International Journal of Analysis and Applications Volume 17, Number 5 (2019), 864-878 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-864 WEAK NON-ASSOCIATIVE STRUCTURES OF GROUPS WITH APPLICATIONS SHAH NAWAZ1, MUHAMMAD GULISTAN1, NAVEED YAQOOB2 AND SEIFEDINE KADRY3,∗ 1Department of Mathematics, Hazara University, Mansehra, KP, Pakistan 2Department of Mathematics, College of science, Al-Zulfi Majmaah University, Al-Zulfi Saudi Arabia 3Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, P.O. Box 11-5020, Beirut 11072809, Lebanon ∗Corresponding author: s.kadry@bau.edu.lb Abstract. Inspiring by the weak symmetry occurring in the Hv-left invertive structures, in this article we have introduce a new class of Hv-LA-groups which is a generalization of LA-hypergroups. We have investigated different types of homomorphisms of Hv-LA-groups. Moreover, we have constructed the Hv- LA-groups. At the end a useful application of weak symmetry related with Hv-left invertive structure has been presented using the chemical redox reaction. 1. Introduction Kazim and naseerudin [1] laid the idea of left almost semigroup (denoted by LA-semigroups). They generalized some handy result of semigroup theory. Afterwards, Mushtaq [2] and other, went further in the detail of the structure and added various beneficial results to the theory of LA-semigroup, see paper [3–9]. An LA-semigroup is midway structure between commutative semigroup and groupoid. Mushtaq and kamran [10] in 1996 proposed the idea of left almost groups. They proved that if G is left almost group and H is left Received 2019-05-15; accepted 2019-06-19; published 2019-09-02. 2010 Mathematics Subject Classification. 20N05. Key words and phrases. hyperoperation; Hv-LA-groups; homomorphisms; chemical redox reaction. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 864 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-864 Int. J. Anal. Appl. 17 (5) (2019) 865 almost subgroup then G/H is a set of left almost group. Hyperstucture notion was initiated by Marty in 1934, when he [11] elucidated hypergroup and embarked in analyze their properties and applied them to a group. Handful papers and books have been written in this direction, see [12–14]. In 1990, in Greece, Thomas Vougiouklis was organized a congress on hyperstructure and name it AHA, but also there had been three more congresses organized in Italy by Corsini, on the same topic with different name. During that congress Vougiouklis [17] introduced the concept of weak structure which is now named Hv-structures. The impetus of Hv-structures is that, the quotient of group with respect to invariant subgroup is a group. Marty from 1934 notify that, the quotient of group with respect to any subgroup is a hypergroup. Finally, the quotient of a group with respect to any partition (equivalently to any equivalence relation) is an Hv-group. Let (G,◦) be group and R be equivalence relation, then (G/R,◦) is an Hv-Group. Several authors have studied different aspects of Hv-structure. For instance, Vougiouklis [18, 19], Spratalis [20–24] and Davvaz [25]. Not long ago, in 2011, Hila and Dine [15] laid the idea of LA-semihypergroups, which is generalization of semigroups, semihypergroups and LA-semigroups. Yaqoob, Corsini and Yousfzai [16] extended the work of Hila and Dine and characterized intera regular left almost hypergroups by their hyperideal by using pure left identity. The idea of Hv-LA-semigroups was given by Gulistan et al., [26] in 2015. Another motivation for the study of hyperstructure comes from chemical reaction, such as chain redox and dismutation reaction were provided different example of weak structures. for detail see papers [27–29]. This very article communicates the novel class of Hv-LA-groups which is a generalization of LA-hypergroups. We have defined various types of homomorphisms. Additionally, we have constructed the Hv-LA-groups, and presented the chemical example by making use of redox reactions. 2. Preliminaries In this section we recall some helping material from different papers, like [16, 19, 26]. Definition 2.1. [16] A hypergroupoid (H,◦) is called LA-semihypergroup, if it satisfies the following law (x◦y) ◦z = (z ◦y) ◦x for all x,y,z ∈H Example: [16] Let H = Z if we define x ◦ y = y − x + 3Z, where x,y ∈ Z. Then (H,◦) becomes an LA-semihypergroup. Definition 2.2. [19] The hyperoperation ∗ : H×H −→ P∗(H) is called weakly associative hyperoperation (abbreviated as WASS) if for any a,b,c ∈H (a∗ b) ∗ c∩a∗ (b∗ c) 6= ∅ Definition 2.3. [19]The hyperoperation is weakly commutative (abbreviated as COW) if for any a,b ∈H a∗ b∩ b∗a 6= φ Int. J. Anal. Appl. 17 (5) (2019) 866 Definition 2.4. [26] Let H be non-empty set and ∗ be hyperoperation on H. Then (H,∗) is called an Hv-LA-semigroup, if it satisfies the weak left invertive law, for all x,y,z ∈H (x∗y) ∗z ∩ (z ∗y) ∗x 6= ∅ Example: [26] Let H = (0,∞) we define x ∗ y = { y x+1 , y x } where x,y ∈ H. Then for all x,y,z ∈ H satisfies (x∗y) ∗z ∩ (z ∗y) ∗x 6= ∅. Hence (H,∗) is an Hv-LA-semigroup. 3. Hv-LA-Groups In this section, we present a new generalize class of non-associative hyperstructures namely Hv-LA-groups and provided different examples and some basic results. Definition 3.1. Let H be a non-empty set and ∗ be hyperoperation on H. Then (H,∗) is called an Hv-LA- group, if it satisfies the following axioms (i) (x∗y) ∗z ∩ (z ∗y) ∗x 6= ∅ for all x,y,z ∈H, (ii) H∗x = H = H∗x for all x ∈H. Example: Let H = {a,b,c} be a finite set. The hyperoperation ∗ is defined as follow ∗ a b c a a {a,b} {a,c} b c {H} {a,b} c b {a,c} {H} Here all elements of H satisfy the weak left invertive law. Also left invertive law is not hold in H, i.e. H = (a∗ b) ∗ c 6= (c∗ b) ∗a = {a,b} Alike, associative law is not hold in H, i.e. H = (c∗ c) ∗a 6= c∗ (c∗a) = {a,c} . Even, weak associative law is not true {b} = (b∗a) ∗a∩ b∗ (a∗a) = {c} = φ Hence (H,∗) is an Hv-LA-group. Int. J. Anal. Appl. 17 (5) (2019) 867 Example: Let H = {a,b,c,d} be a finite set and the hyperoperation is defined in the following table. ∗ a b c d a a b c d b c {a,c} {b,c} d c b {a,c} {a,c} d d d d d {a,b,c} As all elements of H satisfy the weak left invertive law but H do not satisfies the left invertive law, associative law and weak associative law i.e. {a,c} = (b∗ b) ∗ c 6= (c∗ b) ∗ b = {a,b,c} , and {a,c} = (b∗ b) ∗ c 6= b∗ (b∗ c) = {a,b,c} . Also {b} = (b∗a) ∗a∩ b∗ (a∗a) = {c} = φ So (H,∗) is an Hv-LA-group. Now we will discuss some very basic results related with Hv-LA-groups. Lemma 3.1. If (H,∗) is an Hv-LA-group, then (a∗ b) ∗ (c∗d) ∩ (a∗ c) ∗ (b∗d) 6= φ, hold for all a,b,c,d ∈H. Proof: Let us consider x ∈ (a∗ b) ∗ (c∗d) = ((c∗d) ∗ b) ∗a = ((b∗d) ∗ c) ∗a = (a∗ c) ∗ (b∗d) . This implies that x ∈ (a∗ c) ∗ (b∗d). From this we can say that (a∗ b) ∗ (c∗d) ∩ (a∗ c) ∗ (b∗d) 6= φ, hold for all a,b,c,d ∈H. 2 Proposition 3.1. Let (H,◦) be an LA-hypergroup with left identity e and non-empty set A, such that A ⊆ H. If (A ◦ (A ◦ x)) ◦ y ∩ (A ◦ (A ◦ y)) ◦ x 6= ∅ ∀x,y ∈ H and we define a hyperoperation A⊗R on H as xA⊗Ry = (x◦y) ◦A, then (H,A ⊗ R) become an Hv-LA-group. Int. J. Anal. Appl. 17 (5) (2019) 868 Proof: Let x,y,z ∈H, we have (xA⊗Ry)A ⊗ Rz = ((x◦y) ◦A)A ⊗ Rz = (((x◦y) ◦A) ◦z) ◦A = ((z ◦A) ◦ (x◦y)) ◦A = (A◦ (A◦z)) ◦ (y ◦x) = y ◦ ((A◦ (A◦z)) ◦x) and on the other hand (zA⊗Ry)A ⊗ Rx = ((z ◦y) ◦A)A ⊗ Rx = (((z ◦y) ◦A) ◦z) ◦A = ((x◦A) ◦ (z ◦y)) ◦A = (A◦ (A◦x)) ◦ (y ◦z) = y ◦ ((A◦ (A◦x)) ◦z) but, since y◦((A◦(A◦z))◦x)∩y◦((A◦(A◦x))◦z) 6= ∅ ∀ for all x,y,z ∈H. It follows that (xA⊗Ry)A ⊗ Rz∩ (zA⊗Ry)A ⊗ Rx 6= ∅. Next, we have xA⊗RH = (x◦H) ◦A = H also HA ⊗ Rx = (H◦x) ◦A = H. Hence (H,A⊗R) becomes an Hv-LA-group. 2 Next we defined some regular relations on Hv-LA-groups. Definition 3.2. The equivalence relation ρ is called regular on right (on the left ), if for all x of H ( H is an Hv-LA-group), from aρb, it follows that (a◦x) − ρ (b◦x) , (x◦a) − ρ (x◦ b) respectively. Lemma 3.2. The relation ρ = φ−1 ∗φ = {(x,y) ∈H1 ×H1 : φ(x) = φ(y)} is regular on H1. Proof: The ρ is an equivalence relation on H1 obviously. We have to show that ρ is regular on H1. Let x,y,z ∈ H1 such that xρy, this implies that φ(x) = φ(y) =⇒ φ(xz) = φ(yz) and φ(zx) = φ(zy). so (xz)ρ(yz). Thus xρy =⇒ (xz)ρ(yz) and (zx)ρ(zy). Hence ρ = φ−1 ∗φ = {(x,y) ∈H1 ×H1 : φ(x) = φ(y)} is regular on H1. 2 On Hv-LA-group H, we are concerned with equivalence relation for which the family of equivalence classes form an Hv-LA-group under the hyperoperation induced by that on H. For an equivalence relation ρ on H, we may use xρ, and x or ρ(x) to denote the equivalence class of x ∈ H. Moreover, generally if A is a Int. J. Anal. Appl. 17 (5) (2019) 869 non-empty subset of H then Aρ = U{xρ | x ∈ A}. We let H/ρ denote the family {xρ | x ∈H} of class of ρ. The hyperoperation on H induces a hyperoperation ⊗ on H/ρ defined by xρ ⊗yρ = {zρ / z ∈ xρ ◦yρ} Where x, y ∈H. The structure (H/ρ,⊗) is known as quotient structure. Theorem 3.1. Let (H,◦) be Hv-LA-group. Then (H/ρ,⊗) is an Hv-LA-group iff (ρ (x) ◦ρ (y)) ◦ρ (z) ∩ (ρ (z) ◦ρ (y)) ◦ρ (x) 6= φ ∀x,y,z ∈H Proof: In H/ρ, we have let U ∈ (ρ (x) ⊗ρ (y)) ⊗ρ (z) = {u/u ∈ x◦y}⊗z = {t ∈ u◦z,u ∈ x◦y} = {t ∈ (z ◦y) ◦x} =⇒ U ∈ (ρ (x) ⊗ρ (y)) ⊗ρ (z) This implies that (ρ (x) ⊗ρ (y)) ⊗ρ (z) ⊆ (ρ (z) ⊗ρ (y)) ⊗ρ (x) . So (ρ (x) ⊗ρ (y)) ⊗ρ (z) ∩ (ρ (z) ⊗ρ (y)) ⊗ ρ (x) 6= φ. Since x ◦H = H = H ◦ x, =⇒ whence H/ρ ⊗ x = H/ρ = x ⊗H/ρ. Hence (H/ρ,⊗) is an Hv-LA-group. 2 Theorem 3.2. Let (H,◦) be Hv-LA-group and ρ be equivalence relation on H. If ρ is a regular relation, then (H/ρ,⊗) is an Hv-LA-group. Proof: First we show that ⊗ is a well defined on H/ρ, consider x = x1 and y = y1. We check that x⊗y = x1⊗y1. We have xρx1 and yρy1. Since ρ is regular, it follows that (x◦y) ρ (x1 ◦y) , (x1 ◦y) ρ (x1 ◦y1) whence (x◦y) ρ (x1 ◦y1). This implies that for all n ∈ (x◦y) there exists n1 such that nρn1. Which shows that n = n1. It follows that that x⊗y ⊆ x1 ⊗y1 and similarly we obtain converse. Hence ⊗ is well defined . Next we show weak left invertive property of ⊗. Let x,y,z be arbitrary element in H/ρ and l ∈ (x⊗y) ⊗z. This implies that v ∈ x⊗y and l ∈ v⊗z. It means that v1 ∈ x◦y and l1 ∈ v◦z such that vρv1 and lρl1. since ρ is regular relation, it follows that there exists l2 ∈ v1◦z ⊆ (x◦y)◦z ⊆ (z ◦y)◦x ( since H is a Hv-LA-group) such that l1ρl2. From here we obtain that there exist l3 ∈ z ◦y such that l2 ∈ l3 ◦x. we have l = l1 = l2 ∈ l3 ⊗x ⊆ (z ⊗y) ⊗x =⇒ (x⊗y) ⊗z ⊆ (z ⊗y) ⊗x =⇒ (x⊗y) ⊗z ∩ (z ⊗y) ⊗x 6= φ. Int. J. Anal. Appl. 17 (5) (2019) 870 Finally we show the reproductive axiom since x◦H = H = H◦x whence x⊗H/ρ = H/ρ = H/ρ⊗x. Hence (H/ρ,⊗) is a Hv-LA-group. 2 Next we defined the homorphisms of Hv-LA-groups. Definition 3.3. A mapping φ : H1 −→ H2 (where H1 and H2 are Hv-LA-group) is said to be good homo- morphism if it satisfies the following property φ(xy) = φ(x)φ(y) ∀ x,y ∈H1 Example: Let H1 = {a,b,c} and H2 = {l,m,n}, be two Hv-LA-hypergroups with hyperoperation is defined in the following tables respectively, ∗ l m n l {l} {m} {n} m {n} {l,n} {m} n {m} {l,n} {l,n} and ◦ a b c a {a} {b} {c} b {c} {a,c} {b,c} c {b} {a,c} {a,c} The mapping f : H1 −→H2 is defined by f (a) = l , f (b) = m, f (c) = n. Then assuredly f homomorphism is a strong homomorphism. If good homomorphism is 1 − 1 and onto is called isomorphism. If f is an isomorphism, then H1 and H2 are said to be isomorphic, which is denoted by H1 ∼= H2. Definition 3.4. Let (H1,◦) and (H2,∗) be two Hv-LA-hypergroups. The map f : H1 −→ H2 is called inclusion homomorphism if for all x,y ∈H1 satisfies the following property f(x◦y) ⊆ f(x) ∗f(y) Int. J. Anal. Appl. 17 (5) (2019) 871 Example: Let H1 = {l,m,n} and H2 = {a,b,c} be two Hv-LA- hypergroups with hyperoperations defined in the following table: ∗ l m n l {l} {m} {n} m {n} {l,m} {m} n {m} {n} {n,l} and ◦ a b c a {a} {b} {c} b {c} {H} {b} c {b} {c} {H} The mapping f : H1 −→ H2 is defined by f(l) = a , f(m) = b , f(n) = c. Then clearly f is an inclusion homomorphism. Definition 3.5. Let (H1,◦) and (H2,∗) be two Hv-LA-hypergroup. The map f : H1 −→H2 is called weak homomorphism or Hv homomorphism, if for all x,y ∈H1, the condition is hold f(x◦y) ∩f(x) ∗f(y) 6= φ Example: Let H1 = {l,m,n} and H2 = {a,b,c} are two finite sets, where (H1,∗) and (H2,◦) are Hv-LA-hypergroups, the hyperoperation is defined in following tables: ∗ l m n l {l} {m} {n} m {n} {l,m} {m} n {m} {n} {n,l} and ◦ a b c a a {a,b} {a,c} b c {H} {a,b} c b {a,c} {H} . The mapping f : H1 −→ H2 is defined by f(l) = a, f(m) = b, f(n) = c. Then clearly f is a weak homomorphism or Hv-LA-homomorphism. Theorem 3.3. Let φ : H1 −→ H2 be good homomorphism of an Hv-LA groups. Then there exist a monomorphism ψ : H1/ρ −→H2 such that imφ = imψ and diagram Int. J. Anal. Appl. 17 (5) (2019) 872 commutes i.e. ψ ∗ρ• = φ where the mapping ρ• : H1 −→H1/ρ is defined by ρ•(x) = ρ(x) ∀x ∈H1. Proof: Let ψ : H1/ρ −→ H2 is defined as ψ(ρ(x)) = φ(x) ∀x ∈ H1 since φ : H1 −→ H2. First we show that ψ is well defined. For this let ρ(x1),ρ(x2) ∈H1/ρ such that ρ(x1) = ρ(x2) ρ•(x1) = ρ •(x2) ψ(ρ•(x1)) = ψ(ρ •(x2)) φ(x1) = φ(x2) ψ(ρ(x1)) = ψ(ρ(x2)). Next we will show that ψ is one -one. For this let ψ(ρ(x1)), ψ(ρ(x2)) ∈H1/ρ ψ(ρ(x1)) = ψ(ρ(x2)) φ(x1) = φ(x2) ψ(ρ•(x1)) = ψ(ρ •(x2)) ρ•(x1) = ρ •(x2) ρ(x1) = ρ(x2). Finally we show that ψ is homomorphism. Let x,y ∈H1 we have ψ(ρ(x) ∗ρ(y)) = {ψ(ρ(z)) : z ∈ xy} = {φ (z) : z ∈ xy} = φ(xy) = φ (x) φ (y) = ψ(ρ(x)) ∗ψ(ρ(y)) Int. J. Anal. Appl. 17 (5) (2019) 873 Hence ψ is monomorphism and it is easy to prove that imφ = imψ. Now for all x ∈H1, we have (ψ∗ρ•)(x) = ψ((ρ•)(x)) = ψ((ρ)(x)) = φ(x). Hence diagram commutes. 2 Theorem 3.4. Let φ : H1 −→H2 be good homomorphism of an Hv-LA groups. Then H1/ρ ∼= Im φ. Proof: It follows from the Theorem 3.3. 2 Theorem 3.5. Let φ : H1 −→H2 be good homomorphism of an Hv-LA groups. If k is a regular relation on H1 such that k ⊆ ρ, then there exists a unique monomorphism ψ : H1/k −→H2 such that Im φ = Im ψ and diagram commute i.e.ψ ∗k? = φ, where mapping k? : H1 −→H2/k is defined as k?(x) = k (x) ∀x ∈H1. Proof: Straightforward. 2 Theorem 3.6. Let θ and σ be two regular relation on an Hv-LA-group H such that θ ⊆ σ. Then σ/θ is regular relation on H/θ. Proof: We define σ/θ : H/θ ◦ H/θ −→ P?(H/θ) by σ/θ(θ(x)) = θ(x) ∀θ(x) ∈ H/θ. We first show that the mapping is well defined, consider θ (x) = θ (y) =⇒ (x,y) ∈ θ ⊆ σ =⇒ (θ (x) ,θ (y)) ∈ σ/θ and so σ/θ (θ (x)) = σ/θ (θ (y)) . Next we show that σ/θ is an equivalence relation. Let x ∈ H, then (x,x) ∈ σ =⇒ (θ (x) ,θ (y)) ∈ σ/θ, thus σ/θ is reflexive. Also let x,y ∈ H, such that (θ (x) ,θ (y)) ∈ σ/θ. As (x,y) ∈ σ =⇒ (y,x) ∈ σ due to the symmetry of σ. Which implies that (θ (y) ,θ (x)) ∈ σ/θ. Hence σ/θ is symmetric . Again let x,y,z ∈ H, such that (θ (x) ,θ (y)) , (θ (y) ,θ (z)) ∈ σ/θ and (x,y) , (y,z) ∈ σ =⇒ (x,z) ∈ σ due to the transitivity of σ. Which implies that (θ (x) ,θ (y)) ∈ σ/θ. Hence σ/θ is transitive. Thus σ/θ is an equivalence relation. Now we have to show that, it is a regular. For it let x,y,z ∈ H, such that (θ (x)) σ/θ (θ (y)) =⇒ (x,y) ∈ σ =⇒ xσy =⇒ (xz) ρ (yz) =⇒ {θ (µ) : µ ∈ xz}σ/θ{θ (ν) : ν ∈ yz}. Int. J. Anal. Appl. 17 (5) (2019) 874 Which implies that (θ (x) ⊗θ (z)) σ/θ (θ (y) ⊗θ (z)) and similarly we can show that (θ (x)) σ/θ (θ (y)) =⇒ (θ (z) ⊗θ (x)) σ/θ (θ (z) ⊗θ (z)) . Hence σ/θ is regular relation on H/θ. 2 Theorem 3.7. Let θ and σ be two regular relations on an Hv-LA-group H such that θ ⊆ σ. Then (H/θ) / (σ/θ) ∼= H/σ. Proof: Let us define ψ : (H/θ) / (σ/θ) −→ H/σ by ψ (σ/θ (θ (x))) = σ (x) ∀x ∈ H. It is easy to show that this map is bijective. We only show that it is homomorphism. Suppose x,y ∈H, then ψ(σ/θ (θ (x)) ⊗σ/θ (θ (y))) = ψ({σ/θ (θ (z)) : θ (z) ∈ θ (x) ⊗θ (z)}) = ψ({σ/θ (θ (z)) : z ∈ xy}) = {ψ(σ/θ (θ (z)) : z ∈ xy)} = {σ (z) : z ∈ xy} = σ (x) σ (y) = ψ(σ/θ (θ (x))) ⊗ψ(σ/θ (θ (y))) Hence ψ is homomorphism. Thus (H/θ) / (σ/θ) ∼= H/σ. 2 3.1. Construction of Hv-LA-Groups: In this section we present the construction of Hv-LA-groups through any non-empty set having more than two elements. Consider a finite set H, such that |H| > 2. Define the hyperoperation ◦ on H as follows xi ◦xj =   xj for i = 1 xk for j = 1and k ≡ 2 − i mod |H| H for i = j,i 6= 1,j 6= 1 xi otherwise, for i ≺ j or i � j   Then H under the hyperoperation ◦ form an Hv-LA-group. The above construction can be explained with the help of an example. Example: Let H = {x1,x2,x3} be any set and define the binary hyperoperation ◦ defined above in the following cayley,s table: ◦ x1, x2 x3 x1, x1 x2 x3 x2 x3 H x2 x3 x2 x3 H Then clearly H form an Hv-LA-group. One can see that ◦ satisfy the weak left invertive law with reproductive axiom, also ◦ is non-left invertive and non-associative i.e. H = (x3 ◦x3) ◦x2 6= (x2 ◦x3) ◦x3 = x2 Int. J. Anal. Appl. 17 (5) (2019) 875 and H = (x2 ◦x2) ◦x1 6= x2 ◦ (x2 ◦x1) = x2 also it is not WASS (x2 ◦x1) ◦x1 ∩x2 ◦ (x1 ◦x1) = φ. Hence (H,◦) is an Hv-LA-group. The result can easily be generalized to n elements. 3.2. Chemical example of Hv-LA- group: Here in this section we utilize the newly defined structure namely Hv-LA- groups in applications. For this purpose we study chemical reactions. The best example of Hv-LA-group in chemical reaction is a redox reaction. 3.3. Redox reaction: The chemical reaction in which one specie loss the electron and other specie gain the electron. Oxidation mean loss of electron. Reduction mean gain of electron. The redox reaction is a vital for biochemical reaction and industrial process. The electron transfer in cell and oxidation of glucose in the human body are the example of redox reaction. The reaction between hydrogen and fluorine is an example of redox reaction i.e. H2 + F −→ 2HF H2 −→ 2H+ + 2e−( Oxidation) F2 + 2e − −→ 2F (Reduction) Each half reaction has standard reduction potential ( E0 ) which is equal to the potential difference at equi- librium under the standard condition of an electrochemical cell in which the cathode reaction is half reaction considered and anode is a standard hydrogen electrode (SHE). For the redox reaction, the potential of cell is defined as E◦cell = E◦cathod −E◦anode where E◦anode is the standard potential at the anode and E ◦ cathod is the standard potential at the cathode as given in the table of standard electrode potential. Now consider the redox reaction of Mn Mn0 + 2Mn+4 + 2Mn+3 −→ 3Mn+2 + 2Mn+4 Mn0 −→ Mn+2 + Mn+4 + 2e− + 2Mn+3 + 2Mn+4 Manganese having a variable oxidation state of 0, +1, +2, +3, +4, +5, +6, +7. If we take Mn0,Mn+4,Mn+3,Mn+4 together we will get pure redox reaction. The flow chart is given as Int. J. Anal. Appl. 17 (5) (2019) 876 Mn species with different oxidation state react with themselves. All possible reactions are presented in the following table ⊕ Mn0 Mn+1 Mn+2 Mn+3 Mn+4 Mn0 Mn0 { Mn0,Mn+1 } { Mn0,Mn+2 } { Mn0,Mn+3 } { Mn0,Mn+4 } Mn+1 { Mn0,Mn+1 } { Mn0,Mn+2 } { Mn0,Mn+3 } { Mn+2 } { Mn+1,Mn+4 } Mn+2 Mn+1 { Mn0,Mn+3 } { Mn+1,Mn+3 } { Mn+1,Mn+4 } { Mn+2,Mn+4 } Mn+3 { Mn0,Mn+3 } { Mn+1,Mn+3 } { Mn+2,Mn+3 } Mn+3 { Mn+3,Mn+4 } Mn+4 { Mn0,Mn+4 } { Mn+1,Mn+4 } { Mn+2,Mn+4 } { Mn+3,Mn+4 } Mn+4 The standard reduction potentials ( E0 ) for conversion of each oxidation state to another are E0 ( Mn+4/Mn+3 ) = +0.95,E0 ( Mn+3/Mn+2 ) = +1.542,E0 ( Mn+2/Mn+1 ) = −0.59, E0 ( Mn+1/Mn+0 ) = 0.296. If we replace Mn0 = a, Mn+1 = b, Mn+2 = c, Mn+3 = d, Mn+4 = e, then we obtain the following table ⊕ a b c d e a {a} {a,b} {a,c} {a,d} {a,e} b {a,b} {a,c} {a,d} {c} {b,e} c {a,c} {a,d} {b,d} {b,e} {c,e} d {a,d} {b,d} {c,d} {d} {d,e} e {a,e} {b,e} {c,e} {d,e} {e} Int. J. Anal. 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