On Pure Hyperradical in Semihypergroups

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.

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.

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 _{𝑢∈𝑥∘𝑦}𝑢∘𝑧=_{𝑣∈𝑦∘𝑧}𝑥∘𝑣. (1.1) If 𝑥∈𝐻 and 𝐴,𝐵 are nonempty subsets of 𝐻, then 𝐴∘𝐵=_{𝑎∈𝐴,𝑏∈𝐵}𝑎∘𝑏,𝐴∘𝑥=𝐴∘{𝑥},𝑥∘𝐵={𝑥}∘𝐵. (1.2)

A nonempty subset 𝐵 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 subset 𝐼 of a semihypergroup 𝐻 is called a right (left) hyperideal of 𝐻 if for all 𝑥∈𝐻 and 𝑟∈𝐼, 𝑟∘𝑥⊆𝐼(𝑥∘𝑟⊆𝐼). (1.3) 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 element 𝑒 in a semihypergroup 𝐻 is called scalar identity if 𝑥∘𝑒=𝑒∘𝑥={𝑥},∀𝑥∈𝐻. (1.4) An element 0 in a semihypergroup 𝐻 is called zero element if 𝑥∘0=0∘𝑥={0},∀𝑥∈𝐻. (1.5)

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. Let 𝐻 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. Let 𝐻 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: ∘0𝑥𝑦𝑧𝑡10𝑥0000000{}𝑦1,𝑥}𝑥,𝑦,1{0,1,𝑥,𝑧𝐻𝑥0𝑦𝑦𝑧0,𝑦𝑦,𝑡,10,1,𝑦,𝑡{}𝑧{}𝑧𝑡0𝑧𝑧,𝑡𝑧,𝑡0{}𝑡10,𝑡}{0,𝑡}{0,𝑡}{0,𝑡0𝑥𝑦𝑧𝑡1 (2.1) 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: ∘0𝑎𝑏𝑐𝑑𝑒𝑓10𝑎00000000{}𝑐{}𝑒𝑎𝑏0𝑎𝑎,𝑏𝑐,𝑑𝑒,𝑓𝑐0𝑏𝑏𝑑𝑑𝑓𝑓𝑏{}𝑐{}𝑐{}𝑐𝑑0𝑐𝑐,𝑑𝑐,𝑑𝑐,𝑑𝑒0𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑐{}𝑒𝑒𝑓0𝑒𝑒,𝑓𝑐,𝑑𝑒,𝑓10𝑓𝑓𝑑𝑑𝑓𝑓𝑓0𝑎𝑏𝑐𝑑𝑒𝑓1 (2.2) 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. Let 𝐻=[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. Let 𝐴 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. Suppose 𝐴 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. 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).

Proof. (0) and 𝐻 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 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.

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, 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. 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 𝐼₁ and 𝐼₂, 𝐼₁∩𝐼₂⊆𝐼⇒𝐼₁⊆𝐼 or 𝐼₂⊆𝐼.

In Example 2.4, 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. Suppose 𝐼 is purely maximal, and 𝐼₁,𝐼₂ are right pure hyperideals such that 𝐼₁∩𝐼₂⊆𝐼. Suppose 𝐼₁⊆𝐼̸. 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. Let 𝑀 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⊆𝑀̸, 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. 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, ⋃_𝑘𝐽_𝑘 is a semipure hyperideal with 𝑎∉⋃_𝑘𝐽_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 𝐼₁,𝐼₂ are right semipure hyperideals such that 𝐼₁⊆𝐽̸ and 𝐼₂⊆𝐽̸. 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⊆𝐽̸(𝑘=1,2). Hence, 𝑎∈(𝐼₁∩𝐼₂∪𝐽). Since 𝑎∉𝐽, it follows that 𝐼₁∩𝐼₂⊆𝐽̸. 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.

Proof. By Propositions 2.14 and 2.11, there exists a set {𝑃_𝛼∶𝑃_𝛼 is a semipurely prime hyperideal containing 𝐼,𝛼∈Λ}. Hence 𝐼⊆⋂_𝛼∈Λ𝑃_𝛼. To prove ⋂_𝛼∈Λ𝑃_𝛼⊆𝐼, assume there exists an element 𝑥 such that 𝑥∉𝐼. Then by the above proposition, there exists a semipurely prime hyperideal 𝑃_𝛼0 such that 𝐼⊆𝑃_𝛼0, but 𝑥∉𝑃_𝛼0. This implies that 𝑥∉⋂_𝛼∈Λ𝑃_𝛼. This proves the proposition.

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∈Λ𝐾_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. (3)If 𝑃_𝛼 are those purely prime hyperideals which contain 𝐴, then 𝑃(𝐴)=⋂_𝛼𝑃_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.
(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)⊆⋂_𝛼𝑃_𝛼. We prove that ⋂_𝛼𝑃_𝛼⊆𝑃(𝐴). To prove this, assume that 𝑥∉𝑃(𝐴). Since 𝑃(𝐴)=⋂_𝛼𝐾_𝛼, where each 𝐾_𝛼 is a right pure hyperideal containing 𝐴. Hence, 𝑥∉𝐾_𝛼0 for some 𝛼₀. Thus 𝐾_𝛼0 is a proper pure hyperideal which contains 𝐴 but misses 𝑥. Hence, by Proposition 2.13, there exists a purely prime hyperideal 𝑃_𝛼0, such that 𝐴⊆𝐾_𝛼0⊆𝑃_𝛼0 and 𝑥∉𝑃_𝛼0. Hence, 𝑥∉⋂_𝛼𝑃_𝛼, where 𝑃_𝛼's are purely prime hyperideals containing 𝐴. From this we conclude that 𝑃(𝐴)=⋂_𝛼𝑃_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.

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 𝑃(𝐴)=,𝐻(𝐴)={0}.0,𝑑,𝑓 (2.3)