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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2010, Article IDΒ 376985, 12 pages
http://dx.doi.org/10.1155/2010/376985
Research Article

Note on Isomorphism Theorems of Hyperrings

Department of Mathematics, Aditanar College of Arts and Science, Tiruchendur 628216, Tamilnadu, India

Received 11 May 2010; Revised 19 October 2010; Accepted 29 December 2010

Academic Editor: HeinzΒ Gumm

Copyright Β© 2010 Muthusamy Velrajan and Arjunan Asokkumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

There are different notions of hyperrings (𝑅,+,β‹…). 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], DaΕ‘iΔ‡ [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,+,β‹…). If the addition + is a hyperoperation and the multiplication β‹… 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 β‹… 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., π‘₯βˆ˜π‘¦βŠ†π», for every π‘₯,π‘¦βˆˆπ»). 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 𝐡 of a hypergroupoid 𝐻 and for π‘₯∈𝐻, 𝐴∘𝐡, we mean the set β‹ƒπ‘Žβˆˆπ΄π‘βˆˆπ΅(π‘Žβˆ˜π‘), 𝐴∘π‘₯=𝐴∘{π‘₯}, and π‘₯∘𝐡={π‘₯}∘𝐡.

A hypergroupoid (𝐻,∘) is called a semihypergroup if π‘₯∘(π‘¦βˆ˜π‘§)=(π‘₯βˆ˜π‘¦)βˆ˜π‘§ for every π‘₯,𝑦,π‘§βˆˆπ» (the associative axiom). A hypergroupoid (𝐻,∘) is called a quasihypergroup if π‘₯∘𝐻=𝐻∘π‘₯=𝐻 for every π‘₯∈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 π‘₯,π‘¦βˆˆπ», π‘₯+𝑦=𝑦+π‘₯,(ii)for every π‘₯,𝑦,π‘§βˆˆπ», π‘₯+(𝑦+𝑧)=(π‘₯+𝑦)+𝑧,(iii)there exists 0∈𝐻 (called neutral element of 𝐻) such that 0+π‘₯={π‘₯}=π‘₯+0 for all π‘₯∈𝐻,(iv)for every π‘₯∈𝐻, there exists a unique element denoted by βˆ’π‘₯∈𝐻 such that 0∈π‘₯+(βˆ’π‘₯)∩(βˆ’π‘₯)+π‘₯,(v)for every π‘₯,𝑦,π‘§βˆˆπ», π‘§βˆˆπ‘₯+𝑦 implies π‘¦βˆˆβˆ’π‘₯+𝑧 and π‘₯βˆˆπ‘§βˆ’π‘¦.

Example 2.2. Consider the set 𝐻={0,π‘₯,𝑦}. Define a hyperaddition + on 𝐻 as in the following table. Then, (𝐻,+) is a canonical hypergroup. +0π‘₯𝑦0π‘₯0π‘₯𝑦𝑦π‘₯{0,π‘₯}𝑦𝑦𝑦{0,π‘₯,𝑦}(2.1)

The following elementary facts in a canonical hypergroup 𝐻 easily follow from the axioms.(i)βˆ’(βˆ’π‘Ž)=π‘Ž for every π‘Žβˆˆπ»,(ii)0 is the unique element such that for every π‘Žβˆˆπ», there is an element βˆ’π‘Žβˆˆπ» with the property 0βˆˆπ‘Ž+(βˆ’π‘Ž),(iii)βˆ’0=0, (iv)βˆ’(π‘Ž+𝑏)=βˆ’π‘Žβˆ’π‘ for all π‘Ž,π‘βˆˆπ».

For any subset 𝐴 of a canonical hypergroup 𝐻, βˆ’π΄ denotes the set {βˆ’π‘ŽβˆΆπ‘Žβˆˆπ΄}. 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 π‘₯,π‘¦βˆˆπ‘, π‘₯βˆ’π‘¦βŠ†π‘. In particular, for any π‘₯βˆˆπ‘, π‘₯βˆ’π‘₯βŠ†π‘. Since 0∈π‘₯βˆ’π‘₯, it follows that 0βˆˆπ‘.

Definition 2.3. An equivalence relation 𝜌 defined on a canonical hypergroup (𝐻,+) is called strongly regular if for all π‘₯,π‘¦βˆˆπ» and π‘₯πœŒπ‘¦ implies that for every π‘βˆˆπ», for every π‘Žβˆˆπ‘₯+𝑝 and for every π‘βˆˆπ‘¦+𝑝 one has π‘ŽπœŒπ‘.

Definition 2.4. A subcanonical hypergroup 𝐴 of a canonical hypergroup 𝐻 is said to be normal if π‘₯+π΄βˆ’π‘₯βŠ†π΄ for all π‘₯∈𝐴.

Definition 2.5. The heart of a canonical hypergroup 𝐻 is the union of the sums (π‘₯1βˆ’π‘₯1)+(π‘₯2βˆ’π‘₯2)+(π‘₯3βˆ’π‘₯3)+β‹―+(π‘₯π‘›βˆ’π‘₯𝑛), where π‘₯π‘–βˆˆπ» and 𝑛 is a natural number and it is denoted by πœ”π».

Definition 2.6. Let 𝐻1 and 𝐻2 be two canonical hypergroups. A mapping πœ™ from 𝐻1 into 𝐻2 is called a homomorphism from 𝐻1 into 𝐻2 if (i) πœ™(π‘Ž+𝑏)βŠ†πœ™(π‘Ž)+πœ™(𝑏) for all π‘Ž,π‘βˆˆπ»1 and (ii) πœ™(0)=0 hold. The mapping πœ™ is called a good or strong homomorphism if (i) πœ™(π‘Ž+𝑏)=πœ™(π‘Ž)+πœ™(𝑏) for all π‘Ž,π‘βˆˆπ»1 and (ii) πœ™(0)=0 hold.

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

Definition 2.7. Let πœ™ be a homomorphism from canonical hypergroup 𝐻1 into a canonical hypergroup 𝐻2. Then, the set {π‘₯∈𝐻1βˆΆπœ™(π‘₯)=0} is called kernel of πœ™ and is denoted by Kerπœ™, and the set {πœ™(π‘₯)∢π‘₯∈𝐻1} is called Image of πœ™ and is denoted by Imπœ™.

It is clear that Kerπœ™ is a subcanonical hypergroup of 𝐻1 and Imπœ™ 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,+,β‹…), where 𝑅 is a nonempty set with a hyperaddition + and a hypermultiplication β‹… satisfying the following axioms: (1)(𝑅,+) is a canonical hypergroup,(2)(𝑅,β‹…) is a semihypergroup such that π‘₯β‹…0=0β‹…π‘₯=0 for all π‘₯βˆˆπ‘…, (i.e, 0 is a bilaterally absorbing element), (3)The hypermultiplication β‹… is distributive with respect to the hyperoperation +. That is, for every π‘₯,𝑦,π‘§βˆˆπ‘…, π‘₯β‹…(𝑦+𝑧)=π‘₯⋅𝑦+π‘₯⋅𝑧 and (π‘₯+𝑦)⋅𝑧=π‘₯⋅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}β‹…{1}{0,1}0101{0}{0}{0}{0,1}(2.2)

Then, (𝑅,+,β‹…) 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 π‘ŽβˆˆπΌ and π‘Ÿβˆˆπ‘…, π‘Ÿπ‘ŽβŠ†πΌ (resp., π‘Žπ‘ŸβŠ†πΌ). 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 πœ™ from 𝑅1 into 𝑅2 is called a homomorphism if (i) πœ™(π‘Ž+𝑏)βŠ†πœ™(π‘Ž)+πœ™(𝑏); (ii) πœ™(π‘Žπ‘)βŠ†πœ™(π‘Ž)πœ™(𝑏) and (iii) πœ™(0)=0 hold for all π‘Ž,π‘βˆˆπ‘…1. The mapping πœ™ is called a good homomorphism or a strong homomorphism if (i) πœ™(π‘Ž+𝑏)=πœ™(π‘Ž)+πœ™(𝑏); (ii) πœ™(π‘Žπ‘)=πœ™(π‘Ž)πœ™(𝑏) and (iii) πœ™(0)=0 hold for all π‘Ž,π‘βˆˆπ‘…1.

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

Remark 2.13. Let πœ™ be a homomorphism from a hyperring 𝑅1 into a hyperring 𝑅2. Then Kerπœ™ is a hyperideal of 𝑅1 and Imπœ™ 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 π‘Ž,π‘βˆˆπ», if we define a relation π‘ŽβˆΌπ‘ if π‘Žβˆˆπ‘+𝑁, then ~ is an equivalence relation on 𝐻.

Proof. Let π‘Žβˆˆπ». Since π‘Ž=π‘Ž+0βˆˆπ‘Ž+𝑁, the relation ~ is reflexive. Let π‘Ž,π‘βˆˆπ». If π‘Žβˆˆπ‘+𝑁, then π‘Žβˆˆπ‘+𝑛 for some π‘›βˆˆπ‘. That is, π‘βˆˆπ‘Žβˆ’π‘›βŠ†π‘Ž+𝑁. So, ~ is a symmetric relation. Suppose that π‘Ž,𝑏,π‘βˆˆπ» such that π‘ŽβˆΌπ‘ and π‘βˆΌπ‘, then π‘Žβˆˆπ‘+𝑁 and π‘βˆˆπ‘+𝑁. Therefore, π‘Žβˆˆπ‘+𝑛, and π‘βˆˆπ‘+π‘š, for some 𝑛,π‘šβˆˆπ‘. So, π‘Žβˆˆπ‘+π‘š+π‘›βŠ†π‘+𝑁. Hence π‘ŽβˆΌπ‘. 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 π‘₯∈𝐻 by the equivalence relation ~ by π‘₯. It is clear that π‘₯=π‘₯+𝑁.

Proposition 3.3. Let 𝐻 be a canonical hypergroup, and let 𝑁 be a normal subcanonical hypergroup of 𝐻. Then, for π‘₯,π‘¦βˆˆπ‘, the following are equivalent: (1)π‘¦βˆˆπ‘₯+𝑁, (2)π‘₯βˆ’π‘¦βŠ†π‘, (3)(π‘₯βˆ’π‘¦)βˆ©π‘β‰ βˆ….

Proof. (1) implies (2).
Since π‘¦βˆˆπ‘₯+𝑁, we have π‘¦βˆ’π‘₯βŠ†π‘₯+π‘βˆ’π‘₯. Since 𝑁 is normal subcanonical hypergroup of 𝐻, we get π‘₯+π‘βˆ’π‘₯βŠ†π‘. Thus, π‘¦βˆ’π‘₯βŠ†π‘. That is, βˆ’(π‘¦βˆ’π‘₯)βŠ†π‘, and hence π‘₯βˆ’π‘¦βŠ†π‘.
(2) implies (3) is obvious.
(3) implies  (1). Since (π‘₯βˆ’π‘¦)βˆ©π‘β‰ βˆ…, there exists π‘Žβˆˆπ‘₯βˆ’π‘¦ and π‘Žβˆˆπ‘. Therefore, βˆ’π‘¦+π‘₯βŠ†βˆ’π‘¦+π‘Ž+π‘¦βŠ†π‘. If π‘§βˆˆβˆ’π‘¦+π‘₯, then π‘§βˆˆπ‘. Therefore, βˆ’π‘¦βˆˆπ‘§βˆ’π‘₯. That is, π‘¦βˆˆπ‘₯βˆ’π‘§βŠ†π‘₯+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 π‘₯∈𝐻, we have (βˆ’π‘₯)=βˆ’(π‘₯).

Theorem 3.5. Let 𝐻 be a canonical hypergroup, 𝑁 be a subcanonical hypergroup of 𝐻. Then for π‘₯,π‘¦βˆˆπ», the sets 𝐴={π‘§βˆΆπ‘§βˆˆπ‘₯+𝑦}, 𝐡={π‘§βˆΆπ‘§βˆˆπ‘₯+𝑦} and 𝐢={π‘§βˆΆπ‘§βŠ†π‘₯+𝑦} are equal.

Proof. Let π‘§βˆˆπ΄. Then π‘§βˆˆπ‘₯+𝑦. Since π‘₯∈π‘₯ and π‘¦βˆˆπ‘¦ we have π‘§βˆˆπ‘₯+𝑦. Thus π΄βŠ†π΅. Suppose π‘§βˆˆπ΅, then π‘§βˆˆπ‘₯+𝑦. That is, π‘§βˆˆπ‘‘+𝑛 for some π‘‘βˆˆπ‘₯+𝑦 and π‘›βˆˆπ‘. Therefore 𝑧=𝑑, where π‘‘βˆˆπ‘₯+𝑦. Since π‘‘βˆˆπ΄, we get π‘§βˆˆπ΄. Thus π΅βŠ†π΄. Hence 𝐴=𝐡.
If π‘§βˆˆπ΄, then π‘§βˆˆπ‘₯+𝑦. Therefore, π‘§βŠ†π‘₯+𝑦+𝑁=π‘₯+𝑁+𝑦+𝑁=π‘₯+𝑦. Hence π΄βŠ†πΆ. On the other hand if π‘§βˆˆπΆ, then π‘§βŠ†π‘₯+𝑦. Since π‘§βˆˆπ‘§βŠ†π‘₯+𝑦, we get π‘§βˆˆπ‘ +𝑛 for some π‘ βˆˆπ‘₯+𝑦 and π‘›βˆˆπ‘. Thus 𝑧=𝑠. Since π‘ βˆˆπ΄, we get πΆβŠ†π΄. Hence 𝐴=𝐢.

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 {π‘₯∢π‘₯∈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 π‘₯βŠ•π‘¦={π‘§βˆΆπ‘§βˆˆπ‘₯+𝑦} for all π‘₯,π‘¦βˆˆπ»/𝑁, then 𝐻/𝑁 is a canonical hypergroup.

Proof. If π‘₯1,𝑦1,π‘₯2,𝑦2∈𝐻 such that π‘₯1=π‘₯2 and 𝑦1=𝑦2, then π‘₯2∈π‘₯1+𝑁 and 𝑦2βˆˆπ‘¦1+𝑁. Let 𝑧2∈π‘₯2+𝑦2βŠ†(π‘₯1+𝑁)+(𝑦1+𝑁). Since 𝐻 is commutative, 𝑧2βˆˆπ‘§1+𝑖 for some 𝑧1∈π‘₯1+𝑦1 and for some π‘–βˆˆπ‘. That is, 𝑧2+𝑁=𝑧1+𝑁. Hence, π‘₯2βŠ•π‘¦2βŠ†π‘₯1βŠ•π‘¦1. Also, since π‘₯1∈π‘₯2+𝑁 and 𝑦1βˆˆπ‘¦2+𝑁, by a similar argument, we get, π‘₯1βŠ•π‘¦1βŠ†π‘₯2βŠ•π‘¦2. Hence, π‘₯1βŠ•π‘¦1=π‘₯2βŠ•π‘¦2. Thus, hyperaddition βŠ• is well defined.
Let π‘₯,𝑦,π‘§βˆˆπ»/𝑁. If π‘’βˆˆ(π‘₯βŠ•π‘¦)βŠ•π‘§, then π‘’βˆˆπ‘βŠ•π‘§ for some π‘βˆˆπ‘₯βŠ•π‘¦. That is, 𝑒=π‘Ž for some π‘Žβˆˆπ‘+𝑧. Also, 𝑝=𝑏 for some π‘βˆˆπ‘₯+𝑦. Now, π‘Žβˆˆπ‘+π‘§βŠ†π‘+𝑁+𝑧=𝑏+𝑧+𝑁. That is, π‘Žβˆˆπ‘£+𝑁 for some π‘£βˆˆπ‘+π‘§βŠ†(π‘₯+𝑦)+𝑧=π‘₯+(𝑦+𝑧). So, π‘£βˆˆπ‘₯+𝑑 for some π‘‘βˆˆπ‘¦+𝑧. This means that π‘Ž=𝑣 and π‘£βˆˆπ‘₯βŠ•π‘‘. Since π‘‘βˆˆπ‘¦βŠ•π‘§, we have 𝑒=π‘Ž=π‘£βˆˆπ‘₯βŠ•π‘‘βŠ†π‘₯βŠ•(π‘¦βŠ•π‘§). This means that π‘’βˆˆπ‘₯βŠ•(π‘¦βŠ•π‘§). Hence (π‘₯βŠ•π‘¦)βŠ•π‘§βŠ†π‘₯βŠ•(π‘¦βŠ•π‘§). Similarly, we get π‘₯βŠ•(π‘¦βŠ•π‘§)βŠ†(π‘₯βŠ•π‘¦)βŠ•π‘§. Hence, π‘₯βŠ•(π‘¦βŠ•π‘§)=(π‘₯βŠ•π‘¦)βŠ•π‘§. Thus, the hyperaddition is associative.
Consider the element 0=0+π‘βˆˆπ»/𝑁. Now, for any π‘₯∈𝐻, we have π‘₯βŠ•0={π‘§βˆΆπ‘§βˆˆπ‘₯+0}=π‘₯. Similarly, 0βŠ•π‘₯=π‘₯. Thus, 0 is the zero element of 𝐻/𝑁.
Let π‘₯∈𝐻, then π‘₯βŠ•(βˆ’π‘₯)={π‘§βˆΆπ‘§βˆˆπ‘₯+(βˆ’π‘₯)=π‘₯βˆ’π‘₯}. Since 0∈π‘₯βˆ’π‘₯, we get 0∈π‘₯βŠ•(βˆ’π‘₯). Similarly, we can show that 0∈(βˆ’π‘₯)βŠ•π‘₯. Let π‘₯∈𝐻/𝑁, and suppose that π‘¦βˆˆπ»/𝑁 is such that 0βˆˆπ‘¦βŠ•π‘₯, then 0=π‘Ž, where π‘Žβˆˆπ‘¦+π‘₯. That is, π‘¦βˆˆπ‘Žβˆ’π‘₯βŠ†π‘βˆ’π‘₯, and hence 𝑦=βˆ’π‘₯. Thus, the element π‘₯∈𝐻/𝑁 has a unique inverse βˆ’π‘₯∈𝐻/𝑁.
Suppose that π‘§βˆˆπ‘₯βŠ•π‘¦, then 𝑧=π‘Ž, where π‘Žβˆˆπ‘₯+𝑦. This implies π‘₯βˆˆπ‘Žβˆ’π‘¦βŠ†π‘§+π‘βˆ’π‘¦. That is, π‘₯βˆˆπ‘Ÿ+𝑁, where π‘Ÿβˆˆπ‘§βˆ’π‘¦. Thus, π‘₯=π‘Ÿβˆˆπ‘§βŠ•(βˆ’π‘¦). Similarly, we can show π‘¦βˆˆ(βˆ’π‘₯)βŠ•π‘§. Since 𝐻 is commutative, it is obvious that 𝐻/𝑁 is also commutative. Thus, 𝐻/𝑁 is a canonical hypergroup.

Corollary 3.8. Let πœ™ be a strong homomorphism from canonical hypergroup 𝐻1 into a canonical hypergroup 𝐻2, then 𝐻1/Kerπœ™ 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 {π‘₯∈𝐻∢π‘₯βˆ’π‘₯βŠ†π΄} of 𝐻 by 𝑆𝐴.

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

Proof. Let π‘₯∈𝐴. Since 𝐴 is a subcanonical hypergroup of 𝐻, π‘₯βˆ’π‘₯βŠ†π΄. This implies π‘₯βˆˆπ‘†π΄. Therefore, π΄βŠ†π‘†π΄. Since π΄β‰ βˆ…, the set 𝑆𝐴 is nonempty.
Let π‘₯,π‘¦βˆˆπ‘†π΄. For π‘Ÿβˆˆπ‘₯βˆ’π‘¦, we get π‘Ÿβˆ’π‘ŸβŠ†(π‘₯βˆ’π‘¦)βˆ’(π‘₯βˆ’π‘¦)=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)βŠ†π΄+𝐴=𝐴. Hence, π‘Ÿβˆˆπ‘†π΄. That is, π‘₯βˆ’π‘¦βŠ†π‘†π΄. Therefore, 𝑆𝐴 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, π‘₯+𝑦 is a singleton set for all π‘₯,π‘¦βˆˆπ΄.

Example 3.12. The set 𝐻={0,π‘Ž,𝑏,𝑐} with the following hyperoperation + is a canonical hypergroup +0π‘Žπ‘π‘0π‘Ž{0}{π‘Ž}{𝑏}{𝑐}𝑏{π‘Ž}{0,𝑏}{π‘Ž,𝑐}{𝑏}𝑐{𝑏}{π‘Ž,𝑐}{0,𝑏}{π‘Ž}{𝑐}{𝑏}{π‘Ž}{0}(3.1)

In this example {0,𝑐},{0} are subgroups of 𝐻 and πœ”π»={0,𝑏} whereas in the Example 2.2, {0} is the subgroup of 𝐻 and πœ”π»=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 π‘₯,π‘¦βˆˆπ‘†{0}. Consider the set π‘₯+𝑦. If 𝑒,π‘£βˆˆπ‘₯+𝑦, then π‘’βˆ’π‘£βŠ†(π‘₯+𝑦)βˆ’(π‘₯+𝑦)=(π‘₯βˆ’π‘₯)+(π‘¦βˆ’π‘¦)=0+0=0. Hence, 𝑒=𝑣. This means that the set π‘₯+𝑦 has only one element. Thus, 𝑆{0} is a subgroup of 𝐻. Suppose, 𝐴 is any subgroup of 𝐻, then for any π‘₯∈𝐴 that we have π‘₯βˆ’π‘₯=0. That is, π‘₯βˆˆπ‘†{0}. Hence, π΄βŠ†π‘†{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 𝑆𝐴=𝐻.

Proof. Let 𝐴 be normal. Then, for π‘₯∈𝐻, π‘₯+0βˆ’π‘₯βŠ†π΄. That is, π‘₯βˆˆπ‘†π΄. Hence, 𝑆𝐴=𝐻. Conversely, if 𝑆𝐴=𝐻, then for π‘₯∈𝐻, we get π‘₯+π΄βˆ’π‘₯=π‘₯βˆ’π‘₯+π΄βŠ†π΄+𝐴=𝐴. Thus, 𝐴 is normal.

Proposition 3.16. The heart πœ”π» of a canonical hypergroup 𝐻 is a normal subcanonical hypergroup of 𝐻.

Proof. If π‘₯,π‘¦βˆˆπœ”π», then π‘₯∈(π‘₯1βˆ’π‘₯1)+(π‘₯2βˆ’π‘₯2)+(π‘₯3βˆ’π‘₯3)+β‹―+(π‘₯π‘›βˆ’π‘₯𝑛) and π‘¦βˆˆ(𝑦1βˆ’π‘¦1)+(𝑦2βˆ’π‘¦2)+(𝑦3βˆ’π‘¦3)+β‹―+(π‘¦π‘šβˆ’π‘¦π‘š), where π‘₯𝑖,π‘¦π‘—βˆˆπ» and π‘š,𝑛 are natural numbers. Thus π‘₯βˆ’π‘¦βˆˆ(π‘₯1βˆ’π‘₯1)+(π‘₯2βˆ’π‘₯2)+(π‘₯3βˆ’π‘₯3)+β‹―+(π‘₯π‘›βˆ’π‘₯𝑛)+(𝑦1βˆ’π‘¦1)+(𝑦2βˆ’π‘¦2)+(𝑦3βˆ’π‘¦3)+β‹―+(π‘¦π‘šβˆ’π‘¦π‘š)βŠ†πœ”π». Now, for any element β„Žβˆˆπœ”π», there exists natural number 𝑛 and elements π‘₯π‘–βˆˆπ» such that β„Žβˆˆ(π‘₯1βˆ’π‘₯1)+(π‘₯2βˆ’π‘₯2)+(π‘₯3βˆ’π‘₯3)+β‹―+(π‘₯π‘›βˆ’π‘₯𝑛). Then, for any π‘₯∈𝐻, π‘₯+β„Žβˆ’π‘₯=π‘₯βˆ’π‘₯+β„ŽβŠ†π‘₯βˆ’π‘₯+(π‘₯1βˆ’π‘₯1)+(π‘₯2βˆ’π‘₯2)+(π‘₯3βˆ’π‘₯3)+β‹―+(π‘₯π‘›βˆ’π‘₯𝑛)βŠ†πœ”π». Hence, heart πœ”π» 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 πœ”π» of the canonical hypergroup 𝐻.

Proof. Let 𝐴 be a normal subcanonical hypergroup of the canonical hypergroup 𝐻. Then π‘₯+π‘–βˆ’π‘₯βŠ†π΄ for every π‘₯∈𝐻, and π‘–βˆˆπ΄. In particular, when 𝑖=0∈𝐴, we get π‘₯βˆ’π‘₯βŠ†π΄ for every π‘₯∈𝐻. Since 𝐴 is a subcanonical hypergroup of 𝐻, the union of the sums (π‘₯1βˆ’π‘₯1)+(π‘₯2βˆ’π‘₯2)+(π‘₯3βˆ’π‘₯3)+β‹…β‹…β‹…+(π‘₯π‘›βˆ’π‘₯𝑛)βŠ†π΄ for π‘₯π‘–βˆˆπ» and 𝑛 is a natural number. That is, πœ”π»βŠ†π΄. Conversely, assume that subcanonical hypergroup 𝐴 contains the heart πœ”π» of the canonical hypergroup 𝐻. For π‘₯∈𝐻 and π‘–βˆˆπ΄, π‘₯+π‘–βˆ’π‘₯=π‘₯βˆ’π‘₯+π‘–βŠ†πœ”π»+π΄βŠ†π΄+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, πœ”π» is the smallest normal subcanonical hypergroup.

Proposition 3.19. Let 𝐴,𝐡 be subcanonical hypergroups of a canonical hypergroup 𝐻 such that π΄βŠ†π΅, then π‘†π΄βŠ†π‘†π΅.

Proof. Let π‘₯βˆˆπ‘†π΄. Then, π‘₯βˆ’π‘₯βŠ†π΄. That is, π‘₯βˆˆπ‘†π΅. Hence, π‘†π΄βŠ†π‘†π΅.

Proposition 3.20. Let 𝐴,𝐡 be subcanonical hypergroups of a canonical hypergroup 𝐻 such that π΄βŠ†π΅. If 𝐴 is normal, then 𝐡 is also normal.

Proof. If 𝐴 is normal, then by Proposition 3.15, 𝑆𝐴=𝐻. Since π΄βŠ†π΅, by Proposition 3.19, π‘†π΄βŠ†π‘†π΅. Hence, 𝐻=𝑆𝐡. By Proposition 3.15, 𝐡 is normal.

Corollary 3.21. Let 𝐴,𝐡 be subcanonical hypergroups of a canonical hypergroup 𝐻 such that 𝐴 is normal, then the subcanonical hypergroup 𝐴+𝐡 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)πœ”π»=(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, πœ”π» is the smallest normal subcanonical hypergroup of 𝐻. Therefore, (0) is normal if and only if πœ”π»=(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 (π‘₯+𝑁)+𝑁+(βˆ’π‘₯+𝑁)=(π‘₯+π‘βˆ’π‘₯)+π‘βŠ†π‘+𝑁=𝑁 for all π‘₯∈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 πœ™ be a strong homomorphism from a canonical hypergroup 𝐻1 into a canonical hypergroup 𝐻2 with kernel 𝐾. Then, 𝐻1/𝐾 is strongly isomorphic to Imπœ™.

Proof. Define a map π‘“βˆΆπ»1/𝐾→Imπœ™ by 𝑓(π‘₯)=πœ™(π‘₯) for all π‘₯∈𝐻1. Suppose that π‘₯=𝑦, where π‘₯,π‘¦βˆˆπ», then π‘₯βˆˆπ‘¦. That is, π‘₯βˆˆπ‘¦+π‘˜ for some π‘˜βˆˆπΎ. Hence, πœ™(π‘₯)βˆˆπœ™(𝑦+π‘˜)=πœ™(𝑦)+πœ™(π‘˜)=πœ™(𝑦)+0=πœ™(𝑦). So πœ™(π‘₯)=πœ™(𝑦). Hence, 𝑓(π‘₯)=𝑓(𝑦). Thus, the map 𝑓 is well defined.
If π‘₯,π‘¦βˆˆπ»1, then 𝑓π‘₯βŠ•π‘¦ξ€Έ=𝑓=ξ€½π‘“ξ€·π‘§βˆΆπ‘§βˆˆπ‘₯+π‘¦ξ€Ύξ€Έπ‘§ξ€Έξ€ΎβˆΆπ‘§βˆˆπ‘₯+𝑦={πœ™(𝑧)βˆΆπ‘§βˆˆπ‘₯+𝑦}.(4.1)Also,𝑓π‘₯ξ€Έξ€·+𝑓𝑦=πœ™(π‘₯)+πœ™(𝑦)=πœ™(π‘₯+𝑦)={πœ™(𝑧)βˆΆπ‘§βˆˆπ‘₯+𝑦}.(4.2) Thus, 𝑓(π‘₯βŠ•π‘¦)=𝑓(π‘₯)+𝑓(𝑦). Moreover, 𝑓(0)=πœ™(0)=0. Hence, 𝑓 is a strong homomorphism.
Suppose that π‘₯,π‘¦βˆˆπ»1/𝐾 such that 𝑓(π‘₯)=𝑓(𝑦), then πœ™(π‘₯)=πœ™(𝑦). This means that 0βˆˆπœ™(π‘₯)βˆ’πœ™(𝑦)=πœ™(π‘₯βˆ’π‘¦). That is, πœ™(𝑧)=0 for some π‘§βˆˆπ‘₯βˆ’π‘¦. Since πœ™(𝑧)=0, we get π‘§βˆˆπΎ. Now, π‘§βˆˆπ‘₯βˆ’π‘¦β‡’π‘₯βˆˆπ‘§+𝑦⇒π‘₯βˆˆπ‘¦+𝐾. Then, by Proposition 3.3  π‘₯=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 Imπœ™.

Corollary 4.2. Let πœ™ 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/(π‘€βˆ©π‘)β‰…(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 (π‘€βˆ©π‘) is a subcanonical hypergroup.
Define π‘”βˆΆπ‘β†’(𝑀+𝑁)/𝑀 by 𝑔(𝑏)=𝑏+𝑀 for every π‘βˆˆπ‘. For all π‘Ž,π‘βˆˆπ‘, 𝑔(π‘Ž+𝑏)=𝑔({π‘₯∢π‘₯βˆˆπ‘Ž+𝑏})={𝑔(π‘₯)∢π‘₯βˆˆπ‘Ž+𝑏}={π‘₯+π‘€βˆΆπ‘₯βˆˆπ‘Ž+𝑏}=(π‘Ž+𝑀)βŠ•(𝑏+𝑀)=𝑔(π‘Ž)βŠ•π‘”(𝑏). Moreover, 𝑔(0)=0. Thus, 𝑔 is a strong homomorphism.
Now, π‘₯+π‘€βˆˆ(𝑀+𝑁)/𝑀 implies that π‘₯βˆˆπ‘¦+𝑀 for some π‘¦βˆˆπ‘€+𝑁. That is, π‘¦βˆˆπ‘Ž+𝑏 for some π‘Žβˆˆπ‘€, π‘βˆˆπ‘. Since π‘¦βˆˆπ‘+𝑀, we get 𝑦+𝑀=𝑏+𝑀. Thus, 𝑔(𝑏)=𝑏+𝑀=𝑦+𝑀=π‘₯+𝑀. Thus, 𝑔 is onto. Let π‘βˆˆπ‘. Then, π‘βˆˆKer𝑔⇔𝑔(𝑏)=0⇔𝑏+𝑀=0+π‘€β‡”π‘βˆˆπ‘€. Thus, π‘βˆˆKer𝑔 if and only if π‘βˆˆπ‘€βˆ©π‘. Hence, by the First Isomorphism Theorem, 𝑁/(π‘€βˆ©π‘)β‰…(𝑀+𝑁)/𝑀.

Theorem 4.4 (Third Isomorphism Theorem). If 𝑀 and 𝑁 are subcanonical hypergroup of a canonical hypergroup 𝐻 such that π‘€βŠ†π‘, then 𝐻/𝑁≅(𝐻/𝑀)/(𝑁/𝑀).

Proof. Define a map β„ŽβˆΆπ»/𝑁→𝐻/𝑀 by β„Ž(π‘₯+𝑁)=π‘₯+𝑀. Then, β„Ž 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={π‘₯∢π‘₯βˆˆπ‘…} 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 π‘₯βŠ—π‘¦={π‘§βˆΆπ‘§βˆˆπ‘₯𝑦} for all π‘₯,π‘¦βˆˆπ‘…/𝐼, then 𝑅/𝐼 is a hyperring.

Proof. If π‘₯1,𝑦1,π‘₯2,𝑦2βˆˆπ‘… such that π‘₯1=π‘₯2 and 𝑦1=𝑦2, then π‘₯2∈π‘₯1+𝐼 and 𝑦2βˆˆπ‘¦1+𝐼. Let 𝑧2∈π‘₯2𝑦2βŠ†(π‘₯1+𝐼)(𝑦1+𝐼)βŠ†π‘₯1𝑦1+𝐼. Then, 𝑧2βˆˆπ‘§1+𝑖 for some 𝑧1∈π‘₯1𝑦1 and for some π‘–βˆˆπΌ. That is, 𝑧2+𝐼=𝑧1+𝐼 and so π‘₯2βŠ—π‘¦2βŠ†π‘₯1βŠ—π‘¦1. Similarly, we get, π‘₯1βŠ—π‘¦1βŠ†π‘₯2βŠ•π‘¦2. Hence, π‘₯1βŠ•π‘¦1=π‘₯2βŠ—π‘¦2. Thus, hypermultiplication βŠ— is well defined.
Suppose, π‘₯,𝑦,π‘§βˆˆπ‘…/𝐼. Then, ξ€·π‘₯βŠ—π‘¦βŠ—π‘§ξ€Έ=ξ€½π‘₯βŠ—ξ€Ύ=ξ€½π‘ŽβˆΆπ‘Žβˆˆπ‘¦π‘§ξ€Ύ=ξ€½π‘ βˆΆπ‘ βˆˆπ‘₯π‘Ž,π‘Žβˆˆπ‘¦π‘§ξ€Ύ=ξ€½π‘ βˆΆπ‘ βˆˆπ‘₯(𝑦𝑧)ξ€Ύ=ξ€½π‘ βˆΆπ‘ βˆˆ(π‘₯𝑦)𝑧=ξ€½π‘ βˆΆπ‘ βˆˆπ‘π‘§,π‘βˆˆπ‘₯π‘¦ξ€ΎβŠ—π‘ βˆΆπ‘ βˆˆπ‘₯𝑦𝑧=ξ€·π‘₯βŠ—π‘¦ξ€ΈβŠ—π‘§(5.1) Thus, we get π‘₯βŠ—(π‘¦βŠ—π‘§)=(π‘₯βŠ—π‘¦)βŠ—π‘§. Hence, hypermultiplication is associative. Further, ξ€·π‘₯βŠ—π‘¦βŠ•π‘§ξ€Έ=π‘₯βŠ—{𝑝+πΌβˆΆπ‘βˆˆπ‘¦+𝑧}={π‘ž+πΌβˆΆπ‘žβˆˆπ‘₯𝑝,π‘βˆˆπ‘¦+𝑧}={π‘ž+πΌβˆΆπ‘žβˆˆπ‘₯(𝑦+𝑧)}={π‘ž+πΌβˆΆπ‘žβˆˆπ‘₯𝑦+π‘₯𝑧}.(5.2) Also, ξ€·π‘₯βŠ—π‘¦ξ€ΈβŠ•ξ€·π‘₯βŠ—π‘§ξ€Έ={π‘Ž+πΌβˆΆπ‘Žβˆˆπ‘₯𝑦}βŠ•{𝑏+πΌβˆΆπ‘βˆˆπ‘₯𝑧}={𝑐+πΌβˆΆπ‘βˆˆa+𝑏,π‘Žβˆˆπ‘₯𝑦,π‘βˆˆπ‘₯𝑧}={𝑐+πΌβˆΆπ‘βˆˆπ‘₯𝑦+π‘₯𝑧}.(5.3) Hence, π‘₯βŠ—(π‘¦βŠ•π‘§)=(π‘₯βŠ—π‘¦)βŠ•(π‘₯βŠ—π‘§). Similarly, we can show that (π‘₯βŠ•π‘¦)βŠ—π‘§=(π‘₯βŠ—π‘§)βŠ•(π‘₯βŠ—π‘§). Therefore, hypermultiplication is distributive with respect to the hyperaddition. Thus, 𝑅/𝐼 is a hyperring.

Corollary 5.2. Let πœ™ be a strong homomorphism from hyperring 𝑅1 into a hyperring 𝑅2, then 𝑅1/Kerπœ™ 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 πœ™ be a strong homomorphism from a hyperring 𝑅1 into a hyperring 𝑅2 with kernel 𝐾. Then, 𝑅1/𝐾 is strongly isomorphic to Imπœ™.

Proof. Define a map π‘“βˆΆπ‘…1/𝐾→Imπœ™ by 𝑓(π‘₯)=πœ™(π‘₯) for all π‘₯βˆˆπ‘…1.
By Theorem 4.1, this map 𝑓 is a strong isomorphism from canonical hypergroup 𝑅1/𝐾 onto Imπœ™. Now, 𝑓π‘₯βŠ—π‘¦ξ€Έ=𝑓=ξ€½π‘“ξ€·π‘§βˆΆπ‘§βˆˆπ‘₯π‘¦ξ€Ύξ€Έπ‘§ξ€Έξ€Ύπ‘“ξ€·βˆΆπ‘§βˆˆπ‘₯𝑦={πœ™(𝑧)βˆΆπ‘§βˆˆπ‘₯𝑦},π‘₯𝑓𝑦=πœ™(π‘₯)πœ™(𝑦)=πœ™(π‘₯𝑦)={πœ™(𝑧)βˆΆπ‘§βˆˆπ‘₯𝑦}.(5.4)
Thus, 𝑓(π‘₯βŠ—π‘¦)=𝑓(π‘₯)𝑓(𝑦). Hence, 𝑓 is a strong hyperring isomorphism.

Corollary 5.5. Let πœ™ 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)β‰…(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 π‘”βˆΆπ½β†’(𝐼+𝐽)/𝐼 by 𝑔(𝑏)=𝑏+𝐼 for every π‘βˆˆπ½. By Theorem 4.3,𝑔 is strong isomorphism from canonical hypergroup 𝐽 onto the canonical hypergroup (𝐼+𝐽)/𝐼. Now, 𝑔(π‘Žπ‘)=𝑔({π‘₯∢π‘₯βˆˆπ‘Žπ‘})={𝑔(π‘₯)∢π‘₯βˆˆπ‘Žπ‘}={π‘₯+𝐼∢π‘₯βˆˆπ‘Žπ‘}=(π‘Ž+𝐼)(𝑏+𝐼)=𝑔(π‘Ž)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)β‰…(𝐼+𝐽)/𝐼.

Theorem 5.7 (Third Isomorphism Theorem). If 𝐼 and 𝐽 are hyperideals of a hyperring 𝑅 such that πΌβŠ†π½, then 𝑅/𝐽≅(𝑅/𝐼)/(𝐽/𝐼).

Proof. Define a map β„ŽβˆΆπ‘…/𝐽→𝑅/𝑀 by β„Ž(π‘₯+𝐼)=π‘₯+𝐽. Then, β„Ž 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).

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