On Pure Hyperradical in Semihypergroups

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.


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,  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 Γsemihypergroups.
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

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.
Left pure left hyperideals and left pure hyperideals are defined analogously.
Hence, there exists y ∈ A such that x ∈ x • y.This proves that A is a right pure hyperideal.Proof.0 and H are obviously right pure hyperideals.Let I 1 and I 2 be right pure hyperideals and let x ∈ I 1 ∩ I 2 .Since x ∈ I 1 and I 1 is right pure, exists y 1 ∈ I 1 such that x ∈ x • y 1 .Similarly, exists y 2 ∈ I 2 such that x ∈ x • y 2 .Thus, we have it follows that I 1 ∩ I 2 is right pure.The remaining cases of this proposition can be similarly proved.
It follows from the above proposition that if I 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.In Example 2.4, the hyperideal I 6 mentioned above is purely prime hyperideal.

International Journal of Mathematics and Mathematical Sciences 5
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.Let M be a maximal hyperideal of H.We show that L M , the pure part of M, is purely prime.Suppose Proposition 2.13.If I 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 set X ordered by inclusion: X {J : J is a right semipure hyperideal, I ⊆ J, a / ∈ J}.Then, X / ∅, since I ∈ X.Let J k k∈K be a totally ordered subset of X.Clearly, k J k is a semipure hyperideal with a / ∈ k J k .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 I 1 , I 2 are right semipure hyperideals such that I 1 / ⊆J and I 2 / ⊆J.Since I k k 1, 2 and J are semipure, I k ∪ J is a semipure hyperideal such that J ⊆ I k ∪ J.We then claim that an a ∈ I k ∪ J k 1, 2 .Because, if a / ∈ I k ∪ J, then by the maximality of J, we have I k ∪ J ⊆ J.But this contradicts the assumption I k / ⊆J k 1, 2 .Hence, a ∈ I 1 ∩ I 2 ∪ J .Since a / ∈ J, it follows that I 1 ∩ I 2 / ⊆J.Hence, by contrapositivity, we conclude that J is semipurely prime.Proposition 2.14.Let I 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.Let I be a right pure (resp., semipure) hyperideal of H.Then, I is the intersection of purely (resp., semipurely) prime hyperideals of H containing I.

International Journal of Mathematics and Mathematical Sciences
Definition 2.16.Let A be a right hyperideal of H and let {K α : α ∈ Λ} be the set of right pure hyperideals containing A. Then, we define P A α∈Λ K α and call it the pure hyperadical of A. Note that the set {K α : K α is a right pure hyperideal containing A} is nonempty, since H itself belongs to this set.Proposition 2.17.If P 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. 2 This is obvious.
3 Since every pure hyperideal is contained in a purely maximal hyperideal Proposition 2.14 and every purely maximal hyperideal is purely prime Proposition 2.11 , the set {P α : P α is purely prime containing A} is nonempty.Hence, from Part 2 , it follows that P A ⊆ α P α .We prove that α P α ⊆ P A .To prove this, assume that x / ∈ P A .Since P A α K α , where each K α is a right pure hyperideal containing A. Hence, x / ∈ K α 0 for some α 0 .Thus K α 0 is a proper pure hyperideal which contains A but misses x.Hence, by Proposition 2.13, there exists a purely prime hyperideal P α 0 , such that A ⊆ K α 0 ⊆ P α 0 and x / ∈ P α 0 .Hence, x / ∈ α P α , where P α 's are purely prime hyperideals containing A. From this we conclude that P A α P α .
Definition 2.18.Let A 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.
{a, b} c {c, d} e e, f a b {c, d} c {c, d} c {c, d} c d e, f c {c, d} e e, f e f 0

Corollary 2 . 7 .
If A 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).

Definition 2 . 9 .
Let I 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 Example 2.4, I 4 {0, c, d}, I 5 {0, c, d, e, f}, and I 6 {0, a, b, c, d, e, f} are right pure hyperideals of the semihypergroup H, and it is clear that I 6 is purely maximal hyperideal.Definition 2.10.Let I 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 I 1 and I 2 , I 1 ∩ I 2 ⊆ I ⇒ I 1 ⊆ I or I 2 ⊆ I.
• : 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.A hyperstructure 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 and H is called in this case supersemihypergroup of B. 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.
{0, d, f}, and I 3 {0, b, d, f} are right pure right hyperideals of H.I 4{0, c, d} is a two-sided hyperideal of H which is a right pure hyperideal but not a left pure hyperideal.Also, I 4 is a right and left semipure hyperideals.I 5 {0, c, d, e, f} is a two-sided hyperideal of H which is a right and left pure hyperideal. .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.Let A be a two-sided hyperideal of H. Then A is right pure if and only for any right hyperideal

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If P α are those purely prime hyperideals which contain A, then P A α P α .Proof. 1 If the set {K α : α ∈ Λ, K α 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 {K α : K α 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.

Proposition 2.19.
For each hyperideal A, H A is the intersection of semipurely prime hyperideals.Proof.It follows from Proposition 2.15.Note that P A and H A are distinct in general.For example, if A {0, d} of Example 2.4, then