RATIO MATHEMATICA 23 (2012), 3–20 ISSN: 1592-7415

General ω-hyperstructures and
certain applications of those

Jan Chvalina, Šárka Hošková-Mayerová
Brno University of Technology, Faculty of Electrical Engineering and

Communication, Department of Mathematics, Czech Republic

University of Defence, Brno, Faculty of Military Technology,

Department of Mathematics and Physics, Czech Republic

chvalina@feec.vutbr.cz, sarka.mayerova@unob.cz

Abstract

The aim of this paper is to investigate general hyperstructures
construction of which is based on ideas of A. D. Nezhad and R. S.
Hashemi. Concept of general hyperstructures considered by the above
mentioned authors is generalized on the case of hyperstructures with
hyperoperations of countable arity. Specifications of treated concepts
to examples from various fields of the mathematical sturctures theory
are also included.

Key words: Action of a hyperstructure on a set, general n-
hyperstructure, transformation hypergroup, Fredholm integral opera-
tor, ordinary and partial differential operator

MSC2010: Primary: 20N20; Secondary: 37L99, 68Q70

1 Preliminaries

A resent book [7] contains a wealth of applications of hyperstructure the-
ory developed since 1934, see [20]. There are applications to the following
subjects: geometry, hypergraphs, binary relations, lattices, fuzzy sets and
rough sets, automata, cryptography, combinatoric, codes, artificial intelli-
gence, and probability. In section 2 and 3 we give some basic definition and
then, we consider three types of actions. In section 4 we present some ap-
plications. In section 5 there are described some applications of formerly



J. Chvalina, Š. Hošková-Mayerová

investigated hyperstructures and corrected certain mistake from [14]. There
is a connection between automata and dynamical systems realized by infi-
nite automata without outputs and discrete dynamical systems.They are also
shortly called as action of semigroups or groups on a given phase (or state)
set. In connection with non-deterministic automata or with multifunctions
(relations) on algebraic structures and topological spaces seems to be natural
to investigate actions of multistructures on sets of various objects. Various
generalizations of the above mentioned classical concepts are possible. Some
motivating factors are coming from the general system theory [11, 23]; one
illustrating example below is just based on the concept of a general time
system. In [8, 9] there are investigated various binary relation and hyper-
structures.

Notice, the binary relation on a binary hyperstructure, e.g. on a semihy-
pergroup, are quite natural created by inner translations.

We use [1, 3, 7, 10, 21, 22, 26] for terminology and notations which are
not define here. We suppose that the reader is familiar with some well-known
notation such as presentation in hyperstructure theory. The following facts
are some definitions and propositions in the theory of hyperstructure which
we need for formulation of our results and in the proofs of the main results.
For x from an ordered set H we denote by [x)≤ = {y ∈ H | x ≤ y} the upper
end generated by x.

The following lemma is called Ends Lemma.

Lemma 1.1. [3, 6, 24, 25] Let (H,◦,≤) be an ordered semigroup. Let a?b =
[a◦ b)≤ for any a,b ∈ H. The following conditions are equivalent:

1) For any pair a,b ∈ H there exists a pair c,d ∈ H such that b ◦ c ≤ a,
c ◦ d ≤ a.

2) A hypergroupoid (H,?) associated with (H,◦,≤) satisfies the associa-
tivity law and the reproduction axioms, i.e., (H,?) is a hypergroup.

Dually we can define the Beginnings Lemma:

Lemma 1.2. [6] Let (H,◦,≤) be an ordered semigroup. Let a ? b = (a◦ b]≤
for any a,b ∈ H. The following conditions are equivalent:

1) For any pair a,b ∈ H there exists a pair c,d ∈ H such that b ◦ c ≥ a,
c ◦ d ≥ a.

2) A hypergroupoid (H,?) associated with (H,◦,≤) satisfies the associa-
tivity law and the reproduction axioms, i.e., (H,?) is a hypergroup.

4



General ω-hyperstructures and certain applications of those

Quasi-order hypergroups have been introduced and studied by J. Chvalina.
The following definition can be found e.g. in [7, 12, 24].

Definition 1.1. A hypergroup (H,?) such that the following condition are
satisfied:

1) a ∈ a2 = a3 for any a ∈ H,

2) a?b = a2 ∪b2 for any pair a,b ∈ H is called a quasi-order hypergroup.
If moreover the unique square root condition:

3) a,b ∈ H, a2 = b2 implies a = b

is satisfied then (H,?) is called an order hypergroup.

Definition 1.2. [19] A hypergroup (G,?) is called a transposition hyper-
group if it satisfies the transposition axiom: For all a,b,c,d ∈ G the relation
b\a∩ c/d 6= ∅ implies a ? d∩ b ? c 6= ∅.
The sets b \ a = {x ∈ G|a ∈ b ? x},c/d = {x ∈ G|c ∈ x ? d} are called left
and right extensions, respectively.

Definition 1.3. [15, 16, 17] Let X be a set, (G,•) be a (semi)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)} (1)

for each x ∈ X, s,t ∈ G. Then (X,G,π) is called a discrete transformation
(semi)hypergroup or an action of the (semi)hypergroup G on the phase set
X. The mapping π is usually said to be simply an action.

Remark 1.1. [16] The condition (1) used above is called Generalized Mixed
Associativity Condition, shortly GMAC.

2 General hyperstuctures and ω-hyper-

structures

Throughout this paper, the symbol X,Y will denote two non-empty sets,
where P∗(X ∪Y ) denotes the set of all non-empty subsets of X ∪Y .

A general hyperstructure is formed by two non-empty sets X,Y together
with a hyperoperation

∗ : X ×Y −→ P∗(X ∪Y ), (x,y) 7→ x∗y ⊆ (X ∪Y ) r ∅.

5



J. Chvalina, Š. Hošková-Mayerová

Remark 2.1. A general hyperoperation ∗ : X ×Y −→ P∗(X ∪Y ) yields a
map of powersets determined by this hyperoperation. Thus the map
⊗ : P∗(X) ×P∗(Y ) −→ P∗(X ∪Y ) is defined by A⊗B =

⋃
a∈A,b∈B

a∗ b.

Conversely an general hyperoperation on P∗(X)×P∗(Y ) yields a general
hyperoperation on X ×Y , defined by x∗y = {x}⊗{y}.

In the above definition if A ⊆ X, B ⊆ Y, x ∈ X, y ∈ Y, then we define,

A∗y = A∗{y} =
⋃
a∈A

a∗y, x∗B = {x}∗B =
⋃
b∈B

x∗ b,

A⊗B =
⋃

a∈A,b∈B

a∗ b.

Remark 2.2. If X = Y = H, then we obtain the classical hyperstructure
theory. The concept of general hyperstructure with a hyperoperation which is
a mapping

∗ : X ×Y −→ P∗(X ∪Y )
mentioned above (used by A. D. Nezhad and R. S. Hashemi) allows straight-
forward generalization onto case of “ hyperoperation of an arbitrary finite
arity” or direct generalization to hyperoperation of countable arity.

Let us define the general ω-hyperstructure:

Definition 2.1. Let ω be the smallest infinite countable ordinal. As usually,
we consider the smallest infinite ordinal ω as the set of all smaller ordi-
nals, i.e. as the domain of all finite ordinals (non-negative integers). Let
{Xk; k ∈ ω} be a system of non-empty sets. By an general ω-hyperstructure
we mean the pair ({Xk; k ∈ ω},∗ω), where ∗ω :

∏
k∈ω

Xk → P∗
( ⋃
k∈ω

Xk

)
is

a mapping assigning to any sequence {xk}k∈ω ∈
∏
k∈ω

Xk a non-empty subset

∗ω({xk}k∈ω) ⊂
⋃
k∈ω

Xk.

Similarly as above, with this hyperoperation there is associated a mapping
of power sets

⊗ω :
∏
k∈ω

P∗(Xk) →P∗
(⋃
k∈ω

Xk

)
defined by

⊗ω
(
{Ak}k∈ω

)
=
⋃{
∗ω
(
{xk}k∈ω

)
;
(
{xk}k∈ω

)
∈
∏
k∈ω

Ak

}
.

Let us formulate the special case:

6



General ω-hyperstructures and certain applications of those

Definition 2.2. [13] Let n ∈ ω be an arbitrary positive integer, n ≥ 1. Let
{Xk; k = 1, . . . ,n} be a system of non-empty sets. By a general n-hyper-
structure we mean the pair

({Xk; k = 1, . . . ,n},∗n),

where ∗n :
n∏
k=1

Xk → P∗
( n⋃
k=1

Xk

)
is a mapping assigning to any n-tuple

(x1, . . . ,xn) ∈
n∏
k=1

Xk a non-empty subset ∗n(x1, . . . ,xn) ⊂
n⋃
k=1

Xk.

Similarly as above, with this hyperoperation there is associated a mapping
of power sets

⊗n :
n∏
k=1

P∗(Xk) →P∗
( n⋃
k=1

Xk

)
defined by

⊗n(A1, . . . ,An) =
⋃{
∗n(x1, . . . ,xn); (x1, . . . ,xn) ∈

n∏
k=1

Ak

}
.

This construction is based on an idea of Nezhad and Hashemi for n = 2.
Hyperstructures with n-ary hyperoperations are investigated among others
in [2, 27].

Definition 2.3. Let G1(ω) =
(
{Xk; k ∈ ω},∗ω

)
, G2 (ω) =

(
{Yk; k ∈ ω},•ω

)
,

be a pair of general ω-hyperstructures. By a good homomorphism
H : G1(ω) → G2 (ω) we mean any system of mappings H = {hk : Xk → Yk}
such that the following diagram is commutative:

∏
Xk

∗ω−−−→ P∗
( ⋃
k∈ω

Xk
)

Q
k∈ω

hk

y yϕ]∏
Yk

•ω−−−→ P∗
( ⋃
k∈ω

Yk)

(D1)

Here
∏
k∈ω

hk({xk}k∈ω) = {hk(xk)}k∈ω for any sequence {xk}k∈ω and

ϕ] : P∗
( ⋃
k∈ω

Xk
)
→P∗

( ⋃
k∈ω

Yk
)

is the lifting of a mapping ϕ:
⋃
k∈ω

Xk →
⋃
k∈ω

Yk

defined by the mathematical induction. For x ∈ X0 we put ϕ(x) = h0(x).

Suppose ϕ:
k⋃
j=0

Xj →
k⋃
j=0

Yj is well-defined. Then for any x ∈ Xk+1\
k⋃
j=0

Xj

7



J. Chvalina, Š. Hošková-Mayerová

we put ϕ(x) = hk+1(x). Then using mathematical induction the mapping
ϕ:

⋃
k∈ω

Xk →
⋃
k∈ω

Yk is well-defined.

If all mappings hk ∈ H are bijections (or isomorphism if all Hk,Yk are
endowed with some structures) we call the H the isomorphism of ω- hyper-
structures G1(ω), G2(ω).

3 General ω-hyperstructures created by or-

dered sets and by differential operators

As a certain generalization of the general n-hyperstructure from [13],
Example 3.2, we will construct the following structure:

Example 3.1. Consider a countable system of pairwise disjoint ordered sets
(Xk,≤k), k ∈ ω and for x ∈ Xk let us denote [x)k = {y ∈ Xk; x ≤k y}, i.e.
[x)k is the principal end generated by the element x within the ordered set
(Xk,≤k). Further, put

∗ω
(
{xk}k∈ω

)
=
⋃
k∈ω

[xk)k

for any sequence ∗ω
(
{xk}k∈ω

)
∈
∏
kω

Xk. Then ∗ω
(
{xk}k∈ω

)
⊆
⋃
k∈ω

Xk, thus

G(ω) =
(
{Xk; k ∈ ω},∗ω

)
is a general ω-hyperstructure in the sense of the above definition.
If H(ω) =

(
{Yk; k ∈ ω},•ω

)
is a general ω-hyperstructure such that (Yk,�k),

k ∈ ω are pairwise disjoint ordered sets and

•ω({yk}k∈ω) =
⋃
k∈ω

[yk)k ⊆
⋃
k∈ω

Yk

for any sequence {yk}k∈ω ∈
∏
k∈ω

Yk we consider a system hk : (Xk,≤k) →

(Yk,�k), k ∈ ω, of strongly isotone mappings, i.e. for any x ∈ Xk there
holds hk

(
[xk)k

)
=
[
hk(xk)

)
k
. Then denoting H = {hk : Xk → Yk; k ∈ ω}

we obtain that H is a good homomorphism of the general ω-hyperstructure;
G(ω) into the general ω-hyperstructure H(ω).

Indeed, consider an arbitrary sequence {xk}k∈ω ∈
∏
k∈ω

Xk. As above denote

by ϕ: P∗
( ⋃
k∈ω

Xk
)
→P∗(

⋃
k∈ω

Yk) the lifting of the mapping ϕ:
⋃
k∈ω

Xk →
⋃
k∈ω

Yk

8



General ω-hyperstructures and certain applications of those

induced by the system {hk : Xk → Yk; k ∈ ω}—here in such a way that
ϕ|Xk = hk. Then for any sequence {xk}k∈ω ∈

∏
k∈ω

Xk we have

ϕ
(
∗ω({xk}k∈ω)

)
= ϕ

(⋃
k∈ω

[xk)k

)
=
⋃
k∈ω

ϕ
(
[xk)k

)
=
⋃
k∈ω

hk
(
[xk)k

)
=
⋃
k∈ω

[
hk(xk)k

)
= •ω

(
{hk(xk)}k∈ω

)
= •ω

(∏
k∈ω

hk
(
{xk}k∈ω

))
,

i.e. ϕ◦∗ω = •ω ◦
∏
k∈ω

hk, thus the diagram

∏
Xk

∗ω−−−→ P∗
( ⋃
k∈ω

Xk

)
Q

k∈ω
hk

y yϕ]∏
Yk

•ω−−−→ P∗
( ⋃
k∈ω

Yk

) (D2)
is commutative.

From the above example there follows immediately the following assertion.

Proposition 3.1. Let (Xk,≤k), (Yk,�k), k ∈ ω, be two countable collections
of pairwise disjoint ordered sets and G(ω), H(ω) be the corresponding ω-
general hyperstructures. Suppose (Xk,≤k) ∼= (Yk,�k) for each k ∈ ω and
hk : (Xk,≤k) → (Yk,�k) are corresponding order-isomorphisms. Then we
have G(ω) ∼= H(ω).

Example 3.2. Let J ⊂ R be an open interval, Cn(J) be the ring (with
respect to usual addition and multiplication of functions) of all real functions
f : J → R with continuous derivatives up to the order n ≥ 0 including.
Denote

L(p0,p1, . . . ,pn−1) : C
n(J) → Cn(J)

the linear differential operator defined by

L(p0,p1, . . . ,pn−1)(y) =
dny(x)

dxn
+

n−1∑
s=0

ps(x)
dsy(x)

dxs

where y ∈ Cn(J) and ps ∈ Cn(J), s = 0, 1, . . . ,n − 1. In accordance with
[4, 5] we put

LAn(J) = {L(p0, . . . ,pn−1); pk ∈ Cn(J)}.

9



J. Chvalina, Š. Hošková-Mayerová

Instead of L(p1,0,p1,1, . . . ,p1,n−1) we write L(~p1). We put L(~p1) ≤ L(~p2 )
whenever
L(~pj) = L(pj,0, . . . ,pj,n−1), j = 1, 2, p1,s(x) ≤ p2,s(x), s = 0, 1, . . . ,n − 1,
x ∈ J and p1,0(x) ≡ p2,0(x). Defining

∗n
(
L(~p1),L(~p2 ), . . . ,L(~pn)

)
=

n⋃
k=1

{L(~p) ∈ LAk(J); L(~pk) ≤ L(~p)}

for any n-tuple
(
L(~p1),L(~p2 ), . . . ,L(~pn)

)
∈

n∏
k=1

LAk(J) we obtain that

L(n) =
(
{LAk(J); k = 1, 2, . . . ,n},∗n

)
is a general n-hyperstructure.

Of course, LA1(J) is the set of all first-order linear differential operators
of the form L(p0 )(y) = y

′(x) + p0 (x)y, where p0 ∈ C(J) and y ∈ C1(J).
Evidently LAj(J) ∩ LAk(J) = ∅ whenever j 6= k.

It is to be noted that if k,m ∈{1, 2, . . . ,n} are fixed different integers then
setting X = LAk(J), Y = LAm(J) we obtain from the above construction
an example of a general hyperstructure in sense of Nezhad and Hashemi. If,
moreover X = Y = LAn(J) then the resulting general hyperstructure is an
order hypergroup of linear differential n-order operators in the sense of [3],
chap. IV, or [4, 5].

Theorem 3.1. Let ω be the smallest infinite countable ordinal, J ⊆ R be an
open interval. If

L(J; ω) =
(∏
k∈ω

LAk(J),∗ω,P∗
(⋃
k∈ω

LAk(J)
))

is the general ω-hyperstructure of ordinary linear differential operators and

S(J; ω) =
(∏
k∈ω

VAk(J),•ω,P∗
(⋃
k∈ω

VAk(J)
))

is the general ω-hyperstructure of solution spaces of linear ordinary homoge-
neous differential equations associated with L(J; ω). Then we have

L(J; ω) ∼= S(J; ω),
i.e. in the commutative diagram∏

k∈ω
LAk(J)

∗ω−−−→ P∗
( ⋃
k∈ω

LAk(J)
)

Q
k∈ω

Φk

y yϕ]∏
k∈ω

VAk(J)
•ω−−−→ P∗

( ⋃
k∈ω

VAk(J)
) (D3)

arrows
∏
k∈ω

Φk, ϕ
] are bijections.

10



General ω-hyperstructures and certain applications of those

Proof. By [4, 5] we have Φk : LAk(J) → VAk(J) is a group-isomorphism
for any k ∈ ω thus

∏
k∈ω

Φk :
∏
k∈ω

LAk(J) →
∏
k∈ω

VAk(J) is a bijection. Since

{LAk(J); k ∈ ω},{VAk(J); k ∈ ω} are pairwise disjoint families we have
that the mapping ϕ:

⋃
k∈ω

LAk(J) →
⋃
k∈ω

VAk(J) such that ϕ|LAk(J) = Φk,

k ∈ ω is a well-defined bijection hence the bijection P∗
( ⋃
k∈ω

LAk(J)
)
→

P∗
( ⋃
k∈ω

VAk(J)
)

is also well-defined.

Now, for an arbitrary sequence {Ln}n∈ω ∈
∏
k∈ω

LAk(J) we obtain that

•ω
((∏

k∈ω

Φk
)
{Ln}n∈ω

)
= •ω

{
Φn(Ln)

}
n∈ω = •ω{Vn}n∈ω =

= ϕ
(
∗ω
{

Φ−1n (Ln)
}
n∈ω

)
= ϕ]

(
∗ω{Ln}n∈ω

)
,

since the hyperoperation “•ω” is associated with the hyperoperation “∗ω” .
Therefore the diagram D3 in the Theorem 3.1 is commutative.

Let
{

(Sk, ·,≤k); k ∈ ω
}

be a system of quasi-ordered semigroups. Define

a mapping �ω :
∏
k∈ω

Sk →P∗
( ⋃
k∈ω

Sk

)
by the rule

�ω(x1, . . . ,xn) =
⋃
k∈ω

[x2k)≤k

for any sequence {xk}k∈omega ∈
∏
k∈ω

Sk. Then the general ω-hyperstructure is

called the general ω-hyperstructure determined by the Ends Lemma or shortly
EL-determined general ω-hyperstructure.

Corollary of Theorem 3.1
Let ω be the smallest infinite countable ordinal, ω 6= 0, J ⊆ R be an
open interval. Let LEL(J; n) =

(∏
k∈ω

LAk(J),∗n,P∗
( ⋃
k∈ω

LAk(J)
))

be the

EL-determined general ω-hyperstructure of all linear ordinary differential
operators of all orders k ∈ ω.
Let SEL(J; n) =

(∏
k∈ω

VAk(J),∗n,P∗
( ⋃
k∈ω

VAk(J)
))

be the EL-determined

general ω-hyperstructure of solutions of homogeneous linear ordinary differ-
ential equations Ly = 0, L ∈

⋃
k∈ω

LAk(J). Then LEL(J; n) ∼= SEL(J; n).

In the above construction we can use a finite sequence of positive inte-
gers {m1,m2, . . . ,mn} and then define the ω-hyperoperation �ω

(
{xk}k∈ω

)
=⋃

k∈ω
[x
mk
k )≤k for any ω sequence {xk}k∈ω ∈

∏
k∈ω

Sk.

11



J. Chvalina, Š. Hošková-Mayerová

4 General R-hyperstuctures (or L-hyper-

stuctures)

In the paper [13], due to ideas of Nezhad and Hashemi, there are consid-
ered General R-hyperstuctures (or L-hyperstuctures) in the following sense.
A general Right hyperstructure (or Left-hyperstructure) consist of two non-
empty sets X,Y together with a hyperoperation:

∗R : X ×Y −→ P∗(X) or ∗L : X ×Y −→ P∗(Y )
(x,y) 7→ x∗R y ⊆ X, (x,y) 7→ x∗L y ⊆ Y.

To be more precise a general Right hyperstructure (or Left hyperstructure)
is the quadruple (X,Y,P∗(X),∗R) or (X,Y,P∗(X),∗L), shortly general R-
hyperstructure or general L-hyperstructure.

The set of points YRx = {x∗y : y ∈ Y} that can be reached from a given
point x ∈ X by the R-hyperoperation of two non-empty sets X,Y , is called
the R-hyperorbit of x.

If YRx = X for all x ∈ X. Then the set Y is said to be R-hypertransitive
on Y . If YRx = {x}. Then x is called a R-hyperfixed point to the R-
hyperoperation. The set {y ∈ Y : x ∗R y = {x}}, is called R-hyperisotopy
set at x.

Example 4.1. Let X 6= ∅ be an arbitrary set, f : X → X be a mapping,
i.e. the pair (X,f) is a monounary algebra. Put Y = N (the set of all
positive integers) and define ∗fR : X × Y → P

∗(X) by the rule x ∗fR n =
{fk(x); k ∈ N,n ≤ k}. Then the quadruple (X,Y,P∗(X),∗fR) is a general
Right hyperstructure, i.e. R-hyperstructure. (Here, fk is the k-th iteration
of f).

Example 4.2. Let T be a linearly ordered set (i.e. a chain) with the least
element. Then T is called a time scale or time axis. Suppose A 6= ∅ 6= B are
arbitrary sets and S is a binary relation between sets of mappings (impulses)
AT , BT , i.e. S ⊂ AT ×BT . Then the triad (AT ,BT ,S) is called a general
time system with input space AT , the output space BT and with input-output
relation (or the transition relation) S—cf.[23]. Now, denote X = AT , Y =
BT and define ∗SL : X × Y → P

∗(Y ) by x ∗SL y = S(x) = {u ∈ Y ; xS u}
for any pair of time-impulses x: T → A,y : T → B. Then we obtain the
quadruple (X,Y,P∗(X),∗SL) which is a general Left hyperstructure, i.e. a
general L-hyperstructure.

In the above mentioned classical monography [23] as a general system
is considered a relation S ⊆

∏
i∈I
Vi on non-void abstract sets. However—in

12



General ω-hyperstructures and certain applications of those

detail—the index set I is decomposed into two subsets Ix, Iy and by input-
output system is ment a relation S ⊆ X × Y , where X =

∏
i∈Ix

Vi is called

the input object and Y =
∏
i∈Iy

Vi is termed as the output object. (cf. [23],

Definition 1.2).
In the connection with general ω-hyperstructures we introduce input-

output systems with inner evaluated input and output objects. Denote K =
{1, 2, . . . ,n}. By an IEO-general system, i.e. an Input-Output general system
with Inner Evaluated Objects we mean a qudruple (

∏
k∈K

Xk,
∏
k∈K

Yk, R,FR),

where Xk, Yk are non-empty sets, R ⊆
∏
k∈K

Xk ×
∏
k∈K

Yk is a domain full

binary relation, i.e. Dom R =
∏
k∈K

Xk = {x ∈
∏
k∈K

Xk;∃y ∈
∏
k∈K

: x R y},

∗x :
∏
k∈K

Xk → P(
⋃
k∈K

Xk), ∗y :
∏
k∈K

Yk → P(
⋃
k∈K

Yk) are mappings (i.e. n-

ary multioperations)—thus (
∏
k∈K

Xk,∗x),(
∏
k∈K

Yk,∗y) are general n-ary hyper-

structures and FR : P(
⋃
k∈K

Xk) → P(
⋃
k∈K

Yk) is a mapping defined in this

way:
For a non-empty set M ⊂

⋃
k∈K

Xk, i.e. M ∈P(
⋃
k∈K

Xk) we denote M
−
X ={

[x1, . . . , (x1, . . . ,xn)n] ∈
∏
k∈K

Xk;∗X(x1, . . . , (x1, . . . ,xn)n) ∩ M 6= ∅
}
. Then

we put FR(M) = ∗y
(
R(M−X

)
⊂
⋃
k∈K

Yk. If M
−
X = ∅ we define FR(M) = ∅.

(This definition includes the case FR(∅) = ∅). Hence the diagram

∏
k∈K

Xk
∗X−−−→ P

( ⋃
k∈K

Xk
)

R

y yFR∏
k∈K

Yk
∗Y−−−→ P

(⋃
k∈k

Yk)

(D4)

is commutative.
Now, in the case when Xk = A, Yk = B for all k ∈ K, then denoting the

index set K by T (the time set-continuous, which means a linearly ordered
continuum with the minimal element, or discrete N0) we obtain

∏
k∈K

Xk = A
T ,∏

k∈K
Yk = B

T and
⋃
k∈K

Xk = A
⋃
k∈K

Yk = B. Thus the above IEO-general

system turns to the above mentioned general time system (here with an
inner evaluation) (AT ,BT ,R,∗A,∗B), where FR ◦∗A = ∗B ◦R.

13



J. Chvalina, Š. Hošková-Mayerová

By a homomorphism of an IEO-general system (
∏
k∈K

Xk,
∏
k∈K

Yk, R,∗X,∗Y )

into another IEO-general system (
∏
k∈K

Uk,
∏
k∈K

Vk, S,∗U,∗V ) we mean a quadru-

ple of mappings H = [ΦXU, ΦY V , ΦPXU, ΦPY V ], ΦXU :
∏
K

Xk →
∏
K

Uk;

ΦY V :
∏
K

Yk →
∏
K

Vk; ΦPXU : P(
⋃
K

Xk →P(
⋃
K

Uk); ΦPY V : P(
⋃
K

Yk) →P(
⋃
K

Vk)

such that the diagram

∏
k∈K

Uk P
(⋃
k∈K

Uk

)

∏
k∈K

Xk P
(⋃
k∈K

Xk

)

∏
k∈K

Yk P
(⋃
k∈K

Yk

)

∏
k∈K

Vk P
(⋃
k∈K

Vk

)

//
∗

U

��
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�

S

��
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�

FS

__??????????????

ΦXU

//
∗

X

��
�
�
�
�
�
�
�
�
�
�

R

??��������������

ΦP XU

��
�
�
�
�
�
�
�
�
�
�

FR

����
��

��
��

��
��

��

ΦY U

//
∗

Y

��
??

??
??

??
??

??
??

ΦP Y V

//
∗

V

(D5)

is commutative. In concrete cases of modelling special systems, the used
objects and their mappings take a concrete interpretation.

4.1 L-hyperoperation (or R-hyperoperation) of a hy-
perstucture on a non-empty set

In this paragraph, we recall two definitions of a L-hyperoperation (or R-
hyperoperation) of a hyperstucture on a non-empty set. Let us make our
point clear with some examples.

Definition 4.1. [13] Let (G,?) be a hyperstructure and X be a non-empty set.

14



General ω-hyperstructures and certain applications of those

A generalized L-hyperaction of G on X is a L-hyperoperation ψ : G×X −→
P∗(X) such that the following axioms are satisfied:

1) For all g,h ∈ G and x ∈ X, ψ(g ? h,x) ⊆ ψ(g,ψ(h,x)),

2) For all g ∈ G, ψ(g,X) = X.

For any g ∈ G and A ⊆ X, ψ(g,A) =
⋃
x∈A

ψ(g,x), also for any x ∈ X and

B ⊆ G, ψ(B,x) =
⋃
b∈B

ψ(b,x). If in the axiom 1) of definition the equality

holds, the generalized R-hyperaction is called strong.

As application of the above concepts we mention the classical interval
binary hyperoperation on a linearly ordered group. See [18]. In detail if
(G, ·,≤) is a linearly ordered group then we define a binary hyperoperation
∗: G×G →P∗(G) by

a∗ b =
[
min{a,b}

)
≤ ∩

(
max{a,b}

]
≤ =

[
min{a,b}, max{a,b}

]
≤

=
{
x ∈ G; min{a,b}≤ x ≤ max{a,b}

}
(which is a closed interval) where min{a,b}, max{a,b} is the least element,
the greatest element of the set {a,b}, respectively. It is easy to verify that
the obtained hypergroupoid (G,∗) is an extensive commutative hypergroup.
This hypergroup we obtain even in the case if we restrict ourselves onto the
set G+ of all positive elements of the linearly ordered group (G, ·,≤), (cf. the
proof of Proposition 4.1).

Proposition 4.1. Let (G, ·,≤) be a linearly ordered group, G+ be its subset
of all positive elements (i.e. the positive cone) endowed with the interval
binary hyperoperation “∗L”. Define a mapping

ψG : G+ ×G →P
∗(G)

by

ψG(a,b) = (a + b]≤ = {x ∈ G; x ≤ a + b}

for all pairs (a,b) ∈ G+ ×G. Then the quadruple
(
G+,G,P∗(G),ψG

)
is the

generalized L-hyperoperation of the commutative extensive hypergroup (G,∗L)
on the group (G, +,≤).

Proof. For the proof see [13].

15



J. Chvalina, Š. Hošková-Mayerová

5 Homomorphism of transformation

semihypergroups

Definition 5.1. Let (X,G,ψ), (Y,H,η) be two generalized transformation
semihypergroups (GTS). A pair of mappings Φ = [µ,ϕ] such that µ: G → H
is a homomorphism of semihypergroups and ϕ: X → Y is a mapping, is said
to be a homomorphism of GTS (X,G,ψ) into GTS (Y,H,η) if for any pair
[g,x] ∈ G×X the equality

η
(
µ(g),ϕ(x)

)
= ϕ

(
ψ(g,x)

)
is satisfied, i.e. the diagram, where ϕ∗ : P∗(X) →P∗(Y ) is the corresponding
liftation of the mapping ϕ: X → Y ,

G×X ψ−−−→ P∗(X)

µ×ϕ
y yϕ∗

H ×Y ω−−−→ P∗(Y )

(D6)

commutes.

Example 5.1. Let X,Y be equivalent non-empty sets and f : X → X,
h: Y → Y be mappings such that mono-unary algebras (X,f) ∼= (Y,h).
Denote G = {fn; n ∈ N0}, H = {hn,n ∈ N0} and define binary hyperopera-
tions

?: G×G →P∗(G), •: H ×H →P∗(H),
by

fn ?fm = {fk; k ∈ N0,m + n ≤ k} and hn•hm = {hk; k ∈ N0,m + n ≤ k}.

Define mappings ψ : G × X → P∗(X), η : H × Y → P∗(Y ), by the same
rule ψ(fn,x) =

{
fk(x); k ∈ {0,n,n + 1,n + 2, . . .}

}
, ω(hn,y) =

{
hk(y); k ∈

{0,n,n + 1,n + 2, . . .}
}

. Suppose ξ : (X,f) → (Y,h) is an isomorphism and
ϕ: (X,f) → (Y,h) a homomorphism of the mono-unary algebra (X,f) onto
the mono-unary algebra (Y,h). Denote Φ = [µ,ϕ] the pair of mappings such
that µ(fn) = ξ ◦ fn ◦ ξ−1. Then Φ is a II-homomorphism of the generalized
transformation semihypergroup (X,G,ψ) into the GTS (Y,H,ω).

Indeed, for an arbitrary pair [fn,x] ∈ G×X we have

ϕ
(
ψ(fn,x)

)
= ϕ

{
x,fn(x),fn+1(x), . . .

}
=
{
ϕ(x),ϕ

(
fn(x)

)
,ϕ
(
fn+1(x)

)
, . . .

}
=
{
ϕ(x),ϕ

(
hn(x)

)
,ϕ
(
hn+1(x)

)
, . . .

}
= ω

(
hn,ϕ(x)

)
= ω

(
ξ ◦fn ◦ ξ−1,ϕ(x)

)
= ω

(
µ(fn),ϕ(x)

)
= ω

(
µ×ϕ)[fn,x]

)
.

16



General ω-hyperstructures and certain applications of those

The following example of generalized transformation hypergroup is based
on consideration published in [4, 5].

Example 5.2. Let J ⊂ R be an open interval and denote C∞(J) the ring
of all infinitely differentiable functions on J. Let us consider the set LAn(J),
n ∈ N, of linear differential operators of the n-th order in the form

L(p0, . . . ,pn−1) =
dn

dxn
+

n−1∑
k=0

pk(x)
dk

dxk
.

Where pk ∈ C∞(J), k = 0, 1, . . . ,n−1; L(p0, . . . ,pn−1) : C∞(J) −→ C∞(J),
thus

L(p0, . . . ,pn−1)(f) = f
(n)(x)+pn−1(x)f

(n−1)(x)+· · ·+p0 (x)f(x), f ∈ C∞(J).

Let δij stand for the Kronecker symbol δ. For any but fixed m ∈{0, 1, . . . ,n− 1}
we denote by

LAn(J)m =
{
L(p0, . . . ,pn−1)|pk ∈ C∞(J),pm > 0

}
.

Shortly we put p = (p0 (x), . . . ,pn−1(x)),x ∈ J and on the set LAn(J)m we
define a binary operation “◦m” and a binary relation ≤m in this way:

L(p) ◦m L(q) = L(u)

where uk(x) = pm(x)qk(x) + (1 − δkm)pk(x),x ∈ J, 0 ≤ k ≤ n− 1, and

L(p) ≤m L(q)

whenever pk(x) ≤ qk(x),k 6= m,k ∈{0, 1, . . . ,n− 1},pm(x) = qm(x),x ∈ J.
It is easy to verify that (LAn(J)m,◦m,≤m) is an ordered noncommuta-

tive group with the neutral element D(w), where D(w) = (w0, . . . ,wn−1),

wk(x) = δkm. An inverse to any D(q) is D
−1(q) =

(
−q0
qm
, . . . , 1

qm
, . . . ,

−qn−1
qm

)
.

Let (Z, +,≤) be an additive group of all integers with an usual ordering
“≤”. Then by Lemma 1.1 the structure (Z,?), where

?: Z × Z −→P∗(Z)

was defined by k ? l = [k + l)≤ is a hypergroup.

For fixed D(q) ∈ LAn(J)m we define an action ψq : Z × LAn(J)m −→
P∗(LAn(J)m) as follows,

ψq(k,L(p)) = {Lt(q) ◦m L(p)|t ≤ k}.

So (LAn(J)m, Z,ϕq) is a generalized transformation hypergroup.

17



J. Chvalina, Š. Hošková-Mayerová

References

[1] Ashrafi, A. R., Hossein Zadeh, A. and Yavari, M. : Hypergraphs and
join spaces, Italian Journal Pure Appl. Math 12, 185–196 (2002)

[2] Bošnjak, I., Madrász, R. : On power structures, Algebra and Discrete
Math. 2, 14–35 (2003)

[3] Chvalina, J. : Function Graphs, Quasi-ordered Sets and Commutative
Hypergroups, MU Brno (1995) (Czech)

[4] Chvalina, J., Chvalinová, L. : Actions of centralizer hypergroups of n-th
order linear differential operators on rings of smooth functions, Aplimat,
Journal of Appl. Math. I, No. I, 45–53 (2008)

[5] Chvalina, J., Chvalinová, L. : Realizability of the endomorphism monoid
of a semi-cascade formed by solution spaces of linear ordinary n–th order
differential equations, Aplimat, Journal of Appl. Math. III, No. II, 211–
223 (2010)

[6] Chvalina, J., Hošková, Š. : Abelizations of weakly associative hyper-
structures based on their direct squares, Acta Math. Inform. Univ. Os-
traviensis 11, No. 1, 11–23 (2003)

[7] Corsini. P., Leoreanu.V. : Application of Hyperstructure theory, Kluwer
Academic Publishers, (2003)

[8] Cristea, I., Stefanescu, M., Angheluta, C.: About the fundamental re-
lations defined on the hypergroupoids associated with binary relations.
Eur. J. Comb. 32(1): 72-81 (2011)

[9] Cristea, I., Stefanescu, M. : Binary relations and reduced hypergroups.
Discrete Mathematics 308(16): 3537-3544 (2008)

[10] Davvaz, B. : Polygroups with hyperoperators, The Journal of Fuzzy
Mathematics 9, No. 4, 815–823 (2004)

[11] Dehghan Nezhad A., Davvaz, B. : Universal Hyperdynamical Systems,
Bulletin of the Korean Mathematical Society, 47(3), 513–526 (2010)

[12] Hošková, Š. : Order Hypergroups—The unique square root condition for
quasi-order hypergroups, Set-valued Mathematics and Applications 1,
No.1, Global Research Publications, India, 1–7 (2008)

18



General ω-hyperstructures and certain applications of those

[13] Hošková-Mayerová, Š., Chvalina, J., Dehghan Nezhad A.: General ac-
tions of hyperstuctures and some applications, Analele Stiintifice ale
Universitatii ‘’Ovidius”, Constanta, Romania, in print

[14] Hošková, Š., Chvalina, J., Račková, P. : Transposition hypergroups of
Fredholm integral operators and related hyperstructuress—part 2, Jour-
nal of Basic Science, Iran, 4, 55–60 (2008)

[15] Hošková, Š., Chvalina, J. : Action of Hypergroups of the First–order
Partial Differential Operators, In: Aplimat 2007, Bratislava, Slovakia,
177–185 (2007)

[16] Hošková, Š., Chvalina, J. : Discrete transformation hypergroup and
transformation hypergroups with phase tolerance space, Discrete Math-
ematics 308, 4133–4143 (2008)

[17] Hošková, Š. : Discrete transformation hypergroups, In: 4th International
Conference on Aplimat 2005, Bratislava, Slovakia, PT II, 275–278 (2005)

[18] Iwasava, K. : On linearly ordered groups, Journal of the Math. Soc.
Japan, 1, No.1, 1–9 (1948)

[19] Jantosciak, J. : Transposition hypergroups: Noncommutative join spa-
ces, J. Algebra 187, 97–19 (1997)

[20] Marty, F. : Sur une generalization de la notion de groupe, In: 8th
Congress Math. Scandenaves, Stockholm, 45–49 (1934)

[21] Massouros, Ch. G. Mittas, J. : On the theory of generalized M-
polysymmetric hypergroups, Proceedings of the 10th International
Congress on Algebraic Hyperstructures and Applications, Brno, Czech
Republic 2009, pp. 217-228.

[22] Massouros Ch. G., Massouros, G. G. : Operators and Hyperoperators
acting on Hypergroups, Proceedings of the International Conference
on Numerical Analysis and Applied Mathematics ICNAAM 2008 Kos,
American Institute of Physics (AIP) Conference Proceedings, pp. 380-
383.

[23] Mesarovic, M., Tabahara, Y. : General System Theory, A Math. Foun-
dations; Accad. Press London, (1975)

[24] Novák, M. : The notion of subhyperstructure of ”Ends lemma”–based
hyperstructures, Aplimat – Journal of Applied Mathematics III, No. II.,
237–247 (2010)

19



J. Chvalina, Š. Hošková-Mayerová

[25] Novák, M. : Potential of the “Ends Lemma” to create ring-like hyper-
structures from quasi-ordered(semi)groups, South Bohemia Math. let-
ters 17, No 1., 39–50 (2009)

[26] Novák, M. : Some basic properties of EL-hyperstructures, European
Journal of Combinatorics 34, 446-459 (2013)

[27] Pelea, C. : Identities and multialgebras, Italian J. Pure and Appl. Math.,
15, 83–92 (2004)

20