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On the Left and Right Almost Hyperideals of LA-Semihypergroups
International Journal of Fuzzy Logic and Intelligent Systems 2021;21(1):86-92
Published online March 25, 2021
© 2021 Korean Institute of Intelligent Systems.

Shah Nawaz1, Muhammad Gulistan1 , Nasreen Kausar2 , Salahuddin3 , and Mohammad Munir4

1Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
2Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan
3Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia
4Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
Correspondence to: Nasreen Kausar (kausar.nasreen57@gmail.com)
Received December 17, 2020; Revised January 29, 2021; Accepted February 22, 2021.
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
In this paper, we define left almost hyperideals, right almost hyperideals, almost hyperideals, and minimal almost hyperideals. We demonstrate that the intersection of almost hyperideals is not required to be an almost hyperideal, but the union of almost hyperideals is an almost hyperideal, which is completely different from the classical algebraic concept of the ideal theory.
Keywords : LA-semihypergroups, Left almost hyperideals, Right almost hyperideals, Almost hyperideals, Minimal almost hyperideals
1. Introduction

Hyperstructures were introduced in 1934 when Marty [1] defined hypergroups, began to study their properties, and applied them to groups. Several papers and books have been written on hyperstructure theory [2,3]. Currently, a book published on hyperstructures [4] describes its applications in rough set theory, cryptography, automata, code automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs.

Kazim and Naseeruddin [5] introduced the study of left almost semigroups (LA-semigroups). They generalized some useful sequels of semigroup theory. Subsequently, Mushtaq and his colleagues [6–13] further described the structure and included many useful results of theory of an LA-semigroup [7,14–20]. An LA-semigroup is the midway structure between a commutative semigroup and a groupoid. Moreover, it has many interesting properties, which are frequently used in commutative and associative algebraic structures.

Hila and Dine [21] presented the concept of LA-semihypergroups. Their paper created a new research direction in non-associative hyperstructures. Yaqoob et al. [22] extended the work of Hila and Dine [21] and characterized intra-regular LA-semihypergroups by their hyperideals using pure left identities. Subsequently, many researchers studied the structures of LA-semihypergroups from different perspectives [23–30].

Azhar and his colleagues [31–34] developed the structures of partially ordered LA-semihypergroups. Nawaz and his colleagues [35–41] developed and studied many non-associative structures. The concept of almost ideals in semigroups was provided by Grosek and Satko [42,43] in 1980 and 1981, respectively. Various other researchers studied almost ideals for different algebraic structures [44–46].

Recently, for the first time, Suebsung et. al [48] developed the concept of almost hyperideals in semihypergroups (see also [48,49]).

Thus, inspired by the concept by Suebsung et. al [47], we developed structures of almost hyperideals in LA-semihypergroups. These new types of almost hyperideals have two main characteristics. First, they are purely non-associative substructures of LA-semihypergroups. Second, in the construction of these hyperideals, we ensured that a hyperoperation did not yield a single element as in the scenario of the paper by Suebsung et. al [47]. We proved some useful results related to almost hyperideals.

2. LA-Semihypergroups

In this section, we recall some basic concepts from the literature on ideals and LA-semihypergroups that were used in the further development of this article.

Definition 1

A map ◦ : X × X –→ ℘*(X) is called an hyperoperation or join operation on the set X, where X is a non-empty set and ℘*(X) = ℘(X)\{∅} indicates the all non-empty subsets of X. A hypergroupoid is a set (X) for which the binary operation is a hyperoperation.

If A and B are two non-empty subsets of X, then we express the product as follows:

A∘B=∪x1∈A,x2∈Bx1∘x2, x1∘A={x1}∘A         and x1∘B={x1}∘B.

Definition 2 ([21,22])

A hypergroupoid (X, ◦) is called an LA-semihypergroup if for x1, x2, x3 ∈ X,

(x1∘x2)∘x3=(x3∘x2)∘x1.

The law (x1 ◦x2)◦x3 = (x3 ◦x2)◦x1 is known as left invertive law.

Every LA-semihypergroup satisfies the law

(x1∘x2)∘(x3∘x4)=(x1∘x3)∘(x2∘x4)

for all x1, x2, x3, x4 ∈ X. This law is known as medial law (cf. [21]).

Definition 3. [22]

Let X be an LA-semihypergroup. Element e ∈ X is called a

1. Left identity (resp. pure left identity) if for all x1 ∈ X, x1 ∈ e ◦ x1 (resp. x1 = e ◦ x1),

2. Right identity (resp. pure right identity) if for all x1 ∈ X, x1 ∈ x1 ◦ e (resp. z1 = z1 ◦ e),

3. Identity (resp. pure identity) if for all z1 ∈ H, z1 ∈ e ◦ z1 ∩ z1 ◦ e (resp. x1 = e ◦ z1 ∩ z1 ◦ e).

Example 1 ([22])

Let X = {l1, l2, l3, l4} with the binary hyperoperation defined as follows:

◦ l1 l2 l3 l4
l1 l1 l2 l3 l4
l2 l3 {l2, l3} {l2, l3} l4
l3 l2 {l2, l3} {l2, l3} l4
l4 l4 l4 l4 H

Hence X is an LA-semihypergroup because it satisfies the left invertive law. AS the table shows, l1 is a left identity of X.

Lemma 1 ([22])

Let X be an LA-semihypergroup with pure left identity e; then x1 ◦ (x2 ◦ x3) = x2 ◦ (x1 ◦ x3) holds for all x1, x2, x3 ∈ X.

Lemma 2 ([22])

Let X be an LA-semihypergroup with pure left identity e; then (x1 ◦ x2) ◦ (x3 ◦ x4) = (x4 ◦ x2) ◦ (x3 ◦ x1) holds for all x1, x2, x3, x4 ∈ X.

The law (x1 ◦x2)◦(x3 ◦x4) = (x4 ◦x2)◦(x3 ◦x1) is called a paramedial law.

Definition 4 ([47])

Let (H, ◦) be a semihypergroup.

(1) A non-empty subset L of H is called the left almost hyperideal of H if

x∘L∩L≠φ,∀x∈H.

(2) A non-empty subset L of H is called the right almost hyperideal of H if

L∘x∩L≠φ,∀x∈H.

(3) A non-empty subset L of H is called the almost hyper-ideal of H if L is both a left and right almost hyperideal of H.

3. Almost Hyperideals

Here we define left almost hyperideals, right almost hyperideals, almost hyperideals, minimal almost hyperideals, and some interesting properties.

Definition 5

Let (H, *) be an LA-semihypergroup.

(1) A non-empty subset L of H is called the left almost hyperideal of H if

x*L∩L≠φ,∀x∈H.

(2) A non-empty subset R of H is called the right almost hyperideal of H if

R*x∩R≠φ,∀x∈H.

(3) A non-empty subset I ofH is called the almost hyperideal of H if I is both a left and right almost hyperideal of H.

Example 2

Let H = {e, l, m} with the binary operation * defined as follows:

* e l m
e e l m
l m {l, m} m
m l l {l, m}

(H, *) is an LA-semihypergroup. Now, if L = {e, l} ⊆ H with we observe that (L, *) is a left almost hyperideal of (H, *), but it is not a right almost hyperideal as {e, l} * m ∩ {e, l} = φ.

* e l
e e l
l m {l, m}

Remark 1

The structure of left almost hyperideals is not LA-subsemihypergroups as in Example 2, where L = {e, l} ⊆ H is not closed under the same binary operation as H.

Example 3

Consider an LA-semihypergroup H = {a, b, c} under the hyperoperation defined by

◦ a b c
a a b c
b c {b, c} {b, c}
c b {b, c} {b, c}

If R = {b, c} ⊆ H with the same binary operation ◦ that is defined in H, we can observe that R is a right almost hyperideal of H.

Lemma 3

Let (H, *) be a LA-semihypergroup.

(1) Every left hyperideal of H is a left almost hyperideal of H.

(2) Every right hyperideal of H is a left almost hyperideal of H.

(3) Every hyperideal of H is an almost hyperideal of H.

Proof

(1) Assume that L is a left hyperideal of H. Let x ∈ H. Then, x * L ⊆ L. Therefore, x * L ∩ L ≠ φ.

(2) Let R be a right hyperideal of H. Let y ∈ H. Then, R * y ⊆ R. Therefore, R * y ∩ R ≠ φ.

(3) Follows from (1) and (2)

Remark 2

The converse of Lemma 3 may or may not true generally as in the following Example 4.

Example 4

Let H = {x, y, z} be a finite set with hyperoperation * defined as follows:

* x y z
x x x x
y x H z
z x H H

Thus, (H, *) is an LA-semihypergroup. Let L = {x, y} under the same binary operation *, i.e.,

* x y
x x x
y x H

Here, L is an almost hyperideal of H, but it is not a hyperideal of H.

Theorem 1

Let (H, *) be an LA-semihypergroup.

(1) If L1 and L2 are two left almost hyperideals of H, then L1 ∪ L2 is also a left almost hyperideal.

(2) If R1 and R2 are two right almost hyperideals of H, then R1 ∪ R2 is also a right almost hyperideal.

(3) If I1 and I2 are two almost hyperideals of H, then I1 ∪I2 is also an almost hyperideal.

Proof

Same as provided by Suebsung et. al [47].

Remark 3

(1) If L1 and L2 are two left almost hyperideals of H, then L1 ∩ L2 may or may not a left almost hyperideal.

(2) If R1 and R2 are two right almost hyperideals of H, then R ∩ R2 may or may not a right almost hyperideal.

(3) If I1 and I2 are two almost hyperideals of H, then I1 ∩I2 may or may not left almost hyperideal.

Example 5

Let H = {1, 2, 3, 4, 5, 6} be a non-empty set with hyperoperation *, defined as follows: then (H, *) is an LA-semihypergroup. Let L1 = {1, 2, 3, 4} and L2 = {3, 4, 5, 6} be two left almost hyperideals of H. Thus, L1 ∩ L2 = {3, 4} is not a left almost hyperideal of H, as we can observe that 4 * {3, 4} ∩ {3, 4} = φ. However, L1 ∪ L2 = {1, 2, 3, 4, 5, 6} is a left almost hyperideal of H.

* 1 2 3 4 5 6
1 {1, 2, 3, 4} {2, 3, 4, 5, 6} {4, 5, 6} {2, 3, 4, 5, 6} {2, 3, 4, 5, 6} {2, 3, 4, 5, 6}
2 {1, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6} {1, 3, 4, 5, 6} {1, 3, 6} {1, 5, 6} {1, 3, 4, 5, 6}
3 {1, 2, 4, 5, 6} {1, 2} {1, 2, 3, 5, 6} {1, 2, 4, 5, 6} {1, 2, 6} {1, 2, 4, 5, 6}
4 {1, 5, 6} {1, 2, 6} {1, 2, 5, 6} {1, 2, 5, 6} {1, 2, 5, 6} {1, 2, 5, 6}
5 {1, 2, 3, 4, 6} {1, 2, 3, 4, 6} {1, 2, 3, 4, 6} {1, 3, 4, 6} {1, 2, 3, 4, 5, 6} {1, 3, 4, 6}
6 {1, 2, 3, 4, 5} {1, 4, 5} {1, 2, 3, 4, 5} {1, 4, 5} {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}

Lemma 4

If L is a left almost hyperideal of an LA-semihyper-group H with a left identity e; then, aL is a left almost hyper-ideal of H.

Proof

However, if aL is not a left almost hyperideal, i.e.,

x*aL∩aL=φ for all x∈H⇒a*xL∩aL=φ,by (ab)c=b(ac)⇒xL∩L=φ,

this is a contradiction to the given condition. Hence, x * aL ∩ aL ≠ φ and aL is a left almost hyperideal of H.

Lemma 5

If L is a left almost hyperideal of an LA-semihypergroupH with a left identity e, then L2 is a left almost hyperideal of H.

Proof

However, if L2 is not a left almost hyperideal, i.e.,

x*L2∩L2=φ for all x∈H,⇒x*L*L∩L*L⇒(x*L)*L∩L*L⇒L*(x*L)∩L*L=φ, by (ab)c=b(ac)⇒xL∩L=φ,

this is a contradiction to the given condition. Hence, x * L2 ∩ L2 ≠ φ and L2 is a left almost hyperideal of H.

Lemma 6

If I is a left almost hyperideal of H, then I2 is an almost hyperideal of H.

Proof

First, we must demonstrate that I2 is a left-almost hyperideal of H. This, we demonstrate that x * I2 ∩ I2 ≠ φ ⇒ x*I *I∩I *I, as x*I∩I ≠ φ. Hence x*I2∩I2 ≠ φ Therefore, I2 is a left almost ideal of H. Next, we must demonstrate that I2 *x∩I2 ≠ φ ⇒ (I * I) *x∩I *I, as (x * I) *I ∩I *I ≠ by (ab) c = (cb) a. As x * I ∩ I ≠ φ, I2 * x ∩ I2 ≠ φ. Therefore, I2 is a right almost hyperideal of H. Hence, I2 is an almost hyperideal of H.

Definition 6

An almost hyperideal I of an LA-semihypergroup H is minimal if it does not contain any almost hyperideal of H other than itself.

Example 6

Let H = {x1, x2, x3, x4} with the binary hyperoperation defined as follows:

◦ x1 x2 x3 x4
x1 x1 x2 x3 x4
x2 x3 {x2, x3} {x2, x3} x4
x3 x2 {x2, x3} {x2, x3} x4
x4 x4 x4 x4 H

Hence H is an LA-semihypergroup. Let I1 = {x1, x2, x4}, I2 = {x1, x3, x4} and I3 = {x4} are the almost hyperideals of H. Thus, I3 is a minimal almost hyperideal of H.

Theorem 2

Let H be an LA-semihypergroup. If L is a minimal almost left hyperideal of H, then a*L is also a minimal almost left hyperideal of H for every weak idempotent a.

Proof

It is known that a * L is a left almost hyperideal of H. Next, we must demonstrate that a * L is minimal almost left hyperideal of H. Let L be a left almost hyperideal of H, which is properly contained in a*L. We defineK = {i ∈ I : a * i ⊆ L} and let y ∈ K. Let a * i ∈ L =⇒ x * (a * i) ∩ a * i ≠ φ =⇒ x * L ∩ L ≠ φ =⇒ a * i ∈ L, which is contradiction to the minimality of L. Therefore, a * L is minimal almost left hyperideal.

4. Conclusion

In this paper, we have introduced new types of hyperideals in non-associative structures of semihypergroups and indicated some interesting properties. In the future, we aim to obtain more properties of the proposed ideal theory.

Conflict of Interest

No potential conflict of interest relevant to this article was reported.


TABLES
◦ l1 l2 l3 l4
l1 l1 l2 l3 l4
l2 l3 {l2, l3} {l2, l3} l4
l3 l2 {l2, l3} {l2, l3} l4
l4 l4 l4 l4 H

* e l m
e e l m
l m {l, m} m
m l l {l, m}

* e l
e e l
l m {l, m}

◦ a b c
a a b c
b c {b, c} {b, c}
c b {b, c} {b, c}

* x y z
x x x x
y x H z
z x H H

* x y
x x x
y x H

* 1 2 3 4 5 6
1 {1, 2, 3, 4} {2, 3, 4, 5, 6} {4, 5, 6} {2, 3, 4, 5, 6} {2, 3, 4, 5, 6} {2, 3, 4, 5, 6}
2 {1, 3, 4, 5, 6} {1, 2, 3, 4, 5, 6} {1, 3, 4, 5, 6} {1, 3, 6} {1, 5, 6} {1, 3, 4, 5, 6}
3 {1, 2, 4, 5, 6} {1, 2} {1, 2, 3, 5, 6} {1, 2, 4, 5, 6} {1, 2, 6} {1, 2, 4, 5, 6}
4 {1, 5, 6} {1, 2, 6} {1, 2, 5, 6} {1, 2, 5, 6} {1, 2, 5, 6} {1, 2, 5, 6}
5 {1, 2, 3, 4, 6} {1, 2, 3, 4, 6} {1, 2, 3, 4, 6} {1, 3, 4, 6} {1, 2, 3, 4, 5, 6} {1, 3, 4, 6}
6 {1, 2, 3, 4, 5} {1, 4, 5} {1, 2, 3, 4, 5} {1, 4, 5} {1, 2, 3, 4, 5} {1, 2, 3, 4, 5}

◦ x1 x2 x3 x4
x1 x1 x2 x3 x4
x2 x3 {x2, x3} {x2, x3} x4
x3 x2 {x2, x3} {x2, x3} x4
x4 x4 x4 x4 H

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Biographies

Shah Nawaz is a Ph.D. student of Dr. Muhammad Gulistan and has submitted his Ph.D. thesis in the field of non-associative hyperstructures. He has authored some papers on the introduction of new non-associative hyperstructures such as LA-polygroups and LA-hypergroups.

E-mail: shahnawazawan82@gmail.com


Muhammad Gulistan received his M.Phil. degree from Quaid-i- Azam University, Islamabad, in 2011, and his Ph.D. degree from Hazara University, in 2016, where he is currently working as an Assistant Professor in the Department of Mathematics and Statistics. He has supervised many M.Phil. and Ph.D. research students. He has published more than 80 research papers in different well reputed journals. His area of research interests includes cubic sets and their generalizations, non-associative hyperstructures, neutrosophic cubic sets, neutrosophic cubic graphs, and decision making.

E-mail: gulistanmath@hu.edu.pk

ORCID: https://orcid.org/0000-0002-6438-1047


Nasreen Kausar received her Ph.D. degree in Mathematics from the Quaid-i-Azam University in Islamabad, Pakistan. She is currently an assistant Professor of Mathematics at the University of Agriculture Faisalabad, Pakistan. Her research interests include the numerical analysis and numerical solutions of ordinary differential equations (ODEs), partial differential equations (PDEs), and Volterra integral equations. She also has research interests in associative and commutative, non-associative and non-commutative fuzzy algebraic structures and their applications. Department of Mathematics and Statistics University of Agriculture, Faisalabad, Pakistan.

E-mail: kausar.nasreen57@gmail.com

ORCID: https://orcid.org/0000-0002-8659-0747


Salahuddin received his Ph.D. degree for his research work in Mathematics in 2001. He is a faculty member of the Department of Mathematics, Jazan University, Jazan, Saudi Arabia. He is working on a fuzzy set, fuzzy group theory, fuzzy ring and fuzzy ideal theory, variational inequality, and optimization theory.

E-mail: drsalah12@hotmail.com

ORCID: https://orcid.org/0000-0002-0496-3379


Mohammad Munir received his Ph.D. degree in Applied Mathematics from the University of Graz, Graz, Austria, in 2010. He completed his Ph.D. thesis on a project titled “Generalized Sensitivity Functions in Physiological Modelling”. His research interests are in the mathematical modelling of biological systems in the fields of the glucose-insulin dynamics, solute kinetics and hemodialysis using ordinary differential equations (ODEs). Parameter identification, sensitivity analysis and generalized sensitivity analysis are more concentrated areas of his research. His other interests include the applications of the fuzzy sets theory to multi-criteria decision-making (MCDM) problems.

Email: dr.mohammadmunir@gpgc-atd.edu.pk

ORCID: https://orcid.org/0000-0002-4891-2995


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