Biological inheritance on fuzzy hyperlattice ordered group

Abstract

In this manuscript we proposed the concept of fuzzy hyperlattice ordered group. Algebraic hyperstructures represent a natural extension of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Algebraic hyperstructure theory has many applications in other disciplines. The foremost intendment of the manuscript is to contribute some properties of fuzzy hyperlattice ordered group and also an application of fuzzy hyperlattice ordered group on inheritance.

Keywords

Hyperlattice fuzzy hyperlattice lattice ordered group fuzzy lattice ordered group fuzzy hyperlattice ordered group AMS MSC: 03E72,06D72,20N20 03E72 06D72 20N20

1 Introduction

Algebraic hyperstructures were introduced in 1934 by the French mathematician Marty [12]. Since then, hundreds of papers and several books have been written on this topic, there appeared many components of hyperalgebras such as hypergroups, hyperrings, hypermodules and weak hyperstructures [2–5 , 21]. Konstantinidou and Mittas introduced the concept of hyperlattices in [11]. In particular, Rasouli and Davvaz [13] further studied the theory of hyperlattices and obtained some interesting results, which enriched the theory of hyperlattices, also see [14 –18]. On the other hand, the fuzzy mathematics forms a branch of mathematics related to fuzzy set theory and fuzzy logic. It was introduced in 1965 after the publication of Zadeh [22]. Fuzzy sets are sets whose elements have degrees of membership. Many researchers worked on fuzzy set theory, its applications and its extensions. He and Xin [10] introduced the notion of fuzzy hyperlattice and studied connections between fuzzy hyperlattices and hyperlattices. The combination of fuzzy set theory and algebraic systems may provide more new interesting topics which have drawn attention of many mathematicians and computer scientists. There is a considerable amount of works initiated on the relationship between algebraic hyperstructures and fuzzy sets [2, 3]. Since then, we can observe rapid growth in the literature of various fuzzy hyperalgebraic concepts. The contribution of this paper is to enhance the available techniques or to establish some new techniques for the development of fuzzy hyperlattice ordered group which is the extension of fuzzy lattice ordered group and is one of the important structures with many applications. In fuzzy set theory 0 and 1 play a important role so it can be implemented to computer technology and decision making. Comparison of multiple domains is possible through hypersrtucture theory. Hence we can develop wide range of practical applications on the topic fuzzy hyperlattice ordered group.

In Section 2, some basic definitions are examined. In Section 3, considering fuzzy hyperlattice [10] and fuzzy lattice ordered group [20] we have illustrate the fuzzy hyperlattice ordered group and also found some properties on fuzzy hyperlattice ordered group. Section 4 deals with inheritance application.

2 Preliminaries

In this section, we review the fundamentals of algebraic hyperstrutures, hyperlattice and fuzzy algebraic hyperstructures that are useful for further discussions.

Definition 2.1. [4 , 10] Algebraic hyperstructures are a suitable generalizations of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element while in algebraic hyperstructure, the composition of two elements is a set. More exactly, if H is a non-empty set and $P^{*} (H)$ is the family of all non-empty subsets of H, then we consider the maps of the following type $f_{i} : H \times H ⟶ P^{*} (H),$ where i ∈ {1, 2, . . . , n} and n is a positive integer. The maps f_i are called (binary) hyperoperations. For all x, y ∈ H, f_i (x, y) is called the (binary) hyperproduct of x and y. An algebraic system (H, f₁, . . , f_n) is called a (binary) hyperstructure. Usually n = 1 or n = 2.

Let L be non-empty set and $P^{*} (L)$ be the set of all non-empty subsets of L. A hyperoperation on L is a map $\circ : L \times L ⟶ P^{*} (L)$ , which associates a non-empty subset p ∘ q with any pair (p, q) of elements of L × L. The couple (L, ∘) is called a hypergroupoid.

If P and Q are non-empty subsets of L, for all p, q, l ∈ L, then we denote

l ∘ P = {l} ∘ P = ⋃ _p∈Pl ∘ p, P ∘ l = P ∘ {l} = ⋃ _p∈Pp ∘ l;

$P \circ Q = ⋃_{\binom{p \in P}{q \in Q}} p \circ q$ .

Definition 2.2. [7, 10] Let L be a non-empty set endowed with two hyperoperations “⊕" and “⊗". The triple (L, ⊕ , ⊗) is called a hyperlattice if the following identities hold, for all p, q, r ∈ L,

(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. [7] Let (L, ∨ , ∧) be a lattice. Define the following hyperoperations on L for all a, b ∈ L, a ⊗ b = {x ∈ L | a ∨ x = b ∨ x = a ∨ b} and a ⊕ b = {x ∈ L | x ≤ a ∧ b}. Then, (L, ⊗ , ⊕) is a hyperlattice.

Remark 1. [7]

A partially ordered set P is a hyperlattice if

(a) every subset of P has a unique least upper bound (LUB) and

(b) there is a subset L of minimal elements in P such that for each C in L, the set of elements in the ascending chains from C to LUB(P) is a lattice.

Every lattice is a hyperlattice.

Definition 2.3. [15] Let (L, ⋁ , ⋀) be a hyperlattice. We call (L, ≤) is an ordered hyperlattice, if ≤ is an equivalence relation and l ≤ m implies that $\begin{matrix} l ⋁ n \leq m ⋁ n and l ⋀ n \leq m ⋀ n . \end{matrix}$

Definition 2.4. [22] Let X be a set. A function P : X × X ⟶ [0, 1] is called a fuzzy relation in X.

Definition 2.5. [22] Let (X, P) be a fuzzy poset. Then, (X, P) is a fuzzy lattice if and only if l ∨ m and l ∧ m exist for all l, m ∈ X.

Definition 2.6. [10] Let L be a non-empty set endowed with two fuzzy hyper operations ⊗ and ⊕ (Here F^* (L) be the set of all non-zero fuzzy subsets of L. A fuzzy hyperoperation has the mapping in the form ∘ : L × L ⟶ F^* (L)). The triple (L, ⊗ , ⊕) is called a fuzzy hyperlattice if the following identities holds, for all p, q, r ∈ L,

(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: [10] If U and V are two non-zero fuzzy subsets, for all p, l ∈ L, then we define

$(p \circ U) (l) = sup_{t \in L} {(p \circ t) (l) \land U (t)}$ ,

$(U \circ p) (l) = sup_{t \in L} {U (t) \land (t \circ p) (l)}$ .

$(U \circ V) (l) = sup_{t \in L} {U (u) \land (u \circ v) (l) \land V (v)}$ .

Example 2. Let (L, ∧ , ∨) be a lattice. If we define the following two hyperoperations on L: for all a, b ∈ L, a × b = χ_{a,b} and a + b = χ_a∧b then (L, × , +) is a fuzzy hyperlattice.

Definition 2.7. [1] A lattice ordered group is a system G = (G, + , ≤) such that

(G, +) is a group,

(G, ≤) is a lattice,

l ≤ m ⇒ p + l + q ≤ p + m + q for all p, q ∈ G.

Example 3. [1] $(ℤ, +, \leq)$ is a lattice ordered group.

Definition 2.8. [20] A non-empty set G is called a fuzzy lattice ordered group if

(G, +) is a group,

(G, ∨ , ∧) is a fuzzy lattice,

R (p + (l ∨ m) , (p + l) ∨ (p + m)) =1, for all p, l, m ∈ G.

Here R : G × G ⟶ [0, 1] is a fuzzy partial order relation.

Example 4. [20] $(R, +, A)$ is a fuzzy lattice ordered group.

3 Fuzzy hyperlattice ordered group

In this section, we introduce the notion of fuzzy hyperlattice ordered group and discuss some fundamental properties of fuzzy hyperlattice ordered group.

Throughout this section, ⊕, ⊗ , ⋁ , ⋀ are hyperoperations, G is a group, + , * , · are binary operations and define the hyperoperations ⋁, ⋀ on L as follows, for all l, m, n ∈ G, l ⋁ m = {l ∨ m}, l ⋀ m = {l ∧ m}

Definition 3.1. A non-empty set G is called a fuzzy hyperlattice ordered group if

(G, +) is a group,

(G, ⊕ , ⊗) is a fuzzy hyperlattice,

R [{p} + (l ⊕ m) , {p + l} ⊕ {p + m}] =1 and R [{p} + (l ⊗ m) , {p + l} ⊗ {p + m}] =1,

for all p, l, m ∈ G. Here R : G × G ⟶ [0, 1] is a fuzzy partial order relation.

Example 5. (Q⁺ \ {0} , * , ⊗ , ⊕) with the hyperoperations defined by p ⊕ q = χ_{p,q} and p ⊗ q = χ_(p∧q), for all p, q ∈ Q⁺ \ {0} and R : Q⁺ \ {0} × Q⁺ \ {0} ⟶ [0, 1] is a fuzzy partial order relation. Here, (Q⁺ \ {0} , * , ⊗ , ⊕) is an example of fuzzy hyperlattice ordered group.

Proposition 3.2. A fuzzy hyperlattice ordered group is always distributive.

Proposition 3.3. In any fuzzy hyperlattice ordered group G, $\begin{matrix} R [{p ⋀ q} - {p ⋀ r}, {q - r}] \geq 0, \end{matrix}$ for all p, q, r ∈ G.

Proof. We have $\begin{matrix} {p ⋀ q} - {p ⋀ r} & = {p} ⋀ {q - r} \\ \leq {q - r} . \end{matrix}$ Therefore, we obtain R [{p ⋀ q} - {p ⋀ r} , {q - r}] ≥0 .□

Proposition 3.4.In any fuzzy hyperlattice ordered group G, $R [{p \oplus q}, [{p - q} \oplus {0}] + {q}] = 1,$ for all p, q ∈ G.

Proof. We have $\begin{matrix} [{p - q} \oplus {0}] + {q} & = {p - q + q} \oplus {q + 0} \\ = {p \oplus q} . \end{matrix}$ Therefore, we conclude that R [{p ⊕ q} , [{p - q} ⊕ {0}] + {q}] =1.□

Proposition 3.5. In any fuzzy hyperlattice ordered group G, $R [{p \oplus q} - {r}, {p - r} \oplus {q - r}] = 1,$ for all p, q, r ∈ G.

Proof. We have $\begin{matrix} ({p - r} \oplus {q - r}) + {r} & = & {r} + {(p - r) \oplus (q - r)} \\ = & {r + (p - r)} \oplus {r + (q - r)} \\ \Rightarrow {p - r} \oplus {q - r} & = & {p \oplus q} - {r} . \end{matrix}$ Therefore, we get R [{p ⊕ q} - {r} , {p - r} ⊕ {q - r}] =1.□

Definition 3.6. A non-empty set G is called a fuzzy hyperlattice ordered group if for all p, l, m, n ∈ G,

(G, ·) is a group,

(G, ⊕ , ⊗) is a fuzzy hyperlattice,

R [l, m] ≥0 ⇒R [(p · l) ⊕ n, (p · m) ⊕ n] ≥0 and R [l, m] ≥0 ⇒R [(p · l) ⊗ n, (p · m) ⊗ n] ≥0.

Here R : G × G ⟶ [0, 1] is a fuzzy partial order relation since

R (l, m) ≥0 ⇒ l ≤ m,

p · l ≤ p · m,

(p · l) ⊕ n ≤ (p · m) ⊕ n,

R [(p · l) ⊕ n, (p · m) ⊕ n] ≥0.

Theorem 3.7. Both definitions of fuzzy hyperlattice ordered group are equivalent.

Proof. Here, let us define the hyperoperations by $l \oplus m = {l \lor m} and l \otimes m = {l \land m},$ for all l, m ∈ G. It is enough to prove that the following two conditions are equivalent:

R [l, m] ≥0 ⇒R [(p . l) ⊕ n, (p · m) ⊕ n] ≥0,

R [{p} · (l ⊕ m) , {p · l} ⊕ {p · m}] =1,

for all p, l, m, n ∈ G.

(1⇒2): Assume that, R [l, m] ≥0. Then, we have R [{p · l} ⊕ n, {p · m} ⊕ n] ≥0, i.e., l ≤ m implies {p · l} ⊕ n ≤ {p · m} ⊕ n. Hence, {p · l} ≤ {p · m}

Now, we obtain $\begin{matrix} R [{p} \cdot (l \oplus m), {p \cdot l} \oplus {p \cdot m}] \\ = & R [{p} \cdot {m}, {p \cdot m}] \\ = & 1 . \end{matrix}$

(2⇒1): Assume that R [{p} · (l ⊕ m) , {p · l} ⊕ {p · m}] =1 [R (l, m) ≥0, i.e., l ≤ m]. Thus, we get $\begin{matrix} {p} \cdot (l \oplus m) = {p \cdot l} \oplus {p \cdot m} \\ \Rightarrow & {p} \cdot {m} = {p \cdot l} \oplus {p \cdot m} \\ \Rightarrow & {p \cdot l} \leq {p \cdot m} \\ \Rightarrow & {p \cdot l} \oplus n \leq {p \cdot m} \oplus n \\ \Rightarrow & R [{p \cdot l} \oplus n, {p \cdot m} \oplus n] \geq 0 . \end{matrix}$

Theorem 3.8.R [{p} - (q ⋁ r) , {p - q} ⋀ {p - r}] =1

Proof. Let p, q, r ∈ G be arbitrary. Then, we have $\begin{matrix} {q} + {r} \leq q ⋁ r \\ \Rightarrow & - (q ⋁ r) \leq - {q}, - {r} \\ \Rightarrow & {p} - (q ⋁ r) \leq {p} - {q}, {p} - {r} \\ \Rightarrow & {p} - (q ⋁ r) \leq {p - q} ⋀ {p - r} . \end{matrix}$ Also, we have $\begin{matrix} [{p - q} ⋀ {p - r}] + (q ⋁ r) \\ = & [{p - q} ⋀ {p - r}] + {q} ⋁ [{p - q} \\ ⋀ {p - r}] + {r} \\ \leq & [{p - q} + {q}] ⋁ [{p - r} + {r}] \\ \leq & p ⋁ p = {p} . \end{matrix}$

Consequently, we obtain that [{p - q} ⋀ {p - r}] ≤ {p} - (q ⋁ r). This implies that R [{p} - (q ⋁ r) , {p - q} ⋀ {p - r}] =1.□

Theorem 3.9.In any fuzzy hyperlattice ordered group G, we have $R [({p} - {p ⋀ q} + {q}), {q} ⋁ {p}] = 1,$ for all p, q ∈ G.

Proof. We know that, in fuzzy hyperlattice, the hyperoperation ⋁, ⋀ which is idempotent, commutative, associative and satisfies distributive laws, i.e., $\begin{matrix} {p} + {l ⋁ m} = {p + l} ⋁ {p + m}, \\ {l ⋁ m} + {q} = {l + q} ⋁ {m + q}, \end{matrix}$ for all l, m ∈ G. Now, $\begin{matrix} ({p} - {p ⋀ q}) + {q} & = ({p - p} ⋁ {p - q}) + {q} \\ = ({q} ⋁ {p}) . \end{matrix}$ Therefore, we conclude that R [({p} - {p ⋀ q} + {q}) , {q} ⋁ {p}] =1.□

Corollary 3.10. Let G be a fuzzy hyperlattice ordered group, for all p, q ∈ G, the following are equivalent:

R [{p ⋀ q} , {0}] =1,

R [{p + q} , {p ⋁ q}] =1.

Proof. (1⇒2): Assume that R [{p ⋀ q} , {0}] =1, i.e., {p ⋀ q} = {0}. By Theorem 3.9, R [({p} - {p ⋀ q} + {q}) , {q} ⋁ {p}] =1. Then, $\begin{matrix} ({p} - {p ⋀ q} + {q}) = {q} ⋁ {p} \\ ({p + q}) = {p ⋁ q} . \end{matrix}$ Therefore, we have R [{p + q} , {p ⋁ q}] =1.

(2⇒1): Assume that R [{p + q} , {p ⋁ q}] =1, i.e., {p + q}= {p ⋁ q}. By Theorem 3.9, R [({p} - {p ⋀ q} + {q}) , {q} ⋁ {p}] =1. Then,

$\begin{matrix} ({p} - {p ⋀ q} + {q}) = {q} ⋁ {p} \\ ({p + q} - {p ⋁ q}) = {p ⋀ q} \\ {p ⋀ q} = {0} . \end{matrix}$ Therefore, we conclude that R [{p ⋀ q} , {0}] =1 .□

Corollary 3.11. In any fuzzy hyperlattice ordered group G, $R [{p} + {q}, {p ⋁ q} + {p ⋀ q}] = 1,$ for all p, q ∈ G.

Definition 3.12. Two positive elements p, q of fuzzy hyperlattice ordered group G are disjoint if p ⋀ q = {0}. It is denoted by p ⊥ q.

Theorem 3.13. In any fuzzy hyperlattice ordered group G, if p ⊥ q and p ⊥ r, then R [{p} ⋀ {q + r} , {0}] =1 for all p, q, r ∈ G.

Proof. Let G be a fuzzy hyperlattice ordered group and let {p ⋀ q} = {0} and {p ⋀ r} = {0}. We want to show that R [{p} ⋀ {q + r} , {0}] =1. Indeed, $\begin{matrix} {p ⋀ q} = {0} (i . e ., {p} ⋀ {q} = {0}) \\ {r} = ({p} ⋀ {q}) + {r} \\ {r} = ({p + r} ⋀ {q + r}) \end{matrix}$ Now, we have $\begin{matrix} {0} & = {p ⋀ r} = {p} ⋀ {r} \\ = {p} ⋀ {p + r} ⋀ {q + r} \\ = ({p} ⋀ {p}) + ({p} ⋀ {r}) ⋀ {q + r} \\ = {p} ⋀ {q + r} . \end{matrix}$ Therefore, we get R [{p} ⋀ {q + r} , {0}] =1 .□

Theorem 3.14. (Rieze Decomposition Property) Let G be a fuzzy hyperlattice ordered group. Let l, m, n ∈ (G, * , ⋁ , ⋀).

If l, m, n ∈ G⁺ and {l} ≤ {m + n}, then {l} ≤ {m₁ + n₁}, where {0} ≤ {m₁} < {m} and {0} ≤ {n₁} < {n}.

Proof. Let {m₁} = {l ⋀ m} and {n₁} = {- m₁ + l}. Then, {0} ≤ {m₁} < {m}, $\begin{matrix} {n_{1}} & = {- {l ⋀ m} + l} \\ = {{- l ⋁ - m} + l} \\ = {0 ⋁ (- m + l)} \leq {n} \end{matrix}$ Hence, {n₁} ≥ {0}. From this we get, {l} ≤ {m₁+n₁}□

Theorem 3.15. Let {G, · , e, ⋁ , ⋀} be an algebraic system. {G, ·} is a group. {G, ⋁ , ⋀} is a fuzzy hyperlattice (with hyperoperation defined by, l ⋁ m = {l ∨ m}, l ⋀ m = {l ∧ m}) for all l, m, n ∈ G, if the relations $\begin{matrix} R [{(p \cdot l) ⋁ n} ⋁ {(p \cdot m) ⋁ n}, {(p \cdot l) ⋁ (p \cdot m)} ⋁ n] = 1, \\ R [{(p \cdot l) ⋀ n} ⋀ {(p \cdot m) ⋀ n}, {(p \cdot l) ⋀ (p \cdot m)} ⋀ n] = 1 \end{matrix}$ hold, then G is a fuzzy hyperlattice ordered group under the partial order of the fuzzy hyperlattice. If G is a Fuzzy hyperlattice ordered group then that relation holds.

Proof. Let R [l, m] ≥0, i.e., l ≤ m. Then, (p · l) ≤ (p · m). This implies that ((p · l) ⋁ (p · m)) = {(p · l) ∨ (p · m)} = {p · m}. Now, we have $\begin{matrix} {(p \cdot l) ⋁ n} ⋁ {(p \cdot m) ⋁ n} = {(p . l) ⋁ (p \cdot m)} ⋁ {n} \\ {(p \cdot l) ⋁ n} \leq {(p \cdot m) ⋁ n} \\ R [{(p \cdot l) ⋁ n, (p \cdot m) ⋁ n}] \geq 0 . \end{matrix}$ Therefore, G is a fuzzy hyperlattice ordered group. Similarly, let R [l, m] ≥0, i.e., l ≤ m. Then, (p · l) ≤ (p · m). This implies that ((p · l) ⋀ (p · m)) = {(p · l) ∧ (p · m)} = {p · l}. Now, we have $\begin{matrix} {(p \cdot l) ⋀ n} ⋀ {(p \cdot m) ⋀ n} = {(p \cdot l) ⋀ (p \cdot m)} ⋀ {n} \\ {(p \cdot m) ⋀ n} \geq {(p \cdot l) ⋀ n} \\ R [{(p \cdot l) ⋀ n, (p \cdot m) ⋀ n}] \geq 0 \end{matrix}$ Therefore, G is a fuzzy hyperlattice ordered group.

For the converse, let G be a fuzzy hyperlattice ordered group. For all p, l, m, n ∈ G we have R [{(p · l) ⋁ n} , {(p · m) ⋁ n}] ≥0. Therefore, we have $\begin{matrix} {(p \cdot l) ⋁ n} \leq {(p \cdot m) ⋁ n} \\ \Rightarrow & {p \cdot l} \leq {p \cdot m} \\ {(p \cdot l) ⋁ n} ⋁ {(p \cdot m) ⋁ n} \\ = & {(p \cdot m) ⋁ n} \\ \Rightarrow & {(p \cdot l) ⋁ n} ⋁ {(p \cdot m) ⋁ n} \\ = & {((p . l) ⋁ (p \cdot m)) ⋁ n} \end{matrix}$ ⇒R [{(p . l) ⋁ n} ⋁ {(p · m) ⋁ n} , {((p . l) ⋁ (p . m)) ⋁ n}] =1.

Similarly, let G be a fuzzy hyperlattice ordered group. For all p, l, m, n ∈ G, we have R [{(p · l) ⋀ n} , {(p · m) ⋀ n}] ≥0. Therefore, $\begin{matrix} {(p \cdot l) ⋀ n} \leq {(p \cdot m) ⋀ n} \\ \Rightarrow & {p \cdot l} \leq {p \cdot m} \\ {(p \cdot l) ⋀ n} ⋀ {(p \cdot m) ⋀ n} \\ = & {(p \cdot l) ⋀ n} \\ \Rightarrow & {(p \cdot l) ⋀ n} ⋀ {(p \cdot m) ⋀ n} \\ = & {((p \cdot l) ⋀ (p \cdot m)) ⋀ n} \end{matrix}$ ⇒R [{(p · l) ⋀ n} ⋀ {(p · m) ⋀ n} , {((p · l) ⋀ (p · m)) ⋀ n}] =1.□

4 Application

4.1 The dihybrid cross in the pea plant

[6] Mendel crossed varieties of peas that differed in two characteristics (dihybrid crosses) as an addition to his work on monohybrid crosses [8, 19]. For example, he had a homozygous variety of peas that produced round seeds and tall plants. Another homozygous variety produced wrinkled seeds and short plants. When he crossed the two plants, all the F1 progenies had round seeds and tall plants. For example,

P: Round; Tall (RRTT genotype) ⊗Wrinkled; Short (rrtt genotype)

↓

F₁: All Round; Tall (RrTt genotype) and

F₁ ⊗ F₁: Round; Tall (RrTt genotype) ⊗ Round; Tall (RrTt genotype)

↓

F₂: Round; Tall (RRTT genotype)Round; Short (RRtt and Rrtt genotypes) Wrinkled; Tall (rrTT and rrTt genotypes)Wrinkled; Short (rrtt genotype)

Tall and Round are denoted by $P$ , Tall and Wrinkled by $Q$ , Dwarf and Round by $R$ , finally Dwarf and Wrinkled by $S$ . The outcomes of dihybrid cross are shown in Table 1.

Table 1
A diagrammatic explanations of the dihybrid cross in the pea plant

⊗ $P$ $Q$ $R$ $S$

$P$ $P$ , $Q$ , $R$ , $S$ $P$ , $Q$ , $R$ , $S$ $P$ , $Q$ , $R$ , $S$ $P$ , $Q$ , $R$ , $S$

$Q$ $P$ , $Q$ , $R$ , $S$ $Q$ , $S$ $P$ , $Q$ , $R$ , $S$ $Q$ , $S$

$R$ $P$ , $Q$ , $R$ , $S$ $P$ , $Q$ , $R$ , $S$ $R$ , $S$ $R$ , $S$

$S$ $P$ , $Q$ , $R$ , $S$ $Q$ , $S$ $R$ , $S$ $S$

⊗	$P$	$Q$	$R$	$S$
$P$	$P$ , $Q$ , $R$ , $S$	$P$ , $Q$ , $R$ , $S$	$P$ , $Q$ , $R$ , $S$	$P$ , $Q$ , $R$ , $S$
$Q$	$P$ , $Q$ , $R$ , $S$	$Q$ , $S$	$P$ , $Q$ , $R$ , $S$	$Q$ , $S$
$R$	$P$ , $Q$ , $R$ , $S$	$P$ , $Q$ , $R$ , $S$	$R$ , $S$	$R$ , $S$
$S$	$P$ , $Q$ , $R$ , $S$	$Q$ , $S$	$R$ , $S$	$S$

4.2 Inheritance application on fuzzy hyperlattice ordered group

Using the outcomes of dihybrid cross of pea plant, we defined the relation as (λ, μ, ν, e) discussed in Step 1. Our aim is to check the possible relation of outcomes through fuzzy hyperlattice ordered group. (i.e.,) if the relation λ (arbitrary) occurs, the possible outcomes are $(P)$ , $(Q)$ , $(R)$ , $(S)$ and $(R)$ , $(S)$ .From the above we observe $(R)$ , $(S)$ has more possibilities.

Step 1:

The dihybrid cross outcomes in Pea plant is considered as following relation.

λ: $P$ ↔ $Q$ $R$ ↔ $S$

μ: $P$ ↔ $R$ $Q$ ↔ $S$

ν: $P$ ↔ $S$ $R$ ↔ $Q$ .

e: The outcomes equivalent to itself.

Step 2:

Let us take L = {λ, μ, ν, e}. where e is the identity element in L.

Here, * is the binary operation. By Table 2, L forms a group under the binary operation *.

Table 2
Binary operation output of set L

* e λ μ ν

e e λ μ ν

λ λ e ν μ

μ μ ν e λ

ν ν μ λ e

*	e	λ	μ	ν
e	e	λ	μ	ν
λ	λ	e	ν	μ
μ	μ	ν	e	λ
ν	ν	μ	λ	e

Step 3:

We restricted the outcomes through hyperoperations.

Let us define the hyperoperations for all λ, μ, ν, e ∈ L by

If λ ⊕ μ, then λ ⊕ μ= χ_{λ,μ} (x), for all x ∈ L.

Similarly if ν ⊕ μ, then ν ⊕ μ = χ_{ν,μ} (x), for all x ∈ L. Likewise for all elements in L.

If λ ⊗ μ, then λ ⊗ μ= χ_{L} (x), for all x ∈ L.

Similarly if ν ⊗ μ, then ν ⊗ μ = χ_{L} (x), ∀x ∈ L. Likewise for all elements in L.

Step 4:

The group L with the above hyperoperations forms a fuzzy hyperlattice (L, ⊗ , ⊕), i.e., for all λ, μ, ν, e ∈ L,

(λ ⊗ λ) (λ) >0, (λ ⊕ λ) (λ) >0,

(λ ⊗ μ) = (μ ⊗ λ), (λ ⊕ μ) = (μ ⊕ λ),

(λ ⊗ μ) ⊗ ν = λ ⊗ (μ ⊗ ν), (λ ⊕ μ) ⊕ ν = λ ⊕ (μ ⊕ ν),

(λ ⊗ (λ ⊕ μ)) (λ) >0, (λ ⊕ (λ ⊗ μ)) (λ) >0.

Step 5:

L satisfies the condition of fuzzy hyperlattice ordered group, that is

(L, *) is a group,

(L, ⊕ , ⊗) is a fuzzy hyperlattice,

R [{λ} * (μ ⊕ ν) , {λ * μ} ⊕ {λ * ν}] =1 and

R [{λ} * (μ ⊗ ν) , {λ * μ} ⊗ {λ * ν}] =1 ∀ {λ, μ, ν, e} ∈ L.

Here R : L × L ⟶ [0, 1] is a fuzzy partial order relation.

Step 6:

In Tables ??, 4, 5 and 6, 1 represents the possibility, 0 represents the impossibility.

Through Tables 3, 4, 5 and 6, we rendered all the possible combinations for the elements in L. By definition of Fuzzy hyperlattice ordered group, $R [{λ} * (μ \oplus ν), {λ * μ} \oplus {λ * ν}] = 1$ for all λ, μ, ν, e ∈ L. Here, we proceed only for {λ} * (μ ⊕ ν). (Obviously {λ * μ} ⊕ {λ * ν} holds).

Table 3

Fuzzy hyperlattice ordered group- A diagrammatic representation

∘	x = λ	x = μ	x = ν	x = e
λ * (e ⊕ e) (x)	1	0	0	0
λ * (e ⊕ λ) (x)	1	0	0	1
λ * (e ⊕ μ) (x)	1	0	1	0
λ * (e ⊕ ν) (x)	1	1	0	0
λ * (λ ⊕ e) (x)	1	0	0	1
λ * (λ ⊕ λ) (x)	0	0	0	1
λ * (λ ⊕ μ) (x)	0	0	1	1
λ * (λ ⊕ ν) (x)	0	1	0	1
λ * (μ ⊕ e) (x)	1	0	1	0
λ * (μ ⊕ λ) (x)	0	0	1	1
λ * (μ ⊕ μ) (x)	0	0	1	0
λ * (μ ⊕ ν) (x)	0	1	1	0
λ * (ν ⊕ e) (x)	1	1	0	0
λ * (ν ⊕ λ) (x)	0	1	0	1
λ * (ν ⊕ μ) (x)	0	1	1	0
λ * (ν ⊕ ν) (x)	0	1	0	0

Table 4

Fuzzy hyperlattice ordered group - A diagrammatic representation

∘	x = λ	x = μ	x = ν	x = e
μ * (e ⊕ e) (x)	0	1	0	0
μ * (e ⊕ λ) (x)	0	1	1	0
μ * (e ⊕ μ) (x)	0	1	0	1
μ * (e ⊕ ν) (x)	1	1	0	0
μ * (λ ⊕ e) (x)	0	1	1	0
μ * (λ ⊕ λ) (x)	0	0	1	0
μ * (λ ⊕ μ) (x)	0	0	1	1
μ * (λ ⊕ ν) (x)	1	0	1	0
μ * (μ ⊕ e) (x)	0	1	0	1
μ * (μ ⊕ λ) (x)	0	0	1	1
μ * (μ ⊕ μ) (x)	0	0	0	1
μ * (μ ⊕ ν) (x)	1	0	0	1
μ * (ν ⊕ e) (x)	1	1	0	0
μ * (ν ⊕ λ) (x)	1	0	1	0
μ * (ν ⊕ μ) (x)	1	0	0	1
μ * (ν ⊕ ν) (x)	1	0	0	0

Table 5

Fuzzy hyperlattice ordered group - A diagrammatic representation

∘	x = λ	x = μ	x = ν	x = e
ν * (e ⊕ e) (x)	0	0	1	0
ν * (e ⊕ λ) (x)	0	1	1	0
ν * (e ⊕ μ) (x)	1	0	1	0
ν * (e ⊕ ν) (x)	0	0	1	1
ν * (λ ⊕ e) (x)	0	1	1	0
ν * (λ ⊕ λ) (x)	0	1	0	0
ν * (λ ⊕ μ) (x)	1	1	0	0
ν * (λ ⊕ ν) (x)	0	1	0	1
ν * (μ ⊕ e) (x)	1	0	1	0
ν * (μ ⊕ λ) (x)	1	1	0	0
ν * (μ ⊕ μ) (x)	1	0	0	0
ν * (μ ⊕ ν) (x)	1	0	0	1
ν * (ν ⊕ e) (x)	0	0	1	1
ν * (ν ⊕ λ) (x)	0	1	0	1
ν * (ν ⊕ μ) (x)	1	0	0	1
ν * (ν ⊕ ν) (x)	0	0	0	1

Table 6

Fuzzy hyperlattice ordered group- A diagrammatic representation

∘	x = λ	x = μ	x = ν	x = e
e * (e ⊕ e) (x)	0	0	0	1
e * (e ⊕ λ) (x)	1	0	0	1
e * (e ⊕ μ) (x)	1	1	0	1
e * (e ⊕ ν) (x)	0	0	1	1
e * (λ ⊕ e) (x)	1	0	0	1
e * (λ ⊕ λ) (x)	1	0	0	0
e * (λ ⊕ μ) (x)	1	1	0	0
e * (λ ⊕ ν) (x)	1	0	1	0
e * (μ ⊕ e) (x)	0	1	0	1
e * (μ ⊕ λ) (x)	1	1	0	0
e * (μ ⊕ μ) (x)	0	1	0	0
e * (μ ⊕ ν) (x)	0	1	1	0
e * (ν ⊕ e) (x)	0	0	1	1
e * (ν ⊕ λ) (x)	1	0	1	0
e * (ν ⊕ μ) (x)	0	1	1	0
e * (ν ⊕ ν) (x)	0	0	1	0

Similarly, we can construct the tables for the hyperoperation ⊗.

Observation: From Tables 3, 4, 5 and 6, we observe the cases for the possibilities of the relations (λ, μ, ν, e). For example, arbitrarily take a combination from the table λ * (ν ⊕ ν) (x) in which we are not using μ but we discovered that there is a chance for μ.

5 Conclusion

In this investigation, we used our approach to fuzzy algebraic structures and fuzzy algebraic hyperstructures so as to deal with fuzzy hyperlattice Ordered Group, in addition to that some properties of fuzzy hyperlattice Ordered Group are discussed and also an application in inheritance using fuzzy hyperlattice Ordered Group is established. As a future work we planned to extend the theory by introducing morphisms and some more operations on fuzzy hyperlattice ordered group and conjointly planned to contribute some real life applications.

Acknowledgement

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.

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Biological inheritance on fuzzy hyperlattice ordered group

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Fuzzy hyperlattice ordered group

4 Application

4.1 The dihybrid cross in the pea plant

Table 1 A diagrammatic explanations of the dihybrid cross in the pea plant ⊗ P Q R S P P , Q , R , S P , Q , R , S P , Q , R , S P , Q , R , S Q P , Q , R , S Q , S P , Q , R , S Q , S R P , Q , R , S P , Q , R , S R , S R , S S P , Q , R , S Q , S R , S S

Table 2 Binary operation output of set L * e λ μ ν e e λ μ ν λ λ e ν μ μ μ ν e λ ν ν μ λ e

Acknowledgement

References

Table 2
Binary operation output of set L

* e λ μ ν

e e λ μ ν

λ λ e ν μ

μ μ ν e λ

ν ν μ λ e