# On Intra-Regular Left Almost Semihypergroups with Pure Left Identity

Authors: Naveed Yaqoob; Piergiulio Corsini; Faisal Yousafzai
Journal: Journal of Mathematics
 (2013)

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## 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.


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## Body

## 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.IfA and B are two nonempty subsets of H, then we denote
(1)A∘B=⋃a∈A,b∈B‍a∘b,a∘A={a}∘A,a∘B={a}∘B.Definition 2 (see [35]).
A hypergroupoid(H,∘) is called an LA-semihypergroup if for all x,y,z∈H,
(2)(x∘y)∘z=(z∘y)∘x.The law(x∘y)∘z=(z∘y)∘x is called a left invertive law.Example 3.
LetH={x,y,z,w,t} with the binary hyperoperation defined below:
(3)∘xyzwtxxxxxxyx{z,t}z{x,w}{z,t}zxzz{x,w}zwx{x,w}{x,w}w{x,w}tx{y,t}z{x,w}{y,t}
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.
LetH=ℤ. If we define x∘y=y-x+3ℤ, where x,y∈ℤ. Then, (H,∘) becomes an LA-semihypergroup as
(4)(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)+3ℤ=(z∘y)∘x.
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(5)(x∘y)∘(z∘w)=(x∘z)∘(y∘w)
for all x,y,z,w∈H. This law is known as medial law (cf. [35]).Definition 5.
LetH be an LA-semihypergroup. An element e∈H is called(i)
left identity (resp., pure left identity) if for alla∈H, a∈e∘a (resp., a=e∘a),(ii)
right identity (resp., pure right identity) if for alla∈H, a∈a∘e (resp., a=a∘e),(iii)
identity (resp., pure identity) if for alla∈H, a∈e∘a∩a∘e (resp., a=e∘a∩a∘e).Example 6.
LetH={x,y,z,w} with the binary hyperoperation defined below:
(6)∘xyzwxxyzwyz{y,z}{y,z}wzy{y,z}{y,z}wwwwwH
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.
LetH be an LA-semihypergroup with pure left identity e. Then, x∘(y∘z)=y∘(x∘z) holds for all x,y,z∈H.Proof.
LetH be an LA-semihypergroup with pure left identity e. Then, for all x,y,z∈H and by medial law, we have
(7)x∘(y∘z)=(e∘x)∘(y∘z)=(e∘y)∘(x∘z)=y∘(x∘z).
This completes the proof.Lemma 8.
LetH 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.
LetH be an LA-semihypergroup with pure left identity e. Then, for all x,y,z,w∈H and by medial law, we have
(8)(x∘y)∘(z∘w)=((e∘x)∘y)∘((e∘z)∘w)=((y∘x)∘e)∘((w∘z)∘e)=((y∘x)∘(w∘z))∘(e∘e)=((e∘e)∘(w∘z))∘(y∘x)=(e∘(w∘z))∘(y∘x)=(w∘z)∘(y∘x)=(w∘y)∘(z∘x).
This completes the proof.The law(x∘y)∘(z∘w)=(w∘y)∘(z∘x) is called a paramedial law.Definition 9.
LetH 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 subsetI 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-semihypergroupH, we mean a sub LA-semihypergroup B of H such that (B∘H)∘B⊆B.Definition 12.
A sub LA-semihypergroupB 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. IfE(BH) denotes the set of all idempotent subsets of H with pure left identity e, then E(BH) 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-semihypergroupT of H is called an interior hyperideal of H if (H∘T)∘H⊆T.Definition 14.
A nonempty subsetQ of an LA-semihypergroup H is called a quasi-hyperideal of H if Q∘H∩H∘Q⊆Q.Lemma 15.
IfH is an LA-semihypergroup with left identity e, then H∘H=H.Proof.
IfH is an LA-semihypergroup with left identity e, then x∈H implies that
(9)x∈e∘x⊆H∘HandsoH⊆H∘H.
That is, H=H∘H.Corollary 16.
IfH 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 Lemma15.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, letH={x,y,z} with the binary hyperoperation defined below:
(10)∘xyzx{x,z}z{y,z}y{y,z}zzz{y,z}{y,z}{y,z}
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;
(11)b=b∘e=(b∘e)∘e=(e∘e)∘b=e∘b.
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-semihypergroupH is a semihypergroup if and only if a∘(b∘c)=(c∘b)∘a holds for all a,b,c∈H.Proof.
LetH 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,
(12)a∘(b∘c)=(c∘b)∘a∀a,b,c∈H.
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
(13)a∘(b∘c)=(c∘b)∘a=(a∘b)∘c.
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.
LetH 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 elementa 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.
LetH={x,y,z,w} with the binary hyperoperation defined below:
(14)∘xyzwxx{x,w}{x,w}wy{x,w}{y,z}{y,z}wz{x,w}yywwwwww
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.
LetH=ℤ. Define a hyperoperation ∘ on H by
(15)x∘y={x,y}∪2ℤ∀x,y∈H.
Then, for all x,y,z∈H, we have
(16)(x∘y)∘z={x,y,z}∪2ℤ=(z∘y)∘x.
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 elementa 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-semihypergroupH with pure left identity e is intra-regular if H is left (right) invertible.Proof.
LetH 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
(17)a=e∘a=e∘(e∘a)⊆(a′∘a)∘(e∘a)⊆(H∘a)∘(H∘a)=(H∘a)∘((H∘H)∘a)=(H∘a)∘((a∘H)∘H)=(a∘H)∘((H∘a)∘H)=(a∘(H∘a))∘(H∘H)=(a∘(H∘a))∘H=(H∘(a∘a))∘H=(H∘a2)∘H.
This shows that H is intra-regular. The case for right invertible can be seen in a similar way.Theorem 24.
An LA-semihypergroupH with left identity e is intra-regular if H∘a=H or a∘H=H for all a∈H.Proof.
LetH 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
(18)a∈H=(H∘H)∘H=((H∘a)∘(H∘a))∘H=((H∘H)∘(a∘a))∘H=(H∘a2)∘H.
This shows that H is intra-regular.
Leta∈H and assume that a∘H=H holds for all a∈H. Then, by using left invertive law, we have
(19)a∈H=H∘H=(a∘H)∘H=(H∘H)∘a=H∘a.
Thus, H∘a=H holds for all a∈H; therefore, it follows from above that H is intra-regular.Corollary 25.
An LA-semihypergroupH with pure left identity e is intra-regular if H∘a=H or a∘H=H for all a∈H.Corollary 26.
IfH 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.
IfH 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.
LetH 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
(20)b∈(x∘b2)∘y=(x∘(b∘b))∘y=(b∘(x∘b))∘y=(y∘(x∘b))∘b⊆(y∘(x∘((x∘b2)∘y)))∘b=(y∘((x∘b2)∘(x∘y)))∘b=((x∘b2)∘(y∘(x∘y)))∘b=(((x∘y)∘y)∘(b2∘x))∘b=((b∘b)∘(((x∘y)∘y)∘x))∘b=((b∘b)∘((x∘y)∘(x∘y)))∘b=((b∘b)∘(x2∘y2))∘b=((y2∘x2)∘(b∘b))∘b=(b∘((y2∘x2)∘b))∘b⊆(B∘H)∘B.
This shows that (B∘H)∘B=B.The converse is not true in general. For this, let us consider an LA-semihypergroupH={x,y,z,w} with the binary hyperoperation defined below:
(21)∘xyzwxyy{z,w}wyyy{z,w}wz{z,w}{z,w}zwwwwww
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.
IfH 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.
LetH 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
(22)b∈(x∘b2)∘y=((e∘x)∘(b∘b))∘y=((b∘b)∘(x∘e))∘y=(((x∘e)∘b)∘b)∘y⊆(H∘B)∘H.
This shows that (H∘B)∘H=B.The converse is not true in general. For this, let us consider an LA-semihypergroupH={e,x,y,z,w} with the binary hyperoperation defined below:
(23)∘exyzweexyzwxyzz{z,w}wyxzz{z,w}wzz{z,w}{z,w}{z,w}wwwwwww
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.
LetH be an LA-semihypergroup. Then, ∅≠A⊆H is called semiprime if a2⊆Aimplies a∈A.Theorem 30.
An LA-semihypergroupH 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.
LetH 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
(24)a2∘H=(a∘a)∘(H∘H)=(H∘H)∘(a∘a)=H∘a2.
Therefore, by the given assumption, a∈a2∘H. Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have
(25)a∈H∘a∪a2∘H=(H∘a)∘(a2∘H)=(H∘a)∘((a∘a)∘H)=(H∘a)∘((H∘a)∘(e∘a))⊆(H∘a)∘((H∘a)∘(H∘a))=(H∘a)∘((H∘H)∘(a∘a))⊆(H∘a)∘((H∘H)∘(H∘a))=(H∘a)∘((a∘H)∘(H∘H))=(H∘a)∘((a∘H)∘H)=(a∘H)∘((H∘a)∘H)=(a∘(H∘a))∘(H∘H)=(a∘(H∘a))∘H=(H∘(a∘a))∘H=(H∘a2)∘H.
This shows that H is intra-regular.Lemma 31.
IfH is an intra-regular LA-semihypergroup, then H=H2.Proof.
The proof is straightforward.Theorem 32.
For a left invertible LA-semihypergroupH 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
(26)a∈(x∘a2)∘y⊆(H∘a2)∘H=(H∘(a∘a))∘H=(a∘(H∘a))∘H=(a∘(H∘a))∘(H∘H)=(a∘H)∘((H∘a)∘H)=(H∘a)∘((a∘H)∘H)=(H∘a)∘((H∘H)∘a)=(H∘a)∘(H∘a)⊆(H∘R)∘(H∘L)=((H∘H)∘R)∘(H∘L)=((R∘H)∘H)∘(H∘L)⊆R∘L.
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
(27)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.
Thus, we get a2⊆((x∘a2)∘a)∘a for some x∈H.
Now, by using left invertive law, we have(28)(a∘a)∘a′⊆(((x∘a2)∘a)∘a)∘a′(a′∘a)∘a⊆(a′∘a)∘((x∘a2)∘a)a∈(x∘a2)∘a.
This shows that H is intra-regular.Lemma 33.
Every left hyperideal of an intra-regular LA-semihypergroupH with pure left identity is idempotent.Proof.
The proof is straightforward.Theorem 34.
In an LA-semihypergroupH 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-semihypergroupH 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
(29)a∈(x∘a2)∘y=(x∘(a∘a))∘y=(a∘(x∘a))∘y=(y∘(x∘a))∘a⊆(y∘(x∘((x∘a2)∘y)))∘a=(y∘((x∘a2)∘(x∘y)))∘a=((x∘a2)∘(y∘(x∘y)))∘a=((x∘(a∘a))∘(y∘(x∘y)))∘a=((a∘(x∘a))∘(y∘(x∘y)))∘a=((a∘y)∘((x∘a)∘(x∘y)))∘a=((x∘a)∘((a∘y)∘(x∘y)))∘a=((x∘a)∘((a∘x)∘y2))∘a=((y2∘(a∘x))∘(a∘x))∘a=(a∘((y2∘(a∘x))∘x))∘a⊆(A∘H)∘A.
Thus, (A∘H)∘A=A holds. Now, by using left invertive law, paramedial law, medial law, and Lemma 7, we have
(30)a∈(x∘a2∘)∘y=(x∘(a∘a))∘y=(a∘(x∘a))∘y=(y∘(x∘a))∘a⊆(y∘(x∘((x∘a2)∘y)))∘a=(y∘((x∘a2)∘(x∘y)))∘a=((x∘a2)∘(y∘(x∘y)))∘a=((x∘(a∘a))∘(y∘(x∘y)))∘a=((a∘(x∘a))∘(y∘(x∘y)))∘a=(((y∘(x∘y))∘(x∘a))∘a)∘a=(((a∘x)∘((x∘y)∘y))∘a)∘a=(((a∘x)∘(y2∘x))∘a)∘a=(((a∘y2)∘(x∘x))∘a)∘a=(((a∘y2)∘x2)∘a)∘a=(((x2∘y2)∘a)∘a)∘a⊆(((x2∘y2)∘((x∘(a∘a))∘y))∘a)∘a=(((x2∘y2)∘((a∘(x∘a))∘y))∘a)∘a=(((x2∘(a∘(x∘a)))∘(y2∘y))∘a)∘a=(((a∘(x2∘(x∘a)))∘y3)∘a)∘a=(((a∘((x∘x)∘(x∘a)))∘y3)∘a)∘a=(((a∘((a∘x)∘(x∘x)))∘y3)∘a)∘a=((((a∘x)∘(a∘x2))∘y3)∘a)∘a=((((a∘a)∘(x∘x2))∘y3)∘a)∘a=(((y3∘x3)∘(a∘a))∘a)∘a=((a∘((y3∘x3)∘a))∘a)∘a⊆((A∘H)∘A)∘A⊆A∘A=A2.
Hence, A=A2 holds.
(
ii
)
⇒
(
i
) is obvious.Theorem 36.
In an intra-regular LA-semihypergroupH 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
(31)p∘q⊆p∘((x∘q2)∘y)=(x∘q2)∘(p∘y)=(x∘(q∘q))∘(p∘y)=(q∘(x∘q))∘(p∘y)=(q∘p)∘((x∘q)∘y)=(x∘q)∘((q∘p)∘y)=(y∘(q∘p))∘(q∘x)=q∘((y∘(q∘p))∘x)⊆Q∘H.
Now, let q∘p∈Q∘H. Then, by using medial law, paramedial law and Lemma 7, we have
(32)q∘p⊆((x∘q2)∘y)∘p=(p∘y)∘(x∘q2)=(p∘y)∘(x∘(q∘q))=x∘((p∘y)∘(q∘q))=x∘((q∘q)∘(y∘p))=(q∘q)∘(x∘(y∘p))=((x∘(y∘p))∘q)∘q⊆H∘Q.
Hence, Q∘H=H∘Q. As, by using left invertive law, medial law, and Lemma 7, we have
(33)q∈(x∘q2)∘y=(x∘(q∘q))∘y=(q∘(x∘q))∘y=(y∘(x∘q))∘q⊆H∘Q.
Thus, q∈H∘Q∩Q∘H implies that H∘Q∩Q∘H=Q.
(
ii
)
⇒
(
i
) is obvious.Theorem 37.
In an intra-regular LA-semihypergroupH 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
(34)a∈(x∘a2)∘y=(x∘(a∘a))∘y=(a∘(x∘a))∘y=(y∘(x∘a))∘a⊆(y∘(x∘a))∘((x∘a2)∘y)=(((x∘a2)∘y)∘(x∘a))∘y=((a∘x)∘(y∘(x∘a2)))∘y=(((y∘(x∘a2))∘x)∘a)∘y⊆(H∘A)∘H.
Thus, (H∘A)∘H=A.
(
ii
)
⇒
(
i
) is obvious.Theorem 38.
In an intra-regular LA-semihypergroupH 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
(35)a∈(x∘a2)∘y=(x∘(a∘a))∘y=(a∘(x∘a))∘y=(y∘(x∘a))∘a⊆(y∘(x∘((x∘a2)∘y)))∘a=(y∘((x∘a2)∘(x∘y)))∘a=((x∘a2)∘(y∘(x∘y)))∘a=(((x∘y)∘y)∘(a2∘x))∘a=((y2∘x)∘(a2∘x))∘a=(a2∘((y2∘x)∘x))∘a=(a2∘(x2∘y2))∘a=(a∘(x2∘y2))∘a2=(a∘(x2∘y2))∘(a∘a)⊆(A∘H)∘A2.
Thus, (A∘H)∘A2=A. Now, by using left invertive law, medial law, paramedial law, and Lemma 7, we have(36)a∈(x∘a2)∘y=(x∘(a∘a))∘y=(a∘(x∘a))∘y=(y∘(x∘a))a⊆(y∘(x∘a))∘((x∘a2)∘y)=(x∘a2)∘((y∘(x∘a))∘y)=(x∘(a∘a))∘((y∘(x∘a))∘y)=(a∘(x∘a))∘((y∘(x∘a))∘y)=(((y∘(x∘a))∘y)∘(x∘a))∘a=((a∘x)∘(y∘(y∘(x∘a))))∘a=((((x∘a2)∘y)∘x)∘(y∘(y∘(x∘a))))∘a=(((x∘y)∘(x∘a2))∘(y∘(y∘(x∘a))))∘a=(((x∘y)∘y)∘((x∘a2)∘(y∘(x∘a))))∘a=((y2∘x)∘((x∘(a∘a))∘(y∘(x∘a))))∘a=((y2∘x)∘((x∘y)∘((a∘a)∘(x∘a))))∘a=((y2∘x)∘((a∘a)∘((x∘y)∘(x∘a))))∘a=((a∘a)∘((y2∘x)∘((x∘y)∘(x∘a))))∘a=((a∘a)∘((y2∘x)∘((x∘x)∘(y∘a))))∘a=((((x∘x)∘(y∘a))∘(y2∘x))∘(a∘a))∘a=((((a∘y)∘(x∘x))∘(y2∘x))∘(a∘a))∘a=((((x2∘y)∘a)∘(y2∘x))∘(a∘a))∘a=(((x∘y2)∘(a∘(x2∘y)))∘(a∘a))∘a=((a∘((x∘y2)∘(x2∘y)))∘(a∘a))∘a=((a∘(x3∘y3))∘(a∘a))∘a⊆((A∘H)∘A2)∘A⊆A∘A=A2.Hence, A2=A.
(
ii
)
⇒
(
i
) is obvious.Lemma 39.
Every nonempty subsetA 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-semihypergroupH 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
(37)H∘a⊆H∘((x∘a2)∘y)=(x∘a2)∘(H∘y)=(x∘(a∘a))∘(H∘y)=(a∘(x∘a))∘(H∘y)=((H∘y)∘(x∘a))∘a⊆((H∘y)∘(x∘a))∘((x∘a2)∘y)=(x∘a2)∘(((H∘y)∘(x∘a))∘y)=(y∘((H∘y)∘(x∘a)))∘(a2∘x)=a2∘((y∘((H∘y)∘(x∘a)))∘x)=(a∘a)∘((y∘((H∘y)∘(x∘a)))∘x)=(x∘(y∘((H∘y)∘(x∘a))))∘(a∘a)=(x∘(y∘((a∘x)∘(y∘H))))∘(a∘a)=(x∘((a∘x)∘(y∘(y∘H))))∘(a∘a)=((a∘x)∘(x∘(y∘(y∘H))))∘(a∘a)⊆((((x∘a2)∘y)∘x)∘(x∘(y∘(y∘H))))∘(a∘a)=(((x∘y)∘(x∘a2))∘(x∘(y∘(y∘H))))∘(a∘a)=(((a2∘x)∘(y∘x))∘(x∘(y∘(y∘H))))∘(a∘a)=((((y∘x)∘x)∘a2)∘(x∘(y∘(y∘H))))∘(a∘a)=(((y∘(y∘H))∘x)∘(a2∘((y∘x)∘x)))∘(a∘a)=(((y∘(y∘H))∘x)∘(a2∘(x2∘y)))∘(a∘a)=(a2∘(((y∘(y∘H))∘x)∘(x2∘y)))∘(a∘a)=((a∘a)∘(((y∘(y∘H))∘x)∘(x2∘y)))∘(a∘a)=(((x2∘y)∘((y∘(y∘H))∘x))∘(a∘a))∘(a∘a)=(a∘((x2∘y)∘(((y∘(y∘H))∘x)∘a)))∘(a∘a)⊆(A∘H)∘A2⊆A.
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
(38)y∈(a∘H)∘b2=(a∘H)∘(b∘b)=b∘((a∘H)∘b)⊆A∘H⊆A.
Hence, (A∘H)∘A2⊆A; therefore, A is a (1,2)-hyperideal of H.Lemma 41.
Every nonempty subsetA 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-semihypergroupH with pure left identity is minimal if and only if it is the intersection of two minimal two-sided hyperideals of H.Proof.
LetH 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,
(39)H∘(H∘a∩a∘H)∩(H∘a∩a∘H)∘H=H∘(H∘a)∩H∘(a∘H)∩(H∘a)∘H∩(a∘H)∘H⊆(H∘a∩a∘H)∩(H∘a)∘H∩H∘a⊆H∘a∩a∘H.
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
(40)H∘a∩a∘H⊆H∘Q∩Q∘H⊆Q∩Q⊆Q.
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
(41)H∘a⊆H∘Q⊆H∘I⊆I,andgivesH∘a=I.
Thus, H∘a is a minimal two-sided hyperideal of H. Similarly, a∘H is a minimal two-sided hyperideal of H.
Conversely, letQ=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,
(42)H∘Q′∩Q′∘H⊆H∘Q∩Q∘H⊆Q,alsoH∘Q′⊆H∘I⊆I,Q′∘H⊆J∘H⊆J.
Now
(43)H∘(H∘Q′)=(H∘H)∘(H∘Q′)=(Q′∘H)∘(H∘H)=(Q′∘H)∘H=(H∘H)∘Q′=H∘Q′,
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 sinceI 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.



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*Source: 510790-2013-06-02.xml*