# On Pure Hyperradical in Semihypergroups

Authors: Kostaq Hila; Krisanthi Naka
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

This paper deals with a class of algebraic hyperstructures called semihypergroups, which are a generalization of semigroups. In this paper, we introduce pure hyperradical of a hyperideal in a semihypergroup with zero element. For this purpose, we define pure, semipure, and other related types of hyperideals and establish some of their basic properties in semihypergroups.


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## Body

## 1. Introduction and Preliminaries

The applications of mathematics in other disciplines, for example, in informatics, play a key role and they represent, in the last decades, one of the purpose, of the study of the experts of hyperstructures theory all over the world. Hyperstructure theory was introduced in 1934 by the French mathematician Marty [1], at the 8th Congress of Scandinavian Mathematicians, where he defined hypergroups based on the notion of hyperoperation, began to analyze their properties, and applied them to groups. In the following decades and nowadays, a number of different hyperstructures are widely studied from the theoretical point of view and for their applications to many subjects of pure and applied mathematics by many mathematicians. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. Several books have been written on hyperstructure theory, see [2–5]. A recent book on hyperstructures [3] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs and hypergraphs. Another book [4] is devoted especially to the study of hyperring theory. Several kinds of hyperrings are introduced and analyzed. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: e-hyperstructures and transposition hypergroups. Some principal notions about semihypergroups theory can be found in [6–18]. Recently, Davvaz et al. [19–23] introduced the notion of Γ-semihypergroup as a generalization of a semigroup, a generalization of a semihypergroup, and a generalization of a Γ-semigroup. They presented many interesting examples and obtained a several characterizations of Γ-semihypergroups.In this paper, we introduce pure hyperradical of a hyperideal in a semihypergroup with zero element. For this purpose, we define pure, semipure, and other related types of hyperideals, and we establish some of their basic properties in semihypergroups.Recall first the basic terms and definitions from the hyperstructure theory.A map∘:H×H→𝒫*(H) is called hyperoperation or join operation on the set H, where H is a nonempty set and 𝒫*(H)=𝒫(H)∖{∅} denotes the set of all nonempty subsets of H.Ahyperstructure is called the pair (H,∘) where ∘ is a hyperoperation on the set H.A hyperstructure(H,∘) is called a semihypergroup if for all x,y,z∈H, (x∘y)∘z=x∘(y∘z), which means that
(1.1)⋃u∈x∘y‍u∘z=⋃v∈y∘z‍x∘v.
If x∈H and A,B are nonempty subsets of H, then
(1.2)A∘B=⋃a∈A,b∈B‍a∘b,A∘x=A∘{x},x∘B={x}∘B.A nonempty subsetB of a semihypergroup H is called a subsemihypergroup of H if B∘B⊆B and H is called in this case supersemihypergroup of B.Let(H,∘) be a semihypergroup. Then, H is called a hypergroup if it satisfies the reproduction axiom, for all a∈H, a∘H=H∘a=H.A nonempty subsetI of a semihypergroup H is called a right (left) hyperideal of H if for all x∈H and r∈I,
(1.3)r∘x⊆I(x∘r⊆I).
A nonempty subset I of H is called a hyperideal (or two-sided hyperideal) if it is both a left hyperideal and right hyperideal.An elemente in a semihypergroup H is called scalar identity if
(1.4)x∘e=e∘x={x},∀x∈H.
An element 0 in a semihypergroup H is called zero element if
(1.5)x∘0=0∘x={0},∀x∈H.We call a semihypergroup(H,∘) a regular semihypergroup if for every x∈H,x∈x∘y∘x, for some y∈H. Hence, every regular semigroup is a regular semihypergroup. Notice that if (H;∘) is a hypergroup, then for every x∈H,x∘H∘x=H. This implies that every hypergroup is a regular semihypergroup.


## 2. Main Results

In this section, we introduce pure hyperradical of a hyperideal in a semihypergroup with zero element. For this purpose, we define pure, semipure, and other related types of hyperideals, and we establish some of their basic properties. In what follows,H will denote a semihypergroup with scalar identity 1, which contains a zero element.Definition 2.1.
LetH be a semihypergroup. A right hyperideal A of H is called a right pure right hyperideal if for each x∈A, there is an element y∈A such that x∈x∘y. If A is a two-sided hyperideal with the property that for each x∈A, there is an element y∈A such that x∈x∘y, then A is called a right pure hyperideal.
Left pure left hyperideals and left pure hyperideals are defined analogously.Definition 2.2.
LetH be a semihypergroup. A right hyperideal A of H is called a right semipure right hyperideal if for each x∈A, there is an element y belonging to some proper right hyperideal of H such that x∈x∘y. If A is a two-sided hyperideal with the property that for each x∈A, there is an element y belonging to a proper hyperideal of H such that x∈x∘y, then A is called a right semipure hyperideal.
Left semipure left hyperideals and left semipure hyperideals can be similarly defined.Example 2.3.
Let(H,∘) be a semihypergroup on H={0,1,x,y,z,t} with the hyperoperation ∘ given by the following:
(2.1)∘0xyzt10000000x0{1,x}{x,y,1}{0,1,x,z}Hxy0{0,y}y{y,t,1}{0,1,y,t}yz0z{z,t}z{z,t}zt0{0,t}{0,t}{0,t}{0,t}t10xyzt1
It is easy to see that I1={0,t}, I2={0,z,t} are right pure right hyperideals of H.Example 2.4.
Let(H,∘) be a semihypergroup on H={0,1,a,b,c,d,e,f} with the hyperoperation ∘ given by the following:
(2.2)∘0abcdef1000000000a0a{a,b}c{c,d}e{e,f}ab0bbddffbc0c{c,d}c{c,d}c{c,d}cd0ddddddde0e{e,f}c{c,d}e{e,f}ef0ffddfff10abcdef1
Clearly, I1={0,d}, I2={0,d,f}, and I3={0,b,d,f} are right pure right hyperideals of H. I4={0,c,d} is a two-sided hyperideal of H which is a right pure hyperideal but not a left pure hyperideal. Also, I4 is a right and left semipure hyperideals. I5={0,c,d,e,f} is a two-sided hyperideal of H which is a right and left pure hyperideal.Example 2.5.
LetH=[0,1]. Then, H with the hyperoperation x∘y=[0,xy] is a semihypergroup. Let t∈[0,1] and T=[0,t]. Then, T is a semihypergroup, and, moreover, T is a two-sided hyperideal of H which is neither right pure nor left pure, but it is left and right semipure.Proposition 2.6.
LetA be a two-sided hyperideal of H.  Then A is right pure if and only for any right hyperideal B,  B∩A=B∘A.Proof.
SupposeA is a right pure hyperideal of H. Since B is a right hyperideal of H, B∘A⊆B. Also, since A is a left hyperideal, B∘A⊆A. Hence, B∘A⊆B∩A. Let x∈B∩A. Since A is a right pure hyperideal, there exists y∈A such that x∈x∘y. As x∈B and y∈A, x∘y⊆B∘A. Hence, x∈B∘A. This implies that B∩A=B∘A. Conversely, assume B∩A=B∘A, for any right hyperideal B of H. We show that A is right pure. Let x∈A. Then, x∘H=x∘H∩A=x∘H∘A=x∘A. Since x∈x∘H,x∈x∘A. Hence, there exists y∈A such that x∈x∘y. This proves that A is a right pure hyperideal.Corollary 2.7.
IfA is a right pure hyperideal, then A=A∘A.Proposition 2.8.
(0) and H are right pure hyperideals of H. Any union and finite intersection of right pure (resp., semipure) hyperideals is right pure (resp., semipure).Proof.
(0) andH are obviously right pure hyperideals. Let I1 and I2 be right pure hyperideals and let x∈I1∩I2. Since x∈I1 and I1 is right pure, exists y1∈I1 such that x∈x∘y1. Similarly, exists y2∈I2 such that x∈x∘y2. Thus, we have x∈x∘y2⊆(x∘y1)∘y2=x∘(y1∘y2). Since y1∘y2∈I1∩I2, it follows that I1∩I2 is right pure. The remaining cases of this proposition can be similarly proved.It follows from the above proposition that ifI is any hyperideal of H, then I contains a largest pure hyperideal, which is in fact the union of all pure hyperideals contained in I (such hyperideals exist, e.g., (0)) and hence a pure hyperideal. The largest pure hyperideal contained in I is denoted by L(I). Similarly, each hyperideal I contains a largest semipure hyperideal, denoted by H(I). L(I) (resp., H(I)) is called the pure (resp., semipure) part of I.Definition 2.9.
LetI be a right pure (resp., semipure) hyperideal of H. Then, I is called purely (resp., semipurely) maximal if I is a maximal element in the set of proper right pure (resp., semipure) hyperideals.In Example2.4, I4={0,c,d},I5={0,c,d,e,f}, and I6={0,a,b,c,d,e,f} are right pure hyperideals of the semihypergroup H, and it is clear that I6 is purely maximal hyperideal.Definition 2.10.
LetI be a right pure (resp., semipure) hyperideal of H. Then, I is called purely (resp. semipurely) prime if it is proper and if for any right pure (resp., semipure) hyperideals I1 and I2, I1∩I2⊆I⇒I1⊆I or I2⊆I.In Example2.4, the hyperideal I6 mentioned above is purely prime hyperideal.The following propositions are stated for pure and semipure hyperideals simultaneously. However, the proofs are given only for one case, since the proofs are similar for the remaining cases.Proposition 2.11.
Any purely (resp., semipurely) maximal hyperideal is purely (resp., semipurely) prime.Proof.
SupposeI is purely maximal, and I1,I2 are right pure hyperideals such that I1∩I2⊆I. Suppose I1⊈I. Then, I1∪I=H. Now, I2=I2∩H=I2∩(I1∪I)=(I2∩I1)∪(I2∩I)⊆I∪I=I.Proposition 2.12.
The pure (resp. semipure) part of any maximal hyperideal is purely (resp. semipurely) prime.Proof.
LetM be a maximal hyperideal of H. We show that L(M), the pure part of M, is purely prime. Suppose I1∩I2⊆L(M) with I1,I2 pure. If I1⊆M, then I1⊆L(M) and we are done. Suppose I1⊈M, then I1∪M=H. Hence, I2=I2∩H=I2∩(I1∪M)=(I2∩I1∪(I2∩M)⊆M∪M=M. Hence, I2⊆M. This implies that I2⊆L(M), since I2 is pure.Proposition 2.13.
IfI is right pure (resp., semipure) hyperideal of H and a∉I, then there exists a purely (resp., semipurely) prime hyperideal J such that I⊆J and a∉J.Proof.
Consider the setX ordered by inclusion: X={J:J is a right semipure hyperideal, I⊆J,a∉J}. Then, X≠∅, since I∈X. Let (Jk)k∈K be a totally ordered subset of X. Clearly, ⋃kJk is a semipure hyperideal with a∉⋃kJk. Hence, X is inductively ordered. Therefore, by Zorn's Lemma, X has a maximal element J. We will show that J is semipurely prime. Suppose I1,I2 are right semipure hyperideals such that I1⊈J and I2⊈J. Since Ik(k=1,2) and J are semipure, Ik∪J is a semipure hyperideal such that J⊆Ik∪J. We then claim that an a∈Ik∪J(k=1,2). Because, if a∉Ik∪J, then by the maximality of J, we have Ik∪J⊆J. But this contradicts the assumption Ik⊈J(k=1,2). Hence, a∈(I1∩I2∪J). Since a∉J, it follows that I1∩I2⊈J. Hence, by contrapositivity, we conclude that J is semipurely prime.Proposition 2.14.
LetI be a proper right pure (resp. semipure) hyperideal of H. Then, I is contained in a purely (resp., semipurely) maximal hyperideal.Proof.
Consider the set, ordered by inclusion,X={J:J is a proper right semipure hyperideal and J⊇I}. Clearly, X≠∅, since I∈X. Moreover, any J∈X is a proper hyperideal because 1∉J. Hence, any directed union of elements in X is still in X. So, X is inductively ordered. Hence, by Zorn's Lemma, X contains a maximal element J and any proper semipure hyperideal containing J also contains I and so it belongs to X. But such an hyperideal will be J itselft, since J is maximal in X. Therefore, J is semipurely maximal.Proposition 2.15.
LetI be a right pure (resp., semipure) hyperideal of H. Then, I is the intersection of purely (resp., semipurely) prime hyperideals of H containing I.Proof.
By Propositions2.14 and 2.11, there exists a set {Pα:Pα is a semipurely prime hyperideal containing I,α∈Λ}. Hence I⊆⋂α∈ΛPα. To prove ⋂α∈ΛPα⊆I, assume there exists an element x such that x∉I. Then by the above proposition, there exists a semipurely prime hyperideal Pα0 such that I⊆Pα0, but x∉Pα0. This implies that x∉⋂α∈ΛPα. This proves the proposition.Definition 2.16.
LetA be a right hyperideal of H and let {Kα:α∈Λ} be the set of right pure hyperideals containing A. Then, we define P(A)=⋂α∈ΛKα and call it the pure hyperadical of A. Note that the set {Kα:Kα is a right pure hyperideal containing A} is nonempty, since H itself belongs to this set.Proposition 2.17.
IfP(A) is the pure hyperradical of the hyperideal A, then each of the following statement holds true:  (1)
P(A) is either pure or semipure hyperideal containing A.(2)
P(A) is contained in every right pure hyperideal which contains A.(3)
IfPα are those purely prime hyperideals which contain A, then P(A)=⋂αPα.Proof.
(1) If the set{Kα:α∈Λ,Kα is a right pure hyperideal of H containing A} consists of H alone, then P(A)=H. Hence, P(A) is pure in this case. In case the set {Kα:Kα is a right pure hyperideal containing A} has only a finite number of elements, then P(A) is pure by Proposition 2.8. In general, P(A) is semiprime.
(2) This is obvious.
(3) Since every pure hyperideal is contained in a purely maximal hyperideal (Proposition2.14) and every purely maximal hyperideal is purely prime (Proposition 2.11), the set {Pα:Pα is purely prime containing A} is nonempty. Hence, from Part (2), it follows that P(A)⊆⋂αPα. We prove that ⋂αPα⊆P(A). To prove this, assume that x∉P(A). Since P(A)=⋂αKα, where each Kα is a right pure hyperideal containing A. Hence, x∉Kα0 for some α0. Thus Kα0 is a proper pure hyperideal which contains A but misses x. Hence, by Proposition 2.13, there exists a purely prime hyperideal Pα0, such that A⊆Kα0⊆Pα0 and x∉Pα0. Hence, x∉⋂αPα, where Pα's are purely prime hyperideals containing A. From this we conclude that P(A)=⋂αPα.Definition 2.18.
LetA be a hyperideal of H. Then H(A) the semipure part of A, that is, the union of all semipure hyperideals contained in A is called the semipure hyperradical of A.Proposition 2.19.
For each hyperidealA, H(A) is the intersection of semipurely prime hyperideals.Proof.
It follows from Proposition2.15.Note thatP(A) and H(A) are distinct in general. For example, if A={0,d} of Example 2.4, then
(2.3)P(A)={0,d,f},H(A)={0}.



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