EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 1482-1497
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Soft Hyper GR-Algebra

Mark Kenneth C. Engcot1,∗, Gaudencio C. Petalcorin, Jr.2

1 Department of Computer, Information Sciences and Mathematics, School of Arts and
Sciences, University of San Carlos, 6000 Cebu City, Philippines
2 Department of Mathematics and Statistics, College of Science and Mathematics,
Center of Graph Theory, Algebra and Analysis, Premier Research Institute of Science and
Mathematics, Mindanao State University-Iligan Institute of Technology, 9200 Iligan City,
Philippines

Abstract. In this paper, we apply the notion of soft sets to the theory of hyper GR-algebra. Also,
we introduce the concept of soft hyper GR-algebras and some properties of soft hyper GR-ideals.

2020 Mathematics Subject Classifications: 06D72, 14L17

Key Words and Phrases: hyper GR-algebra, soft hyper GR-algebra, soft set

1. Introduction

During the Congress of Scandinavian Mathematics in 1954, Marty [9] introduced the
concept of hyperstructure theory (also known as multialgebra) and defined groups based
on the concept of hyperoperation, which is a generalization of a binary operation in al-
gebra, and did an analysis on the application of its properties to groups. The notion of
hyper GR-algebra was first initiated by Indangan and Petalcorin [3] in 2016. From then,
some studies have been developed to establish some of its properties.

On the other hand, the concept of soft sets was initiated by Molodtsov [10] in 1999
as a new mathematical tool for dealing with uncertainties. It is free from difficulties that
have troubled the usual theoretical approaches. Since then, there are various studies on
soft sets. In 2003, Maji [8] proposed some basic operations on soft sets. Moreover, Ali [2]
in 2009 revised some of these operations and Alcantud [1] in 2015 extended some of the
theories on soft sets.

In this paper we apply the soft set theory to hyper GR-algebras.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4479

Email addresses: mkcengcot@usc,edu.ph (M.K. Engcot),
gaudencio.petalcorin@g.msuiit.edu.ph (G. Petalcorin)

https://www.ejpam.com 1482 © 2022 EJPAM All rights reserved.



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1483

2. Preliminaries

An algebra of type (2,0) is an algebra with a binary operation and a constant element.

Definition 1. [3] Let H be a nonempty set and ⊛ be a hyperoperation on H. Then
(H;⊛, 0) is a called a hyper GR-algebra if it satisfies the following conditions, for all
x, y, z ∈ H:

(i) (HGR1) (x⊛ z)⊛ (y ⊛ z) ≪ x⊛ y;

(ii) (HGR2) (x⊛ y)⊛ z = (x⊛ z)⊛ y;

(iii) (HGR3) x ≪ x;

(iv) (HGR4) 0⊛ (0⊛ x) ≪ x, x ̸= 0; and

(v) (HGR5) (x⊛ y)⊛ z ≪ y ⊛ z.

Example 1. [3] Let H = {0, 1, 2} with hyperoperation ⊛ defined by the Cayley table
below.

⊛ 0 1 2

0 {0} {0} {0}
1 {0, 1, 2} {0, 1} {0, 1}
2 {0, 2} {0, 1, 2} {0, 2}

By routine calculation, we see that (H;⊛, 0) is a hyper GR-algebra.

Example 2. [3] Let H = Z, where Z is the set of integers such that for all x, y,∈ H,
x⊛ y = {0, x, y}. Then H is a hyper GR-algebra.

Definition 2. [3] A hyper GR-algebra H is faithful if for all A,B ⊆ H, 0 ∈ A⊛B implies
A ≪ B.

Example 3. The hyper GR-algebra H = {0, 1, 2} in Example 1 is faithful.

Definition 3. [4] Let H1 and H2 be hyper GR-algebras where ⊛1 and ⊛2 are the hyper-
operations of H1 and H2, respectively,and f : H1 → H2 be function. Then f is called a
hyper GR-algebra hyper homomorphism if

(i) f(01) = 02; and

ii) f(x⊛1 y) = f(x)⊛2 f(y).

A hyper homomorphism f is a hyper monorphism if f is one-to-one and f is a hyper
epimorphism if f is onto; f is called a hyper isomorphism if f is a hyper monorphism and
hyper epimorphism (denoted by ∼=H).



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1484

Definition 4. [3] Let H be a hyper GR-algebra and S be a subset of H containing 0. If
S is a hyper GR-algebra with respect to the hyperoperation ⊛ on H, then we say that H
is a hyper subGR-algebra of H.

Lemma 1. [4] Let f : H → Y be homomorphism of hyper GR-algebras. If S is a hyper
subGR-algebra of H, then f(S) is a hyper subGR-algebra.

Theorem 1. [3] (Hyper SubGR-algebra Criterion)
Let H be a hyper GR-algebra and S be a nonempty subset of H. Then S is a hyper
subGR-algebra of H if and only if x⊛ y ⊆ S, for all x, y ∈ S.

Theorem 2. [3] If {Ii|i ∈

V

} is a nonempty collection of hyper GR-ideals of a hyper
GR-algebra H, then so is

⋂
i∈

VIi.

Definition 5. [6] Let U be an initial universal set and E a set of all possible parameters
under consideration. If A ⊂ E, then a soft set (F,A) over U is defined to be the set of
ordered pairs

(F,A) = {(x, fA(x)) : x ∈ E, fA(x) ∈ P (U)},

where fA : E → P (U) such that fA(x) = ∅ if x /∈ A. The function fA is called the
approximation function of the soft set (F,A). The subscript A in the notion fA indicates
that fA is the approximate function of (F,A). In what follows, let S(U) denote the set of
all soft sets over U by Cagman et al. [7].

Definition 6. [5] Let (F,A) and (G,B) be two soft sets over U . The intersection of
(F,A) and (G,B) is defined to be the soft set (H,C) satisfying the following conditions:

(i) C = A ∩B ̸= ∅

(ii) H(e) = F (e) ∩G(e), for all e ∈ C.

In this case , we write (F,A)
∼
∩ (G,B) = (H,C).

Definition 7. [5] Let (F,A) and (G,B) be two soft sets over a common universe U . Then
the union of (F,A) and (G,B) is defined to be a soft set (H,C) satisfying the following
conditions:

(i) C = A ∪B;

(ii) for all e ∈ C,

H(e) =


F (e), if e ∈ A \B
G(e), if e ∈ B \A
F (e) ∪G(e), if e ∈ A ∩B.

In this case, we write (F,A)
∼
∪ (G,B) = (H,C).



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1485

Definition 8. [5] If (F,A) and (G,B) are two sets over U , then “(F,A) and (G,B)”

denoted by (F,A)
∼
∧ (G,B) is defined by (F,A)

∼
∧ (G,B) = (H,A × B) where H(α, β) =

F (α) ∩G(β) for all (α, β) ∈ A×B.

Definition 9. [5] For two soft sets (F,A) and (G,B) over U , then “(F,A) or (G,B) ”

denoted by (F,A)
∼
∨ (G,B) is defined by (F,A)

∼
∨ (G,B) = (H,A × B) where H(α, β) =

F (α) ∪G(β) for all (α, β) ∈ A×B.

Definition 10. [5] For two soft sets (F,A) and (G,B) over U , we say that (F,A) is a

soft subset of (G,B), denoted by (F,A)
∼
⊂ (G,B), if it satisfies:

(i) A ⊆ B

(ii) For every ϵ ∈ A, F (ϵ) = G(ϵ).

Definition 11. [6] Let (F,A) ∈ S(U) and τ ⊆ U . Then the τ -exclusive set of (F,A) is
defined to be the set e((F,A), τ) = {x ∈ A : fA(x) ⊆ τ}.

From Definition 11, we have the following properties [6]:

1. e((F,A), U) = A,

2. fA(x) = ∩{τ ⊆ U : x ∈ e((F,A), τ)}, ∀x ∈ A, and

3. τ1 ⊆ τ2 implies e((F,A), τ1) ⊆ e((F,A), τ2), ∀τ1, τ2 ⊆ U .

3. Soft Hyper GR-Algebra

Let H be a hyper GR-algebra, A a nonempty set, and R̊ an arbitrary binary relation
between an element of A and an element of P (H), that is, R̊ ⊆ A × P (H). A set-valued
function F : A → P (H) can be defined as F (a) =

⋃
B where B ⊂ H and aR̊B, for all

a ∈ A. Then (F,A) is then a soft set over H.

Definition 12. Let (F,A) be a soft set over a hyper GR-algebra H. Then (F,A) is called
a soft hyper GR-algebra over H if F (a) =

⋃
B⊂H,aR̊B

B is a hyper GR-algebra of H, for all

a ∈ A.

Example 4. Consider H = {0, 1, 2, 3} defined by the Cayley table below.

⊛ 0 1 2 3

0 {0,1} {0,1} {0,1} {0,1}
1 {1} {0,1} {0,1} {0,1}
2 {0,2} {0,2} {0,1,2} {0,1,2}
3 {3} {0,1,3} {0,1,3} {0,1,3}.



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1486

By Definition 1 (H;⊛, 0) is a hyper GR-algebra. We will verify if (H;⊛, 0) is a soft hyper
GR-algebra. Let A = H and define a relation R̊ such that aR̊B if and only if B = an,
where B ⊂ H and a ∈ A, let F : A → P (H) be a set-valued function defined as follows:
F (a) =

⋃
B⊂H,aRB⇔B=an

B, for all a ∈ A, where an = ((((a ⊛ a) ⊛ a) ⊛ a) ⊛ ... ⊛ a). Then

F (0) = {0, 1}, F (1) = {0, 1}, F (2) = {0, 1, 2} and F (3) = {0, 1, 3} . Note that S is
nonempty. Let S = F (0) = F (1) = {0, 1}. Then 0 ⊛ 0 = 0 ⊛ 1 = 1 ⊛ 1 = {0, 1} ∈ S and
1⊛ 0 = {1} ∈ S. Thus, by Theorem 1, F (0) = F (1) is a hyper subGR-algebra. Similarly,
F (2) and F (3) are hyper subGR-algebras. Hence, F (a) is a hyper GR-algebra over H, for
all a ∈ A. Therefore, (F,A) is a soft hyper GR-algebra.

Example 5. Consider the same hyper GR-algebra H = {0, 1, 2, 3} in Example 4. Let
A = {a, b} and R̊ = {(a, {0, 1}), (a, {0, 1, 2}), (b, {0, 1}), (b, {0, 1, 3})}. Then

F (a) =
⋃

B⊂H,aR̊B

B = {0, 1, 2}

and
F (b) =

⋃
B⊂H,bR̊B

B = {0, 1, 3}

which are both hyper subGR-algebra of H. Hence, (F,A) is a soft hyper GR-algebra with
respect to R̊.

Example 6. Consider the same hyper GR-algebra H = {0, 1, 2, 3} in Example 4. Let
A = {a, b} and R̊ = {(a, {1, 3}), (a, {0}), (b, {0}), (b, {2})}. Then

F (a) =
⋃

B⊂H,aR̊B

B = {0, 1, 3}

which is a hyper subGR-algebra. Now,

F (b) =
⋃

B⊂H,aR̊B

B = {0, 2}.

However, 0⊛ 0 = {0, 1} ⊈ F (b). Thus, F (b) is not a hyper subGR-algebra. Hence, (F,A)
is not a soft hyper GR-algebra.

Theorem 3. Let (F,A) be a soft hyper GR-algebra over H. If B ⊆ A, then (F,B) is a
soft hyper GR-algebra over H.

Proof: Since (F,A) is a soft hyper GR-algebra it follows that F (a) is a hyper GR-
algebra for all a ∈ A. Since B ⊆ A, F (a) is a hyper GR-algebra over H for all a ∈ B.
Hence, (F,B) is a soft hyper GR-algebra over H.

Theorem 4. Let (F,A) and (G,B) be two soft hyper GR-algebras over H. If A∩B ̸= ∅,

then the intersection (F,A)
∼
∩ (G,B) is a soft hyper GR-algebra over H.



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1487

Proof: Using Definition 6, we can write (F,A)
∼
∩(G,B) = (D,C), where A∩B = C ̸= ∅

and D(x) = F (x) ∩ G(x) for all x ∈ G. Note that D : C → P (H) is a mapping since
the intersection of two hyper GR-algebra is a hyper GR-algebra, thus, (D,C) is a soft
set over H. Since (F,A) and (G,B) are soft hyper GR-algebras over H, it follows that
D(x) = F (x) or D(x) = G(x) for all x ∈ C. Thus, (D,C) is a soft hyper GR-algebra over

H. Hence, (D,C) = (F,A)
∼
∩ (G,B) is a soft hyper GR-algebra over H.

Theorem 5. Let (F,A) and (G,B) be two soft hyper GR-algebras over H. If A∩B = ∅,

then the union (F,A)
∼
∪ (G,B) is a soft hyper GR-algebra over H.

Proof: Using Definition 7, we can write (F,A)
∼
∪ (G,B) = (J,C), where C = A ∪ B,

and for all x ∈ C,

J(x) =


F (x), if x ∈ A \B
G(x), if x ∈ B \A
F (x) ∪G(x), if x ∈ A ∩B.

Since A∩B = ∅, this implies that either x ∈ A\B or x ∈ B \A for all x ∈ C. If x ∈ A\B,
J(x) = F (x). Thus, (J,C) is a soft hyper GR-algebra over H. If x ∈ B \A, J(x) = G(x).

Thus, (J,C) is a soft hyper GR-algebra over H. Hence, (J,C) = (F,A)
∼
∪ (G,B) is a soft

hyper GR-algebra over H.

Theorem 6. If (F,A) and (G,B) are soft hyper GR-algebras overH. Then (F,A)
∼
∧(G,B)

is a soft hyper GR-algebra over H.

Proof: By Definition 8, (F,A)
∼
∧ (G,B) = (J,A× B). Since F (x) and G(y) are hyper

GR-algebras of H, it follows that the intersection (F ∩ G)(x, y), is also a hyper subGR-
algebra of H. Hence, J(x, y) is a hyper subGR-algebra of H for all (x, y) ∈ A×B, and so

(F,A)
∼
∧ (G,B) = (J,A×B) is a soft hyper GR-algebra over H.

Definition 13. A soft hyper GR-algebra (F,A) over H is said to be trivial (respectively,
whole) if F (a) = {0} (respectively, F (a) = X) for all a ∈ A.

Example 7. Let H = {0, 1}. Define ⊛ as shown in the table below.

⊛ 0 1

0 {0} {0}
1 {0} {0}.

By routine calculations, (H;⊛, 0) is a hyper GR-algebra. Let A = H and let F : A →
P (H) be the set-valued function defined as follows:

F (a) =
⋃

B⊂H,aR̊B⇔0⊛x≪B

B.

Then, F (0) = F (1) = F (2) = {0}. Since {0} is a hyper subGR-algebra of H, (F,A) is a
trivial soft hyper GR-algebra of H.



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1488

Example 8. Let H = {0, 1, 2}. Define ⊛ as shown in the table below:

⊛ 0 1 2

0 {0} {0} {0}
1 {0,1,2} {0,1} {0,1}
2 {0,2} {0,1,2} {0,1,2}.

By routine calculations, (H;⊛, 0) is a hyper GR-algebra. Let A = H and let F : A →
P (H) be the set-valued function defined as follows:

F (a) =
⋃

B⊂H,aR̊B⇔x⊛0≪B

B,

where B ⊂ H and define a relation R̊ such that aR̊B if and only if x⊛ 0 ≪ B and a ∈ A.
Then F (0) = F (1) = F (2) = {0, 1, 2} = H. Since H is a hyper GR-algebra of H, (F,A)
is whole soft hyper GR-algebra of H.

Lemma 2. Let f : H → Y be a homomorphism of hyper GR-algebras. If (F,A) is a soft
hyper GR-algebra over H, then (f(F ), A) is a soft hyper GR-algebra over Y .

Proof: Let a ∈ A. Since F (a) is a hyper subGR-algebra on H and f is a homomor-
phism, it follows that f(F )(a) = f(F (a)) is a hyper subGR-algebra on Y by Lemma 1.
Hence, (f(F ), A) is a soft hyper GR-algebra on Y .

Theorem 7. Let f : H → Y be a homomorphism of hyper GR-algebras and let (F,A) be
a soft hyper GR-algebra over H.

(i) (f(F ), A) is trivial soft hyper Gr-algebra over Y if and only if F (x) ⊆ ker f for all
x ∈ A.

(ii) If f is onto and (F,A) is whole, then (f(F ), A) is a whole soft hyper GR-algebra
over Y .

(iii) If (f(F ), A) is whole and f is one-to-one, then f is onto and (F,A) is a whole soft
hyper GR-algebra over Y .

(iv) If f is bijective, then (F,A) is whole over H if and only if (f(F ), A) is whole over
H.

Proof: (i) Suppose F (a) ⊆ kerf for all a ∈ A. Then f(F )(a) = f(F (a)) = {0Y } for
all a ∈ A. Hence, (f(F ), A) is a trivial soft hyper GR-algebra over Y by Definition 13
and Lemma 2. Conversely, suppose that (f(F ), A) is a trivial soft hyper GR-algebra over
Y . Then f(F )(a) = f(F (a)) = {0Y } for all a ∈ A. This means that F (a) ⊆ ker f , for all
a ∈ A.
(ii) Assume that f is onto and (F,A) is whole. Then F (a) = H for all a ∈ A, and so
f(F )(a) = f(F (a)) = H for all a ∈ A. It follows from Definition 13 and Lemma 2 that
(f(F ), A) is a whole soft hyper GR-algebra.



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1489

(iii) Suppose (f(F ), A) is whole. Then f(F )(a) = f(F (a)) = H for all a ∈ A. Thus,
f(H) = Y since F (a) ⊆ H for all a ∈ A. Hence, f is onto. Now, let a ∈ A and z ∈ H.
Since f(z) ∈ Y = f(F (a)), there exists x ∈ F (a) such that f(x) = f(z). Since f is
one-to-one, it follows that x = z ∈ F (a). Therefore, F (a) = H implying that (F,A) is
whole.
(iv) The proof follows from (ii) and (iii).

Definition 14. Let (F,A) and (G,C) be two soft hyper GR-algebras over H. Then (F,A)

is called a soft hyper subGR-algebra of (G,C), written as (F,A)
∼
< (G,C), if it satisfies

the following:

(i) A ⊆ C

(ii) F (a) is a hyper subGR-algebra of G(a) for all a ∈ A.

Example 9. Consider the hyper GR-algebra in Example 8. For C = H, let G : C → P (H)
be the set-valued function defined by

G(a) =
⋃

B⊂H,aR̊B⇔B≪a

B,

for all a ∈ C. Then
G(0) =

⋃
B⊂H,0R̊B⇔B≪0

B = {0},

G(1) =
⋃

B⊂H,1R̊B⇔B≪1

B = {0, 1, 2}

and
G(2) =

⋃
B⊂H,2R̊B⇔B≪2

B = {0, 1, 2}.

Let A = {0, 1, 2} and F : A → P (H) be the set-valued function defined by

F (a) =
⋃

B⊂H,aR̊B⇔B=an

B,

where an = ((((a⊛ a)⊛ a)⊛ a)⊛ ...⊛ a) for all a ∈ A. Then

F (0) =
⋃

B⊂H,0R̊B⇔B=0n

B = {0},

F (1) =
⋃

B⊂H,1R̊B⇔B=1n

B = {0, 1}

and
F (2) =

⋃
B⊂H,2R̊B⇔B=2n

B = {0, 1, 2}.

Hence, F (0), F (1) and F (2) are soft hyper subGR-algebras on G(0), G(1) and G(2), re-

spectively. Therefore, (F,A)
∼
< (G,C).



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1490

Theorem 8. Let (F,A) and (G,A) be two soft hyper GR-algebras over H. Then (F,A)
∼
<

(G,A) if and only if F (a) ⊆ G(a) for all a ∈ A.

Proof: Let (F,A)
∼
< (G,A). By Definition 14, F (a) is a hyper subGR-algebra of G(a)

for all a ∈ A. This implies that F (a) ⊆ G(a) for all a ∈ A. Conversely, let F (a) ⊆ G(a)
for all a ∈ A. Since (F,A) and (G,A) are soft hyper GR-algebras over H and F (a) ⊆ G(a)
for all a ∈ A, it follows that F (a) is a hyper subGR-algebra of G(a) for all a ∈ A. Hence,

(F,A)
∼
< (G,A).

Theorem 9. Let (F,A) be soft hyper GR-algebra over H and let (G1, C1) and (G2, C2)
be two soft hyper subGR-algebras of (F,A). Then

(i) (G1, C1)
∼
∩ (G2, C2)

∼
< (F,A)

(ii) C1 ∩ C2 = ∅ =⇒ (G1, C1)
∼
∪ (G2, C2)

∼
< (F,A).

Proof: (i) By Definition 6, we can write (G1, C1)
∼
∩(G2, C2) = (G,C), where C = C1∩C2

and G(a) = G1(a)∩G2(a) for all a ∈ C. Since G1(a), G2(a) ⊆ F (a), G1(a)∩G2(a) ⊆ F (a)
for all a ∈ C. Thus, G(a) = G1(a)∩G2(a) is a hyper subGR-algebra of F (a) for all a ∈ C.

(ii) Assume that C1∩C2 = ∅. By Definition 7, we can write (G1, C1)
∼
∪ (G2, C2) = (G,C),

where C = C1 ∪ C2 and

G(x) =


G1(a), if a ∈ C1 \ C2

G2(a), if a ∈ C2 \ C1

G1(a) ∪G2(a), if a ∈ C1 ∩ C2

for all a ∈ C. By the hypothesis, C1, C2 ⊆ A. This implies that C = C1 ∪ C2 ⊆ A.
Also, Gi(a) is a hyper subGR-algebra of F (a) for all a ∈ Ci, i = 1, 2. Since C1 ∩ C2 =
∅, by Theorem 5, G(a) is a hyper subGR-algebra of F (a) for all a ∈ C. Therefore,

(G1, C1)
∼
∪ (G2, C2)

∼
< (F,A).

Definition 15. Let (F,A) be a soft set over hyper GR-algebra H. A soft set (G, I) over

H is called a soft hyper GR-ideal of (F,A), written as (G,M)
∼⋄ (F,A) if the following are

satisfied:

(i) I ⊂ A with I ̸= ∅

(ii) for all a ∈ I, G(a) is a hyper GR-ideal on F (a).

Example 10. Consider the same hyper GR-algebra H = {0, 1, 2, 3} in Example 4. Let
A = H and I = {0, 1}. Suppose F : A → P (H) is defined by F (a) =

⋃
b⊂H,aR̊B

B with

R̊ = {(0, {1}), (0, {0, 2}), (1, {0, 2}), (1, {0, 1, 3})}. Then

F (0) =
⋃

B⊂H,0R̊B

B = {0, 1, 2}



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1491

and
F (1) =

⋃
B⊂H,1R̊B

B = {0, 1, 2, 3}.

DefineG(a) =
⋃

B⊂H,aR̊B⇔B=an
B for some n ∈ N. ThenG(0) = {0, 1}, G(1) = {0, 1}, G(2) =

{0, 1, 2} and G(3) = {0, 1, 3}.

a b a ⊛ b ≪ I a ∈ I

0 0 {0,1} ✓ ✓
0 1 {0,1} ✓ ✓
1 0 {1} ✓ ✓
2 0 {0,2} ×
2 1 {0,2} ×

Table 3.1

a b a ⊛ b ≪ G(1) = {0, 1} a ∈ G(1) = {0, 1}
0 0 {0,1} ✓ ✓
1 0 {1} ✓ ✓
1 1 {0,1} ✓ ✓.

2 0 {0,2} ×
2 1 {0,2} ×
3 0 {3} ×
3 1 {0,1,3} ×

Table 3.2

Note that G(0) is a hyper GR-ideal of F (0) (see Table 3.1) and G(1) is a hyper GR-ideal
of F (1) is a hyper GR-ideal (see Table 3.2). Hence, for all a ∈ I, G(a) is a hyper GR-ideal
of F (a). Therefore, (G, I) is a soft hyper GR-ideal of (F,A).

Theorem 10. Let (F,A) be a soft hyper GR-algebra overH. Suppose (G,M1) and (J,M2)

are two soft hyper GR-ideals of (F,A) such that M1 ∩M2 ̸= ∅. Then (G,M1)
∼
∩ (J,M2)

is a soft hyper GR-ideal of (F,A).

Proof: Suppose (G,M1) and (J,M2) are two soft hyper GR-ideals such that M1∩M2 ̸=
∅. Take M = M1 ∩M2. Clearly, M ⊆ A and by hypothesis M ̸= ∅. By Theorem 2,M
is a hyper GR-ideal. Let m ∈ M . Then this implies that m ∈ M1 and m ∈ M2. Also,

G(m) ⋄ F (m) and J(m) ⋄ F (m). Thus, [G(m) ∩ J(m)] ⋄ F (m). Hence, (G,M1)
∼
∩ (J,M2)

is a soft hyper GR-ideal of (F,A).
If M = M1 = M2, then we have the following Corollary.

Corollary 1. Let (F,A) be a soft hyper GR-algebra over H. For any soft sets (G,M)

and (J,M) over H, we have (G,M)
∼⋄ (F,A) =⇒ (G,M)

∼
∩ (J,M)

∼⋄ (F,A).

Theorem 11. Let (F,A) be a soft hyper GR-algebra over H. For any soft sets (G, I) and
(J,K) with I ∩K = ∅, we have

(G, I)
∼⋄ (F,A), (J,K)

∼⋄ (F,A) =⇒ (G, I)
∼
∪ (J,K)

∼⋄ (F,A).



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1492

Proof: Using Definition 7, we can write (G, I)
∼
∪ (J,K) = (R,U), where U = I ∪ K,

and for all x ∈ U ,

R(x) =


G(x), if x ∈ I \K
J(x), if x ∈ K \ I
G(x) ∪ J(x), if x ∈ I ∩K.

Now, I ∩K = ∅ implies either x ∈ I \K or x ∈ K \ I for all x ∈ U . If x ∈ I \K, then
R(x) = G(x) ⋄ F (x). If x ∈ K \ I, then R(x) = J(x) ⋄ F (x). Thus, R(x) ⋄ F (x) for all

x ∈ U . Hence, (G, I)
∼
∪ (J,K) = (R,U)

∼⋄ (F,A).

Theorem 12. Let (F,A) be a soft hyper GR-algebra over H. If (G,M) and (J,K) are

soft hyper GR-ideals of (F,A), then (G,M)
∼
∧ (J,K) is a soft hyper GR-ideal of (F,A).

Proof: Using Definition 8, we can write (G,M)
∼
∧(J,K) = (R,M×K), where R(x, y) =

G(x) ∩ J(y) for all (x, y) ∈ M ×K. By Theorem 10,

G(x) ⋄ F (x), J(y) ⋄ F (x) =⇒ G(x) ∩ J(y) ⋄ F (x)

=⇒ R(x, y) ⋄ F (x) for all (x, y) ∈ M ×K.

Therefore, (G,M)
∼
∧ (J,K) = (R,M ×K) is a soft hyper GR-ideal of (F,A).

Definition 16. Let S be a hyper subGR-algebra on H. A subset M of H is a hyper
GR-commutative ideal of H related to S denoted by M ⋄hgrc S if it satisfies the following:

(i) 0 ∈ M ;

(ii) (x⊛ y)⊛ z ⊆ M and z ∈ M imply that x⊛ (y ⊛ (y ⊛ x)) ⊆ M for all x, y ∈ S.

Example 11. Consider the same hyper GR-algebra H = {0, 1, 2, 3} in Example 4. Then
S = {0, 1, 3} is a hyper subGR-algebra of H and M = {0, 1, 2} is a hyper GR-commutative
ideal of S.

Definition 17. Let (F,A) be a soft set over a hyper GR-algebra H. A soft set (G,M) over

H is called a soft hyper GR-commutative ideal of (F,A) denoted by (G,M)
∼⋄hgrc (F,A),

if the following are satisfied:

(i) M ⊂ A with M ̸= ∅;

(ii) for all a ∈ M , G(a) ⋄hgrc F (a).

Example 12. Consider the same hyper GR-algebra H = {0, 1, 2, 3} in Example 4. Let
M = {0, 1, 2} and (F,A) be a soft set over H, where A = H. Let F : A → P (H) be
defined by

F (a) =
⋃

B⊂H,aR̊B

B



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1493

with R̊ = {(0, 0), (0, {1, 2}), (1, {0, 1}), (1, {1}), (2, {0, 1}), (2, {0, 3})}. Then

F (0) =
⋃

B⊂H,0R̊B

B = {0, 1, 2}

F (1) =
⋃

B⊂H,1R̊B

B = {0, 1}

and F (2) =
⋃

B⊂H,2R̊B

B = {0, 1, 3}.

Also, let G : A → P (H) be a set-valued function defined by G(a) =
⋃

B⊂H,aR̊B⇔B=an
B,

where an = ((((a ⊛ a) ⊛ a) ⊛ a) ⊛ ... ⊛ a). Then G(0) = G(1) = {0, 1}, G(2) = {0, 1, 2}
and G(3) = {0, 1, 3}. Then G(0) = {0, 1} ⋄hgrc F (0) = {0, 1, 2}, G(1) = {0, 1} ⋄hgrc F (1) =
{0, 1} since they are equal, and G(2) = {0, 1, 2} ⋄hgrc F (2) = {0, 1, 3}. Hence, for all

a ∈ M,G(a)
∼⋄hgrc F (a).

Theorem 13. Let S be a hyper subGR-algebra of a hyper GR-algebra H. If I1 ⋄hgrc S
and I2 ⋄hgrc S, then I1 ∩ I2 ⋄hgrc S.

Proof: Since 0 ∈ I1 and 0 ∈ I2, 0 ∈ I1 ∩ I2. Suppose x, y ∈ S and (x⊛ y)⊛ z ⊆ M =
I1 ∩ I2 with z ∈ I1 ∩ I2. Since I1 ⋄hgrc S and I2 ⋄hgrc S, it follows that (x1 ⊛ y1)⊛ z1 ⊆ M
and (x2⊛ y2)⊛ z2 ⊆ M implying that x1⊛ (y1⊛ (y1⊛x1)) ⊆ M and x2⊛ (y2⊛ (y2⊛x2)),
respectively, for all x1, x2, y1, y2 ∈ S and for all z1, z2 ∈ M . Let x ⊆ I1, y ⊆ I2 and z ⊆ M .
Consider that x1, x2, y1, y2 ⊆ S and for all z1, z2 ⊆ M such that x = x1 ∩ x2, y = y1 ∩ y2
and z = z1 ∩ z2. Then

(x⊛ y)⊛ z = [(x1 ∩ x2)⊛ (y1 ∩ y2)]⊛ (z1 ∩ z2)

= [(x1 ⊛ y1) ∩ (x2 ⊛ y2)]⊛ (z1 ∩ z2)

= [(x1 ⊛ y1)⊛ (z1 ∩ z2)] ∩ [(x2 ⊛ y2)⊛ (z1 ∩ z2)]

=
[
(x′1 ⊛ y′1)⊛ z′1

]
∩
[
(x′2 ⊛ y′2)⊛ z′2

]
⊆ I1 ∩ I2 = M.

By Definition 16, x⊛ (y ⊛ (y ⊛ x)) ⊆ M . Hence, M = I1 ∩ I2 ⋄hgrc S.

Theorem 14. Let (F,A) be a soft hyper GR-algebra over H. For any soft sets (G1,M1)
and (G2,M2) over H, where M1 ∩M2 ̸= ∅ we have

(G1,M1)
∼⋄hgrc (F,A), (G2,M2)

∼⋄hgrc (F,A) =⇒ (G1,M1)
∼
∩ (G2,M2)

∼⋄hgrc (F,A).

Proof: By Definition 6, we write (G1,M1)
∼
∩ (G2,M2) = (G,M) where M = M1 ∩M2

and G(a) = G1(a) ∩ G2(a) for all a ∈ M . Clearly, M ⊆ A. By Theorem 13, G1(a) ∩
G2(a) ⋄hgrc F (a). Hence, (G1,M1)

∼
∩ (G2,M2) = (G,M)

∼⋄hgrc (F,A).

If M = M1 = M2, then we have the following Corollary.



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1494

Corollary 2. Let (F,A) be a soft hyper GR-algebra over H. For any soft sets (G,M)
and (J,M) over H, we have

(G,M)
∼⋄hgrc (F,A), (J,M)

∼⋄hgrc (F,A) =⇒ (G,M)
∼
∩ (J,M)

∼⋄hgrc (F,A).

Theorem 15. Let (F,A) be a soft hyper GR-algebra over H. For any soft sets (G,M)
and (J,N), with M ∩N = ∅, we have

(G,M)
∼⋄hgrc (F,A), (J,N)

∼⋄hgrc (F,A) =⇒ (G,M)
∼
∪ (J,N)

∼⋄hgrc (F,A).

Proof: Using Definition 7, we can write (G,M)
∼
∪ (J,N) = (R,U), where U = M ∪N ,

and for all x ∈ U ,

R(x) =


G(x), if x ∈ M \N
J(x), if x ∈ N \M
G(x) ∪ J(x), if x ∈ M ∩N.

Since M ∩ N = ∅, it implies that either x ∈ M \ N or x ∈ N \ M , for all x ∈ U . If
x ∈ M \ N , R(x) = G(x) ⋄hgrc F (x). If x ∈ N \ M , R(x) = J(x) ⋄hgrc F (x). Thus,

R(x) ⋄hgrc F (x) for all x ∈ U . Hence, (G,M)
∼
∪ (J,N) = (R,U)

∼⋄hgrc (F,A).

Definition 18. Let E be a hyper GR-algebra. Given a hyper subGR-algebra A of E, let
(F,A) ∈ S(U). Then (F,A) is called a union-soft hyper GR-algebra over U if fA satisfies

fA(x⊛ y) ⊆ fA(x) ∪ fA(y),∀x, y ∈ A.

Example 13. Consider the hyper GR-algebra H = {0, 1, 2, 3} defined in Example 4. Let
τ1, τ2, τ3 be subsets of H such that τ1 ⊆ τ2 ⊆ τ3. Define a soft set (F,A) as follows:

(F,A) = {(0, τ1), (1, τ1), (2, τ2), (3, τ3)}.

By routine calculations, (F,A) is a union-soft hyper GR-algebra.

Example 14. Consider the hyper GR-algebra H = {0, 1, 2, 3} defined in Example 4. Let
τ1, τ2, τ3, τ4 be subsets of H such that τ1 ⊊ τ2 ⊊ τ3 ⊊ τ4. Define a soft set (F,A) as follows:

(F,A) = {(0, τ1), (1, τ2), )(2, τ3), (3, τ4)}.

By routine calculations, (F,A) is not a union-soft hyper GR-algebra since fA(0 ⊛ 0) =
τ2 ⊈ fA(0) ∪ fA(0) = τ1.

Theorem 16. Let E be a hyper GR-algebra. Given a hyper subGR-algebra A of E, let
(F,A) ∈ S(U). Then (F,A) is a union-soft hyper GR-algebra over U if and only if the
nonempty τ -exclusive set of (F,A) is a hyper subGR-algebra of A for all τ ⊆ U .



M.K. Engcot, G. Petalcorin / Eur. J. Pure Appl. Math, 15 (4) (2022), 1482-1497 1495

Proof: Assume (F,A) is the union-soft hyper GR-algebra over U . Let τ ⊆ U and
x, y ∈ e((F,A); τ). Then fA(x) ⊆ τ and fA(y) ⊆ τ . It follows from the definition that
fA(x ⊛ y) ⊆ fA(x) ∪ fA(y) ⊆ τ. Hence, x ⊛ y ⊆ e((F,A); τ) and so e((F,A); τ) is a
hyper subGR-algebra. Conversely, suppose that the nonempty τ -exclusive set of (F,A)
is a hyper subGR-algebra of A for all τ ⊆ U . Let x, y ∈ A such that fA(x) = τ1 and
fA(y) = τ2. Take τ = τ1 ∪ τ2. Then x, y ∈ e((F,A); τ) and so x ⊛ y ⊆ e((F,A); τ).
Thus, fA(x ⊛ y) ⊆ τ = τ1 ∪ τ2 = fA(x) ∪ fA(y). Hence, (F,A) is a union-soft hyper
GR-algebra.

Theorem 17. Let E be a hyper GR-algebra. Given a hyper subGR-algebra A of E,
let (F,A) ∈ S(U). Suppose (F,A)∗ ∈ S(U) with approximation function f∗

A defined by
f∗
A : E → P (U),

x 7−→

{
fA(x), if x ∈ e((F,A); τ)

U, otherwise.

If (F,A) is a union-soft hyper GR-algebra over U , then so is (F,A)∗.

Proof: Since (F,A) is a union-soft hyper GR-algebra over U , it follows from Theorem
16 that e((F,A); τ) is a hyper subGR-algebra of A for all τ ⊆ U . Let x, y ∈ A. If
x, y ∈ e((F,A); τ), then x⊛y ⊆ e((F,A); τ) and so f∗

A(x⊛y) = fA(x⊛y) ⊆ fA(x)∪fA(y) =
f∗
A(x) ∪ f∗

A(y). x /∈ e((F,A); τ) or y /∈ e((F,A); τ), then f∗
A(x) = U or f∗

A(y) = U . Hence,
fA(x⊛ y) ⊂ U = f∗

A(x) ∪ f∗
A(y). Therefore, (F,A)∗ is a union-soft hyper GR-algebra over

U .

Definition 19. Let E be a hyper GR-algebra. Given a hyper subGR-algebra A of E, let
(F,A) ∈ S(U). Then (F,A) is called a union-soft hyper GR-ideal over U if fA(x) satisfies
the following:

(i) fA(0) ⊆ fA(x), ∀x ∈ A

(ii) fA(x) ⊆ fA(x⊛ y) ∪ fA(y),∀x, y ∈ A.

Example 15. Consider the hyper GR-algebra H = {0, 1, 2, 3} defined in Example 4. Let
τ1, τ2, τ3, τ4 be subsets of H such that τ1 ⊆ τ2 ⊆ τ3 ⊆ τ4. Define a soft set (F,A) as follows:

(F,A) = {(0, τ1), (1, τ2), (3, τ3), (4, τ4)}.

By routine calculations, (F,A) is a union-soft hyper GR-ideal.

Example 16. Consider the H = {0, 1, 2} defined in the Cayley table below.

⊛ 0 1 2

0 {0} {0} {0}
1 {0,1} {0,1} {0,1}
2 {0,2} {0,1} {0,1,2}.



REFERENCES 1496

By routine calculations, (H;⊛, 0) is hyper GR-algebra. Let τ1, τ2, τ3 be subsets of H
such that τ1 ⊊ τ2 ⊊ τ3. Define a soft set (F,A) as follows:

(F,A) = {(0, τ1), (1, τ2), (2, τ3)}.

By routine calculations, (F,A) is not a union-soft hyper GR-ideal since F (2) = τ3 ⊈
F (2⊛ 1) ∪ F (1) = τ2.

Theorem 18. Let E be a hyper GR-algebra. Given a hyper subGR-algebra A of E,
suppose (F,A) ⊆ S(U). If (F,A) is a union-soft hyper GR-ideal over U , then for all
x, y ∈ A, fA(x) ⊆ [fA(x⊛ y) ∩ fA(x)] ∪ [fA(x⊛ y) ∩ fA(y)] ∪ [fA(x) ∩ fA(y)] ∪ fA(y).

Proof: Note that fA(x) ⊆ fA(x) ∪ fA(y) and fA(x) ⊆ fA(x⊛ y) ∪ fA(y). This implies
that fA(x) = [fA(x⊛ y) ∪ fA(y)] ∩ [fA(x) ∪ fA(y)]. Hence, fA(x) ⊆ [fA(x⊛ y) ∩ fA(x)] ∪
[fA(x⊛ y) ∩ fA(y)] ∪ [fA(x) ∩ fA(y)] ∪ fA(y).

Theorem 19. Let E be a hyper GR-algebra. Given a hyper subGR-algebra A of E,
suppose (F,A) ⊆ S(U). If (F,A) is a union-soft hyper GR-ideal over U , then the nonempty
τ -exclusive set of (F,A) is an ideal of A for all τ ⊆ U .

Proof: Let (F,A) be a union-soft hyper GR-ideal over U . Let τ ⊂ U such that
e((F,A); τ) ̸= ∅. Then for some x ∈ A, fA(x) ⊆ τ . It follows from Definition 19 (i) that
fA(0) ⊆ fA(x) ⊆ τ . Thus, 0 ∈ e((F,A); τ). Now let x, y ∈ A such that x⊛y ⊆ e((F,A); τ)
and y ⊆ e((F,A); τ). Hence, fA(x ⊛ y) ⊆ τ and fA(y) ⊆ τ . By Definition 19(ii) fA(x) ⊆
fA(x⊛ y) ∪ fA(y) ⊆ τ . Hence, x ∈ e((F,A); τ) and so e((F,A); τ) is an ideal of A.

4. Conclusion

In this paper, the notion of soft hyper GR-algebra is presented. Some of its properties
and characterization are also presented such as the soft hyper subGR-algebra, the soft
hyper GR-ideal, the soft hyper GR-commutative ideal and the union-soft hyper GR-ideal.

Acknowledgements

This research is funded by the Commission on Higher Education (CHED), University
of San Carlos, Philippines and Mindanao State University-Iligan Institute of Technology,
Philippines.

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