial sub-
hypergroups in quasiorder hypergroups.

Key words: Hypergroup, transposition hypergroup, join space, closed,
reflexive, invertible, and normal subhypergroup, quasiordered group,
quasiorder hypergroup.

MSC 2010: 20N20.

1 Introduction

We will sum up basic concepts from hypergroup theory and results which
will be needed in the following text.

• Let H 6= ∅. A mapping ∗: H × H → P∗ is called binary hyperoperation
on H. The pair (H,∗) is hypergroupoid.
• A hypergroupoid (H,∗) is extensive if {a, b}⊆ a ∗ b for any a, b ∈ H.
• A hypergroupoid (H,∗) is called hypergroup if the hyperoperation ∗ is

associative, i.e

(x ∗ y) ∗ z = x ∗ (y ∗ z) for any x, y, z ∈ H,



Pavĺına Račková

and the reproduction axiom

a ∗ H = H = H ∗ a for any a ∈ H

is satisfied.

• Let (H,∗) be a hypergroup and S ⊆ H. Assume that a ∗ b ⊆ S for any
a, b ∈ S. Thus (S,∗) is an associative hypergroupoid (so called semihyper-
group). If, moreover, it satisfies the reproduction axiom, that is, it is a
hypergroup, we say that (S,∗) is subhypergroup of (H,∗).

• A hypergroup (H,∗) is called transposition hypergroup if the transposition
axiom is satisfied, that is, for any quadruple of elements a, b, c, d ∈ H the
implication:

If b\a ≈ c/d, then a ∗ d ≈ b ∗ c

holds, where

b\a = {x ∈ H : a ∈ b ∗ x},
c/d = {x ∈ H : c ∈ x ∗ d}

are left and right extensions, respectively.
(For two set A, B, the symbol A ≈ B means that A and B are incident,
i.e. A ∩ B 6= ∅.)
• A commutative transposition hypergroup (H,∗) is called join space.

The following concepts play the key role in the formulation of the main
results.

Definition 1.1 A subhypergroup (S,∗) of a hypergroup (H,∗) is called

• closed if a/b ⊆ S and b\a ⊆ S for any a, b ∈ S,
• invertible if a/b ≈ S implies b/a ≈ S and b\a ≈ S implies a\b ≈ S for

any a, b ∈ H,
• reflexive if a\S = S/a for any a ∈ H,
• normal if a ∗ S = S ∗ a for any a ∈ H.

J. Jantosciak (see [5]) proved that

• invertible subhypergroup of any hypergroup is closed,
• closed and normal subhypergroup of a transposition hypergroup is reflexive,
• invertible and normal subhypergroup of a transposition hypergroup is closed

and reflexive.

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Closed, Reflexive, Invertible, and . . .

2 Hypergroups Induced by Quasiordered

Groups

First the relationship among closed, normal, invertible, and reflexive sub-
hypergroups of hypergroups induced by quasiordering will be tackled.

Definition 2.1 An ordered (quasiordered) group is a triple (H, ·,≤), where
(H, ·) is a group and “≤” is an ordering (quasiordering) on H having the
substitution property on (H, ·), that is for any quadruple a, b, c, d ∈ H such
that a ≤ b, c ≤ d, there is a · c ≤ b · d.

A hyperoperation is naturally induced on each (quasi)ordered group
(H, ·,≤). For x ∈ H, let us denote [x)≤ the principal end generated by x,
that is [x)≤ = {y ∈ H : x ≤ y}. Analogously the principal beginning (x]≤
is defined. For x, y ∈ H, let us denote x ∗ y = [x · y)≤. Then (H,∗) is a
hypergroupoid associated with (H, ·,≤). The following result can be proven
(see [3, 6, 7]):

Theorem 2.1 Let (H, ·,≤) be a quasiordered group and (H,∗) the hyper-
groupoid associated with it. Then (H,∗) is the transposition hypergroup.

Let (G, ·,≤) be a quasiordered group, (G,∗), where a∗b = [a ·b)≤, be the
induced (transposition) hypergroup. Let (S,∗) be its subhypergroup, that
is, [x · y)≤ ⊂ S holds for x, y ∈ S, namely x · y ∈ S.

For the right and left extensions we have:

a/b = {x ∈ G : a ∈ x ∗ b} = {x ∈ G : x · b ≤ a} =
= {x ∈ G : x ≤ a · b−1},

b\a = {x ∈ G : a ∈ b ∗ x} = {x ∈ G : b · x ≤ a} =
= {x ∈ G : x ≤ b−1 · a}.

Theorem 2.2 A subhypergroup (S,∗) is closed iff (S, ·) is a subgroup of the
group (G, ·) and (a]≤ ∪ [a)≤ ⊂ S for any a ∈ S.

Proof. Necessity:
For a ∈ S there is a/a = (e]≤, hence e ∈ S (e is the neutral element).
If a ∈ S, then e/a = (a−1]≤, hence a−1 ∈ S.
If a ∈ S, then a/e = (a]≤, hence (a]≤ ⊂ S. Because (S,∗) is a subhyper-

group we also get a ∗ e = [a)≤ ⊂ S.
Sufficiency: Let a, b ∈ S. Then a · b−1 ∈ S, so (a · b−1]≤ = a/b ⊂ S.

Analogously for b\a. 2

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Pavĺına Račková

Theorem 2.3 A subhypergroup (S,∗) is closed iff it is invertible.

Proof. Each invertible subhypergroup is closed.
Now, let us assume that (S,∗) is closed. If a/b ≈ S, there exists x ∈ S

such that x ≤ a · b−1. Due to the previous theorem a · b−1 ∈ S and also
(a·b−1)−1 = b·a−1 ∈ S, therefore b/a ⊂ S. Especially, b/a ≈ S. Analogously
the statement for b\a can be proven. 2

Theorem 2.4 A subhypergroup (S,∗) is reflexive iff the following property
holds: If for x, y ∈ G the element x · y is covered by an element of S, then
y · x is also covered by an element of S.

Proof. The following equalities hold:

a\S =
⋃
b∈S

a\b =
⋃
b∈S

(a−1 · b]≤ ,

S/a =
⋃
b∈S

b/a =
⋃
b∈S

(b · a−1]≤ .

Hence, x ∈ a\S iff b ∈ S exists such that x ≤ a−1 · b, that is, a · x ≤ b.
Analogously x ∈ S/a iff c ∈ S exists such that x · a ≤ c. This implies the
statement. 2

If (S,∗) is a closed subhypergroup, according to the previous result, the
condition “to be comparable with an element of S” is equivalent with the
condition “to be in S”. Therefore, we get:

Corollary 2.1 A closed subhypergroup (S,∗) is reflexive iff the following
property holds: If for x, y ∈ G the element x · y ∈ S, then also y · x ∈ S.

Theorem 2.5 A subhypergroup (S,∗) is normal iff the following property
holds: If x · y, where x, y ∈ G, covers an element of S, then y · x also covers
an element of S.

Proof. The following equalities hold:

a ∗ S =
⋃
b∈S

a ∗ b =
⋃
b∈S

[a · b)≤ ,

S ∗ a =
⋃
b∈S

b ∗ a =
⋃
b∈S

[b · a)≤ .

Hence, x ∈ a ∗ S iff b ∈ S exists such that a · b ≤ x, that is, b ≤ a−1 · x.
Analogously x ∈ S ∗a iff c ∈ S exists such that c ≤ x ·a−1. This implies the
statement. 2

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Closed, Reflexive, Invertible, and . . .

In case (S,∗) is a closed subhypergroup, analogously to reflexive subhy-
pergroups we have:

Corollary 2.2 A closed subhypergroup (S,∗) is normal iff the following prop-
erty holds: If for x, y ∈ G the element x · y ∈ S, than also y · x ∈ S.

Joining the previous two results we get:

Corollary 2.3 Let (S,∗) be a closed subhypergroup. Then (S,∗) is normal
iff it is reflexive.

3 Quasiorder Hypergroups

Now the relationship among closed, normal, invertible, and reflexive sub-
hypergroups of quasiorder hypergroups will be tackled.

If R is a quasiordering on H we denote R(a) = {x ∈ H : a R x}. Further,
for A ⊆ H we set R(A) =

⋃
a∈A

R(a).

Theorem 3.1 Let (H, R) be a quasiordered set. For any pair a, b ∈ H we
set

a ∗R b = R(a) ∪ R(b) = R({a, b}).

Then (H,∗R) is commutative extensive hypergroup.

For the proof see [3, p. 150, Th. 2.1].

Definition 3.1 A hypergroup (H,∗) is said quasiorder if the following con-
ditions are satisfied:

• a ∈ a3 ⊆ a2,
• a ∗ b = a2 ∪ b2

for any a, b ∈ H.

The following theorem characterizes all quasiorder hypergroups.

Theorem 3.2 A hypergroup (H,∗) is quasiorder iff a quasiordering R on H
exists such that for any a, b ∈ H there is

a ∗ b = R(a) ∪ R(b) = R({a, b}).

For the proof see [1, p. 96].

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Pavĺına Račková

Theorem 3.3 Let (G,∗) be a quasiorder hypergroup. Then any subhyper-
group of (G,∗) is reflexive and normal. Moreover, (G,∗) contains no proper
closed or invertible subhypergroup.

Proof. Suppose that (G,∗) is a quasiorder hypergroup and R is a qua-
siordering from the previous theorem.

The hyperoperation ∗ is commutative, hence any subhypergroup (S,∗) is
reflexive and normal.

Further,

a/b = {x ∈ G : a ∈ b ∗ x} = {x ∈ G : a ∈ R(b) ∪ R(x)}.

Especially a/a = G. If (S,∗) is closed, then necessarily S = G. Therefore,
no proper closed subhypergroup exists.

Analogously, any invertible subhypergroup is closed, therefore, no proper
invertible subhypergroup exists. 2

References

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[2] Hošková, Š., Representation of quasi-order hypergroups. Global Journal
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173–176.

[3] Chvalina, J., Funkcionálńı grafy, kvaziuspořádané množiny a komutativńı
hypergrupy. Brno, MU 1995, 206 pp. (in Czech)

[4] Jantosciak, J., Transposition in hypergroups. Internat. Congress on
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[5] Jantosciak, J., Transposition hypergroups: Noncommutative join spaces.
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[6] Račková, P., Akce polohypergrup a modelováńı hypergrup integrálńımi
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[7] Račková, P., Hypergroups of symmetric matrices. 10th International
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80-7231-688-5.

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