International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 86-92
Published online March 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.1.86
© The Korean Institute of Intelligent Systems
Shah Nawaz^{1}, Muhammad Gulistan^{1 }, Nasreen Kausar^{2 }, Salahuddin^{3} , and Mohammad Munir^{4 }
^{1}Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
^{2}Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan
^{3}Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia
^{4}Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
Correspondence to :
Nasreen Kausar (kausar.nasreen57@gmail.com)
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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
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.
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.
A map ◦ :
If
A hypergroupoid (
The law (
Every LA-semihypergroup satisfies the law
for all
Let
1. Left identity (resp. pure left identity) if for all
2. Right identity (resp. pure right identity) if for all
3. Identity (resp. pure identity) if for all
Let
◦ | ||||
---|---|---|---|---|
{ | { | |||
{ | { | |||
Hence
Let
Let
The law (
Let (
(1) A non-empty subset
(2) A non-empty subset
(3) A non-empty subset
Here we define left almost hyperideals, right almost hyperideals, almost hyperideals, minimal almost hyperideals, and some interesting properties.
Let (
(1) A non-empty subset
(2) A non-empty subset
(3) A non-empty subset
Let
* | |||
---|---|---|---|
{ | |||
{ |
(
* | ||
---|---|---|
{ |
The structure of left almost hyperideals is not LA-subsemihypergroups as in Example 2, where
Consider an LA-semihypergroup
◦ | |||
---|---|---|---|
{ | { | ||
{ | { |
If
Let (
(1) Every left hyperideal of
(2) Every right hyperideal of
(3) Every hyperideal of
(1) Assume that
(2) Let
(3) Follows from (1) and (2)
The converse of Lemma 3 may or may not true generally as in the following Example 4.
Let
* | |||
---|---|---|---|
Thus, (
* | ||
---|---|---|
Here,
Let (
(1) If
(2) If
(3) If
Same as provided by Suebsung et. al [47].
(1) If
(2) If
(3) If
Let
* | 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} |
If
However, if
this is a contradiction to the given condition. Hence,
If
However, if
this is a contradiction to the given condition. Hence,
If
First, we must demonstrate that
An almost hyperideal
Let
◦ | ||||
---|---|---|---|---|
{ | { | |||
{ | { | |||
Hence
Let
It is known that
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.
No potential conflict of interest relevant to this article was reported.
E-mail: shahnawazawan82@gmail.com
E-mail: gulistanmath@hu.edu.pk
ORCID:
E-mail: kausar.nasreen57@gmail.com
ORCID:
E-mail: drsalah12@hotmail.com
ORCID:
Email: dr.mohammadmunir@gpgc-atd.edu.pk
ORCID:
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(1): 86-92
Published online March 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.1.86
Copyright © The Korean Institute of Intelligent Systems.
Shah Nawaz^{1}, Muhammad Gulistan^{1 }, Nasreen Kausar^{2 }, Salahuddin^{3} , and Mohammad Munir^{4 }
^{1}Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
^{2}Department of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan
^{3}Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia
^{4}Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
Correspondence to:Nasreen Kausar (kausar.nasreen57@gmail.com)
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 noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
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.
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.
A map ◦ :
If
A hypergroupoid (
The law (
Every LA-semihypergroup satisfies the law
for all
Let
1. Left identity (resp. pure left identity) if for all
2. Right identity (resp. pure right identity) if for all
3. Identity (resp. pure identity) if for all
Let
◦ | ||||
---|---|---|---|---|
{ | { | |||
{ | { | |||
Hence
Let
Let
The law (
Let (
(1) A non-empty subset
(2) A non-empty subset
(3) A non-empty subset
Here we define left almost hyperideals, right almost hyperideals, almost hyperideals, minimal almost hyperideals, and some interesting properties.
Let (
(1) A non-empty subset
(2) A non-empty subset
(3) A non-empty subset
Let
* | |||
---|---|---|---|
{ | |||
{ |
(
* | ||
---|---|---|
{ |
The structure of left almost hyperideals is not LA-subsemihypergroups as in Example 2, where
Consider an LA-semihypergroup
◦ | |||
---|---|---|---|
{ | { | ||
{ | { |
If
Let (
(1) Every left hyperideal of
(2) Every right hyperideal of
(3) Every hyperideal of
(1) Assume that
(2) Let
(3) Follows from (1) and (2)
The converse of Lemma 3 may or may not true generally as in the following Example 4.
Let
* | |||
---|---|---|---|
Thus, (
* | ||
---|---|---|
Here,
Let (
(1) If
(2) If
(3) If
Same as provided by Suebsung et. al [47].
(1) If
(2) If
(3) If
Let
* | 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} |
If
However, if
this is a contradiction to the given condition. Hence,
If
However, if
this is a contradiction to the given condition. Hence,
If
First, we must demonstrate that
An almost hyperideal
Let
◦ | ||||
---|---|---|---|---|
{ | { | |||
{ | { | |||
Hence
Let
It is known that
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.
◦ | ||||
---|---|---|---|---|
{ | { | |||
{ | { | |||
* | |||
---|---|---|---|
{ | |||
{ |
* | ||
---|---|---|
{ |
◦ | |||
---|---|---|---|
{ | { | ||
{ | { |
* | |||
---|---|---|---|
* | ||
---|---|---|
* | 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} |
◦ | ||||
---|---|---|---|---|
{ | { | |||
{ | { | |||