Note on Isomorphism Theorems of Hyperrings

Abstract

There are different notions of hyperrings (𝑅,+,⋅). In this paper, we extend the isomorphism theorems to hyperrings, where the additions and the multiplications are hyperoperations.

1. Introduction

The theory of hyperstructures was introduced in 1934 by Marty [1] at the 8th Congress of Scandinavian Mathematicians. This theory has been subsequently developed by Corsini [2–4], Mittas [5, 6], Stratigopoulos [7], and by various authors. Basic definitions and propositions about the hyperstructures are found in [3, 4, 8]. Krasner [9] has studied the notion of hyperfields, hyperrings, and then some researchers, namely, Ameri [10], Dašić [11], Davvaz [12], Gontineac [13], Massouros [14], Pianskool et al. [15], Sen and Dasgupta [16], Vougiouklis [8, 17], and others followed him.

Hyperrings are essentially rings with approximately modified axioms. There are different notions of hyperrings (𝑅,+,⋅). If the addition + is a hyperoperation and the multiplication ⋅ is a binary operation, then the hyperring is called Krasner (additive) hyperring [9]. Rota [18] introduced a multiplicative hyperring, where + is a binary operation and the multiplication ⋅ is a hyperoperation. De Salvo [19] studied hyperrings in which the additions and the multiplications were hyperoperations. These hyperrings were also studied by Barghi [20] and by Asokkumar and Velrajan [21–23]. In 2007, Davvaz and Leoreanu-Fotea [24] published a book titled Hyperring Theory and Applications. Davvaz [12] extended that the isomorphism theorems to Krasner hyperrings, provided the hyperideals considered in the isomorphism theorems are normal.

In this paper, we extend the isomorphism theorems to hyperrings, in which both the additions and the multiplications are hyperoperations. Also, it is observed that if 𝐼 is an hyperideal of a hyperring 𝑅 and (𝐼,+) is a normal subcanonical hypergroup of (𝑅,+), then 𝑅/𝐼 is a ring, and hence the quotient hyperrings considered in the isomorphism theorems by Davvaz in [12] are rings.

2. Basic Definitions and Notations

This section provides some basic definitions that have been used in the sequel. A hyperoperation ∘ on a nonempty set 𝐻 is a mapping of 𝐻×𝐻 into the family of nonempty subsets of 𝐻 (i.e., 𝑥∘𝑦⊆𝐻, for every 𝑥,𝑦∈𝐻). The definitions are found in references [3, 4, 8, 24]. A hypergroupoid is a nonempty set 𝐻 equipped with a hyperoperation ∘. For any two nonempty subsets 𝐴 and 𝐵 of a hypergroupoid 𝐻 and for 𝑥∈𝐻, 𝐴∘𝐵, we mean the set ⋃_{𝑎∈𝐴𝑏∈𝐵}(𝑎∘𝑏), 𝐴∘𝑥=𝐴∘{𝑥}, and 𝑥∘𝐵={𝑥}∘𝐵.

A hypergroupoid (𝐻,∘) is called a semihypergroup if 𝑥∘(𝑦∘𝑧)=(𝑥∘𝑦)∘𝑧 for every 𝑥,𝑦,𝑧∈𝐻 (the associative axiom). A hypergroupoid (𝐻,∘) is called a quasihypergroup if 𝑥∘𝐻=𝐻∘𝑥=𝐻 for every 𝑥∈𝐻 (the reproductive axiom). A reproductive semihypergroup is called a hypergroup (in the sense of Marty). A comprehensive review of the theory of hypergroups appears in [3].

Definition 2.1. A nonempty set 𝐻 with a hyperoperation + is said to be a canonical hypergroup if the following conditions hold: (i)for every 𝑥,𝑦∈𝐻, 𝑥+𝑦=𝑦+𝑥,(ii)for every 𝑥,𝑦,𝑧∈𝐻, 𝑥+(𝑦+𝑧)=(𝑥+𝑦)+𝑧,(iii)there exists 0∈𝐻 (called neutral element of 𝐻) such that 0+𝑥={𝑥}=𝑥+0 for all 𝑥∈𝐻,(iv)for every 𝑥∈𝐻, there exists a unique element denoted by −𝑥∈𝐻 such that 0∈𝑥+(−𝑥)∩(−𝑥)+𝑥,(v)for every 𝑥,𝑦,𝑧∈𝐻, 𝑧∈𝑥+𝑦 implies 𝑦∈−𝑥+𝑧 and 𝑥∈𝑧−𝑦.

Example 2.2. Consider the set 𝐻={0,𝑥,𝑦}. Define a hyperaddition + on 𝐻 as in the following table. Then, (𝐻,+) is a canonical hypergroup. +0𝑥𝑦𝑥00𝑥𝑦𝑦𝑥{0,𝑥}𝑦𝑦𝑦{0,𝑥,𝑦}(2.1)

The following elementary facts in a canonical hypergroup 𝐻 easily follow from the axioms.(i)−(−𝑎)=𝑎 for every 𝑎∈𝐻,(ii)0 is the unique element such that for every 𝑎∈𝐻, there is an element −𝑎∈𝐻 with the property 0∈𝑎+(−𝑎),(iii)−0=0, (iv)−(𝑎+𝑏)=−𝑎−𝑏 for all 𝑎,𝑏∈𝐻.

For any subset 𝐴 of a canonical hypergroup 𝐻, −𝐴 denotes the set {−𝑎∶𝑎∈𝐴}. A nonempty subset 𝑁 of a canonical hypergroup of 𝐻 is called a subcanonical hypergroup of 𝐻 if 𝑁 is a canonical hypergroup under the same hyperoperation as that of 𝐻. Equivalently, for every 𝑥,𝑦∈𝑁, 𝑥−𝑦⊆𝑁. In particular, for any 𝑥∈𝑁, 𝑥−𝑥⊆𝑁. Since 0∈𝑥−𝑥, it follows that 0∈𝑁.

Definition 2.3. An equivalence relation 𝜌 defined on a canonical hypergroup (𝐻,+) is called strongly regular if for all 𝑥,𝑦∈𝐻 and 𝑥𝜌𝑦 implies that for every 𝑝∈𝐻, for every 𝑎∈𝑥+𝑝 and for every 𝑏∈𝑦+𝑝 one has 𝑎𝜌𝑏.

Definition 2.4. A subcanonical hypergroup 𝐴 of a canonical hypergroup 𝐻 is said to be normal if 𝑥+𝐴−𝑥⊆𝐴 for all 𝑥∈𝐴.

Definition 2.5. The heart of a canonical hypergroup 𝐻 is the union of the sums (𝑥₁−𝑥₁)+(𝑥₂−𝑥₂)+(𝑥₃−𝑥₃)+⋯+(𝑥_𝑛−𝑥_𝑛), where 𝑥_𝑖∈𝐻 and 𝑛 is a natural number and it is denoted by 𝜔_𝐻.

Definition 2.6. Let 𝐻₁ and 𝐻₂ be two canonical hypergroups. A mapping 𝜙 from 𝐻₁ into 𝐻₂ is called a homomorphism from 𝐻₁ into 𝐻₂ if (i) 𝜙(𝑎+𝑏)⊆𝜙(𝑎)+𝜙(𝑏) for all 𝑎,𝑏∈𝐻₁ and (ii) 𝜙(0)=0 hold. The mapping 𝜙 is called a good or strong homomorphism if (i) 𝜙(𝑎+𝑏)=𝜙(𝑎)+𝜙(𝑏) for all 𝑎,𝑏∈𝐻₁ and (ii) 𝜙(0)=0 hold.

A homomorphism (resp., strong homomorphism) 𝜙 from a canonical hypergroup 𝐻₁ to a canonical hypergroup 𝐻₂ is called an isomorphism (resp., strong isomorphism) if 𝜙 is one to one and onto. If 𝐻₁ is strongly isomorphic to 𝐻₂, then we denote it by 𝐻₁≅𝐻₂.

Definition 2.7. Let 𝜙 be a homomorphism from canonical hypergroup 𝐻₁ into a canonical hypergroup 𝐻₂. Then, the set {𝑥∈𝐻₁∶𝜙(𝑥)=0} is called kernel of 𝜙 and is denoted by Ker𝜙, and the set {𝜙(𝑥)∶𝑥∈𝐻₁} is called Image of 𝜙 and is denoted by Im𝜙.

It is clear that Ker𝜙 is a subcanonical hypergroup of 𝐻₁ and Im𝜙 is a subcanonical hypergroup of 𝐻₂. The definition of a hyperring given below is equivalent to one formulated by De Salvo [19] (see Corsini [3]) and studied by Barghi [20].

Definition 2.8. A hyperring is a triple (𝑅,+,⋅), where 𝑅 is a nonempty set with a hyperaddition + and a hypermultiplication ⋅ satisfying the following axioms: (1)(𝑅,+) is a canonical hypergroup,(2)(𝑅,⋅) is a semihypergroup such that 𝑥⋅0=0⋅𝑥=0 for all 𝑥∈𝑅, (i.e, 0 is a bilaterally absorbing element), (3)The hypermultiplication ⋅ is distributive with respect to the hyperoperation +. That is, for every 𝑥,𝑦,𝑧∈𝑅, 𝑥⋅(𝑦+𝑧)=𝑥⋅𝑦+𝑥⋅𝑧 and (𝑥+𝑦)⋅𝑧=𝑥⋅𝑧+𝑦⋅𝑧.
In a hyperring if the hypermultiplication is a binary operation, then it is called as Krasner or additive hyperring. Also, in the Definition 2.8, if the hyperaddition is a binary operation, then it is called as multiplicative hyperring.

Example 2.9. Let 𝑅={0,1} be a set with two hyperoperations defined as follows: +0101{0}{1}{1}{0,1}⋅0101{0}{0}{0}{0,1}(2.2)

Then, (𝑅,+,⋅) is a hyperring.

Definition 2.10. Let 𝑅 be a hyperring, and let 𝐼 be a nonempty subset of 𝑅. 𝐼 is called a left (resp., right) hyperideal of 𝑅 if (𝐼,+) is a canonical subhypergroup of 𝑅 and for every 𝑎∈𝐼 and 𝑟∈𝑅, 𝑟𝑎⊆𝐼 (resp., 𝑎𝑟⊆𝐼). A hyperideal of 𝑅 is one which is a left as well as a right hyperideal of 𝑅.

If 𝐼,𝐽 are left (resp., right) hyperideals of a hyperring 𝑅, then 𝐼+𝐽, 𝐼∩𝐽 are left (resp., right) hyperideal of 𝑅. If 𝐼,𝐽 are hyperideals of a hyperring 𝑅, then 𝐼+𝐽, 𝐼∩𝐽 are hyperideals of 𝑅.

Definition 2.11. Let 𝑅₁ and 𝑅₂ be two hyperrings. A mapping 𝜙 from 𝑅₁ into 𝑅₂ is called a homomorphism if (i) 𝜙(𝑎+𝑏)⊆𝜙(𝑎)+𝜙(𝑏); (ii) 𝜙(𝑎𝑏)⊆𝜙(𝑎)𝜙(𝑏) and (iii) 𝜙(0)=0 hold for all 𝑎,𝑏∈𝑅₁. The mapping 𝜙 is called a good homomorphism or a strong homomorphism if (i) 𝜙(𝑎+𝑏)=𝜙(𝑎)+𝜙(𝑏); (ii) 𝜙(𝑎𝑏)=𝜙(𝑎)𝜙(𝑏) and (iii) 𝜙(0)=0 hold for all 𝑎,𝑏∈𝑅₁.

Definition 2.12. A homomorphism (resp., strong homomorphism) 𝜙 from hyperring 𝑅₁ into a hyperring 𝑅₂ is said to be an isomorphism (resp., strong isomorphism) if 𝜙 is one to one and onto. If 𝑅₁ is strongly isomorphic to 𝑅₂, then it is denoted by 𝑅₁≅𝑅₂.

Remark 2.13. Let 𝜙 be a homomorphism from a hyperring 𝑅₁ into a hyperring 𝑅₂. Then Ker𝜙 is a hyperideal of 𝑅₁ and Im𝜙 is a hyperideal of 𝑅₂.

3. Canonical Hypergroups

Let 𝑁 be a subcanonical hypergroup of a canonical hypergroup 𝐻. In this section, we construct quotient canonical hypergroup 𝐻/𝑁 and prove that when 𝑁 is normal, 𝐻/𝑁 is an abelian group.

Proposition 3.1. Let 𝐻 be a canonical hypergroup, and let 𝑁 be a subcanonical hypergroup of 𝐻. For any two elements 𝑎,𝑏∈𝐻, if we define a relation 𝑎∼𝑏 if 𝑎∈𝑏+𝑁, then ~ is an equivalence relation on 𝐻.

Proof. Let 𝑎∈𝐻. Since 𝑎=𝑎+0∈𝑎+𝑁, the relation ~ is reflexive. Let 𝑎,𝑏∈𝐻. If 𝑎∈𝑏+𝑁, then 𝑎∈𝑏+𝑛 for some 𝑛∈𝑁. That is, 𝑏∈𝑎−𝑛⊆𝑎+𝑁. So, ~ is a symmetric relation. Suppose that 𝑎,𝑏,𝑐∈𝐻 such that 𝑎∼𝑏 and 𝑏∼𝑐, then 𝑎∈𝑏+𝑁 and 𝑏∈𝑐+𝑁. Therefore, 𝑎∈𝑏+𝑛, and 𝑏∈𝑐+𝑚, for some 𝑛,𝑚∈𝑁. So, 𝑎∈𝑐+𝑚+𝑛⊆𝑐+𝑁. Hence 𝑎∼𝑐. Therefore, the relation ~ is transitive.

Remark 3.2. Let 𝑁 be a subcanonical hypergroup of a canonical hypergroup 𝐻. We denote the equivalence class determined by the element 𝑥∈𝐻 by the equivalence relation ~ by 𝑥. It is clear that 𝑥=𝑥+𝑁.

Proposition 3.3. Let 𝐻 be a canonical hypergroup, and let 𝑁 be a normal subcanonical hypergroup of 𝐻. Then, for 𝑥,𝑦∈𝑁, the following are equivalent: (1)𝑦∈𝑥+𝑁, (2)𝑥−𝑦⊆𝑁, (3)(𝑥−𝑦)∩𝑁≠∅.

Proof. (1) implies (2).
Since 𝑦∈𝑥+𝑁, we have 𝑦−𝑥⊆𝑥+𝑁−𝑥. Since 𝑁 is normal subcanonical hypergroup of 𝐻, we get 𝑥+𝑁−𝑥⊆𝑁. Thus, 𝑦−𝑥⊆𝑁. That is, −(𝑦−𝑥)⊆𝑁, and hence 𝑥−𝑦⊆𝑁.
(2) implies (3) is obvious.
(3) implies  (1). Since (𝑥−𝑦)∩𝑁≠∅, there exists 𝑎∈𝑥−𝑦 and 𝑎∈𝑁. Therefore, −𝑦+𝑥⊆−𝑦+𝑎+𝑦⊆𝑁. If 𝑧∈−𝑦+𝑥, then 𝑧∈𝑁. Therefore, −𝑦∈𝑧−𝑥. That is, 𝑦∈𝑥−𝑧⊆𝑥+𝑁.

Remark 3.4. Let 𝐻 be a canonical hypergroup, and let 𝑁 be a subcanonical hypergroup of 𝐻. When 𝑁 is normal, the equivalence relation defined in the Proposition 3.1 coincides with the the equivalence relation defined by Davvaz [12]. Further, the Propositions 3.1 and 3.3 are true when the hyperaddition on the canonical hypergroup 𝐻 is not commutative. Also, for any 𝑥∈𝐻, we have (−𝑥)=−(𝑥).

Theorem 3.5. Let 𝐻 be a canonical hypergroup, 𝑁 be a subcanonical hypergroup of 𝐻. Then for 𝑥,𝑦∈𝐻, the sets 𝐴={𝑧∶𝑧∈𝑥+𝑦}, 𝐵={𝑧∶𝑧∈𝑥+𝑦} and 𝐶={𝑧∶𝑧⊆𝑥+𝑦} are equal.

Proof. Let 𝑧∈𝐴. Then 𝑧∈𝑥+𝑦. Since 𝑥∈𝑥 and 𝑦∈𝑦 we have 𝑧∈𝑥+𝑦. Thus 𝐴⊆𝐵. Suppose 𝑧∈𝐵, then 𝑧∈𝑥+𝑦. That is, 𝑧∈𝑡+𝑛 for some 𝑡∈𝑥+𝑦 and 𝑛∈𝑁. Therefore 𝑧=𝑡, where 𝑡∈𝑥+𝑦. Since 𝑡∈𝐴, we get 𝑧∈𝐴. Thus 𝐵⊆𝐴. Hence 𝐴=𝐵.
If 𝑧∈𝐴, then 𝑧∈𝑥+𝑦. Therefore, 𝑧⊆𝑥+𝑦+𝑁=𝑥+𝑁+𝑦+𝑁=𝑥+𝑦. Hence 𝐴⊆𝐶. On the other hand if 𝑧∈𝐶, then 𝑧⊆𝑥+𝑦. Since 𝑧∈𝑧⊆𝑥+𝑦, we get 𝑧∈𝑠+𝑛 for some 𝑠∈𝑥+𝑦 and 𝑛∈𝑁. Thus 𝑧=𝑠. Since 𝑠∈𝐴, we get 𝐶⊆𝐴. Hence 𝐴=𝐶.

Remark 3.6. Let 𝐻 be a canonical hypergroup, and let 𝑁 be a subcanonical hypergroup of 𝐻. Then, we denote the collection of all equivalence classes {𝑥∶𝑥∈𝐻} induced by the equivalence relation ~ by 𝐻/𝑁.

Theorem 3.7. Let 𝐻 be a canonical hypergroup, and let 𝑁 be a subcanonical hypergroup of 𝐻. If we define 𝑥⊕𝑦={𝑧∶𝑧∈𝑥+𝑦} for all 𝑥,𝑦∈𝐻/𝑁, then 𝐻/𝑁 is a canonical hypergroup.

Proof. If 𝑥₁,𝑦₁,𝑥₂,𝑦₂∈𝐻 such that 𝑥₁=𝑥₂ and 𝑦₁=𝑦₂, then 𝑥₂∈𝑥₁+𝑁 and 𝑦₂∈𝑦₁+𝑁. Let 𝑧₂∈𝑥₂+𝑦₂⊆(𝑥₁+𝑁)+(𝑦₁+𝑁). Since 𝐻 is commutative, 𝑧₂∈𝑧₁+𝑖 for some 𝑧₁∈𝑥₁+𝑦₁ and for some 𝑖∈𝑁. That is, 𝑧₂+𝑁=𝑧₁+𝑁. Hence, 𝑥₂⊕𝑦₂⊆𝑥₁⊕𝑦₁. Also, since 𝑥₁∈𝑥₂+𝑁 and 𝑦₁∈𝑦₂+𝑁, by a similar argument, we get, 𝑥₁⊕𝑦₁⊆𝑥₂⊕𝑦₂. Hence, 𝑥₁⊕𝑦₁=𝑥₂⊕𝑦₂. Thus, hyperaddition ⊕ is well defined.
Let 𝑥,𝑦,𝑧∈𝐻/𝑁. If 𝑢∈(𝑥⊕𝑦)⊕𝑧, then 𝑢∈𝑝⊕𝑧 for some 𝑝∈𝑥⊕𝑦. That is, 𝑢=𝑎 for some 𝑎∈𝑝+𝑧. Also, 𝑝=𝑏 for some 𝑏∈𝑥+𝑦. Now, 𝑎∈𝑝+𝑧⊆𝑏+𝑁+𝑧=𝑏+𝑧+𝑁. That is, 𝑎∈𝑣+𝑁 for some 𝑣∈𝑏+𝑧⊆(𝑥+𝑦)+𝑧=𝑥+(𝑦+𝑧). So, 𝑣∈𝑥+𝑡 for some 𝑡∈𝑦+𝑧. This means that 𝑎=𝑣 and 𝑣∈𝑥⊕𝑡. Since 𝑡∈𝑦⊕𝑧, we have 𝑢=𝑎=𝑣∈𝑥⊕𝑡⊆𝑥⊕(𝑦⊕𝑧). This means that 𝑢∈𝑥⊕(𝑦⊕𝑧). Hence (𝑥⊕𝑦)⊕𝑧⊆𝑥⊕(𝑦⊕𝑧). Similarly, we get 𝑥⊕(𝑦⊕𝑧)⊆(𝑥⊕𝑦)⊕𝑧. Hence, 𝑥⊕(𝑦⊕𝑧)=(𝑥⊕𝑦)⊕𝑧. Thus, the hyperaddition is associative.
Consider the element 0=0+𝑁∈𝐻/𝑁. Now, for any 𝑥∈𝐻, we have 𝑥⊕0={𝑧∶𝑧∈𝑥+0}=𝑥. Similarly, 0⊕𝑥=𝑥. Thus, 0 is the zero element of 𝐻/𝑁.
Let 𝑥∈𝐻, then 𝑥⊕(−𝑥)={𝑧∶𝑧∈𝑥+(−𝑥)=𝑥−𝑥}. Since 0∈𝑥−𝑥, we get 0∈𝑥⊕(−𝑥). Similarly, we can show that 0∈(−𝑥)⊕𝑥. Let 𝑥∈𝐻/𝑁, and suppose that 𝑦∈𝐻/𝑁 is such that 0∈𝑦⊕𝑥, then 0=𝑎, where 𝑎∈𝑦+𝑥. That is, 𝑦∈𝑎−𝑥⊆𝑁−𝑥, and hence 𝑦=−𝑥. Thus, the element 𝑥∈𝐻/𝑁 has a unique inverse −𝑥∈𝐻/𝑁.
Suppose that 𝑧∈𝑥⊕𝑦, then 𝑧=𝑎, where 𝑎∈𝑥+𝑦. This implies 𝑥∈𝑎−𝑦⊆𝑧+𝑁−𝑦. That is, 𝑥∈𝑟+𝑁, where 𝑟∈𝑧−𝑦. Thus, 𝑥=𝑟∈𝑧⊕(−𝑦). Similarly, we can show 𝑦∈(−𝑥)⊕𝑧. Since 𝐻 is commutative, it is obvious that 𝐻/𝑁 is also commutative. Thus, 𝐻/𝑁 is a canonical hypergroup.

Corollary 3.8. Let 𝜙 be a strong homomorphism from canonical hypergroup 𝐻₁ into a canonical hypergroup 𝐻₂, then 𝐻₁/Ker𝜙 is a canonical hypergroup.

Remark 3.9. Let 𝐻 be a canonical hypergroup, and let 𝐴 be a subcanonical hypergroup of 𝐻. We denote the subset {𝑥∈𝐻∶𝑥−𝑥⊆𝐴} of 𝐻 by 𝑆_𝐴.

Proposition 3.10. Let 𝐻 be a canonical hypergroup, and let 𝐴 be a subcanonical hypergroup of 𝐻. Then, 𝑆_𝐴 is a subcanonical hypergroup of 𝐻 containing 𝐴.

Proof. Let 𝑥∈𝐴. Since 𝐴 is a subcanonical hypergroup of 𝐻, 𝑥−𝑥⊆𝐴. This implies 𝑥∈𝑆_𝐴. Therefore, 𝐴⊆𝑆_𝐴. Since 𝐴≠∅, the set 𝑆_𝐴 is nonempty.
Let 𝑥,𝑦∈𝑆_𝐴. For 𝑟∈𝑥−𝑦, we get 𝑟−𝑟⊆(𝑥−𝑦)−(𝑥−𝑦)=(𝑥−𝑥)+(𝑦−𝑦)⊆𝐴+𝐴=𝐴. Hence, 𝑟∈𝑆_𝐴. That is, 𝑥−𝑦⊆𝑆_𝐴. Therefore, 𝑆_𝐴 is a subcanonical hypergroup of 𝐻 containing 𝐴.

Definition 3.11. Let (𝐻,+) be a canonical hypergroup, and let 𝐴 be a subcanonical hypergroup of 𝐻.  𝐴 is called a subgroup of 𝐻 if (𝐴,+) is a group. That is, 𝑥+𝑦 is a singleton set for all 𝑥,𝑦∈𝐴.

Example 3.12. The set 𝐻={0,𝑎,𝑏,𝑐} with the following hyperoperation + is a canonical hypergroup +0𝑎𝑏𝑐𝑎0{0}{𝑎}{𝑏}{𝑐}𝑏{𝑎}{0,𝑏}{𝑎,𝑐}{𝑏}𝑐{𝑏}{𝑎,𝑐}{0,𝑏}{𝑎}{𝑐}{𝑏}{𝑎}{0}(3.1)

In this example {0,𝑐},{0} are subgroups of 𝐻 and 𝜔_𝐻={0,𝑏} whereas in the Example 2.2, {0} is the subgroup of 𝐻 and 𝜔_𝐻=𝐻.

Proposition 3.13. Let 𝐻 be a canonical hypergroup. Then, 𝑆_{0} is the subgroup of 𝐻 containing all subgroups of 𝐻.

Proof. By the Proposition 3.10, 𝑆_{0} is the subcanonical hypergroup of 𝐻. Let 𝑥,𝑦∈𝑆_{0}. Consider the set 𝑥+𝑦. If 𝑢,𝑣∈𝑥+𝑦, then 𝑢−𝑣⊆(𝑥+𝑦)−(𝑥+𝑦)=(𝑥−𝑥)+(𝑦−𝑦)=0+0=0. Hence, 𝑢=𝑣. This means that the set 𝑥+𝑦 has only one element. Thus, 𝑆_{0} is a subgroup of 𝐻. Suppose, 𝐴 is any subgroup of 𝐻, then for any 𝑥∈𝐴 that we have 𝑥−𝑥=0. That is, 𝑥∈𝑆_{0}. Hence, 𝐴⊆𝑆_{0}. Thus, 𝑆_{0} contains all subgroups of 𝐻.

Corollary 3.14. Let 𝐻 be a canonical hypergroup. Then, 𝐻 is an abelian group if and only if 𝑆_{0}=𝐻.

Proposition 3.15. Let 𝐻 be a canonical hypergroup, and let 𝐴 be a subcanonical hypergroup of 𝐻. Then, 𝐴 is normal if and only if 𝑆_𝐴=𝐻.

Proof. Let 𝐴 be normal. Then, for 𝑥∈𝐻, 𝑥+0−𝑥⊆𝐴. That is, 𝑥∈𝑆_𝐴. Hence, 𝑆_𝐴=𝐻. Conversely, if 𝑆_𝐴=𝐻, then for 𝑥∈𝐻, we get 𝑥+𝐴−𝑥=𝑥−𝑥+𝐴⊆𝐴+𝐴=𝐴. Thus, 𝐴 is normal.

Proposition 3.16. The heart 𝜔_𝐻 of a canonical hypergroup 𝐻 is a normal subcanonical hypergroup of 𝐻.

Proof. If 𝑥,𝑦∈𝜔_𝐻, then 𝑥∈(𝑥₁−𝑥₁)+(𝑥₂−𝑥₂)+(𝑥₃−𝑥₃)+⋯+(𝑥_𝑛−𝑥_𝑛) and 𝑦∈(𝑦₁−𝑦₁)+(𝑦₂−𝑦₂)+(𝑦₃−𝑦₃)+⋯+(𝑦_𝑚−𝑦_𝑚), where 𝑥_𝑖,𝑦_𝑗∈𝐻 and 𝑚,𝑛 are natural numbers. Thus 𝑥−𝑦∈(𝑥₁−𝑥₁)+(𝑥₂−𝑥₂)+(𝑥₃−𝑥₃)+⋯+(𝑥_𝑛−𝑥_𝑛)+(𝑦₁−𝑦₁)+(𝑦₂−𝑦₂)+(𝑦₃−𝑦₃)+⋯+(𝑦_𝑚−𝑦_𝑚)⊆𝜔_𝐻. Now, for any element ℎ∈𝜔_𝐻, there exists natural number 𝑛 and elements 𝑥_𝑖∈𝐻 such that ℎ∈(𝑥₁−𝑥₁)+(𝑥₂−𝑥₂)+(𝑥₃−𝑥₃)+⋯+(𝑥_𝑛−𝑥_𝑛). Then, for any 𝑥∈𝐻, 𝑥+ℎ−𝑥=𝑥−𝑥+ℎ⊆𝑥−𝑥+(𝑥₁−𝑥₁)+(𝑥₂−𝑥₂)+(𝑥₃−𝑥₃)+⋯+(𝑥_𝑛−𝑥_𝑛)⊆𝜔_𝐻. Hence, heart 𝜔_𝐻 is a normal subcanonical hypergroup of 𝐻.

Proposition 3.17. A subcanonical hypergroup 𝐴 of a canonical hypergroup 𝐻 is normal if and only if 𝐴 contains the heart 𝜔_𝐻 of the canonical hypergroup 𝐻.

Proof. Let 𝐴 be a normal subcanonical hypergroup of the canonical hypergroup 𝐻. Then 𝑥+𝑖−𝑥⊆𝐴 for every 𝑥∈𝐻, and 𝑖∈𝐴. In particular, when 𝑖=0∈𝐴, we get 𝑥−𝑥⊆𝐴 for every 𝑥∈𝐻. Since 𝐴 is a subcanonical hypergroup of 𝐻, the union of the sums (𝑥₁−𝑥₁)+(𝑥₂−𝑥₂)+(𝑥₃−𝑥₃)+⋅⋅⋅+(𝑥_𝑛−𝑥_𝑛)⊆𝐴 for 𝑥_𝑖∈𝐻 and 𝑛 is a natural number. That is, 𝜔_𝐻⊆𝐴. Conversely, assume that subcanonical hypergroup 𝐴 contains the heart 𝜔_𝐻 of the canonical hypergroup 𝐻. For 𝑥∈𝐻 and 𝑖∈𝐴, 𝑥+𝑖−𝑥=𝑥−𝑥+𝑖⊆𝜔_𝐻+𝐴⊆𝐴+𝐴=𝐴. Hence, 𝐴 is a normal subcanonical hypergroup.

From Propositions 3.16 and 3.17, we have the following proposition.

Proposition 3.18. In a canonical hypergroup 𝐻, 𝜔_𝐻 is the smallest normal subcanonical hypergroup.

Proposition 3.19. Let 𝐴,𝐵 be subcanonical hypergroups of a canonical hypergroup 𝐻 such that 𝐴⊆𝐵, then 𝑆_𝐴⊆𝑆_𝐵.

Proof. Let 𝑥∈𝑆_𝐴. Then, 𝑥−𝑥⊆𝐴. That is, 𝑥∈𝑆_𝐵. Hence, 𝑆_𝐴⊆𝑆_𝐵.

Proposition 3.20. Let 𝐴,𝐵 be subcanonical hypergroups of a canonical hypergroup 𝐻 such that 𝐴⊆𝐵. If 𝐴 is normal, then 𝐵 is also normal.

Proof. If 𝐴 is normal, then by Proposition 3.15, 𝑆_𝐴=𝐻. Since 𝐴⊆𝐵, by Proposition 3.19, 𝑆_𝐴⊆𝑆_𝐵. Hence, 𝐻=𝑆_𝐵. By Proposition 3.15, 𝐵 is normal.

Corollary 3.21. Let 𝐴,𝐵 be subcanonical hypergroups of a canonical hypergroup 𝐻 such that 𝐴 is normal, then the subcanonical hypergroup 𝐴+𝐵 is also normal.

Corollary 3.22. Let 𝐻 be a canonical hypergroup such that (0) is normal, then all the subcanonical hypergroups are normal.

Theorem 3.23. Let 𝐻 be a canonical hypergroup. Then, the following are equivalent: (i)𝐻 is an abelian group,(ii)(0) is a normal subcanonical hypergroup of 𝐻,(iii)𝜔_𝐻=(0).

Proof. By Corollary 3.14, a canonical hypergroup 𝐻 is an abelian group if and only if 𝑆_{0}=𝐻. By Proposition 3.15, 𝑆_{0}=𝐻 if and only if (0) is a normal subcanonical hypergroup of 𝐻. Hence, a canonical hypergroup 𝐻 is an abelian group if and only if (0) is a normal subcanonical hypergroup of 𝐻.
By Proposition 3.18, 𝜔_𝐻 is the smallest normal subcanonical hypergroup of 𝐻. Therefore, (0) is normal if and only if 𝜔_𝐻=(0).

Corollary 3.24. 𝐻 is an abelian group if and only if all subcanonical hypergroups of 𝐻 are normal.

Theorem 3.25. Let 𝐻 be a canonical hypergroup, and let 𝑁 be a normal subcanonical hypergroup of 𝐻. Then, 𝐻/𝑁 is an abelian group.

Proof. For the quotient canonical hypergroup 𝐻/𝑁, the zero element is 𝑁. Since (𝑥+𝑁)+𝑁+(−𝑥+𝑁)=(𝑥+𝑁−𝑥)+𝑁⊆𝑁+𝑁=𝑁 for all 𝑥∈𝐻, we have {𝑁} is a normal subcanonical hypergroup in 𝐻/𝑁. By Theorem 3.23, 𝐻/𝑁 is an abelian group.

Remark 3.26. If 𝑁 is a normal subcanonical hypergroup of a canonical hypergroup 𝐻, then the relation ~ defined in Proposition 3.1, is a strongly regular equivalence relation. Hence, by Theorem  31 in [3], 𝐻/𝑁 is an abelian group. However, we have proved Theorem 3.25 in a different way.

4. Isomorphism Theorems of Canonical Hypergroups

In this section, we prove the isomorphism theorems of canonical hypergroups.

Theorem 4.1 (First Isomorphism Theorem). Let 𝜙 be a strong homomorphism from a canonical hypergroup 𝐻₁ into a canonical hypergroup 𝐻₂ with kernel 𝐾. Then, 𝐻₁/𝐾 is strongly isomorphic to Im𝜙.

Proof. Define a map 𝑓∶𝐻₁/𝐾→Im𝜙 by 𝑓(𝑥)=𝜙(𝑥) for all 𝑥∈𝐻₁. Suppose that 𝑥=𝑦, where 𝑥,𝑦∈𝐻, then 𝑥∈𝑦. That is, 𝑥∈𝑦+𝑘 for some 𝑘∈𝐾. Hence, 𝜙(𝑥)∈𝜙(𝑦+𝑘)=𝜙(𝑦)+𝜙(𝑘)=𝜙(𝑦)+0=𝜙(𝑦). So 𝜙(𝑥)=𝜙(𝑦). Hence, 𝑓(𝑥)=𝑓(𝑦). Thus, the map 𝑓 is well defined.
If 𝑥,𝑦∈𝐻₁, then 𝑓𝑥⊕𝑦==𝑓𝑓𝑧∶𝑧∈𝑥+𝑦={𝜙(𝑧𝑧)∶∶𝑧𝑧∈∈𝑥𝑥++𝑦𝑦}.(4.1)Also,𝑓𝑥+𝑓𝑦===𝜙𝜙{𝜙((𝑥𝑥(𝑧)+)+∶𝑦𝜙)𝑧(∈𝑦)𝑥+𝑦}.(4.2) Thus, 𝑓(𝑥⊕𝑦)=𝑓(𝑥)+𝑓(𝑦). Moreover, 𝑓(0)=𝜙(0)=0. Hence, 𝑓 is a strong homomorphism.
Suppose that 𝑥,𝑦∈𝐻₁/𝐾 such that 𝑓(𝑥)=𝑓(𝑦), then 𝜙(𝑥)=𝜙(𝑦). This means that 0∈𝜙(𝑥)−𝜙(𝑦)=𝜙(𝑥−𝑦). That is, 𝜙(𝑧)=0 for some 𝑧∈𝑥−𝑦. Since 𝜙(𝑧)=0, we get 𝑧∈𝐾. Now, 𝑧∈𝑥−𝑦⇒𝑥∈𝑧+𝑦⇒𝑥∈𝑦+𝐾. Then, by Proposition 3.3  𝑥=𝑦 and hence 𝑓 is one to one. Clearly, 𝑓 is onto. Thus, 𝑓 is a strong isomorphism. That is, 𝐻₁/𝐾 is strongly isomorphic to Im𝜙.

Corollary 4.2. Let 𝜙 be a strong homomorphism from a canonical hypergroup 𝐻₁ onto a canonical hypergroup 𝐻₂ with kernel 𝐾. Then, 𝐻₁/𝐾 is isomorphic to 𝐻₂.

Theorem 4.3 (Second Isomorphism Theorem). If 𝑀 and 𝑁 are subcanonical hypergroups of a canonical hypergroup 𝐻, then 𝑁/(𝑀∩𝑁)≅(𝑀+𝑁)/𝑀.

Proof. It is clear that we can consider the subcanonical hypergroup 𝑀+𝑁 of the canonical hypergroup 𝐻 as a canonical hypergroup 𝑀+𝑁 for which 𝑀 is a subcanonical hypergroup. Similarly, the subcanonical hypergroup 𝑁 of the canonical hypergroup 𝐻 as a canonical hypergroup 𝑁 for which (𝑀∩𝑁) is a subcanonical hypergroup.
Define 𝑔∶𝑁→(𝑀+𝑁)/𝑀 by 𝑔(𝑏)=𝑏+𝑀 for every 𝑏∈𝑁. For all 𝑎,𝑏∈𝑁, 𝑔(𝑎+𝑏)=𝑔({𝑥∶𝑥∈𝑎+𝑏})={𝑔(𝑥)∶𝑥∈𝑎+𝑏}={𝑥+𝑀∶𝑥∈𝑎+𝑏}=(𝑎+𝑀)⊕(𝑏+𝑀)=𝑔(𝑎)⊕𝑔(𝑏). Moreover, 𝑔(0)=0. Thus, 𝑔 is a strong homomorphism.
Now, 𝑥+𝑀∈(𝑀+𝑁)/𝑀 implies that 𝑥∈𝑦+𝑀 for some 𝑦∈𝑀+𝑁. That is, 𝑦∈𝑎+𝑏 for some 𝑎∈𝑀, 𝑏∈𝑁. Since 𝑦∈𝑏+𝑀, we get 𝑦+𝑀=𝑏+𝑀. Thus, 𝑔(𝑏)=𝑏+𝑀=𝑦+𝑀=𝑥+𝑀. Thus, 𝑔 is onto. Let 𝑏∈𝑁. Then, 𝑏∈Ker𝑔⇔𝑔(𝑏)=0⇔𝑏+𝑀=0+𝑀⇔𝑏∈𝑀. Thus, 𝑏∈Ker𝑔 if and only if 𝑏∈𝑀∩𝑁. Hence, by the First Isomorphism Theorem, 𝑁/(𝑀∩𝑁)≅(𝑀+𝑁)/𝑀.

Theorem 4.4 (Third Isomorphism Theorem). If 𝑀 and 𝑁 are subcanonical hypergroup of a canonical hypergroup 𝐻 such that 𝑀⊆𝑁, then 𝐻/𝑁≅(𝐻/𝑀)/(𝑁/𝑀).

Proof. Define a map ℎ∶𝐻/𝑁→𝐻/𝑀 by ℎ(𝑥+𝑁)=𝑥+𝑀. Then, ℎ is a strong onto homomorphism of canonical hypergroup with kernel 𝑁/𝑀. Therefore, by the First Isomorphism Theorem of canonical hypergroups, 𝐻/𝑁≅(𝐻/𝑀)/(𝑁/𝑀).

5. Isomorphism Theorems of Hyperrings

Let 𝑅 be a hyperring, and let 𝐼 be a hyperideal of 𝑅. Since 𝐼 is a subcanonical hypergroup of 𝑅,𝑅/𝐼={𝑥∶𝑥∈𝑅} is a canonical hypergroup under the hyperaddition defined in the Theorem 3.7. In this section, we define a hypermultiplication on 𝑅/𝐼 and prove that 𝑅/𝐼 is a hyperring.

Theorem 5.1. If we define 𝑥⊗𝑦={𝑧∶𝑧∈𝑥𝑦} for all 𝑥,𝑦∈𝑅/𝐼, then 𝑅/𝐼 is a hyperring.

Proof. If 𝑥₁,𝑦₁,𝑥₂,𝑦₂∈𝑅 such that 𝑥₁=𝑥₂ and 𝑦₁=𝑦₂, then 𝑥₂∈𝑥₁+𝐼 and 𝑦₂∈𝑦₁+𝐼. Let 𝑧₂∈𝑥₂𝑦₂⊆(𝑥₁+𝐼)(𝑦₁+𝐼)⊆𝑥₁𝑦₁+𝐼. Then, 𝑧₂∈𝑧₁+𝑖 for some 𝑧₁∈𝑥₁𝑦₁ and for some 𝑖∈𝐼. That is, 𝑧₂+𝐼=𝑧₁+𝐼 and so 𝑥₂⊗𝑦₂⊆𝑥₁⊗𝑦₁. Similarly, we get, 𝑥₁⊗𝑦₁⊆𝑥₂⊕𝑦₂. Hence, 𝑥₁⊕𝑦₁=𝑥₂⊗𝑦₂. Thus, hypermultiplication ⊗ is well defined.
Suppose, 𝑥,𝑦,𝑧∈𝑅/𝐼. Then, 𝑥⊗𝑦⊗𝑧==𝑥⊗𝑎∶𝑎∈𝑦𝑧=𝑠∶𝑠∈𝑥𝑎,𝑎∈𝑦𝑧=𝑠∶𝑠∈𝑥(𝑦𝑧)=𝑠∶𝑠∈(𝑥𝑦)𝑧=𝑠∶𝑠∈𝑏𝑧,𝑏∈𝑥𝑦=𝑠∶𝑠∈𝑥𝑦⊗𝑧𝑥⊗𝑦⊗𝑧(5.1) Thus, we get 𝑥⊗(𝑦⊗𝑧)=(𝑥⊗𝑦)⊗𝑧. Hence, hypermultiplication is associative. Further, 𝑥⊗𝑦⊕𝑧====𝑥{{{𝑞𝑞𝑞⊗+++{𝐼𝐼𝐼𝑝∶∶∶+𝑞𝑞𝑞𝐼∈∈∈∶𝑥𝑥𝑥𝑝𝑝𝑦(,𝑦∈+𝑝+𝑦𝑥∈𝑧𝑧+)𝑦}}𝑧.+}𝑧}(5.2) Also, 𝑥⊗𝑦⊕𝑥⊗𝑧==={{{𝑐𝑐𝑎+++𝐼𝐼𝐼∶∶∶𝑐𝑐𝑎∈∈∈a𝑥𝑥𝑦+𝑦}+𝑏⊕,𝑥𝑎𝑧{}∈𝑏.𝑥+𝑦𝐼,𝑏∶∈𝑏𝑥∈𝑧}𝑥𝑧}(5.3) Hence, 𝑥⊗(𝑦⊕𝑧)=(𝑥⊗𝑦)⊕(𝑥⊗𝑧). Similarly, we can show that (𝑥⊕𝑦)⊗𝑧=(𝑥⊗𝑧)⊕(𝑥⊗𝑧). Therefore, hypermultiplication is distributive with respect to the hyperaddition. Thus, 𝑅/𝐼 is a hyperring.

Corollary 5.2. Let 𝜙 be a strong homomorphism from hyperring 𝑅₁ into a hyperring 𝑅₂, then 𝑅₁/Ker𝜙 is a hyperring.

Remark 5.3. If 𝑅 is a Krasner hyperring and 𝐼 is a hyperideal of 𝑅, then 𝑅/𝐼 is also a Krasner hyperring. Further if (𝐼,+) is a normal subcanonical hypergroup of 𝑅, then by the Theorems 3.23 and 5.1, 𝑅/𝐼 is a ring. Hence, the quotient hyperrings considered in [12] are just rings. So, in the isomorphism theorems proved in [12], all the quotient hyperrings considered are rings. However, we prove the isomorphism theorems of hyperrings in which the additions and the multiplications are hyperoperations.
If 𝑅 is a hyperring, and 𝐼 is a hyperideal of 𝑅, and (𝐼,+) is a normal subcanonical hypergroup of 𝑅, then 𝑅/𝐼 is a multiplicative hyperring.

Theorem 5.4 (First Isomorphism Theorem). Let 𝜙 be a strong homomorphism from a hyperring 𝑅₁ into a hyperring 𝑅₂ with kernel 𝐾. Then, 𝑅₁/𝐾 is strongly isomorphic to Im𝜙.

Proof. Define a map 𝑓∶𝑅₁/𝐾→Im𝜙 by 𝑓(𝑥)=𝜙(𝑥) for all 𝑥∈𝑅₁.
By Theorem 4.1, this map 𝑓 is a strong isomorphism from canonical hypergroup 𝑅₁/𝐾 onto Im𝜙. Now, 𝑓𝑥⊗𝑦==𝑓𝑓𝑧∶𝑧∈𝑥𝑦𝑓={𝜙(𝑧𝑧)∶∶𝑧𝑧∈∈𝑥𝑥𝑦𝑦},𝑥𝑓𝑦===𝜙𝜙{𝜙((𝑥𝑥(𝑧)𝑦))𝜙∶(𝑦𝑧)∈𝑥𝑦}.(5.4)
Thus, 𝑓(𝑥⊗𝑦)=𝑓(𝑥)𝑓(𝑦). Hence, 𝑓 is a strong hyperring isomorphism.

Corollary 5.5. Let 𝜙 be a strong homomorphism from a hyperring 𝑅₁ onto a hyperring 𝑅₂ with kernel 𝐾. Then, 𝑅₁/𝐾 is strongly isomorphic to 𝑅₂.

Theorem 5.6 (Second Isomorphism Theorem). If 𝐼 and 𝐽 are hyperideals of a hyperring 𝑅, then 𝐽/(𝐼∩𝐽)≅(𝐼+𝐽)/𝐼.

Proof. We can consider the hyperideal 𝐼+𝐽 of the hyperring 𝑅 as a hyperring 𝐼+𝐽 for which 𝐼 is a hyperideal. Similarly, hyperideal 𝐽 of the hyperring 𝑅 as a hyperring 𝐽 for which (𝐼∩𝐽) is a hyperideal.
Define 𝑔∶𝐽→(𝐼+𝐽)/𝐼 by 𝑔(𝑏)=𝑏+𝐼 for every 𝑏∈𝐽. By Theorem 4.3,𝑔 is strong isomorphism from canonical hypergroup 𝐽 onto the canonical hypergroup (𝐼+𝐽)/𝐼. Now, 𝑔(𝑎𝑏)=𝑔({𝑥∶𝑥∈𝑎𝑏})={𝑔(𝑥)∶𝑥∈𝑎𝑏}={𝑥+𝐼∶𝑥∈𝑎𝑏}=(𝑎+𝐼)(𝑏+𝐼)=𝑔(𝑎)𝑔(𝑏). Thus, 𝑔 is strong isomorphism from hyperring 𝐽 onto the hyperring (𝐼+𝐽)/𝐼. Also, from Theorem 4.3,Ker𝑔=𝐼∩𝐽. Hence, by First Isomorphism Theorem of hyperrings, 𝐽/(𝐼∩𝐽)≅(𝐼+𝐽)/𝐼.

Theorem 5.7 (Third Isomorphism Theorem). If 𝐼 and 𝐽 are hyperideals of a hyperring 𝑅 such that 𝐼⊆𝐽, then 𝑅/𝐽≅(𝑅/𝐼)/(𝐽/𝐼).

Proof. Define a map ℎ∶𝑅/𝐽→𝑅/𝑀 by ℎ(𝑥+𝐼)=𝑥+𝐽. Then, ℎ is a strong onto homomorphism of hyperring with kernel 𝐽/𝐼. Therefore, by the First Isomorphism Theorem of hyperrings, 𝑅/𝐽≅(𝑅/𝐼)/(𝐽/𝐼).