# On Hyperideals in Left Almost Semihypergroups

Authors: Kostaq Hila; Jani Dine
Journal: ISRN Algebra
 (2011)

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## Abstract

This paper deals with a class of algebraic hyperstructures called left almost semihypergroups (LA-semihypergroups), which are a generalization of LA-semigroups and semihypergroups. We introduce the notion of LA-semihypergroup, the related notions of hyperideal, bi-hyperideal, and some properties of them are investigated. It is a useful nonassociative algebraic hyperstructure, midway between a hypergroupoid and a commutative hypersemigroup, with wide applications in the theory of flocks, and so forth. We define the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. The topological spaces formation guarantee for the preservation of finite intersection and arbitrary union between the set of hyperideals and the open subsets of resultant topologies.


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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 purposes of the study of the experts of hyperstructures theory all over the world. Hyperstructure theory was introduced in 1934 by a 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 and computer science 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. Some principal notions about hyperstructures and semihypergroups theory can be found in [1–7].The Theory of ideals, in its modern form, is a contemporary development of mathematical knowledge to which mathematicians of today may justly point with pride. Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics and its applications such as in informatics, physics, and others. As an example of applications of the concept of an ideal in informatics, let us mention that ideals of algebraic structures have been used recently to design efficient classification systems, see [8–12].The study of LA-semigroup as a generalization of commutative semigroup was initiated in 1972 by Kazim and Naseeruddin [13]. They have introduced the concept of an LA-semigroup and have investigated some basic but important characteristics of this structure. They have generalized some useful results of semigroup theory. Since then, many papers on LA-semigroups appeared showing the importance of the concept and its applications [13–23]. In this paper, we generalize this notion introducing the notion of LA-semihypergroup which is a generalization of LA-semigroup and semihypergroup, proposing so a new kind of hyperstructure for further studying. It is a useful nonassociative algebraic hyperstructure, midway between a hypergroupoid and a commutative hypersemigroup, with wide applications in the theory of flocks etc. Although the hyperstructure is nonassociative and noncommutative, nevertheless, it possesses many interesting properties which we usually find in associative and commutative algebraic hyperstructures. A several properties of hyperideals of LA-semihypergroup are investigated. In this note, we define the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. The topological spaces formation guarantee for the preservation of finite intersection and arbitrary union between the set of hyperideals and the open subsets of resultant topologies.Recall first the basic terms and definitions from the hyperstructure theory.Definition 1.1.
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.Definition 1.2.
Ahyperstructure is called the pair (H,∘), where ∘ is a hyperoperation on the set H.Definition 1.3.
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.
Ifx∈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.Definition 1.4.
A nonempty subsetB of a semihypergroup H is called a sub-semihypergroup of H if B∘B⊆B, and H is called in this case super-semihypergroup of B.Definition 1.5.
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.Definition 1.6.
A hypergrupoid(H,∘) is called anLA-semihypergroup if, for all x,y,z∈H,
(1.3)(x∘y)∘z=(z∘y)∘x.
Every LA-semihypergroup(H,∘) satisfies the medial law, that is, for all x,y,z,w∈H,
(1.4)(x∘y)∘(z∘w)=(x∘z)∘(y∘w).
In every LA-semihypergroup with left identity, the following law holds:(1.5)(x∘y)∘(z∘w)=(w∘y)∘(z∘x),
for all x,y,z,w∈H.An elemente in anLA-semihypergroupH is called identity if x∘e=e∘x={x},forallx∈H. An element 0 in a semihypergroup H is called zero element if x∘0=0∘x={0},forallx∈H. A subset I of anLA-semihypergroupH is called a right (left) hyperideal if I∘H⊆I(H∘I⊆I) and is called a hyperideal if it is two-sided hyperideal, and if I is a left hyperideal of H, then I∘I=I2 becomes a hyperideal of H. By a bi-hyperideal of anLA-semihypergroupH, we mean a sub-LA-semihypergroup B of H such that (B∘H)∘B⊆B. It is easy to note that each right hyperideal is a bi-hyperideal. If H has a left identity, then it is not hard to show that B2 is a bi-hyperideal of H and B2⊆H∘B2=B2∘H. If E(BH) denotes the set of all idempotents subsets of H with left identity e, then E(BH) forms a hypersemilattice structure, also if C=C2, then (C∘H)∘C∈E(BH). The intersection of any set of bi-hyperideals of anLA-semihypergroupH is either empty or a bi-hyperideal of H. Also the intersection of prime bi-hyperideals of anLA-semihypergroupH is a semiprime bi-hyperideal of H.


## 2. Main Results

Proposition 2.1.
LetH be an LA-semihypergroup  with left identity, T a left hyperideal, and B a bi-hyperideal of H. Then B∘T and T2∘B are bi-hyperideals of H.
Proof.
Using the medial law (1.4), we get
(2.1)((B∘T)∘H)∘(B∘T)=((B∘T)∘B)∘(H∘T)⊆((B∘H)∘B)∘T⊆B∘T,
also
(2.2)(B∘T)∘(B∘T)=(B∘B)∘(T∘T)⊆B∘T.
Hence, B∘T is a bi-hyperideal of H. we obtain
(2.3)((T2∘B)∘H)∘(T2∘B)=((T2∘H)∘(B∘H))∘(T2∘B)⊆(T2∘(B∘H))∘(T2∘B)=(T2∘T2)∘((B∘H)∘B)⊆T2∘B,
also
(2.4)(T2∘B)∘(T2∘B)=(T2∘T2)∘(B∘B)⊆T2∘B.
Hence, T2∘B is a bi-hyperideal of H.Proposition 2.2.
LetH be an LA-semihypergroup  with left identity and B1,B2 two bi-hyperideals of H. Then B1∘B2 is a bi-hyperideal of H.Proof.
Using (1.4), we get
(2.5)((B1∘B2)∘H)∘(B1∘B2)=((B1∘B2)∘(H∘H))∘(B1∘B2)=((B1∘H)∘(B2∘H))∘(B1∘B2)=((B1∘H)∘B1)∘((B2∘H)∘B2)⊆B1∘B2.By the above, ifB1 and B2 are nonempty, then B1∘B2 and B2∘B1 are connected bi-hyperideals. Proposition 2.1 leads us to an easy generalization, that is, if B1,B2,B3,…,Bn are bi-hyperideals of anLA-semihypergroupH with left identity, then (2.6)(⋯((B1∘B2)∘B3)∘⋯)∘Bn,(⋯((B12∘B22)∘B32)∘⋯)∘Bn2
are bi-hyperideals of H, consequently the set 𝒞(HB) of bi-hyperideals forms anLA-semihypergroup.IfH is anLA-semihypergroup with left identity e, then 〈a〉L=H∘a,〈a〉R=a∘H and 〈a〉H=(H∘a)∘H are bi-hyperideals of H. It can be easily shown that 〈a∘b〉L=〈a〉L∘〈b〉L, 〈a∘b〉R=〈a〉R∘〈b〉R, and 〈a∘b〉R=〈b〉L∘〈a〉L. Hence, this implies that 〈a〉R∘〈b〉R=〈b〉L∘〈a〉L and 〈a〉L∘〈b〉L=〈b〉R∘〈a〉R. Also, 〈a〉L∘〈b〉R=〈b〉L∘〈a〉R, 〈a∘a〉L=〈a〉L2, 〈a∘a〉R=〈a〉R2, 〈a∘a〉L=〈a∘a〉R, and 〈a〉L=〈a〉R (if a is an idempotent), consequently 〈a∘a〉L=〈a∘a〉R. It is easy to show that 〈a〉R∘a2=a2∘〈a〉L.Lemma 2.3.
LetH be an LA-semihypergroup  with left identity, and let B be an idempotent bi-hyperideal of H. Then B is a hyperideal of H.
Proof.
By the definition of LA-semihypergroup (1.3), we have
(2.7)B∘H=(B∘B)∘H=(H∘B)∘B=(H∘B2)∘B=(B2∘H)∘B=(B∘H)∘B,
and every right hyperideal in H with left identity is left.Lemma 2.4.
LetH be an LA-semihypergroup  with left identity e, and let B be a proper bi-hyperideal of H. Then e∉B.Proof.
Let us suppose thate∈B. Since h∘b⊆(e∘h)∘b⊆B, using (1.3), we have h∈(e∘e)∘h=(h∘e)∘e⊆(S∘B)∘B⊆B. It is impossible. So, e∉B.It can be easily noted that{x∈H:(x∘a)∘x=e}⊆B.Proposition 2.5.
LetH be an LA-semihypergroup  with left identity, and let A,B be bi-hyperideals of H. Then the following statements are equivalent:  (1)
every bi-hyperideal is idempotent,(2)
A∩B=A∘B,(3)
the hyperideals ofH form a hypersemilattice (LH,∧), where A∧B=A∘B.Proof.
(1)⇒(2). Using Lemma 2.3, it is easy to note that A∘B⊆A∘B. Since A∩B⊆A,B implies (A∩B)2⊆A∘B, hence A∩B⊆A∘B.
(2)⇒(3). A∧B=A∘B=A∩B=B∩A=B∧A and A∧A=A∘A=A∩A=A. Similarly, associativity follows. Hence, (LH,∧) is a hypersemilattice.
(3)⇒(1). A=A∧A=A∘A.A bi-hyperidealB of anLA-semihypergroupH is called a prime bi-hyperideal if B1∘B2⊆B implies either B1⊆B or B2⊆B for every bi-hyperideal B1 and B2 of H. The set of bi-hyperideals of H is totally ordered under the set inclusion if for all bi-hyperideals I,J either I⊆J or J⊆I.Theorem 2.6.
LetH be an LA-semihypergroup  with left identity. Every bi-hyperideal of H is prime if and only if it is idempotent and the set of the bi-hyperideals of H is totally ordered under the set inclusion.Proof.
Let us assume that every bi-hyperideal ofH is prime. Since B2 is a hyperideal and so is prime which implies that B⊆B∘B, hence B is idempotent. Since B1∩B2 is a bi-hyperideal of H (where B1 and B2 are bi-hyperideals of H) and so is prime. Now by Lemma 2.3, either B1⊆B1∩B2 or B2⊆B1∩B2 which further implies that either B1⊆B2 or B2⊆B1. Hence, the set of bi-hyperideals of H is totally ordered under set inclusion.
Conversely, let us assume that every bi-hyperideals ofH is idempotent and the set of bi-hyperideals of H is totally ordered under set inclusion. Let B1,B2 and B be the bi-hyperideals of H with B1∘B2⊆B and without loss of generality assume that B1⊆B2. Since B1 is an idempotent, so B1=B1∘B1⊆B1∘B2⊆B implies that B1⊆B, and, hence, every bi-hyperideal of H is prime.A bi-hyperidealB of anLA-semihypergroupH is called strongly irreducible bi-hyperideal if B1∩B2⊆B implies either B1⊆B or B2⊆B for every bi-hyperideal B1 and B2 of H.Theorem 2.7.
LetH be an LA-semihypergroup  with zero. Let D be the set of all bi-hyperideals of H, and Ω the set of all strongly irreducible proper bi-hyperideals of H, then Γ(Ω)={OB:B∈D} forms a topology on the set Ω, where OB={J∈Ω;B⊆J} and ϕ : Bi-hyperideal(H)→Γ(Ω) preserves finite intersection and arbitrary union between the set of bi-hyperideals of H and open subsets of Ω.Proof.
Since{0} is a bi-hyperideal of H and 0 belongs to every bi-hyperideal of H, then OB={J∈Ω,{0}⊆J}={}, also OH={J∈Ω,H⊆J}=Ω which is the first axiom for the topology. Let {OBα:α∈I}⊆Γ(Ω), then ∪OBα={J∈Ω,Bα⊆J,forsomeα∈I}={J∈Ω,〈∪Bα〉⊆J}=O∪Bα, where 〈∪Bα〉 is a bi-hyperideal of H generated by 〈∪Bα〉. Let OB1 and OB2∈Γ(Ω), if J∈OB1∩OB2, then J∈Ω and B1⊆,B2⊆J. Let us suppose B1∩B2⊆J, this implies that either B1⊆J or B2⊆J. It is impossible. Hence, B1∩B2⊆J which further implies that J∈OB1∩B2. Thus OB1∩OB2⊆OB1∩B2. Now if J∈OB1∩B2, then J∈Ω and B1∩B2⊆J. Thus J∈OB1 and J∈OB2, therefore J∈OB1∩OB2, which implies that OB1∩B2⊆OB1∩OB2. Hence Γ(Ω) is the topology on Ω. Define ϕ : Bi-hyperideal(H)→Γ(Ω) by ϕ(B)=OB, then it is easy to note that ϕ preserves finite intersection and arbitrary union.A hyperidealP of anLA-semihypergroupH is called prime if A∘B⊆P implies that either A⊆P or B⊆P for all hyperideals A and B in H.LetPH denotes the set of proper prime hyperideals of anLA-semihypergroupH absorbing 0. For a hyperideal I of H, we define the sets ΘI={J∈PH:I⊆J} and Γ(PH)={ΘI,IisahyperidealofH}.Theorem 2.8.
LetH be an LA-semihypergroup  with zero. The set Γ(PH) constitutes a topology on the set PH.Proof.
LetΘI1,ΘI2∈Γ(PH), if J∈ΘI1∩ΘI2, then J∈PH and I1⊆J and I2⊆J. Let I1∩I2⊆J which implies that either I1⊆J or I2⊆J, which is impossible. Hence, J∈ΘI1∩I2. Similarly ΘI1∩I2⊆ΘI1∩ΘI2. The remaining proof follows from Theorem 2.7.The assignmentI→ΘI preserves finite intersection and arbitrary union between the hyperideal(H) and their corresponding open subsets of ΘI.LetP be a left hyperideal of anLA-semihypergroupH. P is called quasiprime if for left hyperideals A,B of H such that A∘B⊆P, we have A⊆P or B⊆P.Theorem 2.9.
LetH be an LA-semihypergroup  with left identity e. Then a left hyperideal P of H is quasiprime if and only if (H∘a)∘b⊆P implies that either a∈P or b∈P.Proof.
LetP be a left hyperideal of H. Let us assume that (H∘a)∘b⊆P, then
(2.8)H∘((H∘a)∘b)⊆H∘P⊆P,
that is,
(2.9)H∘((H∘a)∘b)=(H∘a)∘(H∘b).
Hence, either a∈P or b∈P.
Conversely, let us assume thatA∘B⊆P, where A and B are left hyperideal of H such that A⊆P. Then there exists x∈A such that x∉P. Now, by the hypothesis, we have (H∘x)∘y⊆(H∘A)∘B⊆A∘B⊆P for all y∈B. Since x∉P, so by hypothesis, y∈P for all y∈B, we obtain B⊆P. This shows that P is quasiprime.AnLA-semihypergroupH is called an antirectangular if a∈(b∘a)∘b, for all a,b∈H. It is easy to see that H=H∘H. In the following results for an antirectangular LA-semihypergroup H, e∉H.Proposition 2.10.
LetH be an LA-semihypergroup. If A,B are hyperideals of H, then A∘B is a hyperideal.Proof.
Using (1.4), we have
(2.10)(A∘B)∘H=(A∘B)∘(H∘H)=(A∘H)∘(B∘H)⊆A∘B,
also
(2.11)H∘(A∘B)=(H∘H)∘(A∘B)=(H∘A)∘(H∘B)⊆A∘B,
which shows that A∘B is a hyperideal.Consequently, ifI1,I2,…,In are hyperideals of H, then (2.12)(⋯∘((I1∘I2)∘I3)∘⋯∘In),(⋯∘((I12∘I22)∘I32)∘⋯∘In2)
are hyperideals of H and the set ℋ(I) of hyperideals of H form an antirectangular LA-semihypergroup.Lemma 2.11.
LetH be an antirectangular LA-semihypergroup. Any subset of H is left hyperideal if and only if it is right.Proof.
LetI be a right hyperideal of H, then using (1.3), we get h∘i⊆((k∘h)∘k)∘i⊆(i∘k)∘(k∘h)⊆I.
Conversely, let us suppose thatI is a left hyperideal of H, then using (1.3), we have i∘h⊆((t∘i)∘t)∘h⊆(h∘t)∘(t∘i)⊆I.It is fact thatH∘I=I∘H. From the above lemma, we remark that every quasiprime hyperideal becomes prime in an antirectangular LA-semihypergroup.Lemma 2.12.
LetH be an anti-rectangular LA-semihypergroup. If I is a hyperideal of H, then S(a)={x∈H:a∈(x∘a)∘x,fora∈I}⊆I.Proof.
Lety∈S(a), then y∈(y∘a)∘y⊆(H∘I)∘H⊆I. Hence H(a)⊆I. Also, S(a)={x∈H:x∈(x∘a)∘x,fora∈I}⊆I.An hyperidealI of anLA-semihypergroupH is called an idempotent if I∘I=I. AnLA-semihypergroupH is said to be fully idempotent if every hyperideal of H is idempotent.Proposition 2.13.
LetH be an antirectangular LA-semihypergroup, and, A,B be hyperideals of H. Then the following statements are equivalent:  (1)
H is fully idempotent,(2)
A∩B=A∘B,(3)
the hyperideals ofH form a hypersemilattice (LH,∧) where A∧B=A∘B.The proof follows from Proposition2.5.The set of hyperideals ofH is totally ordered under set inclusion if for all hyperideals I,J either I⊆J or J⊆I and denoted by hyperideal(H).Theorem 2.14.
LetH be an antirectangular LA-semihypergroup. Then every hyperideal of H is prime if and only if it is idempotent and hyperideal (H) is totally ordered under set inclusion.Proof.
The proof follows from Theorem2.6.In conclusion, let us mention that it would be interesting to investigate whether it is possible to apply hyperideals of hyperstructures to the construction of classification systems similar to those introduced in [8–12].



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