International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 135-144
Published online June 25, 2021
https://doi.org/10.5391/IJFIS.2021.21.2.135
© The Korean Institute of Intelligent Systems
Mohammad Munir^{1}, Nasreen Kausar^{2}, Salahuddin^{3}, Rukhshanda Anjum^{4}, Qingbing Xu^{5}, and Waqas Ahmad^{6}
^{1}Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
^{2}Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Istanbul, Turkey
^{3}Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia
^{4}Department of Mathematics and Statistics, University of Lahore, Lahore
^{5}Department of Basic Courses, Chuzhou Polytechnic College, Chuzhou, China
^{6}Department of Zoology, Hazara University, Mansehra, 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.
We initially introduce the concepts of an m-right (m-left) hyperideal and an m-hyperideal in a hypergroupoid. The ideas behind an m-factor and a generalized m-factor are then introduced. Next, we demonstrate the existence and important properties of these sub-hyperstructures through theorems and examples. We then define the m-right (m-left) consistent, m-consistent, m-intra-consistent, and m-simple hypergroupoids. Finally, we demonstrate that practical problems in biology, such as ABO blood group genetics, can be studied by defining these hypergroupoid substructures
Keywords: Blood group genetics, m-factors, Genotype, Phenotype, m-factorizable, Hypersemigroups
Among all algebraic structures, a groupoid is the simplest within the entire field of mathematics, consisting of a non-empty set,
This issue motivated us to pursue the study of
Marty [2] introduced hyperstructures in 1934. Later mathematicians then followed up on this study for different algebraic structures and applied them to study various problems in scientific fields. For example, Hasankhani [3] studied the ideals with respect to Greens’s relation in hypersemigroupoids. Kehayopulu [4] characterized hypergroupoids through the properties of fuzzy prime and fuzzy semiprime hyperideals. Suebsung et al. [5] characterized the semihypergroupoids through the properties of almost hyperideals.
The motivation to work on the so-called
This paper is divided into seven sections. In Section 1, we provide the necessary introduction and motivation of our study. Section 2 presents the concepts necessary to be used in further research. In Section 3, we introduce the idea of
In this section, we present some preliminary ideas from the literature on hypergroupoids, which will be necessary to build up the theory and applications of
Following [10] and [5], let
We define a hypergroupoid as an ordered pair (
for all non-empty subsets
We denote the hypergroupoid (
A non-empty subset
For a positive integer
The condition
A hypergroupoid
A hypergroupoid
A hypergroupoid
An element
It should be noted that the identity and regular elements of a hypergroupoid are not weak right or weak left magnifying elements.
We define the principal left hyperideal as the left hyperideal
A non-empty subset
A non-empty set
Every left hyperideal is an
The principal
Consider the hypergroupoid
The product of an
Let
Thus,
The product of two
Straightforward.
The product of an
Because the product is both an
The product of the
Straightforward.
Any finite collection of
If
Because
In the above case, product
For some natural number
Clear.
For the intersection of an
Hypergroupoids can be factorized with respect to their proper hyperideals, proper subhypergroupoids, and proper subsets. In this section, we mainly pursue factorization with respect to the subhypergroupoids and subsets. Thus, we define the factorizable, (
A hypergroupoid
If the identity
Consider the hypergroupoid
If the right hyperideal
Because
If a hypergroupoid
We prove this for the case of the
When
When
Relaxing the condition of the subhypergroupoid on the factors of a hypergroupoid leads us to define the generalized factors. This relaxion results in studying the non-empty subsets of hypergroupoids that form the factorization of the hypergroupoids.
A proper subset
The following lemma is used in the proof of the preceding theorems characterizing factorizable hypergroupoids.
Let
If
Letting
Let
Letting
If
Supposing that
(a) If
(b) Suppose that
If
Although the factors of a hypergroupoid are generalized, but the reverse does not hold.
Now, we present the ideas of (
If there exist two proper subhypergroupoids
If
If there exist two proper subsets
Consider the hypergroupoid
In Section 5, {
In Definitions 4.1 and 4.8, without a loss of generality, by taking
A non-empty proper subset
A hypergroupoid
Because the concepts of maximality and minimality in the context of hyperideals play a vital role in characterizing a hypergroupoid, we define the concepts of the maximal and minimal generalized (
A generalized
A generalized
Based on the pattern of Definitions 4.14 and 4.15, we can define the maximum and minimum factors for a hypergroupoid
A hypergroupoid
Suppose that
Conversely, suppose that
The blood groups of the offspring were determined based on their parents. There are almost 38 blood groups, but two are the most important with regard to blood transfusions. These were the ABO blood groups and ABO/Rh-factor blood groups. The percentages of blood groups in different racial groups world-wide are given in Table 4.
The most well-known and medically important blood types are the ABO blood types discovered by Karl Landsteiner in 1900 A.D. [14]. The ABO blood type of the offspring is controlled by a single gene, each of which is called the ABO gene with three types of alleles, namely
Some red blood cells have an Rh factor, known as an RhD antigen. If the red blood cells contain the RhD antigen, they are RhD positive; otherwise, they are RhD-negative. Combining the effect of the RhD factor with the ABO blood group, an eight-blood-group-type system is obtained that consists of
In this study, we considered the ABO blood groups and studied their genetics through hypergroupoids. The genetics of the ABO/Rh blood group system can be studied using this pattern.
We keep the ABO blood groups in the set
From Table 5, we extract the following information:
The samples {
The hypergroupoid
{
Because {
Because {
{
The theory of the
Not all proper subsets or subhypergroupoids are factors, (
In hypergroupoid
The following hyperoperation ∘
The first ordered pair (
In cases of questioned paternity, the
Defining the hyperoperation ∘
The hypergroupoids (
In the following, we characterize the
A hypergroupoid
The hypergroupoid
If
Let
A hypergroupoid
A hypergroupoid
The following proposition describes the converse of Proposition 6.3.
Let
Suppose
A right (left)
Let
Let
On the patterns of Propositions 6.3 and 6.7, we have the following.
An
Similar.
We presented the concepts of
The method of using hypergroupoids is easier and more applicable than other methods used in studies on the genetics of blood groups. In our research on the ABO example, the positive integers
Percentage of blood groups around the world.
Blood group | AP | BP | ABP | OP | AN | BN | ABN | ON |
---|---|---|---|---|---|---|---|---|
African-American | 24 | 18 | 4 | 47 | 2 | 1 | 0.3 | 4 |
Asian | 27 | 25 | 7 | 39 | 0.5 | 0.4 | 0.1 | 1 |
Caucasian | 33 | 9 | 3 | 37 | 7 | 2 | 1 | 8 |
Latino-American | 29 | 9 | 2 | 53 | 2 | 1 | 0.2 | 4 |
Blood groups depicted as hypergroupoid
∘ | A | B | AB | O |
---|---|---|---|---|
A | {A, O} | {A, B, AB, O} | {A, B, AB} | {A, O} |
B | {A, B, AB, O} | {B, O} | {A, B, AB} | {B, O} |
AB | {A, B, AB} | {A, B, AB} | {A, B, AB} | {A, B} |
O | {A, O} | {B, O} | {A, B} | {O} |
∘ | A | B | AB |
---|---|---|---|
A | {A, O} | {A, B, AB, O} | {A, B, AB} |
B | {A, B, AB, O} | {B, O} | {A, B, AB} |
AB | {A, B, AB} | {A, B, AB} | {A, B, AB} |
Blood groups of impossible children depicted as hypergroupoid
∘ | A | B | AB | O |
---|---|---|---|---|
A | {B, AB} | ∅︀ | {O} | {B, AB} |
B | ∅︀ | {A, AB} | {O} | {A, AB} |
AB | {O} | {O} | {O} | {AB, O} |
O | {B, AB} | {A, AB} | {AB, O} | {A, B, AB} |
International Journal of Fuzzy Logic and Intelligent Systems 2021; 21(2): 135-144
Published online June 25, 2021 https://doi.org/10.5391/IJFIS.2021.21.2.135
Copyright © The Korean Institute of Intelligent Systems.
Mohammad Munir^{1}, Nasreen Kausar^{2}, Salahuddin^{3}, Rukhshanda Anjum^{4}, Qingbing Xu^{5}, and Waqas Ahmad^{6}
^{1}Department of Mathematics, Government Postgraduate College, Abbottabad, Pakistan
^{2}Department of Mathematics, Faculty of Arts and Science, Yildiz Technical University, Istanbul, Turkey
^{3}Department of Mathematics, Jazan University, Jazan, Kingdom of Saudi Arabia
^{4}Department of Mathematics and Statistics, University of Lahore, Lahore
^{5}Department of Basic Courses, Chuzhou Polytechnic College, Chuzhou, China
^{6}Department of Zoology, Hazara University, Mansehra, 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.
We initially introduce the concepts of an m-right (m-left) hyperideal and an m-hyperideal in a hypergroupoid. The ideas behind an m-factor and a generalized m-factor are then introduced. Next, we demonstrate the existence and important properties of these sub-hyperstructures through theorems and examples. We then define the m-right (m-left) consistent, m-consistent, m-intra-consistent, and m-simple hypergroupoids. Finally, we demonstrate that practical problems in biology, such as ABO blood group genetics, can be studied by defining these hypergroupoid substructures
Keywords: Blood group genetics, m-factors, Genotype, Phenotype, m-factorizable, Hypersemigroups
Among all algebraic structures, a groupoid is the simplest within the entire field of mathematics, consisting of a non-empty set,
This issue motivated us to pursue the study of
Marty [2] introduced hyperstructures in 1934. Later mathematicians then followed up on this study for different algebraic structures and applied them to study various problems in scientific fields. For example, Hasankhani [3] studied the ideals with respect to Greens’s relation in hypersemigroupoids. Kehayopulu [4] characterized hypergroupoids through the properties of fuzzy prime and fuzzy semiprime hyperideals. Suebsung et al. [5] characterized the semihypergroupoids through the properties of almost hyperideals.
The motivation to work on the so-called
This paper is divided into seven sections. In Section 1, we provide the necessary introduction and motivation of our study. Section 2 presents the concepts necessary to be used in further research. In Section 3, we introduce the idea of
In this section, we present some preliminary ideas from the literature on hypergroupoids, which will be necessary to build up the theory and applications of
Following [10] and [5], let
We define a hypergroupoid as an ordered pair (
for all non-empty subsets
We denote the hypergroupoid (
A non-empty subset
For a positive integer
The condition
A hypergroupoid
A hypergroupoid
A hypergroupoid
An element
It should be noted that the identity and regular elements of a hypergroupoid are not weak right or weak left magnifying elements.
We define the principal left hyperideal as the left hyperideal
A non-empty subset
A non-empty set
Every left hyperideal is an
The principal
Consider the hypergroupoid
The product of an
Let
Thus,
The product of two
Straightforward.
The product of an
Because the product is both an
The product of the
Straightforward.
Any finite collection of
If
Because
In the above case, product
For some natural number
Clear.
For the intersection of an
Hypergroupoids can be factorized with respect to their proper hyperideals, proper subhypergroupoids, and proper subsets. In this section, we mainly pursue factorization with respect to the subhypergroupoids and subsets. Thus, we define the factorizable, (
A hypergroupoid
If the identity
Consider the hypergroupoid
If the right hyperideal
Because
If a hypergroupoid
We prove this for the case of the
When
When
Relaxing the condition of the subhypergroupoid on the factors of a hypergroupoid leads us to define the generalized factors. This relaxion results in studying the non-empty subsets of hypergroupoids that form the factorization of the hypergroupoids.
A proper subset
The following lemma is used in the proof of the preceding theorems characterizing factorizable hypergroupoids.
Let
If
Letting
Let
Letting
If
Supposing that
(a) If
(b) Suppose that
If
Although the factors of a hypergroupoid are generalized, but the reverse does not hold.
Now, we present the ideas of (
If there exist two proper subhypergroupoids
If
If there exist two proper subsets
Consider the hypergroupoid
In Section 5, {
In Definitions 4.1 and 4.8, without a loss of generality, by taking
A non-empty proper subset
A hypergroupoid
Because the concepts of maximality and minimality in the context of hyperideals play a vital role in characterizing a hypergroupoid, we define the concepts of the maximal and minimal generalized (
A generalized
A generalized
Based on the pattern of Definitions 4.14 and 4.15, we can define the maximum and minimum factors for a hypergroupoid
A hypergroupoid
Suppose that
Conversely, suppose that
The blood groups of the offspring were determined based on their parents. There are almost 38 blood groups, but two are the most important with regard to blood transfusions. These were the ABO blood groups and ABO/Rh-factor blood groups. The percentages of blood groups in different racial groups world-wide are given in Table 4.
The most well-known and medically important blood types are the ABO blood types discovered by Karl Landsteiner in 1900 A.D. [14]. The ABO blood type of the offspring is controlled by a single gene, each of which is called the ABO gene with three types of alleles, namely
Some red blood cells have an Rh factor, known as an RhD antigen. If the red blood cells contain the RhD antigen, they are RhD positive; otherwise, they are RhD-negative. Combining the effect of the RhD factor with the ABO blood group, an eight-blood-group-type system is obtained that consists of
In this study, we considered the ABO blood groups and studied their genetics through hypergroupoids. The genetics of the ABO/Rh blood group system can be studied using this pattern.
We keep the ABO blood groups in the set
From Table 5, we extract the following information:
The samples {
The hypergroupoid
{
Because {
Because {
{
The theory of the
Not all proper subsets or subhypergroupoids are factors, (
In hypergroupoid
The following hyperoperation ∘
The first ordered pair (
In cases of questioned paternity, the
Defining the hyperoperation ∘
The hypergroupoids (
In the following, we characterize the
A hypergroupoid
The hypergroupoid
If
Let
A hypergroupoid
A hypergroupoid
The following proposition describes the converse of Proposition 6.3.
Let
Suppose
A right (left)
Let
Let
On the patterns of Propositions 6.3 and 6.7, we have the following.
An
Similar.
We presented the concepts of
The method of using hypergroupoids is easier and more applicable than other methods used in studies on the genetics of blood groups. In our research on the ABO example, the positive integers
∘ | 1 | 2 | 3 |
---|---|---|---|
1 | {1} | {1, 2} | {2} |
2 | {1} | {2} | {1} |
3 | {1, 2} | {2} | {1} |
∘ | 1 | 2 |
---|---|---|
1 | {1} | {1, 2} |
2 | {1} | {2} |
∘ | 1 | 2 | 3 | 4 |
---|---|---|---|---|
1 | {1} | {2} | {3} | {4} |
2 | {2} | {1, 2} | {4} | {3, 4} |
3 | {3} | {4} | {1, 3} | {3, 4} |
4 | {4} | {3, 4} | {2, 4} | {1, 2, 3, 4} |
Percentage of blood groups around the world.
Blood group | AP | BP | ABP | OP | AN | BN | ABN | ON |
---|---|---|---|---|---|---|---|---|
African-American | 24 | 18 | 4 | 47 | 2 | 1 | 0.3 | 4 |
Asian | 27 | 25 | 7 | 39 | 0.5 | 0.4 | 0.1 | 1 |
Caucasian | 33 | 9 | 3 | 37 | 7 | 2 | 1 | 8 |
Latino-American | 29 | 9 | 2 | 53 | 2 | 1 | 0.2 | 4 |
Blood groups depicted as hypergroupoid
∘ | A | B | AB | O |
---|---|---|---|---|
A | {A, O} | {A, B, AB, O} | {A, B, AB} | {A, O} |
B | {A, B, AB, O} | {B, O} | {A, B, AB} | {B, O} |
AB | {A, B, AB} | {A, B, AB} | {A, B, AB} | {A, B} |
O | {A, O} | {B, O} | {A, B} | {O} |
∘ | A | B |
---|---|---|
A | {A, O} | {A, B, AB, O} |
B | {A, B, AB, O} | {B, O} |
∘ | A | B | AB |
---|---|---|---|
A | {A, O} | {A, B, AB, O} | {A, B, AB} |
B | {A, B, AB, O} | {B, O} | {A, B, AB} |
AB | {A, B, AB} | {A, B, AB} | {A, B, AB} |
Blood groups of impossible children depicted as hypergroupoid
∘ | A | B | AB | O |
---|---|---|---|---|
A | {B, AB} | ∅︀ | {O} | {B, AB} |
B | ∅︀ | {A, AB} | {O} | {A, AB} |
AB | {O} | {O} | {O} | {AB, O} |
O | {B, AB} | {A, AB} | {AB, O} | {A, B, AB} |
∘ | A | B | O |
---|---|---|---|
A | {A, O} | {A, B, AB, O} | {A, O} |
B | {A, B, AB, O} | {B, O} | {B, O} |
O | {A, O} | {B, O} | {O} |