On Intra-Regular Left Almost Semihypergroups with Pure Left Identity

Abstract

We characterize intra-regular LA-semihypergroups by using the properties of their left and right hyperideals, and we investigate some useful conditions for an LA-semihypergroup to become an intra-regular LA-semihypergroup.

1. Introduction

Kazim and Naseeruddin [1] introduced the concept of left almost semigroups (abbreviated as LA-semigroups) and right almost semigroups (abbreviated as RA-semigroups). They generalized some useful results of semigroup theory. Later, Mushtaq [2, 3] and others further investigated the structure and added many useful results to the theory of LA-semigroups; see also [4–9]. An LA-semigroup is the midway structure between a commutative semigroup and a groupoid. It nevertheless possesses many interesting properties which we usually find in commutative and associative algebraic structures. Mushtaq and Yusuf produced useful results [10] on locally associative LA-semigroups in 1979. In this structure, they defined powers of an element and congruences using these powers. There are several results which have been added to the theory of LA-semigroups by Mushtaq, Shabir, Aslam, Davvaz, Madad, Hila, Chinram, Holgate, Jezek, Protic, and many other researchers.

Hyperstructure theory was introduced in 1934, when Marty [11] defined hypergroups, 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. Nowadays, hyperstructures have a lot of applications to several domains of mathematics and computer science, and they are studied in many countries of the world. 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. A lot of papers and several books have been written on hyperstructure theory; see [12, 13]. A recent book on hyperstructures [14] points out on their applications in rough set theory, cryptography, codes, automata, probability, geometry, lattices, binary relations, graphs, and hypergraphs. Another book [15] is devoted especially to the study of hyperring theory. The volume ends with an outline of applications in chemistry and physics, analyzing several special kinds of hyperstructures: 𝑒-hyperstructures and transposition hypergroups. Many authors studied different aspects of semihypergroups, for instance, Aslam et al. [16–20], Corsini et al. [21, 22], Davvaz et al. [23, 24], Hasankhani [25], Hila et al. [26], Leoreanu [27], Onipchuk [28], and Yaqoob et al. [29–34].

Recently, Hila and Dine [35] introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semigroups. They investigated several properties of hyperideals of LA-semihypergroup and defined the topological space and studied the topological structure of LA-semihypergroups using hyperideal theory.

In this paper, we will prove some results on intra-regular LA-semihypergroups.

2. Basic Definitions and Examples

In this section, we recall certain definitions and results needed for our purpose.

Definition 1. 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 hypergroupoid is a set 𝐻 together with a (binary) hyperoperation.

If 𝐴 and 𝐵 are two nonempty subsets of 𝐻, then we denote 𝐴∘𝐵=_{𝑎∈𝐴,𝑏∈𝐵}{𝑎}{𝑎}𝑎∘𝑏,𝑎∘𝐴=∘𝐴,𝑎∘𝐵=∘𝐵. (1)

Definition 2 (see [35]). A hypergroupoid (𝐻,∘) is called an LA-semihypergroup if for all 𝑥,𝑦,𝑧∈𝐻, 𝑥∘𝑦∘𝑧=𝑧∘𝑦∘𝑥. (2)

The law (𝑥∘𝑦)∘𝑧=(𝑧∘𝑦)∘𝑥 is called a left invertive law.

Example 3. Let 𝐻={𝑥,𝑦,𝑧,𝑤,𝑡} with the binary hyperoperation defined below: ∘𝑥𝑦𝑧𝑤𝑡𝑥𝑦𝑥𝑥𝑥𝑥𝑥𝑥{}𝑧{}𝑧𝑧,𝑡𝑥,𝑤}{𝑧,𝑡{}𝑧𝑤𝑥𝑧𝑧𝑥,𝑤𝑥{}𝑤{}𝑡𝑥,𝑤}{𝑥,𝑤𝑥,𝑤𝑥𝑦,𝑡𝑧{𝑥,𝑤}𝑦,𝑡 (3) Clearly, 𝐻 is not a semihypergroup because {𝑦,𝑧,𝑡}=(𝑡∘𝑡)∘𝑦≠𝑡∘(𝑡∘𝑦)={𝑦,𝑡}. Thus, 𝐻 is an LA-semihypergroup because the elements of 𝐻 satisfy the left invertive law.

Example 4. Let 𝐻=ℤ. If we define 𝑥∘𝑦=𝑦−𝑥+3ℤ, where 𝑥,𝑦∈ℤ. Then, (𝐻,∘) becomes an LA-semihypergroup as =𝑥∘𝑦∘𝑧=𝑧−𝑥∘𝑦+3ℤ=𝑧−𝑦−𝑥+3ℤ+3ℤ=𝑧−𝑦+𝑥−3ℤ+3ℤ=𝑥−𝑦+𝑧−3ℤ+3ℤ=𝑥−𝑦−𝑧+3ℤ+3ℤ=𝑥−𝑧∘𝑦+3ℤ𝑧∘𝑦∘𝑥. (4) This implies that (𝑥∘𝑦)∘𝑧=(𝑧∘𝑦)∘𝑥 holds for all 𝑥,𝑦,𝑧∈𝐻=ℤ, and also it is clear that (𝑥∘𝑦)∘𝑧≠𝑥∘(𝑦∘𝑧). Hence (𝐻,∘) is an LA-semihypergroup.

Every LA-semihypergroup satisfies the law 𝑥∘𝑦∘(𝑧∘𝑤)=(𝑥∘𝑧)∘𝑦∘𝑤 (5) for all 𝑥,𝑦,𝑧,𝑤∈𝐻. This law is known as medial law (cf. [35]).

Definition 5. Let 𝐻 be an LA-semihypergroup. An element 𝑒∈𝐻 is called(i)left identity (resp., pure left identity) if for all 𝑎∈𝐻, 𝑎∈𝑒∘𝑎 (resp., 𝑎=𝑒∘𝑎),(ii)right identity (resp., pure right identity) if for all 𝑎∈𝐻, 𝑎∈𝑎∘𝑒 (resp., 𝑎=𝑎∘𝑒),(iii)identity (resp., pure identity) if for all 𝑎∈𝐻, 𝑎∈𝑒∘𝑎∩𝑎∘𝑒 (resp., 𝑎=𝑒∘𝑎∩𝑎∘𝑒).

Example 6. Let 𝐻={𝑥,𝑦,𝑧,𝑤} with the binary hyperoperation defined below: ∘𝑥𝑦𝑧𝑤𝑥𝑦𝑥𝑦𝑧𝑤𝑧𝑤𝑧𝑦,𝑧𝑦,𝑧𝑦𝑤𝑤𝑦,𝑧𝑦,𝑧𝑤𝑤𝑤𝐻 (6) Clearly 𝐻 is an LA-semihypergroup because the elements of 𝐻 satisfy the left invertive law. Here, 𝑥 is a pure left identity because for all 𝑎∈𝐻, 𝑎=𝑥∘𝑎. And, in Example 6, one can see that 𝑡 is a left identity but not a pure left identity.

Lemma 7. Let 𝐻 be an LA-semihypergroup with pure left identity 𝑒. Then, 𝑥∘(𝑦∘𝑧)=𝑦∘(𝑥∘𝑧) holds for all 𝑥,𝑦,𝑧∈𝐻.

Proof. Let 𝐻 be an LA-semihypergroup with pure left identity 𝑒. Then, for all 𝑥,𝑦,𝑧∈𝐻 and by medial law, we have =()∘=∘()().𝑥∘𝑦∘𝑧𝑒∘𝑥𝑦∘𝑧𝑒∘𝑦𝑥∘𝑧=𝑦∘𝑥∘𝑧 (7) This completes the proof.

Lemma 8. Let 𝐻 be an LA-semihypergroup with pure left identity 𝑒. Then, (𝑥∘𝑦)∘(𝑧∘𝑤)=(𝑤∘𝑦)∘(𝑧∘𝑥) holds for all 𝑥,𝑦,𝑧,𝑤∈𝐻.

Proof. Let 𝐻 be an LA-semihypergroup with pure left identity 𝑒. Then, for all 𝑥,𝑦,𝑧,𝑤∈𝐻 and by medial law, we have ∘()=()∘))=∘))=∘()∘()=)∘(∘=((∘=()∘=∘().𝑥∘𝑦𝑧∘𝑤𝑒∘𝑥∘𝑦((𝑒∘𝑧∘𝑤𝑦∘𝑥∘𝑒((𝑤∘𝑧∘𝑒𝑦∘𝑥𝑤∘𝑧𝑒∘𝑒((𝑒∘𝑒𝑤∘𝑧))𝑦∘𝑥𝑒∘𝑤∘𝑧))𝑦∘𝑥𝑤∘𝑧𝑦∘𝑥𝑤∘𝑦𝑧∘𝑥 (8) This completes the proof.

The law (𝑥∘𝑦)∘(𝑧∘𝑤)=(𝑤∘𝑦)∘(𝑧∘𝑥) is called a paramedial law.

Definition 9. Let 𝐻 be an LA-semihypergroup. A nonempty subset 𝐴 of 𝐻 is called a sub LA-semihypergroup of 𝐻 if 𝑥∘𝑦⊆𝐴 for every 𝑥,𝑦∈𝐴.

Definition 10. A subset 𝐼 of an LA-semihypergroup 𝐻 is called a right (left) hyperideal of 𝐻 if 𝐼∘𝐻⊆𝐼 (𝐻∘𝐼⊆𝐼) and is called a hyperideal if it is two-sided hyperideal.

Definition 11. By a bi-hyperideal of an LA-semihypergroup 𝐻, we mean a sub LA-semihypergroup 𝐵 of 𝐻 such that (𝐵∘𝐻)∘𝐵⊆𝐵.

Definition 12. A sub LA-semihypergroup 𝐵 of 𝐻 is called a (1,2)-hyperideal of 𝐻 if (𝐵∘𝐻)∘𝐵²⊆𝐵.

It is easy to note that each right hyperideal is a bi-hyperideal. If 𝐸(𝐵_𝐻) denotes the set of all idempotent subsets of 𝐻 with pure left identity 𝑒, then 𝐸(𝐵_𝐻) forms a hypersemilattice structure. The intersection of any set of bi-hyperideals of an LA-semihypergroup 𝐻 is either empty or a bi-hyperideal of 𝐻.

Definition 13. A sub LA-semihypergroup 𝑇 of 𝐻 is called an interior hyperideal of 𝐻 if (𝐻∘𝑇)∘𝐻⊆𝑇.

Definition 14. A nonempty subset 𝑄 of an LA-semihypergroup 𝐻 is called a quasi-hyperideal of 𝐻 if 𝑄∘𝐻∩𝐻∘𝑄⊆𝑄.

Lemma 15. If 𝐻 is an LA-semihypergroup with left identity 𝑒, then 𝐻∘𝐻=𝐻.

Proof. If 𝐻 is an LA-semihypergroup with left identity 𝑒, then 𝑥∈𝐻 implies that 𝑥∈𝑒∘𝑥⊆𝐻∘𝐻andso𝐻⊆𝐻∘𝐻. (9) That is, 𝐻=𝐻∘𝐻.

Corollary 16. If 𝐻 is an LA-semihypergroup with pure left identity 𝑒, then 𝐻∘𝐻=𝐻 and 𝐻=𝑒∘𝐻=𝐻∘𝑒.

Proof. The proof is similar to the proof of Lemma 15.

In an LA-semigroup, every right identity becomes a left identity. But in an LA-semihypergroup, every right identity needs not to be a left identity. For this, let 𝐻={𝑥,𝑦,𝑧} with the binary hyperoperation defined below: ∘𝑥𝑦𝑧𝑥{}𝑧𝑦𝑥,𝑧𝑦,𝑧𝑧𝑦,𝑧𝑧𝑧𝑦,𝑧𝑦,𝑧𝑦,𝑧 (10) Clearly, 𝐻 is not a semihypergroup because {𝑦,𝑧}=(𝑦∘𝑦)∘𝑧≠𝑦∘(𝑦∘𝑧)={𝑧}. Thus, 𝐻 is an LA-semihypergroup because the elements of 𝐻 satisfy the left invertive law. Here, 𝑥 is a right identity but not a left identity. However, if an LA-semihypergroup 𝐻 has a pure right identity 𝑒, then 𝑒 becomes a pure left identity. For this, consider 𝑏∈𝐻 and 𝑒 be a pure right identity of 𝐻 then; 𝑏=𝑏∘𝑒=(𝑏∘𝑒)∘𝑒=(𝑒∘𝑒)∘𝑏=𝑒∘𝑏. (11) This shows that in an LA-semihypergroup, every pure right identity becomes a pure left identity. Every LA-semihypergroup with pure right identity becomes a commutative hypermonoid.

Theorem 17. An LA-semihypergroup 𝐻 is a semihypergroup if and only if 𝑎∘(𝑏∘𝑐)=(𝑐∘𝑏)∘𝑎 holds for all 𝑎,𝑏,𝑐∈𝐻.

Proof. Let 𝐻 be a semihypergroup. Then, we have (𝑎∘𝑏)∘𝑐=𝑎∘(𝑏∘𝑐) for all 𝑎,𝑏,𝑐∈𝐻, but (𝑎∘𝑏)∘𝑐=(𝑐∘𝑏)∘𝑎; thus, 𝑎∘(𝑏∘𝑐)=(𝑐∘𝑏)∘𝑎∀𝑎,𝑏,𝑐∈𝐻. (12) On the other hand, suppose that 𝑎∘(𝑏∘𝑐)=(𝑐∘𝑏)∘𝑎 holds for all 𝑎,𝑏,𝑐∈𝐻. Since 𝐻 is an LA-semihypergroup, therefore 𝑎∘(𝑏∘𝑐)=(𝑐∘𝑏)∘𝑎=(𝑎∘𝑏)∘𝑐. (13) Thus, 𝐻 is a semihypergroup. This completes the proof.

3. Intra-Regular LA-Semihypergroups

In this section, we will characterize intra-regular LA-semihypergroup by using the properties of left and right hyperideals.

Definition 18. Let 𝐻 be an LA-semihypergroup and 𝑎∈𝐻. Then, 𝑎 is said to be regular if there exist an element 𝑥∈𝐻 such that 𝑎∈(𝑎∘𝑥)∘𝑎. The LA-semihypergroup 𝐻 is called regular if every element of 𝐻 is regular.

Definition 19. An element 𝑎 of an LA-semihypergroup 𝐻 is called an intra-regular element if there exist 𝑥,𝑦∈𝐻 such that 𝑎∈(𝑥∘𝑎²)∘𝑦 and 𝐻 is called intra-regular, if every element of 𝐻 is intra-regular.

Example 20. Let 𝐻={𝑥,𝑦,𝑧,𝑤} with the binary hyperoperation defined below: ∘𝑥𝑦𝑧𝑤𝑥𝑥{}𝑤𝑦𝑥,𝑤}{𝑥,𝑤{𝑤𝑧𝑥,𝑤}𝑦,𝑧𝑦,𝑧{}𝑤𝑥,𝑤𝑦𝑦𝑤𝑤𝑤𝑤𝑤 (14) Clearly, 𝐻 is an LA-semihypergroup because the elements of 𝐻 satisfy the left invertive law. Here, 𝐻 is intra-regular because, 𝑥∈(𝑦∘𝑥²)∘𝑧, 𝑦∈(𝑧∘𝑦²)∘𝑧, 𝑧∈(𝑦∘𝑧²)∘𝑦, 𝑤∈(𝑥∘𝑤²)∘𝑧.

Example 21. Let 𝐻=ℤ. Define a hyperoperation ∘ on 𝐻 by 𝑥∘𝑦=𝑥,𝑦∪2ℤ∀𝑥,𝑦∈𝐻. (15) Then, for all 𝑥,𝑦,𝑧∈𝐻, we have 𝑥∘𝑦∘𝑧=𝑥,𝑦,𝑧∪2ℤ=𝑧∘𝑦∘𝑥. (16) This implies that (𝐻,∘) is a an LA-semihypergroup. Since 𝑥∈(𝑥∘𝑥)∘𝑥={𝑥}∪2ℤ and also 𝑥∈(𝑥∘(𝑥∘𝑥))∘𝑥={𝑥}∪2ℤ. Thus, (𝐻,∘) is a regular as well as intra-regular LA-semihypergroup.

Definition 22. An element 𝑎 of an LA-semihypergroup 𝐻 with left identity 𝑒 is called a left (right) invertible if there exist 𝑥∈𝐻 such that 𝑒∈𝑥∘𝑎 (𝑒∈𝑎∘𝑥) and 𝑎 is called invertible if it is both a left and a right invertible. An LA-semihypergroup 𝐻 is called a left (right) invertible if every element of 𝐻 is a left (right) invertible and 𝐻 is called invertible if it is both a left and a right invertible.

Theorem 23. Every LA-semihypergroup 𝐻 with pure left identity 𝑒 is intra-regular if 𝐻 is left (right) invertible.

Proof. Let 𝐻 be a left invertible LA-semihypergroup with pure left identity 𝑒. Then, for 𝑎∈𝐻, there exist 𝑎^′∈𝐻 such that 𝑒∈𝑎^′∘𝑎. Now, by using left invertive law, and medial law, Lemmas 7 and 15, we have 𝑎=𝑒∘𝑎=𝑒∘(𝑒∘𝑎)⊆𝑎^′∘()⊆()∘()=()∘))=()∘))=()∘))=((∘()=((((∘𝑎𝑒∘𝑎𝐻∘𝑎𝐻∘𝑎𝐻∘𝑎((𝐻∘𝐻∘𝑎𝐻∘𝑎((𝑎∘𝐻∘𝐻𝑎∘𝐻((𝐻∘𝑎∘𝐻𝑎∘𝐻∘𝑎))𝐻∘𝐻𝑎∘𝐻∘𝑎))∘𝐻=𝐻∘𝑎∘𝑎))∘𝐻=𝐻∘𝑎²∘𝐻. (17) This shows that 𝐻 is intra-regular. The case for right invertible can be seen in a similar way.

Theorem 24. An LA-semihypergroup 𝐻 with left identity 𝑒 is intra-regular if 𝐻∘𝑎=𝐻 or 𝑎∘𝐻=𝐻 for all 𝑎∈𝐻.

Proof. Let 𝐻 be an LA-semihypergroup such that 𝐻∘𝑎=𝐻 holds for all 𝑎∈𝐻. Then, 𝐻=𝐻². Let 𝑎∈𝐻, therefore, by using medial law, we have ())∘(=)∘(𝑎∈𝐻=𝐻∘𝐻∘𝐻=((𝐻∘𝑎𝐻∘𝑎))∘𝐻((𝐻∘𝐻𝑎∘𝑎))∘𝐻=𝐻∘𝑎²∘𝐻. (18) This shows that 𝐻 is intra-regular.
Let 𝑎∈𝐻 and assume that 𝑎∘𝐻=𝐻 holds for all 𝑎∈𝐻. Then, by using left invertive law, we have 𝑎∈𝐻=𝐻∘𝐻=(𝑎∘𝐻)∘𝐻=(𝐻∘𝐻)∘𝑎=𝐻∘𝑎. (19) Thus, 𝐻∘𝑎=𝐻 holds for all 𝑎∈𝐻; therefore, it follows from above that 𝐻 is intra-regular.

Corollary 25. An LA-semihypergroup 𝐻 with pure left identity 𝑒 is intra-regular if 𝐻∘𝑎=𝐻 or 𝑎∘𝐻=𝐻 for all 𝑎∈𝐻.

Corollary 26. If 𝐻 is an LA-semihypergroup such that 𝑎∘𝐻=𝐻 holds for all 𝑎∈𝐻, then 𝐻∘𝑎=𝐻 holds for all 𝑎∈𝐻.

Theorem 27. If 𝐻 is an intra-regular LA-semihypergroup with pure left identity 𝑒, then (𝐵∘𝐻)∘𝐵=𝐵, where 𝐵 is a bi- (generalized bi-) hyperideal of 𝐻.

Proof. Let 𝐻 be an intra-regular LA-semihypergroup with pure left identity 𝑒. Then, clearly, (𝐵∘𝐻)∘𝐵=𝐵. Now let 𝑏∈𝐵. Since 𝐻 is intra-regular, so there exist 𝑥,𝑦∈𝐻 such that 𝑏∈(𝑥∘𝑏²)∘𝑦. Now, by using Lemma 7, left invertive law, paramedial law and medial law, we have 𝑏∈𝑥∘𝑏²((((=()⊆∘𝑦=𝑥∘𝑏∘𝑏))∘𝑦=𝑏∘𝑥∘𝑏))∘𝑦𝑦∘𝑥∘𝑏∘𝑏𝑦∘𝑥∘𝑥∘𝑏²=∘𝑦∘𝑏𝑦∘𝑥∘𝑏²∘=𝑥∘𝑦∘𝑏𝑥∘𝑏²∘=∘𝑏𝑦∘𝑥∘𝑦∘𝑏𝑥∘𝑦∘𝑦²=()∘=()∘∘=()∘𝑥∘𝑥∘𝑏𝑏∘𝑏𝑥∘𝑦∘𝑦∘𝑥∘𝑏𝑏∘𝑏𝑥∘𝑦𝑥∘𝑦∘𝑏𝑏∘𝑏²∘𝑦²=𝑦∘𝑏₂∘𝑥²∘()=𝑦𝑏∘𝑏∘𝑏𝑏∘²∘𝑥²∘𝑏∘𝑏⊆(𝐵∘𝐻)∘𝐵. (20) This shows that (𝐵∘𝐻)∘𝐵=𝐵.

The converse is not true in general. For this, let us consider an LA-semihypergroup 𝐻={𝑥,𝑦,𝑧,𝑤} with the binary hyperoperation defined below: ∘𝑥𝑦𝑧𝑤𝑥{}𝑤𝑦𝑦𝑦𝑧,𝑤{}𝑤𝑧𝑦𝑦𝑧,𝑤{}𝑤𝑧,𝑤}{𝑧,𝑤𝑧𝑤𝑤𝑤𝑤𝑤 (21) Clearly, 𝐻 is an LA-semihypergroup because the elements of 𝐻 satisfy the left invertive law. It is easy to see that {𝑧,𝑤} is a bi-hyperideal of 𝐻 such that (𝐵∘𝐻)∘𝐵=𝐵, but 𝐻 has no pure left identity, and also 𝐻 is not an intra-regular because 𝑥∈𝐻 is not an intra-regular.

Theorem 28. If 𝐻 is an intra-regular LA-semihypergroup with pure left identity 𝑒, then (𝐻∘𝐵)∘𝐻=𝐵, where 𝐵 is an interior hyperideal of 𝐻.

Proof. Let 𝐻 be an intra-regular LA-semihypergroup with pure left identity 𝑒. Then, clearly, (𝐻∘𝐵)∘𝐻=𝐵. Now, let 𝑏∈𝐵. Since 𝐻 is an intra-regular, so there exist 𝑥,𝑦∈𝐻 such that 𝑏∈(𝑥∘𝑏²)∘𝑦. Now, by using paramedial law and left invertive law, we have 𝑏∈𝑥∘𝑏²)∘()∘(=)))()∘𝑦=((𝑒∘𝑥𝑏∘𝑏))∘𝑦=((𝑏∘𝑏𝑥∘𝑒))∘𝑦(((𝑥∘𝑒∘𝑏∘𝑏∘𝑦⊆𝐻∘𝐵∘𝐻. (22) This shows that (𝐻∘𝐵)∘𝐻=𝐵.

The converse is not true in general. For this, let us consider an LA-semihypergroup 𝐻={𝑒,𝑥,𝑦,𝑧,𝑤} with the binary hyperoperation defined below: ∘𝑒𝑥𝑦𝑧𝑤𝑒𝑥𝑒𝑥𝑦𝑧𝑤{}𝑤𝑦𝑦𝑧𝑧𝑧,𝑤{}𝑤𝑧𝑥𝑧𝑧𝑧,𝑤𝑧{}𝑤𝑤𝑧,𝑤}{𝑧,𝑤}{𝑧,𝑤𝑤𝑤𝑤𝑤𝑤 (23) Clearly, 𝐻 is an LA-semihypergroup because the elements of 𝐻 satisfy the left invertive law. It is easy to see that {𝑧,𝑤} is an interior hyperideal of 𝐻 with pure left identity 𝑒 such that (𝐻∘{𝑧,𝑤})∘𝐻={𝑧,𝑤}, but 𝐻 is not an intra-regular because 𝑥∈𝐻, is not an intra-regular.

Definition 29. Let 𝐻 be an LA-semihypergroup. Then, ∅≠𝐴⊆𝐻 is called semiprime if 𝑎²⊆𝐴implies 𝑎∈𝐴.

Theorem 30. An LA-semihypergroup 𝐻 with pure left identity is intra-regular if 𝐿∪𝑅=𝐿∘𝑅, where 𝐿 and 𝑅 are the left and right hyperideals of 𝐻, respectively, such that 𝑅 is semiprime.

Proof. Let 𝐻 be an LA-semihypergroup with pure left identity. Then, clearly, 𝐻∘𝑎 and 𝑎²∘𝐻 are the left and right hyperideals of 𝐻 such that 𝑎∈𝐻∘𝑎 and 𝑎²⊆𝑎²∘𝐻, because by using paramedial law, we have 𝑎²∘𝐻=(𝑎∘𝑎)∘(𝐻∘𝐻)=(𝐻∘𝐻)∘(𝑎∘𝑎)=𝐻∘𝑎². (24) Therefore, by the given assumption, 𝑎∈𝑎²∘𝐻. Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have 𝑎∈𝐻∘𝑎∪𝑎²∘𝐻=(𝐻∘𝑎)∘𝑎²=()∘))=()∘)∘(⊆()∘)∘(=()∘)∘(⊆()∘)∘(=()∘)∘(=()∘))=()∘))=((∘()=((((=∘𝐻𝐻∘𝑎((𝑎∘𝑎∘𝐻𝐻∘𝑎((𝐻∘𝑎𝑒∘𝑎))𝐻∘𝑎((𝐻∘𝑎𝐻∘𝑎))𝐻∘𝑎((𝐻∘𝐻𝑎∘𝑎))𝐻∘𝑎((𝐻∘𝐻𝐻∘𝑎))𝐻∘𝑎((𝑎∘𝐻𝐻∘𝐻))𝐻∘𝑎((𝑎∘𝐻∘𝐻𝑎∘𝐻((𝐻∘𝑎∘𝐻𝑎∘𝐻∘𝑎))𝐻∘𝐻𝑎∘𝐻∘𝑎))∘𝐻=𝐻∘𝑎∘𝑎))∘𝐻𝐻∘𝑎²∘𝐻. (25) This shows that 𝐻 is intra-regular.

Lemma 31. If 𝐻 is an intra-regular LA-semihypergroup, then 𝐻=𝐻².

Proof. The proof is straightforward.

Theorem 32. For a left invertible LA-semihypergroup 𝐻 with pure left identity, the following conditions are equivalent:(i)𝐻 is intra-regular,(ii)𝑅∩𝐿=𝑅∘𝐿, where 𝑅 and 𝐿 are any right and left hyperideals of 𝐻, respectively.

Proof. (i)⇒(ii): assume that 𝐻is an intra-regular LA-semihypergroupwith pure left identity and let 𝑎∈𝐻. Then, there exist 𝑥,𝑦∈𝐻 such that 𝑎∈(𝑥∘𝑎²)∘𝑦. Let 𝑅 and 𝐿 be any right and left hyperideals of 𝐻, respectively. Then, obviously, 𝑅∘𝐿⊆𝑅∩𝐿. Now, let 𝑎∈𝑅∩𝐿 imply that 𝑎∈𝑅 and 𝑎∈𝐿. Now by using medial law, left invertive law, and Lemma 7, we have 𝑎∈𝑥∘𝑎²∘𝑦⊆𝐻∘𝑎²=((((=((∘()=()∘))=()∘))=()∘))=()∘()⊆()∘()=))∘()=))∘()∘𝐻𝐻∘𝑎∘𝑎))∘𝐻=𝑎∘𝐻∘𝑎))∘𝐻𝑎∘𝐻∘𝑎))𝐻∘𝐻𝑎∘𝐻((𝐻∘𝑎∘𝐻𝐻∘𝑎((𝑎∘𝐻∘𝐻𝐻∘𝑎((𝐻∘𝐻∘𝑎𝐻∘𝑎𝐻∘𝑎𝐻∘𝑅𝐻∘𝐿((𝐻∘𝐻∘𝑅𝐻∘𝐿((𝑅∘𝐻∘𝐻𝐻∘𝐿⊆𝑅∘𝐿. (26) This shows that 𝑅∩𝐿=𝑅∘𝐿.
(ii)⇒(i)∶ let 𝐻 be a left invertible LA-semihypergroup with pure left identity. Then, for 𝑎∈𝐻. There, exists 𝑎^′∈𝐻 such that 𝑒∈𝑎^′∘𝑎. Since 𝑎²∘𝐻 is a right hyperideal and also a left hyperideal of 𝐻such that 𝑎²⊆𝑎²∘𝐻, therefore, by using the given assumption, medial law, left invertive law, and Lemma 7, we have 𝑎²⊆𝑎²∘𝐻∩𝑎²∘𝐻=𝑎²∘𝐻∘𝑎²∘𝐻=𝑎²∘𝑎²∘𝐻∘𝐻=𝑎²∘(𝐻∘𝐻)∘𝑎²=(𝑎∘𝑎)∘𝐻∘𝑎²=𝐻∘𝑎²∘𝑎∘𝑎. (27) Thus, we get 𝑎²⊆((𝑥∘𝑎²)∘𝑎)∘𝑎 for some 𝑥∈𝐻.
Now, by using left invertive law, we have (𝑎∘𝑎)∘𝑎^′⊆𝑥∘𝑎²∘𝑎∘𝑎∘𝑎^′𝑎^′∘𝑎∘𝑎⊆𝑎^′∘𝑎∘𝑥∘𝑎²∘𝑎𝑎∈𝑥∘𝑎²∘𝑎. (28) This shows that 𝐻 is intra-regular.

Lemma 33. Every left hyperideal of an intra-regular LA-semihypergroup 𝐻 with pure left identity is idempotent.

Proof. The proof is straightforward.

Theorem 34. In an LA-semihypergroup 𝐻 with pure left identity, the following conditions are equivalent:(i)𝐻 is intra-regular,(ii)𝐴=(𝐻∘𝐴)², where 𝐴 is any left hyperideal of 𝐻.

Proof. (i)⇒(ii): let 𝐴 be a left hyperideal of an intra-regular LA-semihypergroup 𝐻 with left identity. Then, 𝐻∘𝐴⊆𝐴 and, by Lemma 33, (𝐻∘𝐴)²=𝐻∘𝐴⊆𝐴. Now 𝐴=𝐴∘𝐴⊆𝐻∘𝐴=(𝐻∘𝐴)², which implies that 𝐴=(𝐻∘𝐴)².
(ii)⇒(i): let 𝐴 be a left hyperideal of 𝐻. Then, 𝐴=(𝐻∘𝐴)²⊆𝐴², which implies that 𝐴 is idempotent; hence, 𝐻 is intra-regular.

Theorem 35. In an intra-regular LA-semihypergroup 𝐻 with pure left identity, the following conditions are equivalent:(i)𝐴 is a bi- (generalized bi-) hyperideal of 𝐻,(ii)(𝐴∘𝐻)∘𝐴=𝐴 and 𝐴²=𝐴.

Proof. (i)⇒(ii): let 𝐴 be a bi-hyperideal of an intra-regular LA-semihypergroup 𝐻 with pure left identity. Then, (𝐴∘𝐻)∘𝐴⊆𝐴. Let 𝑎∈𝐴. Then, since 𝐻 is intra-regular, so there exist 𝑥,𝑦∈𝐻 such that 𝑎∈(𝑥∘𝑎²)∘𝑦. Now, by using medial, left invertive law, and Lemma 7, we have 𝑎∈𝑥∘𝑎²((=((()⊆∘𝑦=𝑥∘𝑎∘𝑎))∘𝑦𝑎∘𝑥∘𝑎))∘𝑦=𝑦∘𝑥∘𝑎∘𝑎𝑦∘𝑥∘𝑥∘𝑎²=∘𝑦∘𝑎𝑦∘𝑥∘𝑎²∘=𝑥∘𝑦∘𝑎𝑥∘𝑎²∘=((∘=((∘=∘()∘=()∘∘=)∘)𝑦∘𝑥∘𝑦∘𝑎𝑥∘𝑎∘𝑎))𝑦∘𝑥∘𝑦∘𝑎𝑎∘𝑥∘𝑎))𝑦∘𝑥∘𝑦∘𝑎𝑎∘𝑦𝑥∘𝑎𝑥∘𝑦∘𝑎𝑥∘𝑎𝑎∘𝑦𝑥∘𝑦∘𝑎(𝑥∘𝑎(𝑎∘𝑥∘𝑦²=𝑦∘𝑎₂∘(∘(=𝑦𝑎∘𝑥)𝑎∘𝑥)∘𝑎𝑎∘²∘()⊆()𝑎∘𝑥∘𝑥∘𝑎𝐴∘𝐻∘𝐴. (29) Thus, (𝐴∘𝐻)∘𝐴=𝐴 holds. Now, by using left invertive law, paramedial law, medial law, and Lemma 7, we have 𝑎∈𝑥∘𝑎²∘((=((()⊆∘𝑦=𝑥∘𝑎∘𝑎))∘𝑦𝑎∘𝑥∘𝑎))∘𝑦=𝑦∘𝑥∘𝑎∘𝑎𝑦∘𝑥∘𝑥∘𝑎²=∘𝑦∘𝑎𝑦∘𝑥∘𝑎²∘=𝑥∘𝑦∘𝑎𝑥∘𝑎²∘=((∘=((∘=∘(=)∘=()∘𝑦𝑦∘𝑥∘𝑦∘𝑎𝑥∘𝑎∘𝑎))𝑦∘𝑥∘𝑦∘𝑎𝑎∘𝑥∘𝑎))𝑦∘𝑥∘𝑦∘𝑎𝑦∘𝑥∘𝑦𝑥∘𝑎)∘𝑎∘𝑎(𝑎∘𝑥𝑥∘𝑦∘𝑦∘𝑎∘𝑎𝑎∘𝑥²=∘𝑥∘𝑎∘𝑎𝑎∘𝑦²∘()=𝑥∘𝑥∘𝑎∘𝑎𝑎∘𝑦²∘𝑥²=𝑥∘𝑎∘𝑎₂∘𝑦²⊆𝑥∘𝑎∘𝑎∘𝑎₂∘𝑦²∘(=𝑥(𝑥∘𝑎∘𝑎))∘𝑦∘𝑎∘𝑎₂∘𝑦²∘((=𝑥𝑎∘𝑥∘𝑎))∘𝑦∘𝑎∘𝑎₂∘(𝑎∘(𝑥∘𝑎))∘𝑦²=𝑥∘𝑦∘𝑎∘𝑎𝑎∘²∘(𝑥∘𝑎)∘𝑦³=()∘(∘𝑎∘𝑎𝑎∘((𝑥∘𝑥𝑥∘𝑎)))∘𝑦³=()∘(∘𝑎∘𝑎𝑎∘((𝑎∘𝑥𝑥∘𝑥)))∘𝑦³=()∘∘𝑎∘𝑎𝑎∘𝑥𝑎∘𝑥²∘𝑦³=()∘∘𝑎∘𝑎𝑎∘𝑎𝑥∘𝑥²∘𝑦³=𝑦∘𝑎∘𝑎₃∘𝑥³∘()=𝑦𝑎∘𝑎∘𝑎∘𝑎𝑎∘³∘𝑥³⊆))∘𝑎∘𝑎∘𝑎((𝐴∘𝐻∘𝐴∘𝐴⊆𝐴∘𝐴=𝐴². (30) Hence, 𝐴=𝐴² holds.
(ii)⇒(i) is obvious.

Theorem 36. In an intra-regular LA-semihypergroup 𝐻 with pure left identity, the following conditions are equivalent:(i)𝐴 is a quasi-hyperideal of 𝐻,(ii)𝐻∘𝑄∩𝑄∘𝐻=𝑄.

Proof. (i)⇒(ii): let 𝑄 be a quasi hyperideal of an intra-regular LA-semihypergroup 𝐻 with pure left identity. Then, 𝐻∘𝑄∩𝑄∘𝐻⊆𝑄. Let 𝑞∈𝑄, then since 𝐻 is intra-regular, so there exist 𝑥,𝑦∈𝐻 such that 𝑞∈(𝑥∘𝑞²)∘𝑦. Let 𝑝∘𝑞⊆𝐻∘𝑄. Then, by using medial law, paramedial law and Lemma 7, we have 𝑝∘𝑞⊆𝑝∘𝑥∘𝑞²∘𝑦=𝑥∘𝑞²∘=∘=∘=∘=∘=∘𝑝∘𝑦𝑥∘𝑞∘𝑞𝑝∘𝑦𝑞∘𝑥∘𝑞𝑝∘𝑦𝑞∘𝑝𝑥∘𝑞∘𝑦𝑥∘𝑞𝑞∘𝑝∘𝑦𝑦∘𝑞∘𝑝𝑞∘𝑥=𝑞∘𝑦∘𝑞∘𝑝∘𝑥⊆𝑄∘𝐻. (31) Now, let 𝑞∘𝑝∈𝑄∘𝐻. Then, by using medial law, paramedial law and Lemma 7, we have 𝑞∘𝑝⊆𝑥∘𝑞²∘𝑦∘𝑝=𝑝∘𝑦∘𝑥∘𝑞²=∘∘∘=∘=𝑝∘𝑦𝑥∘𝑞∘𝑞=𝑥∘𝑝∘𝑦𝑞∘𝑞=𝑥∘𝑞∘𝑞𝑦∘𝑝𝑞∘𝑞𝑥∘𝑦∘𝑝𝑥∘𝑦∘𝑝∘𝑞∘𝑞⊆𝐻∘𝑄. (32) Hence, 𝑄∘𝐻=𝐻∘𝑄. As, by using left invertive law, medial law, and Lemma 7, we have 𝑞∈𝑥∘𝑞²=∘𝑦=𝑥∘𝑞∘𝑞∘𝑦=𝑞∘𝑥∘𝑞∘𝑦𝑦∘𝑥∘𝑞∘𝑞⊆𝐻∘𝑄. (33) Thus, 𝑞∈𝐻∘𝑄∩𝑄∘𝐻 implies that 𝐻∘𝑄∩𝑄∘𝐻=𝑄.
(ii)⇒(i) is obvious.

Theorem 37. In an intra-regular LA-semihypergroup 𝐻 with pure left identity, the following conditions are equivalent:(i)𝐴 is an interior hyperideal of 𝐻,(ii)(𝐻∘𝐴)∘𝐻=𝐴.

Proof. (i)⇒(ii): let 𝐴 be an interior hyperideal of an intra-regular LA-semihypergroup 𝐻 with pure left identity. Then, (𝐻∘𝐴)∘𝐻⊆𝐴. Let 𝑎∈𝐴, then since 𝐻 is intra-regular, so there exist 𝑥,𝑦∈𝐻 such that 𝑎∈(𝑥∘𝑎²)∘𝑦. Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have 𝑎∈𝑥∘𝑎²((=((()⊆()∘∘𝑦=𝑥∘𝑎∘𝑎))∘𝑦𝑎∘𝑥∘𝑎))∘𝑦=𝑦∘𝑥∘𝑎∘𝑎𝑦∘𝑥∘𝑎𝑥∘𝑎²=∘𝑦𝑥∘𝑎²∘()=()∘∘𝑦𝑥∘𝑎∘𝑦𝑎∘𝑥𝑦∘𝑥∘𝑎²=∘𝑦𝑦∘𝑥∘𝑎²⊆()∘𝑥∘𝑎∘𝑦𝐻∘𝐴∘𝐻. (34) Thus, (𝐻∘𝐴)∘𝐻=𝐴.
(ii)⇒(i) is obvious.

Theorem 38. In an intra-regular LA-semihypergroup 𝐻 with pure left identity, the following conditions are equivalent.(i)𝐴 is a (1,2)-hyperideal of 𝐻,(ii)(𝐴∘𝐻)∘𝐴²=𝐴 and 𝐴²=𝐴.

Proof. (i)⇒(ii): let 𝐴 be a (1,2)-hyperideal of an intra-regular LA-semihypergroup 𝐻 with pure left identity. Then, (𝐴∘𝐻)∘𝐴²⊆𝐴 and 𝐴²⊆𝐴. Let 𝑎∈𝐴. Then, since 𝐻 is intra-regular, so there exist 𝑥,𝑦∈𝐻 such that 𝑎∈(𝑥∘𝑎²)∘𝑦. Now, by using left invertive law, medial law, paramedial law and Lemma 7, we have 𝑎∈𝑥∘𝑎²((=((()⊆∘𝑦=𝑥∘𝑎∘𝑎))∘𝑦𝑎∘𝑥∘𝑎))∘𝑦=𝑦∘𝑥∘𝑎∘𝑎𝑦∘𝑥∘𝑥∘𝑎²=∘𝑦∘𝑎𝑦∘𝑥∘𝑎²∘=𝑥∘𝑦∘𝑎𝑥∘𝑎²∘=∘𝑎𝑦∘𝑥∘𝑦∘𝑎𝑥∘𝑦∘𝑦²=𝑦∘𝑥∘𝑎₂∘𝑥∘𝑎²=𝑎∘𝑥∘𝑎₂∘𝑦²=𝑎∘𝑥∘𝑥∘𝑎₂∘𝑥²∘𝑦²=𝑥∘𝑎𝑎∘²∘𝑦²∘𝑎²=𝑎∘𝑥²∘𝑦²∘(𝑎∘𝑎)⊆(𝐴∘𝐻)∘𝐴². (35) Thus, (𝐴∘𝐻)∘𝐴²=𝐴. Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have𝑎∈𝑥∘𝑎²((=((()𝑎⊆()∘∘𝑦=𝑥∘𝑎∘𝑎))∘𝑦𝑎∘𝑥∘𝑎))∘𝑦=𝑦∘𝑥∘𝑎𝑦∘𝑥∘𝑎𝑥∘𝑎²=∘𝑦𝑥∘𝑎²∘()=((∘()=((∘()=()∘()=()∘()=𝑦∘𝑥∘𝑎∘𝑦𝑥∘𝑎∘𝑎))𝑦∘𝑥∘𝑎∘𝑦𝑎∘𝑥∘𝑎))𝑦∘𝑥∘𝑎∘𝑦𝑦∘𝑥∘𝑎∘𝑦𝑥∘𝑎∘𝑎𝑎∘𝑥𝑦∘𝑦∘𝑥∘𝑎∘𝑎𝑥∘𝑎²∘()=∘∘𝑦∘𝑥𝑦∘𝑦∘𝑥∘𝑎∘𝑎𝑥∘𝑦𝑥∘𝑎²∘()=∘𝑦∘𝑦∘𝑥∘𝑎∘𝑎𝑥∘𝑦∘𝑦𝑥∘𝑎²∘(=𝑦𝑦∘𝑥∘𝑎)∘𝑎₂∘((∘()=𝑦∘𝑥𝑥∘𝑎∘𝑎))𝑦∘𝑥∘𝑎∘𝑎₂∘∘)∘(=𝑦∘𝑥𝑥∘𝑦((𝑎∘𝑎𝑥∘𝑎))∘𝑎₂∘)∘∘()=()∘𝑦∘𝑥(𝑎∘𝑎𝑥∘𝑦𝑥∘𝑎∘𝑎𝑎∘𝑎²∘∘(=)∘𝑦∘𝑥𝑥∘𝑦𝑥∘𝑎)∘𝑎(𝑎∘𝑎²∘)∘=()∘∘𝑦∘𝑥(𝑥∘𝑥𝑦∘𝑎∘𝑎𝑥∘𝑥𝑦∘𝑎²∘()=∘()∘𝑦∘𝑥𝑎∘𝑎∘𝑎𝑎∘𝑦𝑥∘𝑥²∘()=𝑥∘𝑥𝑎∘𝑎∘𝑎₂∘𝑦∘𝑎∘𝑦²∘()=∘𝑥𝑎∘𝑎∘𝑎𝑥∘𝑦²∘𝑎∘𝑥²∘()=∘𝑦𝑎∘𝑎∘𝑎𝑎∘𝑥∘𝑦²∘𝑥²∘()=𝑥∘𝑦𝑎∘𝑎∘𝑎𝑎∘³∘𝑦³∘()⊆()𝑎∘𝑎∘𝑎𝐴∘𝐻∘𝐴²∘𝐴⊆𝐴∘𝐴=𝐴². (36)Hence, 𝐴²=𝐴.
(ii)⇒(i) is obvious.

Lemma 39. Every nonempty subset 𝐴 of an intra-regular LA-semihypergroup 𝐻 with pure left identity is a left hyperideal of 𝐻 if and only if it is a right hyperideal of 𝐻.

Proof. The proof is straightforward.

Theorem 40. In an intra-regular LA-semihypergroup 𝐻 with pure left identity, the following conditions are equivalent:(i)𝐴 is a (1,2)-hyperideal of 𝐻,(ii)𝐴 is a two-sided hyperideal of 𝐻.

Proof. (i)⇒(ii): assume that 𝐻 is intra-regular LA-semihypergroup with pure left identity, and let 𝐴 be a (1,2)-hyperideal of 𝐻. Then, (𝐴∘𝐻)∘𝐴²⊆𝐴. Let 𝑎∈𝐴. Then, since 𝐻 is intra-regular, so there exist 𝑥,𝑦∈𝐻 such that 𝑎∈(𝑥∘𝑎²)∘𝑦. Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have 𝐻∘𝑎⊆𝐻∘𝑥∘𝑎²∘𝑦=𝑥∘𝑎²∘=((∘=((∘=∘(⊆∘()∘𝐻∘𝑦𝑥∘𝑎∘𝑎))𝐻∘𝑦𝑎∘𝑥∘𝑎))𝐻∘𝑦𝐻∘𝑦𝑥∘𝑎)∘𝑎𝐻∘𝑦𝑥∘𝑎𝑥∘𝑎²=∘𝑦𝑥∘𝑎²∘∘()=∘(∘𝑎𝐻∘𝑦𝑥∘𝑎∘𝑦𝑦∘𝐻∘𝑦𝑥∘𝑎)²∘𝑥=𝑎²∘∘()=()∘∘()=∘()∘()=)∘∘()=()∘∘()=()∘∘()⊆𝑦∘𝐻∘𝑦𝑥∘𝑎∘𝑥𝑎∘𝑎𝑦∘𝐻∘𝑦𝑥∘𝑎∘𝑥𝑥∘𝑦∘𝐻∘𝑦𝑥∘𝑎𝑎∘𝑎𝑥∘𝑦∘(𝑎∘𝑥𝑦∘𝐻𝑎∘𝑎𝑥∘𝑎∘𝑥𝑦∘𝑦∘𝐻𝑎∘𝑎𝑎∘𝑥𝑥∘𝑦∘𝑦∘𝐻𝑎∘𝑎𝑥∘𝑎²∘∘()=∘∘𝑦∘𝑥𝑥∘𝑦∘𝑦∘𝐻𝑎∘𝑎𝑥∘𝑦𝑥∘𝑎²∘∘()=𝑎𝑥∘𝑦∘𝑦∘𝐻𝑎∘𝑎₂∘∘∘()=∘𝑥𝑦∘𝑥𝑥∘𝑦∘𝑦∘𝐻𝑎∘𝑎𝑦∘𝑥∘𝑥∘𝑎²∘∘()=∘𝑎𝑥∘𝑦∘𝑦∘𝐻𝑎∘𝑎𝑦∘𝑦∘𝐻∘𝑥²∘∘()=∘𝑎𝑦∘𝑥∘𝑥𝑎∘𝑎𝑦∘𝑦∘𝐻∘𝑥²∘𝑥²∘()=𝑎∘𝑦𝑎∘𝑎₂∘𝑦∘𝑦∘𝐻∘𝑥∘𝑥²∘()=()∘∘𝑥∘𝑦𝑎∘𝑎𝑎∘𝑎𝑦∘𝑦∘𝐻∘𝑥²∘()=𝑥∘𝑦𝑎∘𝑎₂∘∘()∘()=𝑥∘𝑦𝑦∘𝑦∘𝐻∘𝑥𝑎∘𝑎𝑎∘𝑎𝑎∘²∘∘()⊆()∘𝑦𝑦∘𝑦∘𝐻∘𝑥∘𝑎𝑎∘𝑎𝐴∘𝐻∘𝐴²⊆𝐴. (37) Hence, 𝐴 is a left hyperideal of 𝐻, and, by Lemma 39, 𝐴 is a two-sided hyperideal of 𝐻.
(ii)⇒(i): let 𝐴 be a two-sided hyperideal of 𝐻. Let 𝑦∈(𝐴∘𝐻)∘𝐴². Then, 𝑦∈(𝑎∘𝐻)∘𝑏² for some 𝑎,𝑏∈𝐴. Now, by using Lemma 7, we have 𝑦∈(𝑎∘𝐻)∘𝑏²=()∘()))𝑎∘𝐻𝑏∘𝑏=𝑏∘((𝑎∘𝐻∘𝑏⊆𝐴∘𝐻⊆𝐴. (38) Hence, (𝐴∘𝐻)∘𝐴²⊆𝐴; therefore, 𝐴 is a (1,2)-hyperideal of 𝐻.

Lemma 41. Every nonempty subset 𝐴 of an intra-regular LA-semihypergroup 𝐻 with pure left identity is a two-sided hyperideal of 𝐻 if and only if it is a quasi hyperideal of 𝐻.

Proof. The proof is straightforward.

Theorem 42. A two-sided hyperideal of an intra-regular LA-semihypergroup 𝐻 with pure left identity is minimal if and only if it is the intersection of two minimal two-sided hyperideals of 𝐻.

Proof. Let 𝐻 be an intra-regular LA-semihypergroup and 𝑄 a minimal two-sided hyperideal of 𝐻. Let 𝑎∈𝑄. As 𝐻∘(𝐻∘𝑎)⊆𝐻∘𝑎 and 𝐻∘(𝑎∘𝐻)⊆𝑎∘(𝐻∘𝐻)=𝑎∘𝐻, which shows that 𝐻∘𝑎 and 𝑎∘𝐻 are left hyperideals of 𝐻. So, by Lemma 39, 𝐻∘𝑎 and 𝑎∘𝐻 are two-sided hyperideals of 𝐻. Now, ()∩()()()∩()()⊆()∩()𝐻∘𝐻∘𝑎∩𝑎∘𝐻𝐻∘𝑎∩𝑎∘𝐻∘𝐻=𝐻∘𝐻∘𝑎∩𝐻∘𝑎∘𝐻𝐻∘𝑎∘𝐻∩𝑎∘𝐻∘𝐻𝐻∘𝑎∩𝑎∘𝐻𝐻∘𝑎∘𝐻∩𝐻∘𝑎⊆𝐻∘𝑎∩𝑎∘𝐻. (39) This implies that 𝐻∘𝑎∩𝑎∘𝐻 is a quasi hyperideal of 𝐻, so by using Lemma 41, 𝐻∘𝑎∩𝑎∘𝐻 is a two-sided hyperideal of 𝐻. Also since 𝑎∈𝑄, we have 𝐻∘𝑎∩𝑎∘𝐻⊆𝐻∘𝑄∩𝑄∘𝐻⊆𝑄∩𝑄⊆𝑄. (40) Now, since 𝑄 is minimal, so 𝐻∘𝑎∩𝑎∘𝐻=𝑄, where 𝐻∘𝑎 and 𝑎∘𝐻 are minimal two-sided hyperideals of 𝐻. Let, 𝐼 be any two-sided hyperideal of 𝐻 such that 𝐼⊆𝐻∘𝑎. Then, 𝐼∩𝑎∘𝐻⊆𝐻∘𝑎∩𝑎∘𝐻⊆𝑄, which implies that 𝐼∩𝑎∘𝐻=𝑄. Thus 𝑄⊆𝐼. Therefore, we have 𝐻∘𝑎⊆𝐻∘𝑄⊆𝐻∘𝐼⊆𝐼,andgives𝐻∘𝑎=𝐼. (41) Thus, 𝐻∘𝑎 is a minimal two-sided hyperideal of 𝐻. Similarly, 𝑎∘𝐻 is a minimal two-sided hyperideal of 𝐻.
Conversely, let 𝑄=𝐼∩𝐽 be a two-sided hyperideal of 𝐻, where 𝐼 and 𝐽 are minimal two-sided hyperideals of 𝐻. Then, by using Lemma 41, 𝑄 is a quasi hyperideal of 𝐻; that is, 𝐻∘𝑄∩𝑄∘𝐻⊆𝑄. Let 𝑄^′ be a two-sided hyperideal of 𝐻 such that 𝑄^′⊆𝑄. Then, 𝐻∘𝑄^′∩𝑄^′∘𝐻⊆𝐻∘𝑄∩𝑄∘𝐻⊆𝑄,also𝐻∘𝑄^′𝑄⊆𝐻∘𝐼⊆𝐼,_′∘𝐻⊆𝐽∘𝐻⊆𝐽. (42) Now 𝐻∘𝐻∘𝑄^′=(𝐻∘𝐻)∘𝐻∘𝑄^′=𝑄^′∘()=𝑄∘𝐻𝐻∘𝐻_′=()∘𝐻∘𝐻𝐻∘𝐻∘𝑄^′=𝐻∘𝑄^′, (43) which implies that 𝐻∘𝑄^′ is a left hyperideal and, hence, a two-sided hyperideal by Lemma 39. Similarly, 𝑄^′∘𝐻 is a two-sided hyperideal of 𝐻.
But since 𝐼 and 𝐽 are minimal two-sided hyperideals of 𝐻, therefore, 𝐻∘𝑄^′=𝐼 and 𝑄^′∘𝐻=𝐽. But 𝑄=𝐼∩𝐽, which implies that 𝑄=𝐻∘𝑄^′∩𝑄^′∘𝐻⊆𝑄^′. This gives us that 𝑄=𝑄^′ and, hence, 𝑄 is minimal.