Ratio Mathematica
Vol. 33, 2017, pp. 103-114

ISSN: 1592-7415
eISSN: 2282-8214

On P-Hv-Structures in a Two-Dimensional
Real Vector Space

Ioanna Iliou∗, Achilles Dramalidis†

‡doi:10.23755/rm.v33i0.381

Abstract

In this paper we study P-Hv-structures in connection with Hv-structures,
arising from a specific P-hope in a two-dimensional real vector space. The
visualization of these P-Hv-structures is our priority, since visual thinking
could be an alternative and powerful resource for people doing mathematics.
Using position vectors into the plane, abstract algebraic properties of these
P-Hv-structures are gradually transformed into geometrical shapes, which
operate, not only as a translation of the algebraic concept, but also, as a
teaching process.
Keywords: Hyperstructures; Hv-structures; hopes; P-hyperstructures.
2010 AMS subject classifications: 20N20.

∗Democritus University of Thrace, School of Education, 68100 Alexandroupolis, Greece; iil-
iou@eled.duth.gr
†Democritus University of Thrace, School of Education 68100 Alexandroupolis, Greece; adra-

mali@psed.duth.gr
‡ c©Ioanna Iliou and Achilles Dramalidis. Received: 31-10-2017. Accepted: 26-12-2017.

Published: 31-12-2017.

103



Ioanna Iliou and Achilles Dramalidis

1 Introduction
In a set H 6= ∅, a hyperoperation (abbr. hyperoperation=hope) (·) is defined:

· : H ×H → P(H)−{∅} : (x,y) 7→ x ·y ⊂ H

and the (H, ·) is called hyperstructure.
It is abbreviated by WASS the weak associativity: (xy)z∩x(yz) 6= ∅,∀x,y,z ∈ H
and by COW the weak commutativity: xy ∩yx 6= ∅,∀x,y ∈ H.

The largest class of hyperstructures is the one which satisfy the weak proper-
ties. These are called Hv-structures introduced by T. Vougiouklis in 1990 [13],
[14] and they proved to have a lot of applications on several applied sciences such
as linguistics, biology, chemistry, physics, and so on. The Hv-structures satisfy
the weak axioms where the non-empty intersection replaces the equality. The
Hv-structures can be used in models as an organized devise.

The hyperstructure (H, ·) is called Hv -group if it is WASS and the reproduc-
tion axiom is valid, i.e., xH = Hx = H, ∀x ∈ H.
It is called commutative Hv-group if the commutativity is valid and it is called
Hv- commutative group if it is COW.

The motivation for the Hv-structures [13] is that the quotient of a group with
respect to any partition (or equivalently to any equivalence relation), is an Hv-
group. The fundamental relation β* is defined in Hv-groups as the smallest equiv-
alence so that the quotient is a group [14].

In a similar way more complicated hyperstructures are defined [14].
One can see basic definitions, results, applications and generalizations on both

hyperstructure and Hv-structure theory in the books and papers [1], [2], [3], [10],
[12], [14], [18].

The element e ∈ H, is called left unit element if x ∈ ex,∀x ∈ H, right unit
element if x ∈ xe,∀x ∈ H and unit element if x ∈ xe∩x ∈ ex,∀x ∈ h.
An element x′ ∈ H is called left inverse of x ∈ H if there exists a unit e ∈ H,
such that e ∈ x′x, right inverse of x ∈ H if e ∈ xx′ and inverse of x ∈ H if
e ∈ x′x∩xx′.
By El∗ is denoted the set of the left unit elements, by E

r
∗ the set of the right unit

elements and by E∗ the set of the unit elements with respect to hope (*) [7].
By Il∗(x,e) is denoted the set of the left inverses, by I

r
∗(x,e) the set of the right

inverses and by I∗(x,e) the set of the inverses of the element x ∈ H associated
with the unit e ∈ H with respect to hope (*) [7].

The class of P-hyperstructures was appeared in 80’s to represent hopes of con-
stant length [16], [18]. Then many applications appeared [1], [2], [4], [5], [6], [8],
[9], [15].
Vougiouklis introduced the following definition:

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On P-Hv-Structures in a Two-Dimensional Real Vector Space

Definition 1.1. Let (G, ·) be a semigroup and P ⊂ GP 6= ∅. Then the following
hyperoperations can be defined and they are called P-hyperoperations: ∀x,y ∈ G

P∗ : xP∗y = xPy,

P∗r : xP
∗
r y = (xy)P

P∗l : xP
∗
l y = P(xy).

The (G,P∗),(G,P∗r ), (G,P
∗
l ) are called P-hyperstructures.

One, combining the above definitions gets that the most usual case is if (G, ·) is
semigroup, then xPy = xP∗y = xPy and (G,P) is a semihypergroup, but we do
not know about (G,Pr) and (G,P l). In some cases, depending on the choice of
P, (G,Pr) and (G,P l) can be associative or WASS. (G,P), (G,Pr) and (G,P l)
can be associative or WASS.

In this paper we define in the IR2 a hope which is originated from geometry.
This geometrically motivated hope in IR2 constructs Hv-structures and P-HV-
structures in which the existence of units and inverses are studied. One using
the above Hv-structures and P-Hv-structures into the plane can easily combine
abstract algebraic properties with geometrical figures [11].

2 P-Hv-structures on IR2

Let us introduce a coordinate system into the IR2. We place a given vector −→p
so that its initial point P determines an ordered pair (a1,a2). Conversely, a point
P with coordinates (a1,a2) determines the vector

−→p =
−→
OP , where O the origin

of the coordinate system. We shall refer to the elements x,y,z, . . . of the set IR2,
as vectors whose initial point is the origin. These vectors are very well known as
position vectors.
In [7] Dramalidis introduced and studied a number of hyperoperations originated
from geometry. Among them he introduced in IR2 the hyperoperation (⊕) as
follows:

Definition 2.1. For every x,y ∈ IR2

⊕ : IR2 × IR2 → P(IR2)−{∅} : (x,y) 7→ x⊕y =

= [0,x + y] = {µ(x + y)/µ ∈ [0,1]}⊂ IR2

From geometrical point of view and for x,y linearly independent position vectors,
the set x⊕y is the main diagonal of the parallelogram having vertices 0,x,x+y,y.

105



Ioanna Iliou and Achilles Dramalidis

Proposition 2.1. The hyperstructure (IR2,⊕) is a commutative Hv-group.

Now, let P be the set P = [0,p] = {λp/λ ∈ [0,1]} ⊂ IR2, where p is a fixed
point of the plane. Geometrically, P is a line segment.

Consider the P-hyperoperation (P∗r(⊕)):

Definition 2.2. For every x,y ∈ IR2

P∗r(⊕) : IR
2 × IR2 → P(IR2)−{∅} : (x,y) 7→ xP∗r(⊕)y = (x⊕y)⊕P ⊂ IR

2

Obviously, (P∗r(⊕)) is commutative and geometrically, for x,y linearly independent
position vectors, the set xP∗r(⊕)y is the closed region of the parallelogram with
vertices 0,x + y,x + y + p,p.

Proposition 2.2. The hyperstructure (R2,P∗r(⊕)) is a commutative P-Hv-group.

106



On P-Hv-Structures in a Two-Dimensional Real Vector Space

Proof. Obviously, xP∗r(⊕)R
2 = R2P∗r(⊕)x = R

2,∀x ∈ R2.
For x,y,z ∈ R2
(xP∗r(⊕)y)P

∗
r(⊕)z = {[(x⊕y)P ]⊕z}⊕P = [0,z,x + y + z,x + y + z + 2p,p]

xP∗r(⊕)(yP
∗
r(⊕)z) = {x⊕ [(y ⊕z)⊕P ]}⊕P =

= [0,x + y + z,x + y + z + 2p,x + y + 2p,x + 2p,p]

So,

(xP∗r(⊕)y)P
∗
r(⊕)z ∩xP

∗
r(⊕)(yP

∗
r(⊕)z) 6= ∅,∀x,y,z ∈ R

2.2

Proposition 2.3. EP∗
r(⊕)

= [−p,0] = {−λp/λ ∈ [0,1]}

Proof. Let e ∈ ElP∗
r(⊕)
⇔ xeP∗r(⊕)x,∀x ∈ R

2 ⇔ x{µλe+µλx+µνp/µ,ν,λ[0,1]}.
That means that,
µλ = 1 and µλe + µνp = 0 ⇔ e + µνp = 0 ⇔ e = −µνp,−1 ≤ −µν ≤ 0,
then e ∈ [−p,0]. So, ElP∗

r(⊕)
= [−p,0] and according to commutativity ErP∗

r(⊕)
=

[−p,0] = EP∗
r(⊕)

= [−p,0].

107



Ioanna Iliou and Achilles Dramalidis

Proposition 2.4. I(P∗r(⊕))(x,e) = {
1
µλ
e−x− ν

λ
p/µ,λ ∈ (0,1],ν ∈ [0,1]}, where

e ∈ EP∗
r(⊕)

.

Proof. Let e ∈ EP∗
r(⊕)

and x′ ∈ IlP∗
r(⊕)

(x,e) ⇔ e ∈ x′P∗r(⊕)x ⇔ e{µλx
′ +

µλx + µνp/λ,µ,ν[0,1]}.
That means there exist λ1,µ1,ν1[0,1] :

e = µ1λ1x
′ + µ1λ1x + µ1ν1p ⇒ x′ =

1

µ1λ1
e−x−

ν1
λ1
p,µ1,λ1 6= 0.

So, x′ ∈{ 1
µλ
e−x− ν

λ
p/µ,λ ∈ (0,1],ν ∈ [0,1]}.

Since (P∗r(⊕)) is commutative, we get I(P
∗
r(⊕))(x,e) = {

1
µλ
e − x − ν

λ
p/µ,λ ∈

(0,1],ν ∈ [0,1]}.

The P-hyperoperation P∗l(⊕) = P ⊕ (x⊕y) is identical to (P
∗
r(⊕)). But the P-

hyperoperation P∗(⊕) = x⊕P⊕y is different and even more P
∗l
(⊕) = (x⊕P)⊕y 6=

x⊕ (P ⊕y) = xP∗r(⊕)y, since (⊕) is not associative.2

108



On P-Hv-Structures in a Two-Dimensional Real Vector Space

Definition 2.3. For every x,y ∈ IR2

P∗l(⊕) : R
2 ×R2 → (R2) : (x,y) 7→ xP∗l(⊕)y = (x⊕P)⊕y

More specifically,

xP∗l(⊕)y = {λκx + λy + λκµp/λ,κ,µ ∈ [0,1]},∀x,y ∈ R
2.

Geometrically, for x,y linearly independent position vectors, the set xP∗l(⊕) y is the
closed region of the quadrilateral with vertices 0,x + y,x + y + p,y. On the
other hand the set yP∗l(⊕)x is the closed region of the quadrilateral with vertices
0,x,x + y,x + y + p. So,

(xP∗l(⊕)y)∩ (yP
∗l
(⊕)x) = [0,x + y,x + y + p] 6= ∅,∀x,y ∈ R

2.

Proposition 2.5. The hyperstructure (R2,P∗l(⊕)) is a P-Hv- commutative group.

Proof. Obviously, xP∗l(⊕)R
2 = R2P∗l(⊕)x = R

2,∀x ∈ R2.
For x,y,zR2

(xP∗l(⊕)y)P
∗l
(⊕)z = {[(x⊕P)⊕y]P}⊕z ≡ [O,z,x+y+z,x+2p+y+z,y+p+z]

xP∗l(⊕)(yP
∗l
(⊕)z) = (x⊕P)⊕[(y⊕P)⊕z] ≡ [O,x,x+y+z,x+2p+y+z,y+p+z]

109



Ioanna Iliou and Achilles Dramalidis

So,
(xP∗l(⊕)y)P

∗l
(⊕)z ∩xP

∗l
(⊕)(yP

∗l
(⊕)z) 6= ∅,x,y,z ∈ R

2.

2

Proposition 2.6. i) El
P∗l
(⊕)

= R2

ii) Er
P∗l
(⊕)

= [0,−p] = {−νp/ν ∈ [0,1]} = EP∗l
(⊕)

Proof.

i) Notice that x ∈ eP∗l(⊕)x = [0,e+x,e+x+p,x],∀x,e ∈ R
2. So, El

P∗l
(⊕)

= R2.

ii) Let e ∈ Er
P∗l
(⊕)
⇔ x ∈ xP∗l(⊕)e,∀x ∈ R

2 ⇔ x ∈{λκx+λe+λκµp/λ,κ,µ ∈
[0,1]}. Then, there exist µ1,λ1,κ1 ∈ [0,1] : x = λ1κ1x + λ1e + λ11µ1p ⇔
e = 1

/
λ1[(1 − λ11)x − λ1κ1µ1p],λ1 6= 0. The last one is valid ∀x ∈ R2,

so by setting x = 0 we get e = −κ1µ1p. Since µ1,κ1 ∈ [0,1] there exists
ν1 ∈ [0,1] : ν1 = κ1µ1 ⇒ e = −ν1p ⇒

e ∈{−νp/ν ∈ [0,1]} = [0,−p].

Since Er
P∗l
(⊕)
⊂ R2 = El

P∗l
(⊕)

we get El
P∗l
(⊕)
∩ Er

P∗l
(⊕)

= {−νp/ν[0,1]} =
EP∗l

(⊕)
.2

Proposition 2.7. α) Ir
P∗l
(⊕)

(x,e) = {−κx − (ν
λ
+ κµ)p/κ,µ,ν ∈ [0,1],λ ∈

(0,1]},e ∈ Er
P∗l
(⊕)
.

β) Ir
P∗l
(⊕)

(x,e) = {e
λ
−κx−κµp/κ,µ ∈ [0,1],λ ∈ (0,1]},e ∈ El

P∗l
(⊕)

γ) Il
P∗l
(⊕)

(x,e) = {−x
κ
− ( ν

λκ
+ µ)p/κ,λ ∈ (0,1],µ ∈ (0,1]},e ∈ Er

P∗l
(⊕)
.

110



On P-Hv-Structures in a Two-Dimensional Real Vector Space

δ) Il
P∗l
(⊕)

(x,e) = {1
κ
(e
λ
−x)−µp/κ,λ ∈ (0,1],µ ∈ [0,1]},e ∈ El

P∗l
(⊕)

Proof.
α) Let e ∈ Er

P∗l
(⊕)

= [0,−p] and x′ ∈ Ir
P∗l
(⊕)

(x,e), then

e ∈ xP∗l(⊕)x
′ ⇒ e ∈{λκx+λx′ +λκµp/κ,λ,µ ∈ [0,1]}. That means there

exist κ1,λ1,µ1 ∈ [0,1] :
e = λ1κ1x + λ1x

′ + λ1κ1µ1p ⇒ x′ = eλ1 −κ1x−κ1µ1p,λ1 6= 0.
But, e ∈{−νp/ν[0,1]}⇒3 ν1 ∈ [0,1] : e = −ν1p.
So, x′ = −ν1

λ1
p−κ1x−κ1µ1p,λ1 6= 0 ⇒ x′ = −κ1x(ν1λ1 + κ1µ1)p,λ1 6= 0.

Then we get x′ ∈{−κx− (ν
λ
+ κµ)p/κ,µ,ν ∈ [0,1],λ ∈ (0,1]}.

β) Similarly as above.

γ) Similarly as above.

δ) Similarly as above.

2

Definition 2.4. For every x,y ∈ IR2
∗r
(⊕) : R

2 ×R2 → (R2) : (x,y) 7→ x∗r(⊕))y = x⊕ (P ⊕y)
More specifically,

x∗r(⊕)y = {λx + λκy + λκµp/λ,κ,µ ∈ [0,1]},∀x,y ∈ R
2

Geometrically, for x,y linearly independent position vectors, the set xP∗r(⊕)y is the
closed region of the quadrilateral with vertices 0,x,x + y,x + y + p. On the
other hand the set yP∗r(⊕)x is the closed region of the quadrilateral with vertices
0,x + y,x + y + p,y. So,

(xP∗r(⊕)y)∩ (yP
∗r
(⊕)x) = [0,x + y,x + y + p] 6= ∅,∀x,y ∈ R

2.

111



Ioanna Iliou and Achilles Dramalidis

Proposition 2.8. The hyperstructure (R2,P∗r(⊕)) is a P-Hv- commutative group.

Proof. Obviously, xP∗r(⊕)R
2 = R2P∗r(⊕)x = R

2,∀x ∈ R2.
For x,y,z ∈ R2
(xP∗r(⊕)y)P

∗r
(⊕)z = [(x⊕(P⊕y)]⊕(P⊕z) ≡ [O,x,x+z,x+y+z,x+y+z+2p]

xP∗r(⊕)(yP
∗r
(⊕)z) = x⊕{P⊕[y⊕(P⊕z)]}≡ [O,x,x+y+z,x+y+z+2p,y+p+x]

So,
[(xP∗r(⊕)y)P

∗r
(⊕)z]∩ [xP

∗r
(⊕)(yP

∗r
(⊕)z)] 6= ∅,∀x,y,z ∈ R

2.2

The following, are respective propositions of the Propositions 2.6. and 2.7. :

Proposition 2.9. i) ErP∗r
(⊕)

= R2

ii) ElP∗r
(⊕)

= [0,−p] = {−νp/ν ∈ [0,1]} = EP∗r
(⊕)
.

Proposition 2.10. α) IrP∗r
(⊕)

(x,e) = {1
κ
(e
λ
−x)−µp/κ,λ ∈ (0,1],µ ∈ [0,1]},e ∈

ErP∗r
(⊕)

β) IrP∗r
(⊕)

(x,e) = {−x
κ
− ( ν

λκ
+ µ)p/κ,λ ∈ (0,1],µ ∈ (0,1]},e ∈ ElP∗r

(⊕)
.

γ) IlP∗r
(⊕)

(x,e) = {e
λ
−κx−κµp/κ,µ ∈ [0,1],λ ∈ (0,1]},e ∈ ErP∗r

(⊕)

δ) IlP∗r
(⊕)

(x,e) = {−κx− (ν
λ
+ κµ)p/κ,µ,ν ∈ [0,1],λ ∈ (0,1]},e ∈ ElP∗r

(⊕)
.

Remark 2.1. Notice that,

α)x∗l(⊕)y = y
∗r
(⊕)x,∀x,y ∈ R

2

β)x∗r(⊕)y = y
∗l
(⊕)x,∀x,y ∈ R

2

112



On P-Hv-Structures in a Two-Dimensional Real Vector Space

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