Fuzzy hyperlattice ordered δ - group and its application on ABO blood group system

Abstract

This article deals with a fuzzy hypercompositional structure called fuzzy hyperlattice ordered δ- group $(FHLO δ - G)$ , the extension of the fuzzy hypercompositional structure namely fuzzy hyperlattice ordered group (FHLOG). Using $FHLO δ - G$ , we can involve one additional non-empty set δ with FHLOG, which helps to develop new results and applications. The structural characteristics and properties of $FHLO δ - G$ are analysed. Furthermore, an application of $FHLO δ - G$ for $ABO$ blood group system is proposed.

Keywords

Lattice ordered group fuzzy lattice ordered group fuzzy hyperlattice fuzzy hyperlattice ordered group 𝒜ℬ𝒪 blood group system

6 Introduction

Algebraic hyperstructures were introduced in 1934 by Marty [13] as a generalization of classical algebraic structure. In an algebraic hyperstructure, the composition of two elements forms a set whereas in a classical algebraic structure, the composition of two elements forms an element again. Many concepts were developed under the topic of algebraic hyperstructures such as weak hyperstructures, hypergroups, hyperrings, hypermodules (see [5–7 , 28]). As an example of algebraic hyperstructures, hyperlattices were initiated by Konstantinidou [11] and were studied further by Rasouli and Davvaz [12]. For more details about hyperlattices, we refer to [20 , 22–25]. On the other hand, Zadeh introduced fuzzy mathematics in 1965 [30]. The fuzzy set theory dominated research works because of its applications to different areas of Science. The combination of algebraic structures (hyperstructures) and fuzzy set theory showed a rapid growth [5 , 27]. Correspondingly, He and Xin [18] proposed fuzzy hyperlattice. The concept of fuzzy hyperlattice ordered group was developed and studied by Preethi et al. [14 –16] where they framed an application on fuzzy hyperlarttice ordered group using biological inheritance.

Contribution:

In this paper, we extend the concept of FHLOG by introducing the concept of $FHLO δ - G$ . FHLOG has many applications but it deals with a particular FHLOG structure only and it is not possible to include one more non-empty set with this structure. The proposed topic helps to involve a non-empty set δ with the FHLOG. So that we can develop efficient applications. In this regard, we establish some properties of $FHLO δ - G$ and present an interesting application of $FHLO δ - G$ in $ABO$ blood group system.

The paper is organized as follows: After an Introduction, in Section 2, some basic definitions are examined. In Section 3, the concept of $FHLO δ - G$ is introduced and some of its properties are studied. In Section 4, an application of $FHLO δ - G$ in $ABO$ blood group system is discussed.

7 Preliminaries

In this section, we review the fundamentals of algebraic hyperstrutures, hyperlattices, and fuzzy algebraic hyperstructures that are used throughout the paper. For more details, we refer to [7 , 18].

If $S$ is a non-empty set and $P^{*} (S)$ is the family of all non-empty subsets of $S$ then we consider the maps of the following type: $g_{i} : S \times S ⟶ P^{*} (S),$ where i ∈ {1, 2, . . . , k} and k is a positive integer. For every i ∈ {1, 2, . . . , k}, the map g_i is called (binary) hyperoperations. An algebraic system $(S, g_{1}, . ., g_{k})$ is called a (binary) hyperstructure. Usually, we have k = 1 or k = 2.

Let $F$ be a non-empty set and $P^{*} (F)$ be the set of all non-empty subsets of $F$ . A hyperoperation on $F$ is a map $\circ : F \times F ⟶ P^{*} (F)$

Let C and D are non-empty subsets of $F$ . then for all $c, d, e \in F$ , we denote

e ∘ C = {e} ∘ C = ⋃ _c∈Ce ∘ c,

C ∘ e = C ∘ {e} = ⋃ _c∈Cc ∘ e;

$C \circ D = ⋃_{\binom{c \in C}{d \in D}} c \circ d$ .

Definition 2.1. [9, 18] Let $F$ be a non-empty set, “□ x” and “⊞” be hyperoperations on $F$ . Then the triple $(F, ⊠, ⊞)$ is a hyperlattice if the following conditions are satisfied. For all $p, q, r \in F$ ,

(idempotent laws) p ∈ p ⊠ p and p ∈ p⊞p;

(commutative laws) p ⊠ q = q ⊠ p and p⊞q = q⊞p;

(associative laws) (p ⊠ q) ⊠ r = p ⊠ (q ⊠ r) and (p⊞q) ⊞r = p⊞ (q⊞r);

(absorption laws) p ∈ p ⊠ (p⊞q) and p ∈ p⊞ (p ⊠ q).

Example 1. [9] Let $(F, \lor, \land)$ be a lattice and define the hyperoperations on $F$ as follows. For all $p, q \in F$ , $p ⊠ q = {x \in F | p \lor x = q \lor x = p \lor q}$ and $p ⊞ q = {x \in F | x \leq p \land q}$ . Then $(F, ⊠, ⊞)$ is a hyperlattice.

Definition 2.2. [22] Let $(F, ⋁, ⋀)$ be a hyperlattice and ≤ is a partial order relation on $F$ . Then $(F, \leq)$ is an ordered hyperlattice if p ≤ q then $\begin{matrix} p ⋁ r \leq q ⋁ r and p ⋀ r \leq q ⋀ r . \end{matrix}$

Definition 2.3. [30] Let $F$ be a set. A function $P : F \times F ⟶ [0, 1]$ is a fuzzy relation in $F$ .

Definition 2.4. [18] Let $F$ be a non-empty set and ⊠ and ⊞ are two fuzzy hyperoperations on $F$ . (Here $F^{*} (F)$ is the set of all non-zero fuzzy subsets of $F$ . A fuzzy hyperoperation is a map $\circ : F \times F ⟶ F^{*} (F)$ ). Then the triple $(F, ⊠, ⊞)$ is called a fuzzy hyperlattice if the following conditions are satisfied. For all $p, q, r \in F$ ,

(p ⊠ p) (p) >0 and (p⊞p) (p) >0;

(p ⊠ q) = (q ⊠ p) and (p⊞q) = (q⊞p);

(p ⊠ q) ⊠ r = p ⊠ (q ⊠ r) and (p⊞q) ⊞r = p⊞ (q⊞r);

(p ⊠ (p⊞q)) (p) >0 and (p⊞ (p ⊠ q)) (p) >0.

Note: [18] Let C and D are two non-zero fuzzy subsets. Then for all

c, l \in F

we define the following.

$(c \circ C) (l) = sup_{t \in F} {(c \circ t) (l) \land C (t)}$ , $(C \circ c) (l) = sup_{t \in F} {C (t) \land (t \circ c) (l)}$ .

$(C \circ D) (l) = sup_{c, d \in F} {C (c) \land (c \circ d) (l) \land D (d)}$ .

Example 2. Let $(F, \land, \lor)$ be a lattice with fuzzy hyperoperations on $F$ defined by: ∀ $a, b \in F$ , a × b = χ_{a,b} and a + b = χ_a∧b. Then $(F, \times, +)$ is a fuzzy hyperlattice.

Definition 2.5. [4] A lattice ordered group is a system $F = (F, +, \leq)$ such that

$(F, +)$ is a group,

$(F, \leq)$ is a lattice,

r ≤ s ⇒ p + r + q ≤ p + s + q ∀ $p, q \in F$ .

Definition 2.6. [27] A non-empty set $F$ is called a fuzzy lattice ordered group if the following conditions are satisfied.

$(F, +)$ is a group,

$(F, \lor, \land)$ is a fuzzy lattice,

R (p + (r ∨ s) , (p + r) ∨ (p + s)) =1 for all $p, r, s \in F$ .

Here,

R : F \times F ⟶ [0, 1]

is the fuzzy partial order relation.

Definition 2.7. [14] A non-empty set $F$ is called a fuzzy hyperlattice ordered group if the following conditions are satisfied.

$(F, +)$ is a group,

$(F, ⊞, ⊠)$ is a fuzzy hyperlattice,

R [{p} + (r⊞s) , {p + r} ⊞ {p + s}] =1 and R [{p} + (r ⊠ s) , {p + r} ⊠ {p + s}] =1,

for all

p, r, s \in F

. Here,

R : F \times F ⟶ [0, 1]

is the fuzzy partial order relation.

Definition 2.8. Let $F$ be a group, δ be any set if,i) $da \in F$ , ii)d (ab) = (da) b = a (db) ∀ $a, b \in F$ , d ∈ δ. Then $F$ is called δ-group.

8 Fuzzy Hyperlattice ordered δ - Group $(FHLO δ - G)$

In this section, we propose the concept of fuzzy hyperlattice ordered δ- group $(FHLO δ - G)$ and investigate some fundamental properties of it.

Throughout this section, ⊞, ⊠ are fuzzy hyperoperations, $F$ is a δ-group, + , * , · are binary operations and the hyperoperations ⋁, ⋀ on $F$ are defined as follows. For all $p, q \in F$ , p ⋁ q = {p ∨ q}, p ⋀ q = {p ∧ q}.

Definition 3.1. A non-empty set $F$ is called a fuzzy hyperlattice ordered δ-group $(FHLO δ - G)$ if,

$(F, +)$ is a δ-group,

$(F, ⊞, ⊠)$ is a fuzzy hyperlattice,

R [δ {u} + (p⊞q) , {δu + p} ⊞ {δu + q}] =1 and R [δ {u} + (p ⊠ q) , {δu + p} ⊠ {δu + q}] =1,

for all

u, p, q \in F

. Here,

R : F \times F ⟶ [0, 1]

is the fuzzy partial order relation.

Example 3. Let the fuzzy hyperoperations on $(ℤ, +, ⊠, ⊞)$ be defined by u⊞v = χ_{u,v} and $u ⊠ v = χ_{{u, v, ℤ}}$ for all $u, v \in ℤ$ , δ = {1}, and $R : ℤ \times ℤ ⟶ [0, 1]$ be the fuzzy partial order relation. Then $(ℤ, +, ⊠, ⊞)$ is an example of $FHLO δ - G$ .

Proposition 3.2. A $FHLO δ - G$ is always distributive.

Proposition 3.3. Let $F$ be a $FHLO δ - G$ . Then $\begin{matrix} R [δ ({u ⋀ v} - {u ⋀ w}), δ {v - w}] \geq 0, \end{matrix}$ ∀ $u, v, w \in F$ .

Proof. We have $\begin{matrix} δ ({u ⋀ v} - {u ⋀ w}) & = δ ({u} ⋀ {v - w}) \\ = δ {u} ⋀ δ {v - w} \\ \leq δ {v - w} . \end{matrix}$ Therefore, R [δ ({u ⋀ v} - {u ⋀ w}) , δ {v - w}] ≥0.

□

Proposition 3.4. Let $F$ be a $FHLO δ - G$ . Then $R [δ {u ⊞ v}, (δ {u - v} ⊞ {0}) + δ {v}] = 1,$ ∀ $u, v \in F$ .

Proof. We have

(δ {u - v} ⊞ {0}) + δ {v} $\begin{matrix} = (δ {u - v} + δ {v}) ⊞ ({0} + δ {v}) \\ = δ {u} ⊞ δ {v} \\ = δ {u ⊞ v} . \end{matrix}$ Therefore, R [δ {u⊞v} , (δ {u - v} ⊞ {0}) + δ {v}] =1.□

Proposition 3.5. Let $F$ be a $FHLO δ - G$ . Then $R [δ {u ⊞ v} - δ {w}, δ ({u - w} ⊞ {v - w})] = 1,$ ∀ $u, v, w \in F$ .

Proof. We have

(δ ({u - w} ⊞ {v - w})) + δ {w} $\begin{matrix} = δ {w} + (δ {u - w} ⊞ δ {v - w}) \\ = (δ {w} + δ {u - w}) ⊞ (δ {w} + δ {v - w}) \\ = δ {u} ⊞ δ {v} \end{matrix}$ ⇒δ ({u - w} ⊞ {v - w}) = δ {u⊞v} - δ {w}

Therefore, R [δ {u⊞v} - δ {w} , δ ({u - w} ⊞ {v - w})] =1.□

Theorem 3.6. R [δ {u} - δ {v ⋁ w} , δ ({u - v} ⋀ {u - w})] =1

Proof. Let $u, v, w \in F$ be arbitrary. Then, we have $\begin{matrix} δ {v}, δ {w} \leq δ {v ⋁ w} \\ \Rightarrow & - δ {v ⋁ w} \leq - δ {v}, - δ {w} \\ \Rightarrow & δ {u} - δ {v ⋁ w} \leq δ {u} - δ {v}, δ {u} - δ {w} \\ \Rightarrow & δ {u} - δ {v ⋁ w} \leq δ {u - v} ⋀ δ {u - w} . \end{matrix}$

Also, we have $\begin{matrix} δ ({u - v} ⋀ {u - w}) + δ {v ⋁ w} \\ = & (δ ({u - v} ⋀ {u - w}) + δ {v}) ⋁ \\ (δ ({u - v} ⋀ {u - w}) + δ {w}) \\ \leq & (δ {u - v} + δ {v}) ⋁ (δ {u - w} + δ {w}) \\ \leq & δ {u} ⋁ δ {u} = δ {u} . \end{matrix}$ Consequently, we obtain that δ ({u - v} ⋀ {u - w}) ≤ δ {u} - δ {v ⋁ w}.

Therefore, R [δ {u} - δ {v ⋁ w} , δ {u - v} ⋀ δ {u - w}] =1.□

Theorem 3.7. Let $F$ be a $FHLO δ - G$ . Then $R [(δ {u} - δ {u ⋀ v} + δ {v}), δ {v} ⋁ δ {u}] = 1,$ ∀ $u, v \in F$ .

Proof. We know that the hyperoperations ⋁, ⋀ which is idempotent, commutative, associative, and satisfies distributive laws. Now,

(δ {u} - δ {u ⋀ v}) + δ {v} $\begin{matrix} = ((δ {u} - δ {u}) ⋁ (δ {u} - δ {v})) + δ {v} \\ = (0 ⋁ (δ {u} - δ {v})) + δ {v} \\ = δ {v} ⋁ (δ {u} - δ {v} + δ {v}) \\ = δ {v} ⋁ δ {u} . \end{matrix}$ Therefore, R [(δ {u} - δ {u ⋀ v} + δ {v}) , δ {v} ⋁ δ {u}] =1.□

Corollary 3.8. Let $F$ be a $FHLO δ - G$ . Then for all $u, v \in F$ , the following statements are equivalent.

R [δ {u ⋀ v} , {0}] =1,

R [δ {u + v} , δ {u ⋁ v}] =1.

Proof. (1⇒2): Assume that R [δ {u ⋀ v} , {0}] =1, i.e., δ {u ⋀ v} = {0}. By Theorem 3.7, R [(δ {u} - δ {u ⋀ v} + δ {v}) , δ {v} ⋁ δ {u}] =1. Then, $\begin{matrix} (δ {u} - δ {u ⋀ v} + δ {v}) = δ {v} ⋁ δ {u} \\ δ {u + v} = δ {u ⋁ v} . \end{matrix}$ Therefore, R [δ {u + v} , δ {u ⋁ v}] =1.

(2⇒1): Assume that R [δ {u + v} , δ {u ⋁ v}] =1, i.e., δ {u + v}= δ {u ⋁ v}. By Theorem 3.7, R [(δ {u} - δ {u ⋀ v} + δ {v}) , δ {v} ⋁ δ {u}] =1. Then

$\begin{matrix} δ {u} - δ {u ⋀ v} + δ {v} & = δ {v} ⋁ δ {u} \\ δ {u + v} - δ {u ⋁ v} & = δ {u ⋀ v} \\ δ {u ⋀ v} & = {0} . \end{matrix}$ Therefore, R [δ {u ⋀ v} , {0}] =1 .□

Corollary 3.9. In any $FHLO δ - G$ , $F$ , $R [δ {u} + δ {v}, δ {u ⋁ v} + δ {u ⋀ v}] = 1,$ ∀ $u, v \in F$ .

Definition 3.10. Let $F$ be a $FHLO δ - G$ . Then two positive elements u, v of $F$ are disjoint if δ {u ⋀ v} = {0}. We denote it by δu ⊥ δv.

Theorem 3.11. Let $F$ be a $FHLO δ - G$ . If δu ⊥ δv and δu ⊥ δw then R [δ {u} ⋀ δ {v + w} , {0}] =1 for all $u, v, w \in F$ .

Proof. Let $F$ be a $FHLO δ - G$ , δ {u ⋀ v} = {0}, and δ {u ⋀ w} = {0}. Now, $\begin{matrix} δ {u ⋀ v} & = {0} (i . e ., δ {u} ⋀ δ {v} = {0}) \\ δ {w} & = (δ {u} ⋀ δ {v}) + δ {w} \\ δ {w} & = (δ {u + w} ⋀ δ {v + w}) \end{matrix}$ Now, we have $\begin{matrix} {0} & = δ {u ⋀ w} = δ {u} ⋀ δ {w} \\ = δ {u} ⋀ (δ {u + w} ⋀ δ {v + w}) \\ = (δ {u} ⋀ δ {u + w}) ⋀ δ {v + w} \\ = ((δ {u} ⋀ δ {u}) + (δ {u} ⋀ δ {w})) ⋀ δ {v + w} \\ = δ {u} ⋀ δ {v + w} . \end{matrix}$ Therefore, R [δ {u} ⋀ δ {v + w} , {0}] =1.□

9 Application

9.1 ABO Blood Group

The ABO system is the most important blood group system in transfusion medicine. This system consists of two antigens A, B and antibodies against these antigens. In 1900, Karl Landsteiner, the Austrian physician observed this system. The International Society of Blood Transfusion (ISBT) recently perceives 29 blood group systems (BGS). Each BGS contains a single gene or a cluster of 2 or 3 closely linked genes of related sequence. Each BGS is a genetically discrete entity. The ABO blood type is under a single gene (the ABO gene) control. It has three alleles: I^A, I^B, i. Glycosyltransferase is encoded by the gene which is an enzyme where red blood cell antigen’s carbhohydrate contents are modified. The gene is on the ninth chromosome (9q34) [8 , 29].

Each blood type has antigens on the surfaces of their blood cells. Also, it has genotype as shown in Table 1. All parent pairings and the possible blood type of child are shown in Table 2.

Table 1
Antigen and genotype for the blood types

Blood Type Antigen Genotype

$A$ $A$ I^AI^A or I^Ai

$B$ $B$ I^BI^B or I^Bi

$AB$ $A, B$ I ^A I ^B

$O$ No antigen ii

Blood Type	Antigen	Genotype
$A$	$A$	I^AI^A or I^Ai
$B$	$B$	I^BI^B or I^Bi
$AB$	$A, B$	I ^A I ^B
$O$	No antigen	ii

Table 2

Possibilities for child blood types from parent pairings

Parent 1	$A$	$A$	$O$	$O$	$O$	$AB$	$AB$	$AB$	$AB$	$B$
Parent 2	$B$	$A$	$B$	$A$	$O$	$AB$	$B$	$A$	$O$	$B$
Child Blood Type \|
$O$	✓	✓	✓	✓	✓	×	×	×	×	✓
$A$	✓	✓	×	✓	×	✓	✓	✓	✓	×
$B$	✓	×	✓	×	×	✓	✓	✓	✓	✓
$AB$	✓	×	×	×	×	✓	✓	✓	×	×

9.2

ABO

Blood Group System on Fuzzy Hyperlattice Ordered δ-Group

Step 1:

Let us consider Table: 2 (i.e., all parent pairings) by the following relations. Under the relations, all possible parent pairings are included.

$L$ : $O$ ↔ $A;$ $B$ ↔ $AB$

$M$ : $O$ ↔ $B;$ $A$ ↔ $AB$

$N$ : $O$ ↔ $AB;$ $B$ ↔ $A$ .

$E$ : Equivalent to itself.

Step 2:

Let $E$ be the identity in $F$ and let us take $F = {L, M, N, E}$ .

Here, * is the binary operation. By Table 3, $F$ forms a group under *.

Let $δ = {L}$ . The group $F$ satisfies the conditions of δ-group under *.

Table 3
Binary operation output of set $F$

* $E$ $L$ $M$ $N$

$E$ $E$ $L$ $M$ $N$

$L$ $L$ $E$ $N$ $M$

$M$ $M$ $N$ $E$ $L$

$N$ $N$ $M$ $L$ $E$

*	$E$	$L$	$M$	$N$
$E$	$E$	$L$	$M$	$N$
$L$	$L$	$E$	$N$	$M$
$M$	$M$	$N$	$E$	$L$
$N$	$N$	$M$	$L$	$E$

Step 3:

Let us define the fuzzy hyperoperations on $F$ as follows. For all $L, M, N, E \in F$ by

If $L ⊞ M$ , then $L ⊞ M$ = $χ_{{L, M}} (x)$ , ∀ $x \in F$ .

Similarly if $N ⊞ M$ , then $N ⊞ M = χ_{{N, M}} (x)$ , ∀ $x \in F$ . Likewise ∀ $L, M, N, E \in F$ .

If $L ⊠ M$ , then $L ⊠ M$ = $χ_{{F}} (x)$ , ∀ $x \in F$ .

Similarly if $N ⊠ M$ , then $N ⊠ M = χ_{{F}} (x)$ , $\forall x \in F$ . Likewise ∀ $L, M, N, E \in F$ .

Step 4:

The δ-group $F$ forms a fuzzy hyperlattice $(F, ⊞, ⊠)$ with the above fuzzy hyperoperations, i.e., for all $L, M, N, E \in F$ ,

$(L ⊠ L) (L) > 0$ , $(L ⊞ L) (L) > 0$ ,

$(L ⊠ M) = (M ⊠ L)$ , $(L ⊞ M) = (M ⊞ L)$ ,

$(L ⊠ M) ⊠ N = L ⊠ (M ⊠ N)$ , $(L ⊞ M) ⊞ N = L ⊞ (M ⊞ N)$ ,

$(L ⊠ (L ⊞ M)) (L) > 0$ , $(L ⊞ (L ⊠ M)) (L) > 0$ .

Step 5:

$F$ satisfies the conditions of fuzzy hyperlattice ordered δ-group, that is

$(F, *)$ is a δ-group,

$(F, ⊞, ⊠)$ is a fuzzy hyperlattice,

$R [δ {L} * (M ⊞ N), {δ L * M} ⊞ {δ L * N}] = 1$ and $R [δ {L} * (M ⊠ N), {δ L * M} ⊠ {δ L * N}] = 1$ $\forall {L, M, N, E} \in F$ and $δ = {L}$ .

Here,

R : F \times F ⟶ [0, 1]

is the fuzzy partial order relation.

Step 6:

In Tables 4, 5, 6, and 7, the possibility is denoted by 1 and impossibility is denoted by 0.

All the possible combinations for the elements in $F$ are rendered through Tables 4, 5, 6, and 7. By definition of Fuzzy hyperlattice ordered δ-group, $R [δ {L} * (M ⊞ N), {δ L * M} ⊞ {δ L * N}] = 1$ for all $L, M, N, E \in F$ . Here, we construct the table only for ${δ L} * (M ⊞ N)$ . (Obviously ${δ L * M} ⊞ {δ L * N}$ holds).

Similarly, we can develop for the fuzzy hyperoperation ⊠.

Table 4

$FHLO δ - G$ - A representation

∘	$x = L$	$x = M$	$x = N$	$x = E$
$δ L * (E ⊞ E) (x)$	0	0	0	1
$δ L * (E ⊞ L) (x)$	1	0	0	1
$δ L * (E ⊞ M) (x)$	0	1	0	1
$δ L * (E ⊞ N) (x)$	0	0	1	1
$δ L * (L ⊞ E) (x)$	1	0	0	1
$δ L * (L ⊞ L) (x)$	1	0	0	0
$δ L * (L ⊞ M) (x)$	1	1	0	0
$δ L * (L ⊞ N) (x)$	1	0	1	0
$δ L * (M ⊞ E) (x)$	0	1	0	1
$δ L * (M ⊞ L) (x)$	1	1	0	0
$δ L * (M ⊞ M) (x)$	0	1	0	0
$δ L * (M ⊞ N) (x)$	0	1	1	0
$δ L * (N ⊞ E) (x)$	0	0	1	1
$δ L * (N ⊞ L) (x)$	1	0	1	0
$δ L * (N ⊞ M) (x)$	0	1	1	0
$δ L * (N ⊞ N) (x)$	0	0	1	0

Table 5

$FHLO δ - G$ - A representation

∘	$x = L$	$x = M$	$x = N$	$x = E$
$δ M * (E ⊞ E) (x)$	0	0	1	0
$δ M * (E ⊞ L) (x)$	0	1	1	0
$δ M * (E ⊞ M) (x)$	1	0	1	0
$δ M * (E ⊞ N) (x)$	0	0	1	1
$δ M * (L ⊞ E) (x)$	0	1	1	0
$δ M * (L ⊞ L) (x)$	0	1	0	0
$δ M * (L ⊞ M) (x)$	1	1	0	0
$δ M * (L ⊞ N) (x)$	0	1	0	1
$δ M * (M ⊞ E) (x)$	1	0	1	0
$δ M * (M ⊞ L) (x)$	1	1	0	0
$δ M * (M ⊞ M) (x)$	1	0	0	0
$δ M * (M ⊞ N) (x)$	1	0	0	1
$δ M * (N ⊞ E) (x)$	0	0	1	1
$δ M * (N ⊞ L) (x)$	0	1	0	1
$δ M * (N ⊞ M) (x)$	1	0	0	1
$δ M * (N ⊞ N) (x)$	0	0	0	1

Table 6

$FHLO δ - G$ - A representation

∘	$x = L$	$x = M$	$x = N$	$x = E$
$δ N * (E ⊞ E) (x)$	0	1	0	0
$δ N * (E ⊞ L) (x)$	0	1	1	0
$δ N * (E ⊞ M) (x)$	0	1	0	1
$δ N * (E ⊞ N) (x)$	1	1	0	0
$δ N * (L ⊞ E) (x)$	0	1	1	0
$δ N * (L ⊞ L) (x)$	0	0	1	0
$δ N * (L ⊞ M) (x)$	0	0	1	1
$δ N * (L ⊞ N) (x)$	1	0	1	0
$δ N * (M ⊞ E) (x)$	0	1	0	1
$δ N * (M ⊞ L) (x)$	0	0	1	1
$δ N * (M ⊞ M) (x)$	0	0	0	1
$δ N * (M ⊞ N) (x)$	1	0	0	1
$δ N * (N ⊞ E) (x)$	1	1	0	0
$δ N * (N ⊞ L) (x)$	1	0	1	0
$δ N * (N ⊞ M) (x)$	1	0	0	1
$δ N * (N ⊞ N) (x)$	1	0	0	0

Table 7

$FHLO δ - G$ - A representation

∘	$x = L$	$x = M$	$x = N$	$x = E$
$δ E * (E ⊞ E) (x)$	1	0	0	0
$δ E * (E ⊞ L) (x)$	1	0	0	1
$δ E * (E ⊞ M) (x)$	1	0	1	0
$δ E * (E ⊞ N) (x)$	1	1	0	0
$δ E * (L ⊞ E) (x)$	1	0	0	1
$δ E * (L ⊞ L) (x)$	0	0	0	1
$δ E * (L ⊞ M) (x)$	0	0	1	1
$δ E * (L ⊞ N) (x)$	0	1	0	1
$δ E * (M ⊞ E) (x)$	1	0	1	0
$δ E * (M ⊞ L) (x)$	0	0	1	1
$δ E * (M ⊞ M) (x)$	0	0	1	0
$δ E * (M ⊞ N) (x)$	0	1	1	0
$δ E * (N ⊞ E) (x)$	1	1	0	0
$δ E * (N ⊞ L) (x)$	0	1	0	1
$δ E * (N ⊞ M) (x)$	0	1	1	0
$δ E * (N ⊞ N) (x)$	0	1	0	0

Observation: From Tables 4, 5, 6, and 7, the possibilities of the relations ( $L$ , $M$ , $N, E)$ are checked. If arbitrarily the relation $N$ comes means, the possible blood type for the child is $(A), (B)$ and $(O), (A), (B), (AB)$ (i.e.,) $(A), (B)$ blood type has more possibilities.

Also, if we take an arbitrarily combination from any table $δ E * (N ⊞ N) (x)$ , the relation $M$ is not used in the combination but it is possible through this combination. We can obtain many possibilities by assigning different elements to δ.

10 Conclusion

In this research, we introduced the notion of fuzzy hyperlattice ordered δ-group as a new fuzzy hypercompositional structure. We used our accession to fuzzy hyperlattice ordered group (FHLOG) so as to accord with $FHLO δ - G$ . As well, some properties of $FHLO δ - G$ were discussed and an application of $FHLO δ - G$ in $ABO$ blood group system was developed. We hope that the proposed work helps to broaden the applications of FHLOG. As a future work, we will extend the theory by introducing morphisms on $FHLO δ - G$ and check for other applications.

Footnotes

Acknowledgment

The article has been written with the joint financial support of RUSA-Phase 2.0 grant sanctioned vide letter No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt. 09.10.2018, UGC-SAP (DRS-I) vide letter No.F.510/8/DRS-I/2016(SAP-I) Dt. 23.08.2016, DST-PURSE 2nd Phase programme vide letter No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FST - level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17 Dt. 20.12.2018.

References

Al-Tahan

, Davvaz

, Algebraic hyperstructures associated to Biological inheritance, Mathematical Biosciences, Elsevier285 (2017), 112–118.

Al-Tahan

, Davvaz

, n-ary hyperstructures associated to the genotypes of F2-offspring, International Journal of Biomathematics10(8) (2017), 1750118–1750135.

Al-Tahan

, Davvaz

, A new relationship between intuitionistic fuzzy sets and genetics, Journal of Classification (Springer)36 (2019), 494–512.

Birkhoff

, Lattice ordered group, Annals of Mathematics, Second Series43(2) (1942), 298–331.

Corsini

, Leoreanu-Fotea

, Applications of Hyperstructure Theory, Kluwer, Dordrecht, 2003.

Davvaz

, Cristea

, Fuzzy Algebraic Hyperstructures-An Introduvtion, Springer, 2015.

Davvaz

, Leoreanu-Fotea

, Hyperring Theory and Applications, International Academic Press, USA, 2007.

Geoff

, Imelda

, Essential Guide to Blood Groups, Blackwell2007.

Guo

X.Z.

, Xin

X.L.

, Hyperlattices, Pure Appl Math20 (2004), 40–43.

10.

Hartl

D.L.

, Jones

E.W.

, Genetics, Principles and Analysis, Jones and Bartlett Publishers, 1998.

11.

Konstantinidou

, Mittas

, An introduction to the theory of hyperlattices, Mathematica Balkanica7 (1977), 187–193.

12.

Rasouli

, Davvaz

, Lattices derived from hyperlattices, Comm Algebra38(8) (2010), 2720–2737.

13.

Marty

, Sur une generalisation de la notion de groupe, in 8th Congres des Mathematiciens Scandinaves (1934), 45–49.

14.

Preethi

, Vimala

, Bijan Davvaz and S.Rajareega, Biological inheritance on fuzzy hyperlattice ordered group, Journal of Intelligent & Fuzzy Systems38(5) (2020), 6457–6464. DOI: 10.3233/JIFS-179726.

15.

Preethi

, Vimala

, Rajareega

, A systematic study in the applications of fuzzy hyperlattice, AIMS Mathematics6(2) (2020), 1695–1705.

16.

Preethi

, Vimala

, Rajareega

, Some properties on fuzzy hyperlattice ordered group, International Journal of Advanced Science and Technology28(9) (2019), 124–132.

17.

, Xin

, Zhan

, On Rough hyperideals in hyperlattices, Journal of Applied Mathematics Volume 2013, Article ID 915217.

18.

, Xin

, Fuzzy hyperlattices, Computers and Mathematics with Applications62 (2011), 4682–4690.

19.

Rajareega

, Vimala

and Preethi

, Complex Intuitionistic Fuzzy Soft Lattice Ordered Group and Its Weighted Distance Measures Mathematics8 705 (2020).

20.

Rasouli

, Davvaz

, Construction and spectral topology on Hyperlattices, Mediterranean Journal of Mathematics7 (2010), 249–262.

21.

Sabeena Begam

, Vimala

and Preethi

, A Novel Study on the Algebraic Applications of Special Class of Lattice Ordered Multi-Fuzzy Soft Sets, Journal of Discrete Mathematical Sciences and Cryptography22(5) (2019), 883–899.

22.

Soltani

, Lashkenari and B. Davvaz, Ordered join hyperlattices, U.P.B. Scientific Bulletin, Series A78(4) (2016), 35–44.

23.

Soltani

, Lashkenari and B. Davvaz, Complete join hyperlattices, Indian J Pure Appl Math46(5) (2015), 633–645.

24.

Soltani Lashkenari

and Davvaz

, Principal, compact and regular elements in hyperlattices, Advances and Applications in Mathematical Sciences16(5) (2017), 163–178.

25.

Soltani Lashkenari

and Davvaz

, Almost principal ideals and tensor product of hyperlattices, Analele Univ. “Ovidius” din Constanta, Math Series25(2) (2017), 171–183.

26.

Tamarin

R.H.

, Principles of Genetics, 7th Ed., The McGraw-Hill Companies, 2001.

27.

Vimala

, Fuzzy lattice ordered group, International Journal of Scientific and Engineering Research5(9) (2014).

28.

Vougiouklis

, Hyperstructures and Their Representations, Hadronic Press, Inc115 Palm Harber, USA, 1994.

29.

Yazer

, Olsson

, Palcic

, The cis-AB blood group phenotype: fundamental lessons in glycobiology, Transfusion Medicine Reviews20(3) (2006), 207–217.

30.

Zadeh

L.A.

, Fuzzy sets, Information and Control8(3) (1965), 338–353.

*	$E$	$L$	$M$	$N$
$E$	$E$	$L$	$M$	$N$
$L$	$L$	$E$	$N$	$M$
$M$	$M$	$N$	$E$	$L$
$N$	$N$	$M$	$L$	$E$

*	$E$	$L$	$M$	$N$
$E$	$E$	$L$	$M$	$N$
$L$	$L$	$E$	$N$	$M$
$M$	$M$	$N$	$E$	$L$
$N$	$N$	$M$	$L$	$E$

Fuzzy hyperlattice ordered δ - group and its application on ABO blood group system

Abstract

Keywords

6 Introduction

7 Preliminaries

8 Fuzzy Hyperlattice ordered δ - Group ( FHLO δ - G )

9 Application

9.1 ABO Blood Group

Table 1 Antigen and genotype for the blood types Blood Type Antigen Genotype A A I A I A or I A i B B I B I B or I B i AB A , B I A I B O No antigen ii

Table 3 Binary operation output of set F * E L M N E E L M N L L E N M M M N E L N N M L E

Footnotes

Acknowledgment

References

8 Fuzzy Hyperlattice ordered δ - Group $(FHLO δ - G)$

Table 1
Antigen and genotype for the blood types

Blood Type Antigen Genotype

$A$ $A$ I^AI^A or I^Ai

$B$ $B$ I^BI^B or I^Bi

$AB$ $A, B$ I ^A I ^B

$O$ No antigen ii

Table 3
Binary operation output of set $F$

* $E$ $L$ $M$ $N$

$E$ $E$ $L$ $M$ $N$

$L$ $L$ $E$ $N$ $M$

$M$ $M$ $N$ $E$ $L$

$N$ $N$ $M$ $L$ $E$

*	$E$	$L$	$M$	$N$
$E$	$E$	$L$	$M$	$N$
$L$	$L$	$E$	$N$	$M$
$M$	$M$	$N$	$E$	$L$
$N$	$N$	$M$	$L$	$E$