<figure><img src="image_1.png" /></figure>

HAL
open science

# Chemical hyperstructures for Astatine, Tellurium and for Bismuth

Madeline Al Tahan, Bijan Davvaz

<figure><img src="image_2.png" /></figure>

## To cite this version:

Madeline Al Tahan, Bijan Davvaz. Chemical hyperstructures for Astatine, Tellurium and for Bismuth. Bull. Comput. Appl. Math., 2019, 7 (1), pp.9-25. hal-03580845

HAL Id: hal-03580845

https://hal.science/hal-03580845

Submitted on 18 Feb 2022

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_3.png" /></figure>

Distributed under a Creative Commons Attribution - ShareAlike 4.0 International License
# Chemical hyperstructures for Astatine, Tellurium and for Bismuth

Madeline Al Tahan<sup>1</sup>, Bijan Davvaz<sup>23</sup>

CompAMa Vol.7, pp.9-25, 2019 - Received June 1, 2018 - Accepted March 28, 2019

## Abstract

The concept of weak algebraic hyperstructures constitutes a generalization of the well-known algebraic hyperstructures. The aim of this paper is to present some of the results, methods and ideas about chemical examples of weak algebraic hyperstructures. More precisely, we give chemical hyperstructures of chemical reactions for: a set of Astatine (At), a set of Tellurium (Te) and a set of Bismuth (Bi).

**Keywords:** Hypergroup, $H_v$-semigroup, oxidation state, redox reaction.

## Resumen

El concepto de hiperestructuras algebraicas débiles constituye una generalización de las hiperestructuras algebraicas conocidas. El objetivo de este trabajo es presentar algunos de los resultados, métodos e ideas sobre ejemplos químicos de hiperestructuras algebraicas débiles. Más precisamente, ofrecemos hiperestructuras químicas de reacciones químicas para: un conjunto de Ástato (At), un conjunto de Telurio (Te) y un conjunto de Bismuto (Bi).

**Palabras claves:** Hipergrupos, $H_v$-semigrupo, estado de oxidación, reacción redox.

<footnote n="1"><sup>1</sup>Department of Mathematics, Lebanese International University, Lebanon. (made-line.tahan@liu.edu.lb).</footnote>

<footnote n="2"><sup>2</sup>Department of Mathematics, Yazd University, Yazd, Iran (davvaz@yazd.ac.ir).</footnote>

<footnote n="3"><sup>3</sup>Corresponding author.</footnote>
# 1 Introduction

In 1935, F. Marty [1] introduced algebraic hyperstructures which are generalized concepts of algebraic structures. A generalization of algebraic hyperstructures was introduced in 1990 by T. Vougiouklis [2] where he defined weak hyperstructures. Many researchers such as Corsini [3], Corsini and Leoreanu [4], Davvaz [5, 6], Davvaz and Leoreanu-Fotea [7], and Vougiouklis [8] wrote books related to hyperstructure theory and their applications. One of motivations for the study of hyperstructures comes from chemical reactions. In [9], Davvaz and Dehghan-Nezhad provided examples of hyperstructures associated with chain reactions. In [10], Davvaz et al. introduced examples of weak hyperstructures associated with dismutation reactions. In [11], Davvaz et al. investigated the examples of hyperstructures and weak hyperstructures associated with redox reactions. Also, see [12–18]. In [15]. Chung et al. investigated mathematical structures of chemical reactions for three consecutive oxidation states of elements. In [12], the authors presented three different examples of weak hyperstructures associated to elektrochemical cells. Moreover, Chun in [13] presented chemical hyperstructures of chemical reactions for a set of Titanium.

In this paper, we consider the redox reactions: Astatine and Tellurium and Bismuth as algebraic hyperstructures and it is organized as follows: After an Introduction, Section 2 presents some definitions and concepts related to (weak) hyperstructures that are used throughout the paper. Section 3 presents a new example of chemical hyperstructures (and non-isomorphic to previous examples). More precisely, it considers redox reactions of Astatine as algebraic hyperstructures. Finally, Sections 3 and 4 consider redox reactions of Tellurium and Bismuth as algebraic hyperstructures.

# 2 Preliminaries

In this section, we present some definitions and concepts related to (weak) hyperstructures that are used throughout the paper.

**Definition 1.** Let $H$ be a non-empty set. Then, a mapping $\circ : H \times H \rightarrow \mathcal{P}^*(H)$ is called a binary hyperoperation on $H$, where $\mathcal{P}^*(H)$ is the family of all non-empty subsets of $H$. The couple $(H, \circ)$ is called a hypergroupoid.

In the above definition, if $A$ and $B$ are two non-empty subsets of $H$ and $x \in H$, then we define:
$$A \circ B = \bigcup_{\substack{a \in A \\ b \in B}} a \circ b, x \circ A = \{x\} \circ A \text{ and } A \circ x = A \circ \{x\}.$$

$H_v$-structures were introduced by T. Vougiouklis as a generalization of the well-known algebraic hyperstructures [2, 19], also see [8, 20]. Some axioms of classical algebraic hyperstructures are replaced by their corresponding weak axioms in $H_v$-structures. Most of $H_v$-structures are used in representation theory.

A hypergroupoid $(H, \circ)$ is called $H_v$-semigroup if $(x \circ (y \circ z)) \cap ((x \circ y) \circ z) \neq \emptyset$ for all $x, y, z \in H$. A subset $K$ of an $H_v$-semigroup is an $H_v$-subsemigroup if $K$ is an $H_v$-semigroup. An element $x \in H$ is called *idempotent* if $x^2 = x \circ x = x$ and an element $e \in H$ is called an *identity* of $(H, \circ)$ if $x \in x \circ e \cap e \circ x$, for all $x \in H$. The hypergroupoid $(H, \circ)$ is said to be *commutative* if $x \circ y = y \circ x$, for all $x, y \in H$. A hypergroupoid $(H, \circ)$ is called a *semihypergroup* if for every $x, y, z \in H$, we have $x \circ (y \circ z) = (x \circ y) \circ z$ and is called a *quasihypergroup* if for every $x \in H$, $x \circ H = H \circ x$. This condition is called the reproduction axiom. The couple $(H, \circ)$ is called a *hypergroup* if it is a semihypergroup and a quasihypergroup. Two hypergroupoids $(G, \circ)$ and $(H, \star)$ are said to be *isomorphic* if there exists a bijective function $f: G \to H$ such that for all $x, y \in G$, we have $f(x \circ y) = f(x) \star f(y)$.

### 3 Redox reactions of Astatine as algebraic hyperstructures

Inspired by the work done by Chun and Davvaz et al. in [13, 16], we apply their methods to Astatine.

Astatine, denoted as At, is a highly radioactive element and it is the heaviest known halogen. The Latimer diagram of Astatine in basic solution is given as follows [21]:

$$At0_3^- \rightarrow_{0.5} At0^- \rightarrow_0 At_2 \rightarrow_{0.2} At^-.$$

Astatine has four different oxidation states: +5, +3, 0 and -1. We denote $At0_3^-$ by $At^{5+}$ and $At0^-$ by $At^+$. We compute the potential differences of the following reduction:

$$(1) \quad At^{5+} \rightarrow At_2, \quad E = \frac{0.5(4)+0(1)}{5} = 0.4,$$

$$(2) \quad At^{5+} \rightarrow At^-, \quad E = \frac{0.5(4)+0(1)+0.2(1)}{6} = 0.367,$$
$$ (3) \quad At^{+} \rightarrow At^{-}, E = \frac{0(1)+0.2(1)}{2} = 0.1. $$

Let $\{x, x', y, y'\} \subseteq \{At^{5+}, At^{+}, At_2, At^{-}\}$ such that $x \rightarrow_{\alpha} x'$ and $y \rightarrow_{\beta} y'$ where $\alpha, \beta$ are potential differences. We get the following redox reaction:

$$ x + x' \rightarrow_{\alpha+\beta} y + y'. $$

If $E = \alpha + \beta > 0$ then our redox reaction is spontaneous. Otherwise, it is not spontaneous.

**Remark 1.** The redox reaction $At^{+} + At^{+} \rightarrow At^{5+} + At_2$ is not spontaneous as $E = -0.5 < 0$.

We consider the set $G = \{At^{5+}, At^{+}, At_2, At^{-}\}$ of Astatine. The following are all possible spontaneous redox combinations for $G$.

$$ At^{-} + At^{-} \rightarrow At^{-} + At^{-} [0] $$

$$ At^{-} + At_{2} \rightarrow \begin{cases} At^{+} + At^{-} & [0.1] \\ At_{2} + At^{-} & [0] \end{cases} $$

$$ At^{-} + At^{+} \rightarrow At^{+} + At^{-} [0] $$

$$ At^{-} + At^{5+} \rightarrow \begin{cases} At^{+} + At^{+}[0.4] \\ At_{2} + At^{+}[0.3] \\ At_{2} + At_{2}[0.2] \\ At^{5+} + At^{+}[0.133] \\ At^{5+} + At_{2}[0.033] \\ At^{5+} + At^{-}[0] \end{cases} $$

$$ At_{2} + At_{2} \rightarrow \begin{cases} At^{-} + At^{+}[0.2] \\ At_{2} + At_{2}[0] \end{cases} $$

$$ At_{2} + At^{+} \rightarrow \begin{cases} At^{+} + At^{-}[0.1] \\ At^{+} + At_{2}[0] \end{cases} $$

$$ At_{2} + At^{5+} \rightarrow \begin{cases} At^{+} + At^{+}[0.5] \\ At^{+} + At_{2}[0.4] \\ At^{+} + At^{-}[0.367] \\ At^{5+} + At_{2}[0] \end{cases} $$
$$At^+ + At^+ \longrightarrow At^+ + At^+ [0]$$

$$At^+ + At^{5+} \longrightarrow At^{5+} + At^+ [0]$$

$$At^{5+} + At^{5+} \longrightarrow At^{5+} + At^{5+} [0].$$

Let $G = \{At^{-}, At_{2}, At^{+}, At^{5+}\}$ and define “$\oplus$” on $G$ as follows:

For all $x, y \in G$, $x \oplus y = z$, where $z$ is in the product of the spontaneous redox reaction with greatest potential difference that occurs between $x$ and $y$. Then we obtain the following table for $(G, \oplus)$.

<table><thead><tr><th>$\oplus$</th><th>$At^{-}$</th><th>$At_{2}$</th><th>$At^{+}$</th><th>$At^{5+}$</th></tr></thead><tbody><tr><th>$At^{-}$</th><td>$At^{-}$</td><td>$\{At^{-}, At^{+}\}$</td><td>$\{At^{-}, At^{+}\}$</td><td>$At^{+}$</td></tr><tr><th>$At_{2}$</th><td></td><td>$\{At^{-}, At^{+}\}$</td><td>$\{At^{-}, At^{+}\}$</td><td>$At^{+}$</td></tr><tr><th>$At^{+}$</th><td></td><td></td><td>$At^{+}$</td><td>$\{At^{+}, At^{5+}\}$</td></tr><tr><th>$At^{5+}$</th><td></td><td></td><td></td><td>$At^{5+}$</td></tr></tbody></table>

By setting $a = At^{-}, b = At_{2}, c = At^{+}, d = At^{5+}$, we get the following table.

<table><thead><tr><th>$\oplus$</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><th>a</th><td>a</td><td>$\{a, c\}$</td><td>$\{a, c\}$</td><td>c</td></tr><tr><th>b</th><td>$\{a, c\}$</td><td>$\{a, c\}$</td><td>$\{a, c\}$</td><td>c</td></tr><tr><th>c</th><td>$\{a, c\}$</td><td>$\{a, c\}$</td><td>c</td><td>$\{c, d\}$</td></tr><tr><th>d</th><td>c</td><td>c</td><td>$\{c, d\}$</td><td>d</td></tr></tbody></table>

**Remark 2.** $(G, \oplus)$ is not a quasi-hypergroup.

**Theorem 1.** $(G, \oplus)$ is a commutative $H_{v}$-semigroup.

*Proof.* Let $x, y, z \in G$. We consider the following cases:
* Case $(x, y, z) = (a, a, a)$. We have $x \oplus (y \oplus z) = (x \oplus y) \oplus z = \{a\}$.

* Case $(x, y, z) = (d, d, d)$. $x \oplus (y \oplus z) = (x \oplus y) \oplus z = \{d\}$.

* Case $(x, y, z) \neq (a, a, a)$ and $(x, y, z) \neq (d, d, d)$. It is easy to see that $c \in (x \oplus (y \oplus z)) \cap ((x \oplus y) \oplus z)$. $\square$

**Remark 3.** $(G, \oplus)$ is not a semihypergroup. This is clear as $\{c\} = b \oplus (d \oplus d) \neq (b \oplus d) \oplus d = \{c, d\}$.

**Proposition 1.** $(G, \oplus)$ has two $H_v$-subsemigroups up to isomorphisms.

*Proof.* One can easily see that $(\{a, c\}, \oplus), (\{c, d\}, \oplus)$ and $(\{a, c, d\}, \oplus)$ are the only $H_v$-subsemigroups of $G$. Moreover, $(\{a, c\}, \oplus)$ and $(\{c, d\}, \oplus)$ are isomorphic hypergroups. $\square$

**Remark 4.** From the table of $(G, \oplus)$, we deduce that $At^+$ is most likely to occur in nature.

Let $G = \{At^{-}, At_{2}, At^{+}, At^{5+}\}$ and define “$\otimes$" on $G$ as follows:

For all $x, y \in G$, $x \oplus y = z$, where $z$ is in the product of any spontaneous redox reaction that occurs between $x$ and $y$.

Then we obtain the following table for $(G, \oplus)$.

<table><thead><tr><th>&#8855;</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><th>a</th><td>a</td><td>{a, b, c}</td><td>{a, c}</td><td>G</td></tr><tr><th>b</th><td>{a, b, c}</td><td>{a, b, c}</td><td>{a, b, c}</td><td>G</td></tr><tr><th>c</th><td>{a, c}</td><td>{a, b, c}</td><td>c</td><td>{c, d}</td></tr><tr><th>d</th><td>G</td><td>G</td><td>{c, d}</td><td>d</td></tr></tbody></table>

**Proposition 2.** $(G, \otimes)$ is a commutative quasi-hypergroup with identity.

*Proof.* Since all elements of $G$ are present in every row and column of the table corresponding to $(G, \otimes)$, it follows that $(G, \otimes)$ is a quasi-hypergroup. Moreover, every element in $G$ is an identity. $\square$

**Proposition 3.** $(\{a, c\}, \oplus)$ and $(\{c, d\}, \oplus)$ are hypergroups.
*Proof.* The proof results from having $(\{a, c\}, \otimes)$ and $(\{c, d\}, \otimes)$ the form of the Biset hypergroup. $\square$

**Proposition 4.** ($G, \otimes$) is a semihypergroup.

*Proof.* Since $(\{a, c\}, \otimes)$ and $(\{c, d\}, \otimes)$ are hypergroups, it suffices to consider the following cases to prove associativity:

* $x \otimes (y \otimes z) = (x \otimes y) \otimes z$ for $x \in \{a, c\}$ and $\{y, z\} \not\subseteq \{a, c\}$;

* $b \otimes (y \otimes z) = (b \otimes y) \otimes z$ for $\{y, z\} \not\subseteq G$;

* $d \otimes (y \otimes z) = (d \otimes y) \otimes z$ for $\{y, z\} \not\subseteq \{c, d\}$.

* $a \otimes (b \otimes z) = \begin{cases} G, & z = d; \\ \{a, b, c\}, & \text{otherwise.} \end{cases} = (a \otimes b) \otimes z$.

* $a \otimes (d \otimes z) = (a \otimes d) \otimes z = G$.

* $a \otimes (y \otimes b) = \begin{cases} G, & y = d; \\ \{a, b, c\}, & \text{otherwise.} \end{cases} = (a \otimes y) \otimes b$.

* $c \otimes (b \otimes z) = \begin{cases} G, & z = d; \\ \{a, b, c\}, & \text{otherwise.} \end{cases} = (c \otimes b) \otimes z$.

* $c \otimes (y \otimes b) = \begin{cases} G, & y = d; \\ \{a, b, c\}, & \text{otherwise.} \end{cases} = (c \otimes y) \otimes b$.

* $c \otimes (a \otimes d) = (c \otimes a) \otimes d = G$ and $c \otimes (d \otimes a) = (c \otimes d) \otimes a = G$.

* $b \otimes (y \otimes z) = \begin{cases} G, & \text{if } y = d \text{ or } z = d; \\ \{a, b, c\}, & \text{otherwise.} \end{cases} = (b \otimes y) \otimes z$.

* $d \otimes (y \otimes z) = \begin{cases} d, & y = z = d; \\ \{c, d\}, & y = z = c \text{ or } y = c, z = d \text{ or } y = d, z = c; \\ G, & \text{otherwise.} \end{cases} = (d \otimes y) \otimes z$.

$\square$

**Theorem 2.** ($G, \otimes$) is a commutative hypergroup.

*Proof.* The proof results from Propositions 2 and 4. $\square$
We present now some chemical interpretation for our results.

**Remark 5.** From the table of $(G, \otimes)$, we deduce that $At^+$ is most likely to occur in nature and $At^{5+}$ is least likely to occur in nature.

**Remark 6.** $(G, \otimes)$ is not isomorphic to that presented by Chun and Davvaz in [13, 16] as our hypergroup contains three idempotents only whereas their hypergroups contain four idempotents. This may be due to the differences in the Latimer diagrams of each where in the Latimer diagram of Titanium, the potential difference decreases as the oxidation number decreases whereas in the Latimer diagram of Astatine, the potential difference decreases and then increases.

**Remark 7.** Having each element $x$ in $(G, \otimes)$ an identity implies that $y \in G$ is present in the product of at least one of spontaneous reactions that that occurs between $x$ and $y$.

# 4 Redox reactions of Tellurium as algebraic hyperstructures

Tellurium, denoted as Te, is a brittle, mildly toxic, rare, silver-white metalloid and it is chemically related to selenium and sulfur. It is occasionally found in native form as elemental crystals. Tellurium is far more common in the universe as a whole than on Earth. The Latimer diagram of Tellurium in basic solution is given as follows [21]:

$$TeO_2(OH)_4^{2-} \rightarrow_{0.07} TeO_3^{2-} \rightarrow_{-0.42} Te \rightarrow_{-1.14} Te^{2-}.$$

Tellurium has four different oxidation states: +6, +4, 0 and -2. We denote $TeO_2(OH)_4^{2-}$ by $Te^{6+}$ and $TeO_3^{2-}$ by $Te^{4+}$. We compute the potential differences of the following reduction:

$$(1) \quad Te^{6+} \rightarrow Te, E = \frac{0.07(2)-0.42(4)}{6} = -0.257,$$

$$(2) \quad Te^{6+} \rightarrow Te^{2-}, E = \frac{0.07(2)-0.42(4)-1.14(2)}{8} = -0.4775,$$

$$(3) \quad Te^{4+} \rightarrow Te^{2-}, E = \frac{-0.42(4)-1.14(2)}{6} = -0.66.$$
Let {$x, x', y, y'$} ⊆ {$Te^{6+}, Te^{4+}, Te, Te^{2-}$} such that $x \to_{\alpha} x'$ and $y \to_{\beta} y'$ where $\alpha, \beta$ are potential differences. We get the following redox reaction:

$$x + x' \to_{\alpha+\beta} y + y'$$

If $E = \alpha + \beta > 0$ then our redox reaction is spontaneous. Otherwise, it is not spontaneous.

We consider the set $H = \{Te^{6+}, Te^{4+}, Te, Te^{2-}\}$ of Tellurium. The following are all possible spontaneous redox combinations for $H$.

$$Te^{2-} + Te^{2-} \to Te^{2-} + Te^{2-} [0]$$

$$Te^{2-} + Te \to Te^{2-} + Te [0]$$

$$Te^{2-} + Te^{4+} \to \begin{cases} Te + Te[0.72] \\ Te^{4+} + Te[0.22] \\ Te^{6+} + Te[0.0575] \\ Te^{4+} + Te^{2-}[0] \end{cases}$$

$$Te^{2-} + Te^{6+} \to \begin{cases} Te + Te^{4+}[1.21] \\ Te^{4+} + Te^{4+}[0.73] \\ Te + Te[0.883] \\ Te^{6+} + Te^{2-}[0] \end{cases}$$

$$Te + Te \to Te + Te [0]$$

$$Te + Te^{4+} \to Te^{4+} + Te [0]$$

$$Te + Te^{6+} \to \begin{cases} Te^{4+} + Te^{4+}[0.49] \\ Te^{6+} + Te^{4+}[0.327] \\ Te^{4+} + Te[0.163] \\ Te^{6+} + Te[0] \end{cases}$$

$$Te^{4+} + Te^{4+} \to Te^{4+} + Te^{4+} [0]$$

$$Te^{4+} + Te^{6+} \to Te^{6+} + Te^{4+} [0]$$

$$Te^{6+} + Te^{6+} \to Te^{6+} + Te^{6+} [0].$$
For all $x, y \in H$, $x \oplus_1 y = z$, where $z$ is in the product of the spontaneous redox reaction with greatest potential difference that occurs between $x$ and $y$.

Then we obtain the following table for $(H, \oplus_1)$.

<table><thead><tr><th>$\oplus_1$</th><th>$Te^{2-}$</th><th>$Te$</th><th>$Te^{4+}$</th><th>$Te^{6+}$</th></tr></thead><tbody><tr><td>$Te^{2-}$</td><td>$Te^{2-}$</td><td>$\{Te^{2-}, Te\}$</td><td>$Te$</td><td>$\{Te, Te^{4+}\}$</td></tr><tr><td>$Te$</td><td></td><td>$Te$</td><td>$\{Te, Te^{4+}\}$</td><td>$Te^{4+}$</td></tr><tr><td>$Te^{4+}$</td><td></td><td></td><td>$Te^{4+}$</td><td>$\{Te^{4+}, Te^{6+}\}$</td></tr><tr><td>$Te^{6+}$</td><td></td><td></td><td></td><td>$Te^{6+}$</td></tr></tbody></table>

By setting $a = Te^{2-}$, $b = Te$, $c = Te^{4+}$, $d = Te^{6+}$, we get the following table.

<table><thead><tr><th>$\oplus_1$</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><td>a</td><td>a</td><td>$\{a, b\}$</td><td>b</td><td>$\{b, c\}$</td></tr><tr><td>b</td><td>$\{a, b\}$</td><td>b</td><td>$\{b, c\}$</td><td>c</td></tr><tr><td>c</td><td>b</td><td>$\{b, c\}$</td><td>c</td><td>$\{c, d\}$</td></tr><tr><td>d</td><td>$\{b, c\}$</td><td>c</td><td>$\{c, d\}$</td><td>d</td></tr></tbody></table>

**Theorem 3.** $(H, \oplus_1)$ is a commutative $H_v$-semigroup.

*Proof.* $(H, \oplus_1)$ is isomorphic to the $H_v$-semigroup of Titanium presented in Theorem 3.2, [13]. □

Let $H = \{Te^{6+}, Te^{4+}, Te, Te^{2-}\}$ and define "$\oplus_1$" on $H$ as follows:

For all $x, y \in H$, $x \oplus_1 y = z$, where $z$ is in the product of the any spontaneous redox reaction that occurs between $x$ and $y$.

Then we obtain the following table for $(H, \oplus_1)$.
<table><thead><tr><th>$\otimes_1$</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><th>a</th><td>a</td><td>{a, b}</td><td>H</td><td>H</td></tr><tr><th>b</th><td>{a, b}</td><td>b</td><td>{b, c}</td><td>{b, c, d}</td></tr><tr><th>c</th><td>H</td><td>{b, c}</td><td>c</td><td>{c, d}</td></tr><tr><th>d</th><td>H</td><td>{b, c, d}</td><td>{c, d}</td><td>d</td></tr></tbody></table>

**Theorem 4.** ($H, \otimes_1$) is a commutative hypergroup.

*Proof.* ($H, \otimes_1$) is isomorphic to the hypergroup of Titanium presented in Theorem 3.4, [13]. □

Next, we present some chemical interpretation for our results.

**Remark 8.** From the table of ($H, \otimes_1$), we deduce that $Te$ and $Te^{4+}$ are most likely to occur in nature and $Te^{2-}$ is least likely to occur in nature.

**Remark 9.** The hyperstructures of Astatine are not isomorphic to that of Tellurium. This may be due the differences in the Latimer diagram of each. In the Latimer diagram of Tellurium, the potential difference decreases as the oxidation number decreases whereas in the Latimer diagram of Astatine, the potential difference decreases and the increases.

# 5 Redox reactions of Bismuth as algebraic hyperstructures

Bismuth, denoted by Bi, is a pentavalent post-transition metal and one of the pnictogens, chemically resembles its lighter homologs arsenic and antimony. Elemental bismuth may occur naturally, although its sulfide and oxide form important commercial ores. It is a brittle metal with a silvery white color when freshly produced, but surface oxidation can give it a pink tinge. Bismuth is slightly radioactive and the most naturally diamagnetic element, and has one of the lowest values of thermal conductivity among metals. The Latimer diagram of Bismuth in acidic solution is given as follows [21]:

$$Bi_2O_5 \rightarrow_{1.6} BiO^+ \rightarrow_{0.317} Bi \rightarrow_{-0.97} BiH_3.$$
Bismuth has four different oxidation states: +5, +3, 0 and -3. We denote $Bi_2O_5$ by $Bi^{5+}$, $BiO^+$ by $Bi^{3+}$ and $BiH_3$ by $Bi^{3-}$. We compute the potential differences of the following:

$$ (1) \quad Bi^{5+} \rightarrow Bi, E = \frac{1.6(2)+0.317(3)}{5} = 0.8302, $$

$$ (2) \quad Bi^{5+} \rightarrow Bi^{3-}, E = \frac{1.6(2)+0.317(3)-0.97(3)}{8} = 0.155125, $$

$$ (3) \quad Bi^{3+} \rightarrow Bi^{3-}, E = \frac{0.317(3)-0.97(3)}{6} = -0.3265. $$

Let $\{x, x', y, y'\} \subseteq \{Bi^{5+}, Bi^{3+}, Bi, Bi^{2-}\}$ such that $x \rightarrow_{\alpha} x'$ and $y \rightarrow_{\beta} y'$ where $\alpha, \beta$ are potential differences. We get the following redox reaction:

$$ x + x' \rightarrow_{\alpha+\beta} y + y'. $$

If $E = \alpha + \beta > 0$ then our redox reaction is spontaneous. Otherwise, it is not spontaneous.

We consider the set $K = \{Bi^{5+}, Bi^{3+}, Bi, Bi^{2-}\}$ of Bismuth. The following are all possible spontaneous redox combinations for $K$.

$$ Bi^{3-} + Bi^{3-} \rightarrow Bi^{3-} + Bi^{3-} [0] $$

$$ Bi^{3-} + Bi \rightarrow Bi^{3-} + Bi [0] $$

$$ Bi^{3-} + Bi^{3+} \rightarrow \begin{cases} Bi + Bi[1.287] \\ Bi^{3+} + Bi[0.6435] \\ Bi^{5+} + Bi[0.161875] \\ Bi^{3+} + Bi^{3-}[0] \end{cases} $$

$$ Bi^{3-} + Bi^{5+} \rightarrow \begin{cases} Bi + Bi^{3+}[2.57] \\ Bi^{3+} + Bi^{3+}[1.9265] \\ Bi + Bi[1.802] \\ Bi^{5+} + Bi^{3-}[0] \end{cases} $$

$$ Bi + Bi \rightarrow Bi + Bi [0] $$

$$ Bi + Bi^{3+} \rightarrow Bi^{3+} + Bi [0] $$
$$ Bi + Bi^{5+} \rightarrow \begin{cases} Bi^{3+} + Bi^{3+}[1.283] \\ Bi^{5+} + Bi^{3+}[0.768] \\ Bi^{3+} + Bi[0.515] \\ Bi^{5+} + Bi[0] \end{cases} $$

$$ Bi^{3+} + Bi^{3+} \rightarrow Bi^{3+} + Bi^{3+} [0] $$

$$ Bi^{3+} + Bi^{5+} \rightarrow Bi^{5+} + Bi^{3+} [0] $$

$$ Bi^{5+} + Bi^{5+} \rightarrow Bi^{5+} + Bi^{5+} [0] $$

For all $x, y \in K$, $x \oplus_2 y = z$, where $z$ is in the product of the spontaneous redox reaction with greatest potential difference that occurs between $x$ and $y$.

Then we obtain the following table for $(K, \oplus_2)$.

<table><thead><tr><th>$\oplus_2$</th><th>$Bi^{3-}$</th><th>$Bi$</th><th>$Bi^{3+}$</th><th>$Bi^{5+}$</th></tr></thead><tbody><tr><td>$Bi^{3-}$</td><td>$Bi^{3-}$</td><td>{$Bi^{3-}$, $Bi$}</td><td>$Bi$</td><td>{$Bi$, $Bi^{3+}$}</td></tr><tr><td>$Bi$</td><td></td><td>$Bi$</td><td>{$Bi$, $Bi^{3+}$}</td><td>$Bi^{3+}$</td></tr><tr><td>$Bi^{3+}$</td><td></td><td></td><td>$Bi^{3+}$</td><td>{$Bi^{3+}$, $Bi^{5+}$}</td></tr><tr><td>$Bi^{5+}$</td><td></td><td></td><td></td><td>$Bi^{5+}$</td></tr></tbody></table>

By setting $a = Bi^{3-}$, $b = Bi$, $c = Bi^{3+}$, $d = Bi^{5+}$, we get the following table.

<table><thead><tr><th>$\oplus_2$</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><td>a</td><td>a</td><td>{a, b}</td><td>b</td><td>{b, c}</td></tr><tr><td>b</td><td>{a, b}</td><td>b</td><td>{b, c}</td><td>c</td></tr><tr><td>c</td><td>b</td><td>{b, c}</td><td>c</td><td>{c, d}</td></tr><tr><td>d</td><td>{b, c}</td><td>c</td><td>{c, d}</td><td>d</td></tr></tbody></table>

**Theorem 5.** $(K, \oplus_2)$ is a commutative $H_v$-semigroup.
*Proof.* The proof follows from Theorem 3 and having $(K, \oplus_2)$ isomorphic to $(H, \oplus_1)$. $\square$

Let $K = \{Bi^{5+}, Bi^{3+}, Bi, Bi^{3-}\}$ and define "$\otimes_2$" on $K$ as follows: For all $x, y \in H$, $x \otimes_2 y = z$, where $z$ is in the product of the any spontaneous redox reaction that occurs between $x$ and $y$. Then we obtain the following table for $(K, \otimes_2)$.

<table><thead><tr><th>$\otimes_2$</th><th>a</th><th>b</th><th>c</th><th>d</th></tr></thead><tbody><tr><th>a</th><td>a</td><td>{a, b}</td><td>K</td><td>K</td></tr><tr><th>b</th><td>{a, b}</td><td>b</td><td>{b, c}</td><td>{b, c, d}</td></tr><tr><th>c</th><td>K</td><td>{b, c}</td><td>c</td><td>{c, d}</td></tr><tr><th>d</th><td>K</td><td>{b, c, d}</td><td>{c, d}</td><td>d</td></tr></tbody></table>

**Theorem 6.** $(K, \otimes_2)$ is a commutative hypergroup.

*Proof.* The proof follows from Theorem 4 and having $(K, \otimes_2)$ isomorphic to $(H, \otimes_1)$. $\square$

**Remark 10.** The hyperstructures of Astatine are not isomorphic to that of Bismuth.

We present now some chemical interpretation for our results.

**Remark 11.** From the table of $(K, \otimes_2)$, we deduce that $Bi$ and $Bi^{3+}$ are most likely to occur in nature and $Bi^{3-}$ is least likely to occur in nature.

**Remark 12.** The hyperstructures of Bismuth are isomorphic to that of Tellurium and Titanium. This may be due the similarity in the Latimer diagram of each where the potential difference in the Latimer diagrams of each decreases as the oxidation number decreases.
# 6 Conclusion

This paper dealt with two non-isomorphic hyperstructures of three different elements with four different oxidation states. For future work, it will be interesting to generalize our work and investigate all algebraic hyperstructures for chemical reactions for four consecutive oxidation states of an arbitrary element.

# References

[1] F. Marty. Sur une generalization de la notion de group. In *Proc. 8th Congress Math. Scandenaves, Stockholm, Sweden*, volume 1, pages 45-49, 1934.

[2] T. Vougiouklis. The fundamental relation in hyperrings. the general hyperfield. In *Proc. Fourth Int. Congress on Algebraic Hyperstructures and Appl. (AHA 1990)*, World Scientific, 1991, volume 1, pages 203-211, 1991. https://www.researchgate.net/publication/265697147_The_fundamental_relation_in_hyperrings_The_general_hyperfield.

[3] P. Corsini. *Prolegomena of Hypergroup Theory, Second edition*. Rivista di matematica pura ed applicata. Aviani editore, Udine, Tricesimo, Italy, 1993. ISBN 8877720255.

[4] P. Corsini and V. Leoreanu. *Applications of Hyperstructures Theory*, volume 5 of *Advances in Mathematics*. Springer US, 2003. ISBN 978-1-4757-3714-1, doi: 10.1007/978-1-4757-3714-1.

[5] B. Davvaz. *Polygroup Theory and Related Systems*. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. ISBN 978-981-4425-32-2, doi: 10.1142/8593.

[6] B. Davvaz. *Semihypergroup Theory*. Elsevier/Academic Press, London, 2016. ISBN 978-0-12-809815-8, doi: 10.1016/C2015-0-06681-3.

[7] B. Davvaz and V. Leoreanu-Fotea. *Hyperring Theory and Applications*. International Academic Press, USA, 2007. ISBN 9781574850642.
[8] T. Vougiouklis. *Hyperstructures and Their Representations*. monographs in mathematics. Hadronic Press, Inc, 115, Palm Harber, USA, 1994. ISBN 9780911767865.

[9] B. Davvaz and A. Dehgan Nezhad. Chemical examples in hypergroups. *Ratio Matematica*, 14:71–74, 2003. http://eiris.it/ojs/index.php/ratiomathematica/article/view/16.

[10] B. Davvaz, A. Dehghan Nezhad, and A. Benvidi. Chemical hyperalgebra: Dismutation reactions. MATCH Commun. Math. Comput. Chem., 67(1):55–63, 2012. http://match.pmf.kg.ac.rs/electronic_versions/Match67/n1/match67n1_55-63.pdf.

[11] B. Davvaz, A. Dehghan Nezhad, and M. Mazloum-Ardakani. Chemical hyperalgebra: Redox reactions. MATCH Commun. Math. Comput. Chem., 71(2):323–331, 2014. http://match.pmf.kg.ac.rs/electronic_versions/Match71/n2/match71n2_323-331.pdf.

[12] M. Al-Tahan and B. Davvaz. Weak chemical hyperstructures associated to electrochemical cells. *Iranian Journal of Mathematical Chemistry*, 9(1):65–67, 2018. doi: 10.22052/ijmc.2017.88790.1294.

[13] K.M. Chun. Chemical hyperstructures of chemical reactions for iron and idium. *Journal of the Chungcheong Mathematical Society*, 27(2):319–325, 2014. doi: 10.14403/jcms.2014.27.2.319.

[14] S.C. Chun. Chemical hyperstructures for vanadium. *Journal of the Chungcheong Mathematical Society*, 27(2):309–317, 2014. doi: 10.14403/jcms.2014.27.2.309.

[15] S.C. Chun, K.M. Chun, N.J. Kim, S.Y. Jeong, J. Sim, H. Lee, and H. Maeng. Chemical hyperalgebras for three consecutive oxidation states of elements. MATCH Commun. Math. Comput. Chem., 72(2):389–402, 2014. http://match.pmf.kg.ac.rs/electronic_versions/Match72/n2/match72n2_389-402.pdf.
[16] B. Davvaz. Weak algebraic hyperstructures as a model for interpretation of chemical reactions. *Iranian Journal of Mathematical Chemistry*, 7(2):267-283, 2016. doi: 10.22052/ijmc.2016.13975.

[17] B. Davvaz, A. Dehghan Nezhad, and A. Benvidi. Chain reactions as experimental examples of ternary algebraic hyperstructures. *MATCH Commun. Math. Comput. Chem.*, 65(2):491-499, 2011. http://match.pmf.kg.ac.rs/electronic_versions/Match65/n2/match65n2_491-499.pdf.

[18] B. Davvaz and A. Dehghan Nezhad. Dismutation reactions as experimental verifications of ternary algebraic hyperstructures. *MATCH Commun. Math. Comput. Chem.*, 68(2):551-559, 2011. http://match.pmf.kg.ac.rs/electronic_versions/Match68/n2/match68n2_551-559.pdf.

[19] T. Vougiouklis. $h_v$-groups defined on the same set. *Discrete Math.*, 155(1-3):259-265, 1996. doi: 10.1016/0012-365X(94)00390-5.

[20] B. Davvaz. A brief survey of the theory of $h_v$-structures. In *Proc. 8th International Congress on Algebraic Hyperstructures and Applications, 1-9 Sep 2002, Samothraki, Greece, Spanidis Press*, volume 1, pages 39-70, September 2003. https://www.researchgate.net/publication/265334031_A_briefendiagram of the theory of H_v-structures.

[21] B. Douglas, D.H. McDaniel, and J.J. Alexander. *Concepts and Models of Inorganic Chemistry, 2nd ed.* John Wiely & Sons, Inc., 1985. ISBN 978-0471629788.