Redox reaction on homomorphism of fuzzy hyperlattice ordered group

Abstract

This paper introduces the concept of homomorphism on fuzzy hyperlattice ordered group $(FHLOG)$ . It studies how the binary and the fuzzy hyperoperations of a $FHLOG$ can be transformed into the binary and the fuzzy hyperoperations of another $FHLOG$ . The notion of fuzzy hypercongruence relation on $FHLOG$ is also defined. The paper also establishes the redox reaction of copper, gold and americium forms three $FHLOG$ s. Besides, homomorphism and composition function of $FHLOG$ s using the redox reactions are developed. Therefore, the paper develops a relation among three different metal’s redox reactions in which the binary and the fuzzy hyperoperations, are preserved.

Keywords

Lattice ordered group fuzzy lattice ordered group fuzzy hyperlattice fuzzy hyperlattice ordered group homomorphism redox reactions

1 Introduction

As a generalization of classical algebraic structures, Marty, the French mathematician, introduced algebraic hyperstructures in 1934 [10]. Various concepts such as weak hyperstructures, hypergroups, hyperrings, hypermodules have been published under the topic algebraic hyperstructures [3–6 , 29]. Hyperlattices were introduced by Konstantinidou [9] and further analysed by Rasouli and Davvaz [20], also see [21 , 24–27]. Congruence on hyperstructures is studied by P. Cordero et al. [2]. Congruence relations and homomorphisms of hyperlattice were initiated by B. B. N. Koguep and C.Lele [8]. On the other hand, fuzzy mathematics was introduced by Zadeh in 1965 [30]. Extensive researchers worked on the applications and extensions of fuzzy set theory. Murali studied fuzzy congruence relations on [12]. More research topics are developed under the combination of algebraic structures and fuzzy set theory [19 , 28]. Many research pieces were initiated on the fuzzy sets and the algebraic hyperstructures combination [3, 4]. As a consequence, the notion of fuzzy hyperlattice was initiated by He and Xin [18]. Preethi et al. [13 –15] developed and studied $FHLOG$ and developed a biological application on $FHLOG$ .

From the beginning, the effects of oxidation-reduction (redox) reactions were identified in chemistry. The redox reactions form a system in chemistry. In brief, some entity is given or taken between two reacting chemicals. Before the nature of the reactions was clarified, the redox reactions had been put into practice. Redox reactions are the most familiar and essential chemical reactions in everyday life [11].

Objective: The main objectives of the paper are;

To extend the theory of $FHLOG$ .

To develop homomorphism function, composition function and fuzzy hypercongruence on $FHLOG$ .

To develop an application for homomorphism and composition functions of $FHLOG$ in Redox reactions.

Contribution:

Fuzzy hyperlattice ordered groups have many applications in various fields. In this paper, we extended the concept of $FHLOG$ and established the relation among three $FHLOG$ s by using the homomorphism and composition functions on $FHLOG$ . Also introduced the concept called fuzzy hypercongruence on $FHLOG$ . In addition to that, we framed $FHLOG$ s from redox reactions of three different metals. From this $FHLOG$ s, we have created relations among the redox reactions of three different metals as an application of the homomorphism and composition functions of $FHLOG$ s.

In Section 2, some basic definitions are examined. In Section 3, homomorphism function, composition function and fuzzy hypercongruence on $FHLOG$ are introduced and analysed. Section 4 has five subsections. Subsections 4.1,4.2,4.3 deals with the formation of $FHLOG$ form the redox reactions of copper, gold, americium. Subsections 4.4 and 4.5 deals with the application in chemistry using homomorphism and composition functions of fuzzy hyperlattice ordered groups.

2 Preliminaries

This section has recalled the fundamentals of algebraic hyperstructures, hyperlattice and fuzzy hyperlattice and $FHLOG$ that are useful for further discussions.

[5 , 18] In algebraic hyperstructure, the composition of two elements forms a set. If a non-empty set $A$ and $P^{*} (A)$ is the family of all non-empty subsets of $A$ , then consider the maps $g_{i} : A \times A ⟶ P^{*} (A),$ where i ∈ {1, 2, . . . , h} and h is a positive integer. The maps g_i are called (binary) hyperoperations.

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

If R and T are non-empty subsets of $S$ , ∀ $r, s, t \in S$ , then

t ∘ R = {t} ∘ R = ⋃ _r∈Rt ∘ r, R ∘ t = R ∘ {t} = ⋃ _r∈Rr ∘ t;

$R \circ T = ⋃_{\binom{r \in R}{s \in T}} r \circ s$ .

Definition 2.1. [7, 18] Let a non-empty set $S$ with two hyperoperations “⊕” and “⊗”. The triple $(S, \oplus, \otimes)$ is called a hyperlattice if, ∀ $r, s, t \in S$ ,

(idempotent laws) r ∈ r ⊗ r and r ∈ r ⊕ r;

(commutative laws) r ⊗ s = s ⊗ r and r ⊕ s = s ⊕ r;

(associative laws) (r ⊗ s) ⊗ t = r ⊗ (s ⊗ t) and (r ⊕ s) ⊕ t = r ⊕ (s ⊕ t);

(absorption laws) r ∈ r ⊗ (r ⊕ s) and r ∈ r ⊕ (r ⊗ s).

Definition 2.2. [24] Let consider a hyperlattice $(S, ⋁, ⋀)$ . $(S, \leq)$ is an ordered hyperlattice, if ≤ is a partial order relation and r ≤ s implies that $\begin{matrix} r ⋁ t \leq s ⋁ t and r ⋀ t \leq s ⋀ t . \end{matrix}$

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

Definition 2.4. [18] Let a non-empty set $S$ with two fuzzy hyper operations ⊗ and ⊕ (Here $F^{*} (S)$ be the set of all non-zero fuzzy subsets of $S$ . A fuzzy hyperoperation has the map $\circ : S \times S ⟶ F^{*} (S)$ ). The triple $(S, \otimes, \oplus)$ is called a fuzzy hyperlattice if, ∀ $r, s, t \in S$ ,

(r ⊗ r) (r) >0 and (r ⊕ r) (r) >0;

(r ⊗ s) = (s ⊗ r) and (r ⊕ s) = (s ⊕ r);

(r ⊗ s) ⊗ t = r ⊗ (s ⊗ t) and (r ⊕ s) ⊕ t = r ⊕ (s ⊕ t);

(r ⊗ (r ⊕ s)) (r) >0 and (r ⊕ (r ⊗ s)) (r) >0.

Note: [18] If R and T are two non-zero fuzzy subsets, ∀

r, s \in S

, then

$(r \circ R) (s) = sup_{t \in S} {(r \circ t) (s) \land R (t)}$ , $(R \circ r) (s) = sup_{t \in S} {R (t) \land (t \circ r) (s)}$ .

$(R \circ T) (s) = sup_{u, v \in S} {R (u) \land (u \circ v) (s) \land T (v)}$ .

Example 1. Let $(S, \land, \lor)$ be a lattice. If the fuzzy hyperoperations on $S$ defined by r × s = χ_{r,s} and r + s = χ_r∧s, ∀ $r, s \in S$ then $(S, \times, +)$ is a fuzzy hyperlattice.

Definition 2.5. [1] Let $S = (S, +, \leq)$ is a lattice ordered group, such that

$(S, +)$ is a group,

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

l ≤ m ⇒ r + l + s ≤ r + m + s ∀ $r, s \in S$ .

Definition 2.6. [28] A non-empty set $S$ is called a fuzzy lattice ordered group if

$(S, +)$ is a group,

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

R (r + (s ∨ t) , (r + s) ∨ (r + t)) =1, ∀ $r, s, t \in S$ .

Here

R : S \times S ⟶ [0, 1]

is a fuzzy partial order relation.

Definition 2.7. [13] A non-empty set $S$ is called a fuzzy hyperlattice ordered group if

$(S, +)$ is a group,

$(S, \oplus, \otimes)$ is a fuzzy hyperlattice,

R [{r} + (s ⊕ t) , {r + s} ⊕ {r + t}] =1 and R [{r} + (s ⊗ t) , {r + s} ⊗ {r + t}] =1,

∀

r, s, t \in S

. Here

R : S \times S ⟶ [0, 1]

is a fuzzy partial order relation.

3 Homomorphism on fuzzy hyperlattice ordered group

This section initiated the homomorphism function, kernal, composition function, fuzzy equivalence relation, fuzzy hypercongruence relation, and kernal relation on $FHLOG$ . Also, some theorems, which helps to analyse the concepts, were developed.

Throughout this paper, *₁, * ₂, * ₃ are binary operations; ⊕₁, ⊕ ₂, ⊕ ₃, ⊗ ₁, ⊗ ₂, ⊗ ₃ are fuzzy hyperoperations.

Definition 3.1. Let $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ and $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ be two $FHLOG$ s. A map $H : S_{1} \to S_{2}$ is a weak homomorphism of $FHLOG$ s. If,

$H (p *_{1} q) = H (p) *_{2} H (q)$ , $\forall p, q \in S_{1}$ .

$H (p \oplus_{1} q) \subseteq H (p) \oplus_{2} H (q)$ , $\forall p, q \in S_{1}$ .

$H (p \otimes_{1} q) \subseteq H (p) \otimes_{2} H (q)$ , $\forall p, q \in S_{1}$ .

Definition 3.2. Let $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ and $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ be two $FHLOG$ s. A map $H : S_{1} \to S_{2}$ is a homomorphism of $FHLOG$ s. If,

$H (p *_{1} q) = H (p) *_{2} H (q)$ , $\forall p, q \in S_{1}$ .

$H (p \oplus_{1} q) = H (p) \oplus_{2} H (q)$ , $\forall p, q \in S_{1}$ .

$H (p \otimes_{1} q) = H (p) \otimes_{2} H (q)$ , $\forall p, q \in S_{1}$ .

Remark 1.

If the homomorphism $H$ of $FHLOG$ s is surjective then $H$ is an epimorphism of $FHLOG$ s.

If the homomorphism $H$ is injective of $FHLOG$ s then $H$ is an monomorphism of $FHLOG$ s.

If the homomorphism $H$ is bijective of $FHLOG$ s then $H$ is an isomorphism of $FHLOG$ s.

Example 2. (i) Let the set $ℤ$ forms a group under addition. The structure $(ℤ, +, \oplus_{1}, \otimes_{1})$ with the fuzzy hyperoperations defined by p ⊕ ₁q = χ_{p,q} and $p \otimes_{1} q = χ_{{p, q, ℤ}}$ , $\forall p, q \in ℤ$ and $R : ℤ \times ℤ \to [0, 1]$ is a fuzzy partial order relation.

Here $(ℤ, +, \oplus_{1}, \otimes_{1})$ is an example of $FHLOG$ .

(ii) Let the set $𝔽 = {1, - 1}$ and it forms a group under multiplication.

The structure $(𝔽, *, \oplus_{2}, \otimes_{2})$ with the fuzzy hyperoperations defined by p ⊕ ₂q = χ_{p,q} and $p \otimes_{2} q = χ_{{p, q, 𝔽}}$ , $\forall p, q \in 𝔽$ and $R : 𝔽 \times 𝔽 \to [0, 1]$ is a fuzzy partial order relation.

Here $(𝔽, *, \oplus_{2}, \otimes_{2})$ is an example of $FHLOG$ .

Now consider (i),(ii) of Example 2 and a map $H : Z \to F$ and the map defined by, $\forall p \in ℤ$ $H (p) = {\begin{matrix} 1 & if p is even or 0 \\ - 1 & if p is odd \end{matrix}$ is a homomorphism of $FHLOG$ s.

Proof. $\forall p, q \in ℤ$ ,

(i) For p & q are even:

$H (2 p + 2 q) = H (2 (p + q)) = 1 = H (2 p) * H (2 q)$

For p & q are even & odd:

$H (2 p + q) = - 1 = (1) (- 1) = H (2 p) * H (q)$

For p & q are odd:

$H (p + q) = 1 = (- 1) (- 1) = H (p) * H (p)$ ∴ $H (p + q) = H (p) * H (q)$ , $\forall p, q \in ℤ$ .

(ii)For p & q are even:

$H (2 p \oplus_{1} 2 q) = H (χ_{{2 p, 2 q}}) = χ_{{H (2 p), H (2 q)}} = χ_{{1, 1}}$ .

$H (2 p) \oplus_{2} H (2 q) = (1 \oplus_{2} 1) = χ_{{1, 1}}$ .

For p & q are even & odd:

$H (2 p \oplus_{1} q) = H (χ_{{2 p, q}}) = χ_{{H (2 p), H (q)}} = χ_{{1, - 1}}$ .

$H (2 p) \oplus_{2} H (q) = (1 \oplus_{2} - 1) = χ_{{1, - 1}}$ .

For p & q are odd:

$H (p \oplus_{1} q) = H (χ_{{p, q}}) = χ_{{H (p), H (q)}} = χ_{{- 1, - 1}}$ .

$H (p) \oplus_{2} H (q) = (- 1 \oplus_{2} - 1) = χ_{{- 1, - 1}}$ .

∴ $H (p \oplus_{1} q) = H (p) \oplus_{2} H (q)$ , $\forall p, q \in ℤ$ .

(iii)For p & q are even:

$H (2 p \otimes_{1} 2 q) = H (χ_{{2 p, 2 q, Z}}) = χ_{{H (2 p), H (2 q), H (Z)}} = χ_{{1, 1, 1, - 1}} = χ_{{1, - 1}}$ .

$H (2 p) \otimes_{2} H (2 q) = (1 \otimes_{2} 1) = χ_{{1, 1, F}} = χ_{{1, - 1}}$ .

For p & q are even & odd:

$H (2 p \otimes_{1} q) = H (χ_{{2 p, q, Z}}) = χ_{{H (2 p), H (q), H (Z}} = χ_{{1, - 1, 1, - 1}} = χ_{{1, - 1}}$ .

$H (2 p) \otimes_{2} H (q) = (1 \otimes_{2} - 1) = χ_{{1, - 1, F}} = χ_{{1, - 1}}$ .

For p & q are odd:

$H (p \otimes_{1} q) = H (χ_{{p, q, Z}}) = χ_{{H (p), H (q), H (Z}} = χ_{{- 1, - 1, 1, - 1}} = χ_{{1, - 1}}$ .

$H (p) \otimes_{2} H (q) = (1 \otimes_{2} - 1) = χ_{{1, - 1, F}} = χ_{{1, - 1}}$ .

∴ $H (p \otimes_{1} q) = H (p) \otimes_{2} H (q)$ , $\forall p, q \in ℤ$ . □

Proposition 3.3. Let $H$ be a homomorphism from a $FHLOG$ $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ to a $FHLOG$ $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ and let $e_{1} \in S_{1}$ . Then, $H (e_{1}) = e_{2}$ . Here e₁ & e₂ are the identities of $S_{1}$ & $S_{2}$ respectively.

Definition 3.4. Let $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ and $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ be two $FHLOG$ s. The map $H : S_{1} \to S_{2}$ is the homomorphism of $FHLOG$ s then the kernal of $H$ is defined by ker $H = {x \in S_{1} | H (x) = e_{2}}$ where e₂ is the identity element of $S_{2}$ .

Theorem 3.5. Every Isomorphism of $FHLOG$ s is a homomorphism of $FHLOG$ s with $\ker (H) = {e}$ .

Theorem 3.6. The homomorphism composition of $FHLOG$ s is a homomorphism of $FHLOG$ . (i.e.,) If $H_{1} : S_{1} \to S_{2}$ and $H_{2} : S_{2} \to S_{3}$ are homomorphisms of $FHLOG$ s, then the composition function $H_{2} \circ H_{1} : S_{1} \to S_{3}$ is a homomorphism of $FHLOG$ s.

Proof. Let $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ , $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ and $(S_{3}, *_{3}, \oplus_{3}, \otimes_{3})$ be three $FHLOG$ s. The proof is straight forward, $\forall p, q \in S_{1}$ , we have (i) $\begin{matrix} (H_{2} \circ H_{1}) (p *_{1} q) & = H_{2} (H_{1} (p *_{1} q)) \\ = H_{2} (H_{1} (p) *_{2} H_{1} (q)) \\ = H_{2} (H_{1} (p)) *_{3} H_{2} (H_{1} (q)) \\ = H_{2} \circ H_{1} (p) *_{3} H_{2} \circ H_{1} (q) . \end{matrix}$ (ii) $\begin{matrix} (H_{2} \circ H_{1}) (p \oplus_{1} q) & = H_{2} (H_{1} (p \oplus_{1} q)) \\ = H_{2} (H_{1} (p) \oplus_{2} H_{1} (q)) \\ = H_{2} (H_{1} (p)) \oplus_{3} H_{2} (H_{1} (q)) \\ = H_{2} \circ H_{1} (p) \oplus_{3} H_{2} \circ H_{1} (q) . \end{matrix}$ (iii) Similarly $(H_{2} \circ H_{1}) (p \otimes_{1} q) = H_{2} \circ H_{1} (p) \otimes_{3} H_{2} \circ H_{1} (q)$ . ∴ The homomorphism composition of $FHLOG$ s is a homomorphism of $FHLOG$ . □

Definition 3.7. Let $(S, *, \oplus, \otimes)$ is a $FHLOG$ . A fuzzy partial order relation R on $S$ is a fuzzy subset $S \times S$ (i.e., $R : S \times S \to [0, 1]$ ). Consider a fuzzy relation ω on $S$ is a fuzzy subset of $S \times S$ (i.e., $ω : S \times S \to [0, 1]$ ). Now the fuzzy relation ω is said to be a fuzzy equivalence relation on $FHLOG$ , if,

(i) $ω (p, p) = {Sup}_{q, r \in S} ω (q, r)$

(ii)ω (q, p) = ω (p, q)

(iii) $ω (p, q) \geq {Sup}_{r \in S} \min (ω (p, r), ω (r, q))$

In order to develop fuzzy hypercongruence on $FHLOG$ , $(S, *, \oplus, \otimes)$ , we need the following notion,

For any $M, N \in F^{*} (S)$ ,ω be a fuzzy equivalence relation on $(S, *, \oplus, \otimes)$ , we say $M \bar{ω} N$ if,

∀, p ∈ M, ∃ q ∈ N such that pωq; ∀, r ∈ N, ∃ s ∈ M such that rωs.

∀ $p \in S$ , if M (p) >0 then $\exists q \in S$ such that N (q) >0 and pωq; ∀ $r \in S$ , if N (r) >0 then $\exists s \in S$ such that M (s) >0 and rωs.

Definition 3.8. Let $(S, *, \oplus, \otimes)$ is a $FHLOG$ . The equivalence relation ω on $H$ is said to be a fuzzy hypercongruence relation on $(S, *, \oplus, \otimes)$ if for all, $p, p^{'}, q, q^{'} \in S$ , pωp′ and qωq′, ⇒ $(p * q) \bar{ω} (p^{'} * q^{'})$ ; $(p \oplus q) \bar{ω} (p^{'} \oplus q^{'})$ ; $(p \otimes q) \bar{ω} (p^{'} \otimes q^{'})$ .

Theorem 3.9. Let $(S, *, \oplus, \otimes)$ is a $FHLOG$ and let ω be a fuzzy hypercongruence relation on $S$ , $\forall p, q \in S$ then, (i)[p] _ω * [q] _ω ⊆ [p * q] _ω. (ii) [p] _ω ⊗ [q] _ω ⊆ [p ⊗ q] _ω. (iii) [p] _ω ⊕ [q] _ω ⊆ [p ⊕ q] _ω. Where [p] is an equivalence class of p.

Proof. (i) Let m ∈ [p] _ω * [q] _ω then, ∃ m₁ ∈ [p] _ω and m₂ ∈ [q] _ω such that m ∈ m₁ * m₂. Since, pωm₁, qωm₂, By Definition 3.8 $(p * q) \bar{ω} (m_{1} * m_{2})$ So, m ∈ m₁ * m₂ ∃ n ∈ p * q such that mωn ⇒ m ∈ [p * q] _ω. ∴ [p] _ω * [q] _ω ⊆ [p * q] _ω. (ii) Let m₁ ∈ [p] _ω and m₂ ∈ [q] _ω. (i.e., $[p]_{ω} = {m_{1} \in S | m_{1} ω p}$ and $[q]_{ω} = {m_{2} \in S | m_{2} ω q}$ . Now consider ([p] _ω ⊗ [q] _ω) (m₁) >0 Let m₁ = p, m₂ = q and for some $p^{'}, q^{'} \in S$ , Sup(([p′] _ω ⊗ [q′] _ω) (p) |m₁ = p) ≥ ([p] _ω ⊗ [q] _ω) (m₁) >0. ⇒ Sup(([p′] _ω ⊗ [q′] _ω) (p)) >0. ∴ ∀ $m_{1} \in S$ , if ([p] _ω ⊗ [q] _ω) (m₁) >0 then ∃ $p \in S$ such that (([p′] _ω ⊗ [q′] _ω) (p)) >0 and pωm₁. Similarly, ∀ $m_{2} \in S$ , if ([p] _ω ⊗ [q] _ω) (m₂) >0 then ∃ $q \in S$ such that (([p′] _ω ⊗ [q′] _ω) (q)) >0 and qωm₂. By fuzzy hypercongruence relation we have, $(p \otimes q) \bar{ω} (m_{1} \otimes m_{2})$ (1) Now suppose, m ∈ [p] _ω ⊗ [q] _ω and m₁ ∈ [p] _ω and m₂ ∈ [q] _ω such that m ∈ m₁ ⊗ m₂ ∃ n ∈ p ⊗ q such that mωn, (by (1)) ⇒ m ∈ [p ⊗ q] _ω ∴ [p] _ω ⊗ [q] _ω ⊆ [p ⊗ q] _ω. Similarly, [p] _ω ⊕ [q] _ω ⊆ [p ⊕ q] _ω. □

Definition 3.10. Let $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ and $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ be two $FHLOG$ s. The map $H : S_{1} \to S_{2}$ is the homomorphism of $FHLOG$ s. The kernal relation on $H$ is defined as, $\forall (p, q) \in \ker H$ iff $H (p) = H (q)$ i.e., ker $H = {(p, q) \in S_{1} \times S_{1} | H (p) = H (q)}$

Lemma 3.11. Let $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ and $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ be two $FHLOG$ s. The map $H : S_{1} \to S_{2}$ is the homomorphism of $FHLOG$ s. Then ker $H$ is an equivalence relation on $S_{1}$ . i.e., pωq iff $H (p) = H (q)$ , $\forall (p, q) \in \ker H$ .

Theorem 3.12. The kernal relation of any homomorphism between $FHLOG$ s is a fuzzy hypercongruence relation.

Proof. Let $(S_{1}, *_{1}, \oplus_{1}, \otimes_{1})$ and $(S_{2}, *_{2}, \oplus_{2}, \otimes_{2})$ be two $FHLOG$ s. The map $H : S_{1} \to S_{2}$ is the homomorphism of $FHLOG$ s. $\forall p, p^{'}, q, q^{'} \in S_{1}$ . Let pωp′ and qωq′ then $H (p) = H (p^{'})$ and $H (q) = H (q^{'})$ . Let m ∈ (p * ₁q) then $H (m) \in H (p *_{1} q)$ $\begin{matrix} H (p *_{1} q) & = H (p) *_{2} H (q) \\ = H (p^{'}) *_{2} H (q^{'}) \\ = H (p^{'} *_{1} q^{'}) \end{matrix}$ ⇒ ∃ m′ ∈ (p′ * ₁q′) such that $H (m) = H (m^{'})$ (i.e.,) ∃ m′ ∈ (p′ * ₁q′) such that mωm′. Similarly, for n ∈ (p′ * ₁q′) ∃ n′ ∈ (p * ₁q) such that nωn′. ∴ $(p *_{1} q) \bar{ω} (p^{'} *_{1} q^{'})$ . Now, $\forall p, p^{'}, q, q^{'} \in S_{1}$ $\begin{matrix} H (p \otimes_{1} q) & = H (p) \otimes_{2} H (q) \\ = H (p^{'}) \otimes_{2} H (q^{'}) \\ = H (p^{'} \otimes_{1} q^{'}) \end{matrix}$ ⇒ (p ⊗ ₁q) ω (p′ ⊗ ₁q′). $\forall m \in S_{1}$ , let (p ⊗ ₁q) (m) >0 then $H (p \otimes_{1} q) (H (m)) = Sup {(p^{'} \otimes_{1} q^{'}) (m^{'}) | H (m^{'}) = H (m), m^{'} \in S_{1}} \geq (p \otimes_{1} q) (m) > 0$ . ∴ $H (p^{'} \otimes_{1} q^{'}) (H (m)) > 0$ ⇒ $Sup {(p^{'} \otimes_{1} q^{'}) (m^{'}) | H (m^{'}) = H (m), m^{'} \in S_{1}} > 0$ . This means $\exists m^{'} \in S_{1}$ such that (p′ ⊗ ₁q′) (m′) >0 and $H (m^{'}) = H (m)$ . ⇒ (p′ ⊗ ₁q′) (m′) >0 and mωm′. Conversely, $\forall n \in S_{1}$ . Let (p′ ⊗ ₁q′) (n) >0 then there exists $n^{'} \in S_{1}$ such that (p ⊗ ₁q) (n′) >0 and nωn′. Hence $(p \otimes_{1} q) \bar{ω} (p^{'} \otimes_{1} q^{'})$ Similarly, $(p \oplus_{1} q) \bar{ω} (p^{'} \oplus_{1} q^{'})$ □

4 Application using redox reactions

An oxidation-reduction reaction is otherwise called a redox reaction. Redox reaction is a chemical process that involves the transfer of electrons between two reacting species. The redox reactions can occur over a wide range of chemical reactions encountered in the various industrial processes and day-to-day life [11].

This section deals with the application of $FHLOG$ in redox reactions. Also demonstrates how the application of homomorphism and composition functions on $FHLOG$ helps to frame a relation among the redox reactions of different metals.

Principle of Redox reactions: Most redox reaction processes involve the transfer of electrons, hydrogen atoms, or oxygen atoms with all three processes sharing the following two characteristics: (1) In oxidation reaction a reciprocal reduction occurs, i.e., They are coupled and (2) Electron goes from one unit of matter to another, i.e., a characteristic net chemical change. The reciprocity and the net change are the essential and fundamental aspects of a redox reaction.

4.1 $FHLOG$ from redox reaction on Copper

Copper $(C u)$ is a ductile metal with high electrical and thermal conductivity. It is a conductor of electricity and heat. It is a building material and a significant ingredient of several metal alloys [23]. It has four oxidation states 0,1,2,3. All possible products in reactions between $C$ u oxidation states are shown in Table 1.

Table 1
Redox reaction on $C u$

∘ $C u$ $C u^{+}$ $C u^{2 +}$ $C u^{3 +}$

$C u$ $C u$ $C u, C u^{+}$ $C u, C u^{2 +}$ $C u^{+}, C u^{2 +}$

$C u^{+}$ $C u, C u^{+}$ $C u^{+}$ $C u^{+}, C u^{2 +}$ $C u^{2 +}$

$C u^{2 +}$ $C u, C u^{2 +}$ $C u^{+}, C u^{2 +}$ $C u^{2 +}$ $C u^{2 +}, C u^{3 +}$

$C u^{3 +}$ $C u^{+}, C u^{2 +}$ $C u^{2 +}$ $C u^{2 +}, C u^{3 +}$ $C u^{3 +}$

∘	$C u$	$C u^{+}$	$C u^{2 +}$	$C u^{3 +}$
$C u$	$C u$	$C u, C u^{+}$	$C u, C u^{2 +}$	$C u^{+}, C u^{2 +}$
$C u^{+}$	$C u, C u^{+}$	$C u^{+}$	$C u^{+}, C u^{2 +}$	$C u^{2 +}$
$C u^{2 +}$	$C u, C u^{2 +}$	$C u^{+}, C u^{2 +}$	$C u^{2 +}$	$C u^{2 +}, C u^{3 +}$
$C u^{3 +}$	$C u^{+}, C u^{2 +}$	$C u^{2 +}$	$C u^{2 +}, C u^{3 +}$	$C u^{3 +}$

Now consider the following relations,

α₁: $C u$ ↔ $C u^{+}$ $C u^{2 +}$ ↔ $C u^{3 +}$ β₁: $C u$ ↔ $C u^{2 +}$ $C u^{+}$ ↔ $C u^{3 +}$ γ₁: $C u$ ↔ $C u^{3 +}$ $C u^{2 +}$ ↔ $C u^{+}$ . e₁: Equivalent to itself. Let $C = {e_{1}, α_{1}, β_{1}, γ_{1}}$ .

Here, *₁ is the binary operation. By Table 2, $C$ forms a group under *₁. Let the fuzzy hyperoperations for all $α_{1}, β_{1}, γ_{1}, e_{1} \in C$ defined by

If α₁ ⊕ ₁β₁, then α₁ ⊕ ₁β₁= χ_{{α₁,β₁}} (x), for all $x \in C$ . Similarly if γ₁ ⊕ ₁β₁, then γ₁ ⊕ ₁β₁ = χ_{{γ₁,β₁}} (x), for all $x \in C$ . Likewise for all elements in $C$ .

If α₁ ⊗ ₁β₁, then α₁ ⊗ ₁β₁= $χ_{{α_{1}, β_{1}, C}} (x)$ , for all $x \in C$ . Similarly if γ₁ ⊗ ₁β₁, then $γ_{1} \otimes_{1} β_{1} = χ_{{γ_{1}, β_{1}, C}} (x)$ , $\forall x \in C$ . Likewise for all elements in $C$ .

and

R : C \times C \to [0, 1]

is a fuzzy partial order relation. Now the structure,

(C, *_{1}, \oplus_{1}, \otimes_{1})

satisfies the following conditions, ∀

α_{1}, β_{1}, γ_{1}, e_{1} \in C

$C, *_{1}$ is a group.

$(C, \oplus_{1}, \otimes_{1})$ is a fuzzy hyperlattice. (i.e.,)

(α₁ ⊗ ₁α₁) (α₁) >0, (α₁ ⊕ ₁α₁) (α₁) >0,

(α₁ ⊗ ₁β₁) = (β₁ ⊗ ₁α₁), (α₁ ⊕ ₁β₁) = (β₁ ⊕ ₁α₁),

(α₁ ⊗ ₁β₁) ⊗ ₁γ₁ = α₁ ⊗ ₁ (β₁ ⊗ ₁γ₁), (α₁ ⊕ ₁β₁) ⊕ ₁γ₁ = α₁ ⊕ ₁ (β₁ ⊕ ₁γ₁),

(α₁ ⊗ ₁ (α₁ ⊕ ₁β₁)) (α₁) >0, (α₁ ⊕ ₁ (α₁ ⊗ ₁β₁)) (α₁) >0.

R [{α₁} * ₁ (β₁ ⊕ ₁γ₁) , {α₁ * ₁β₁} ⊕ ₁ {α₁ * ₁γ₁}] =1 R [{α₁} * ₁ (β₁ ⊗ ₁γ₁) , {α₁ * ₁β₁} ⊗ ₁ {α₁ * ₁γ₁}] =1

Here

(C, *_{1}, \oplus_{1}, \otimes_{1})

is an example of

FHLOG

Table 2

Binary operation *₁

*₁	e ₁	α ₁	β ₁	γ ₁
e ₁	e ₁	α ₁	β ₁	γ ₁
α ₁	α ₁	e ₁	γ ₁	β ₁
β ₁	β ₁	γ ₁	e ₁	α ₁
γ ₁	γ ₁	β ₁	α ₁	e ₁

4.2

FHLOG

from redox reaction on Gold

Gold $(A u)$ is a malleable,soft, dense, shiny and ductile metal. This chemical element is the least reactive [23]. It has four oxidation states 0,1,2,3. All possible products in reactions between $A$ u oxidation states are shown in Table 3

Table 3
Redox reaction on $A u$

∘ $A u$ $A u^{+}$ $A u^{2 +}$ $A u^{3 +}$

$A u$ $A u$ $A u, A u^{+}$ $A u^{+}$ $A u, A u^{3 +}$

$A u^{+}$ $A u, A u^{+}$ $A u^{+}$ $A u^{+}, A u^{2 +}$ $A u^{+}, A u^{3 +}$

$A u^{2 +}$ $A u^{+}$ $A u^{+}, A u^{2 +}$ $A u^{2 +}$ $A u^{2 +}, A u^{3 +}$

$A u^{3 +}$ $A u, A u^{3 +}$ $A u^{+}, A u^{3 +}$ $A u^{2 +}, A u^{3 +}$ $A u^{3 +}$

∘	$A u$	$A u^{+}$	$A u^{2 +}$	$A u^{3 +}$
$A u$	$A u$	$A u, A u^{+}$	$A u^{+}$	$A u, A u^{3 +}$
$A u^{+}$	$A u, A u^{+}$	$A u^{+}$	$A u^{+}, A u^{2 +}$	$A u^{+}, A u^{3 +}$
$A u^{2 +}$	$A u^{+}$	$A u^{+}, A u^{2 +}$	$A u^{2 +}$	$A u^{2 +}, A u^{3 +}$
$A u^{3 +}$	$A u, A u^{3 +}$	$A u^{+}, A u^{3 +}$	$A u^{2 +}, A u^{3 +}$	$A u^{3 +}$

Now consider the following relations, α₂: $A u$ ↔ $A u^{+}$ $A u^{2 +}$ ↔ $A u^{3 +}$ β₂: $A u$ ↔ $A u^{2 +}$ $A u^{+}$ ↔ $A u^{3 +}$ γ₂: $A u$ ↔ $A u^{3 +}$ $A u^{2 +}$ ↔ $A u^{+}$ . e₂: Equivalent to itself. Let $A = {e_{2}, α_{2}, β_{2}, γ_{2}}$ .

Here, *₂ is the binary operation. By Table 4, $A$ forms a group under *₂.

Table 4

Binary operation *₂

*₂	e ₂	α ₂	β ₂	γ ₂
e ₂	e ₂	α ₂	β ₂	γ ₂
α ₂	α ₂	e ₂	γ ₂	β ₂
β ₂	β ₂	γ ₂	e ₂	α ₂
γ ₂	γ ₂	β ₂	α ₂	e ₂

Now the fuzzy hyperoperations for all $α_{2}, β_{2}, γ_{2}, e_{2} \in A$ defined by

If α₂ ⊕ ₂β₂, then α₂ ⊕ ₂β₂= χ_{{α₂,β₂}} (x), for all $x \in A$ . Similarly if γ₂ ⊕ ₂β₂, then γ₂ ⊕ ₂β₂ = χ_{{γ₂,β₂}} (x), for all $x \in A$ . Likewise for all elements in $A$ .

If α₂ ⊗ ₂β₂, then α₂ ⊗ ₂β₂= $χ_{{α_{2}, β_{2}, A}} (x)$ , for all $x \in A$ . Similarly if γ₂ ⊗ ₂β₂, then $γ_{2} \otimes_{2} β_{2} = χ_{{γ_{2}, β_{2}, A}} (x)$ , $\forall x \in A$ . Likewise for all elements in $A$ .

R : A \times A \to [0, 1]

is a fuzzy partial order relation. Here

(A, +_{2}, \oplus_{2}, \otimes_{2})

is an example of

FHLOG

4.3

FHLOG

from redox reaction on Americium

In actinide series, Americium $(A m)$ is a transuranic radioactive chemical element. Naturally, Americium occurs in uranium minerals [23]. It has four oxidation states 0, 2, 3 and 4. All possible products in reactions between $A$ m oxidation states are shown in Table 5

Table 5
Redox reaction on $A m$

∘ $A m$ $A m^{2 +}$ $A m^{3 +}$ $A m^{4 +}$

$A m$ $A m$ $A m, A m^{2 +}$ $A m, A m^{3 +}$ $A m, A m^{4 +}$

$A m^{2 +}$ $A m, A m^{2 +}$ $A m^{2 +}$ $A m^{2 +}, A m^{3 +}$ $A m^{3 +}$

$A m^{3 +}$ $A m, A m^{3 +}$ $A m^{2 +}, A m^{3 +}$ $A m^{3 +}$ $A m^{3 +}, A m^{4 +}$

$A m^{4 +}$ $A m, A m^{4 +}$ $A m^{3 +}$ $A m^{3 +}, A m^{4 +}$ $A m^{4 +}$

∘	$A m$	$A m^{2 +}$	$A m^{3 +}$	$A m^{4 +}$
$A m$	$A m$	$A m, A m^{2 +}$	$A m, A m^{3 +}$	$A m, A m^{4 +}$
$A m^{2 +}$	$A m, A m^{2 +}$	$A m^{2 +}$	$A m^{2 +}, A m^{3 +}$	$A m^{3 +}$
$A m^{3 +}$	$A m, A m^{3 +}$	$A m^{2 +}, A m^{3 +}$	$A m^{3 +}$	$A m^{3 +}, A m^{4 +}$
$A m^{4 +}$	$A m, A m^{4 +}$	$A m^{3 +}$	$A m^{3 +}, A m^{4 +}$	$A m^{4 +}$

Now consider the following relations, α₃: $A m$ ↔ $A m^{2 +}$ $A m^{3 +}$ ↔ $A m^{4 +}$ β₃: $A m$ ↔ $A m^{3 +}$ $A m^{2 +}$ ↔ $A m^{4 +}$ γ₃: $A m$ ↔ $A m^{4 +}$ $A m^{3 +}$ ↔ $A m^{2 +}$ . e₃: Equivalent to itself. Let $S = {e_{3}, α_{3}, β_{3}, γ_{3}}$ .

Here, *₃ is the binary operation. By Table 6, $S$ forms a group under *₃.

Table 6

Binary operation *₃

*₃	e ₃	α ₃	β ₃	γ ₃
e ₃	e ₃	α ₃	β ₃	γ ₃
α ₃	α ₃	e ₃	γ ₃	β ₃
β ₃	β ₃	γ ₃	e ₃	α ₃
γ ₃	γ ₃	β ₃	α ₃	e ₃

Let the fuzzy hyperoperations for all $α_{3}, β_{3}, γ_{3}, e_{3} \in S$ defined by

If α₃ ⊕ ₃β₃, then α₃ ⊕ ₃β₃= χ_{{α₃,β₃}} (x), for all $x \in S$ . Similarly if γ₃ ⊕ ₃β₃, then γ₃ ⊕ ₃β₃ = χ_{{γ₃,β₃}} (x), for all $x \in S$ . Likewise for all elements in $S$ .

If α₃ ⊗ ₃β₃, then α₃ ⊗ ₃β₃= $χ_{{α_{3}, β_{3}, S}} (x)$ , for all $x \in S$ . Similarly if γ₃ ⊗ ₃β₃, then $γ_{3} \otimes_{3} β_{3} = χ_{{γ_{3}, β_{3}, S}} (x)$ , $\forall x \in S$ . Likewise for all elements in $S$ .

and

R : S \times S \to [0, 1]

is a fuzzy partial order relation. Here

(S, +_{3}, \oplus_{3}, \otimes_{3})

is an example of

FHLOG

4.4 Homomorphism of two

FHLOG

Now consider the $FHLOG$ s of Subsections 4.1, 4.2, 4.3 and a map $H_{1} : C \to A$ and the map defined by, $\forall α_{1}, β_{1} \in C$ $H_{1} (α_{1}) = α_{2}$ , $H_{1} (β_{1}) = β_{2}$ , $H_{1} (γ_{1}) = γ_{2}$ and $H_{1} (e_{1}) = e_{2}$ (i) $H_{1} (α_{1} *_{1} β_{1}) = H_{1} (γ_{1}) = γ_{2}$ $H_{1} (α_{1}) *_{2} H_{1} (β_{1}) = α_{2} *_{2} β_{2} = γ_{2}$ ∴ $H_{1} (α_{1} *_{1} β_{1}) = H_{1} (α_{1}) *_{2} H_{1} (β_{1})$ . (ii) $\begin{matrix} H_{1} (α_{1} \oplus_{1} β_{1}) & = H_{1} (χ_{{α_{1}, β_{1}}}) \\ = χ_{{H_{1} (α_{1}), H_{1} (β_{1})}} \\ = χ_{{α_{2}, β_{2}}} \\ H_{1} (α_{1}) \oplus_{2} H_{1} (β_{1}) & = α_{2} \oplus_{2} β_{2} \\ = χ_{{α_{2}, β_{2}}} \end{matrix}$ ∴ $H_{1} (α_{1} \oplus_{1} β_{1}) = H_{1} (α_{1}) \oplus_{2} H_{1} (β_{1})$ . (iii) $\begin{matrix} H_{1} (α_{1} \otimes_{1} β_{1}) & = H_{1} (χ_{{α_{1}, β_{1}, C}}) \\ = χ_{{H_{1} (α_{1}), H_{1} (β_{1}), H_{1} (C)}} \\ = χ_{{α_{2}, β_{2}, γ_{2}, e_{2}}} \\ H_{1} (α_{1}) \otimes_{2} H_{1} (β_{1}) & = α_{2} \oplus_{2} β_{2} \\ = χ_{{α_{2}, β_{2}, A}} \\ = χ_{{α_{2}, β_{2}, γ_{2}, e_{2}}} \end{matrix}$ ∴ $H_{1} (α_{1} \otimes_{1} β_{1}) = H_{1} (α_{1}) \otimes_{2} H_{1} (β_{1})$ . Similarly the map $H_{2} : A \to S$ defined by, $\forall α_{2}, β_{2} \in A$ $H_{2} (α_{2}) = α_{3}$ , $H_{2} (β_{2}) = β_{3}$ , $H_{2} (γ_{2}) = γ_{3}$ and $H_{2} (e_{2}) = e_{3}$ is a homomorphism of two $FHLOG$ s $(A, *_{2}, \oplus_{2}, \otimes_{2})$ and $(S, *_{3}, \oplus_{3}, \otimes_{3})$

4.5 Composition of three $FHLOG$ s:

By Subsections 4.1, 4.2, 4.3, $(C, *_{1}, \oplus_{1}, \otimes_{1})$ , $(A, *_{2}, \oplus_{2}, \otimes_{2})$ and $(S, *_{3}, \oplus_{3}, \otimes_{3})$ are $FHLOG$ s. By Subsection 4.4, the maps $H_{1} : C \to A$ and $H_{2} : A \to S$ are the homomorphisms of $FHLOG s$ . Now consider the composition function, $H_{2} \circ H_{1} : C \to S$ , $\forall α_{1}, β_{1} \in C$ , we have (i) $(H_{2} \circ H_{1}) (α_{1} *_{1} β_{1})$ $\begin{matrix} = H_{2} (H_{1} (α_{1} *_{1} β_{1})) \\ = H_{2} (H_{1} (α_{1}) *_{2} H_{1} (β_{1})) \\ = H_{2} (H_{1} (α_{1})) *_{3} H_{2} (H_{1} (β_{1})) \\ = H_{2} \circ H_{1} (α_{1}) *_{3} H_{2} \circ H_{1} (β_{1}) . \end{matrix}$ (ii) $(H_{2} \circ H_{1}) (α_{1} \oplus_{1} β_{1})$ $\begin{matrix} = H_{2} (H_{1} (α_{1} \oplus_{1} β_{1})) \\ = H_{2} (H_{1} (α_{1}) \oplus_{2} H_{1} (β_{1})) \\ = H_{2} (H_{1} (α_{1})) \oplus_{3} H_{2} (H_{1} (β_{1})) \\ = H_{2} \circ H_{1} (α_{1}) \oplus_{3} H_{2} \circ H_{1} (β_{1}) . \end{matrix}$ (iii) Similarly $(H_{2} \circ H_{1}) (α_{1} \otimes_{1} β_{1}) = H_{2} \circ H_{1} (α_{1}) \otimes_{3} H_{2} \circ H_{1} (β_{1})$ . ∴ The homomorphism composition of $FHLOG$ s is a homomorphism of $FHLOG$ . Observation: $FHLOG$ s were developed using the redox reactions on copper, gold and americium. Homomorphisms and composition function were also studied for the $FHLOG$ s obtained from the redox reactions on copper, gold and americium. Through which, it was possible to relate three redox reactions of entirely three different metals. By experimenting this, it was also possible to relate one application of $FHLOG$ with another application of $FHLOG$ .

5 Conclusion

This paper has achieved the extension of $FHLOG$ by introducing homomorphism function, composition function and fuzzy hypercongruence of $FHLOG$ s. An application using homomorphisms and composition function of $FHLOG$ s has also been obtained from the redox reaction of three different metals such as copper, gold and americium. As future work, the researchers planned to frame an algorithm using $FHLOG$ , which helps to broader the application of $FHLOG$ in real life.

Footnotes

Acknowledgments

The article has been written with the joint 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

Birkhoff

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

Cabrera

I.P.

, Cordero

, Gutiérrez

, et al., Congruence relations on some hyperstructures, Ann Math Artif Intell56 (2009), 361–370.

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.

Davvaz

, Vougiouklis

, A Walk Through Weak Hyperstructures- H_v-Structure, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.

Guo

X.Z.

, Xin

X.L.

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

Koguep

B.B.N.

, Lele

, On hyperlattices: congruence relations, ideals and homomorphism, Afr Mat30 (2019), 101–111.

Konstantinidou

, Mittas

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

10.

Marty

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

11.

Mortimer

C.E.

, Chemistry: A Conceptual Approach, 4th ed., D. Van Nostrad Co, New York, 1979.

12.

Murali

, Fuzzy congruence relations, Fuzzy Sets and Systems41(3) (1991), 359–369.

13.

Preethi

, Vimala

, Davvaz

, Rajareega

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

14.

Preethi

, Vimala

, Rajareega

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

15.

Preethi

, Vimala

, Rajareega

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

16.

, Xin

, Zhan

, (Fuzzy) hyperlattices and fuzzy preordered lattices, Journal of Intelligent & Fuzzy Systems26 (2014), 2369–2381.

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

, Preethi

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

20.

Rasouli

, Davvaz

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

21.

Rasouli

, Davvaz

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

22.

Begam

S.S.

, Vimala

, 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.

23.

Sethi

M.S.

, Satake

, Chemistry of Transition Elements, South Asian Publishers, New Delhi, 1988.

24.

Lashkenari

A.S.

, Davvaz

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

25.

Lashkenari

A.S.

, Davvaz

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

26.

Lashkenari

A.S.

, Davvaz

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

27.

Lashkenari

A.S.

, Davvaz

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

28.

Vimala

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

29.

Vougiouklis

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

30.

Zadeh

L.A.

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

Redox reaction on homomorphism of fuzzy hyperlattice ordered group

Abstract

Keywords

1 Introduction

2 Preliminaries

3 Homomorphism on fuzzy hyperlattice ordered group

4 Application using redox reactions

4.1 FHLOG from redox reaction on Copper

Table 1 Redox reaction on C u ∘ C u C u + C u 2 + C u 3 + C u C u C u , C u + C u , C u 2 + C u + , C u 2 + C u + C u , C u + C u + C u + , C u 2 + C u 2 + C u 2 + C u , C u 2 + C u + , C u 2 + C u 2 + C u 2 + , C u 3 + C u 3 + C u + , C u 2 + C u 2 + C u 2 + , C u 3 + C u 3 +

Table 3 Redox reaction on A u ∘ A u A u + A u 2 + A u 3 + A u A u A u , A u + A u + A u , A u 3 + A u + A u , A u + A u + A u + , A u 2 + A u + , A u 3 + A u 2 + A u + A u + , A u 2 + A u 2 + A u 2 + , A u 3 + A u 3 + A u , A u 3 + A u + , A u 3 + A u 2 + , A u 3 + A u 3 +

Table 5 Redox reaction on A m ∘ A m A m 2 + A m 3 + A m 4 + A m A m A m , A m 2 + A m , A m 3 + A m , A m 4 + A m 2 + A m , A m 2 + A m 2 + A m 2 + , A m 3 + A m 3 + A m 3 + A m , A m 3 + A m 2 + , A m 3 + A m 3 + A m 3 + , A m 4 + A m 4 + A m , A m 4 + A m 3 + A m 3 + , A m 4 + A m 4 +

4.5 Composition of three FHLOG s:

5 Conclusion

Footnotes

Acknowledgments

References

4.1 $FHLOG$ from redox reaction on Copper

4.5 Composition of three $FHLOG$ s: