Left almost semihyperrings characterized by their hyperideals

1.   Introduction

The study of left almost semigroups (briefly, $ LA $-semigroups), as a generalization of commutative semigroups, was first introduced in 1972 by Kazim and Naseeruddin ^[25]. It is also called an Abel-Grassmann's groupoid (briefly, $ AG $-groupoid) ^[32]. An $ LA $-semigroup is a non-associative and non-commutative algebraic structure midway between a groupoid and a commutative semigroup. Mushtaq and Yousuf ^[27] examined some basic results of the structure of $ LA $-semigroups, for examples, a commutative monoid is an $ LA $-semigroup with right identity, every left cancellative $ LA $-semigroup is right cancellative and every right cancellative $ LA $-monoid is left cancellative. On $ LA $-semigroups, regularities are interesting and essential properties to investigate. Khan and Asif ^[20] classified intra-regular $ LA $-semigroups based on the features of their fuzzy ideals in 2010. Abdullah, Aslam and Amin ^[2] began discussing regular $ LA $-semigroup categorizations in terms of interval $ (\alpha, \beta) $-fuzzy ideals. Also, Khan, Jun and Yousafzai ^[22] used their fuzzy left ideals and fuzzy right ideals to characterize right regular $ LA $-semigroups. In 2016, Khan, Yousafzai and Khan ^[24] defined a class of $ (m, n) $-regular $ LA $-semigroups based on their $ (m, n) $-ideals. Several characterizations of weakly regular $ LA $-semigroups by using the smallest ideals and fuzzy ideals of $ LA $-semigroups were investigated by Yousafzai, Iampan and Tang ^[46]. Furthermore, Sezer ^[36] has developed soft sets to characterize regular, intra-regular, completely regular, weakly regular and quasi-regular $ LA $-semigroups. Recently, various properties of $ LA $-semigroups have been studied by many mathematicians (see, e.g., ^{[4,8,14,15,44,47]}). Additionally, the notion of left almost semirings (briefly, $ LA $-semirings), which is a generalization of left almost rings (briefly, $ LA $-rings) ^[37], has been considered different properties by some mathematicians (see, e.g., ^[12,13,33]). Moreover, the concept of left almost was studied in other algebraic structures (for example, in ordered $ LA $-$ (\Gamma) $-semigroups ^[5,7,19,45], in gamma $ LA $-rings and gamma $ LA $-semigroups ^[23], in $ LA $-polygroups ^[3,40,42]).

The concept of hyperstructures was introduced by Marty ^[26] in the 8$ ^{\rm th} $ Congress of Scandinavian Mathematicians. There are many authors expanded the concept of hyperstructures (see, e.g., (^{[9,10,11,28,29,31,38,39]}). Hila and Dine ^[18] introduced the notion of left almost semihypergroups (briefly, $ LA $-semihypergroups) which is a generalization of $ LA $-semigroups and commutative semihypergroups. It is a useful non-associative algebraic hyperstructure, midway between a hypergroupoid and a commutative semihypergroup, with wide applications in the theory of flocks etc. In 2013, Yaqoob, Corsini and Yousafzai ^[41] used the properties of their left and right hyperideals to characterize intra-regular $ LA $-semihypergroups. Then, the class of regular $ LA $-semihypergroups was characterized in terms of $ (\in_{\Gamma}, \in_{\Gamma}\vee q_{\Delta}) $-cubic (resp., left, right, two-sided, bi, generalized bi, interior, quasi)-hyperideals of $ LA $-semihypergroups by Gulistan, Khan, Yaqoob and Shahzad ^[16]. In addition, Khan, Farooq, Izhar and Davvaz ^[21] studied into some properties of fuzzy left and right hyperideals in regular and intra-regular $ LA $-semihypergroups. In terms of soft interior hyperideals, Abbasi, Khan, Talee and Khan ^[1] gave different essential characterizations of left regular $ LA $-semihypergroups. On the other hand, Yaqoob and Gulistan ^[43] introduced the concept of ordered $ LA $-semihypergroups which is a generalization of $ LA $-semihypergroups. Next, the results of fuzzy hyperideals and generalized fuzzy hyperideals of ordered $ LA $-semihypergroups were then examined by Azhar, Gulistan, Yaqoob and Kadry (see, ^[6,17]).

In 2018, Nawaz, Rehman and Gulistan ^[30] defined the idea of left almost semihyperrings (briefly, $ LA $-semihyperrings), as a generalization of $ LA $-semirings, and studied at some of their basic properties. In 2020, Rahman, Hidayat and Alghofari ^[34] applied the concept of fuzzy sets to define the new algebraic structure, namely, fuzzy left almost semihyperrings, and they have shown that the set of all fuzzy subsets in $ LA $-semihyperrings is also $ LA $-semihyperrings. In this paper, we are interesting in the classes of weakly regular $ LA $-semihyperrings and regular $ LA $-semihyperrings. Then, we give some characterizations of weakly regular $ LA $-semihyperrings and regular $ LA $-semihyperrings in terms of their hyperideals.

2.   Preliminaries

Firstly, we recall some of the basic concepts and properties, which are necessary for this paper. Let $ H $ be a nonempty set. Then, the map $ \circ:H\times H\rightarrow \mathcal{P}^*(H) $ is called a hyperoperation on $ H $ where $ \mathcal{P}^*(H) = \mathcal{P}(H)\setminus\{\emptyset\} $ denotes the set of all nonempty subsets of $ H $. A hypergroupoid is called the pair $ (H, \circ) $, where $ \circ $ is a hyperopartion on a nonempty set $ H $. If $ x\in H $ and $ A, B $ are two nonempty subsets of $ H $, then we denote

A hypergroupoid $ (H, \circ) $ is called an $ LA $-semihypergroup ^[18] if for all $ x, y, z\in H, (x\circ y)\circ z = (z\circ y)\circ x $, which means that

This law is known as a left invertive law.

For any nonempty subsets $ A, B $ and $ C $ of an $ LA $-semihypergroup $ (H, \circ) $, we have that

The following the notion, which appears in ^[30], will be considering in this study.

A hyperstructure $ (S, +, \cdot) $ is called an $ LA $-semihyperring if it satisfies the following conditions:

$(i) $ $ (S, +) $ is an $ LA $-semihypergroup;

$(ii)$ $ (S, \cdot) $ is an $ LA $-semihypergroup;

$(iii)$ $ x\cdot (y+z) = x\cdot y+ x\cdot z $ and $ (y+z)\cdot x = y\cdot x+z\cdot x $ for all $ x, y, z\in S $.

Example 2.1. ^[30] Let $ S = \{a, b, c\} $ be a set with the hyperoperations $ + $ and $ \cdot $ on $ S $ defined as follows:

Then, $ (S, +, \cdot) $ is an $ LA $-semihyperring.

For more convenient, we say an $ LA $-semihyperring $ S $ instead of an $ LA $-semihyperring $ (S, +, \cdot) $ and we write $ xy $ instead of $ x\cdot y $ for any $ x, y\in S $.

In an $ LA $-semihyperring $ S $, the medial law $ (xy)(zw) = (xz)(yw) $ holds for all $ x, y, z, w\in S $. An element $ e $ of an $ LA $-semihyperring $ S $ is called a left identity (resp., pure left identity) if for all $ x\in S $, $ x\in ex $ (resp., $ x = ex $). If an $ LA $-semihyperring $ S $ contains a pure left identity $ e $, then it is unique. In an $ LA $-semihyperring $ S $ with a pure left identity $ e $, the paramedial law $ (xy)(zw) = (wy)(zx) $ holds for all $ x, y, z, w\in S $.

An element $ a $ of an $ LA $-semihyperring $ S $ with a left identity (resp., pure left identity) $ e $ is called a left invertible (resp., pure left invertible) if there exists $ x\in S $ such that $ e\in xa $ (resp., $ e = xa $). An $ LA $-semihyperring $ S $ is called a left invertible (resp., pure left invertible) if every element of $ S $ is a left invertible (resp., pure left invertible).

We observe that if an element $ e $ is a pure left identity of an $ LA $-semihyperring $ S $, then $ e $ is a left identity. But the converse is not true in general, as the following example.

Example 2.2. Let $ S = \{a, b, c\} $ be a set with the hyperoperations $ + $ and $ \cdot $ on $ S $ defined as follows:

Then, $ (S, +, \cdot) $ is an $ LA $-semihyperring ^[35]. One can see that $ b $ is a left identity, but it is not a pure left identity.

Lemma 2.3. ^[30] Let $ S $ be an $ LA $-semihyperring with a pure left identity $ e $. Then $ x(yz) = y(xz) $ for all $ x, y, z\in S $.

For any $ LA $-semihyperring $ S $, the following law holds $ (AB)(CD) = (AC)(BD) $ for all nonempty subsets $ A, B, C, D $ of $ S $. If an $ LA $-semihyperring $ S $ contains the pure left identity $ e $, then $ (AB)(CD) = (DB)(CA) $ and $ A(BC) = B(AC) $ for every nonempty subsets $ A, B, C, D $ of $ S $.

Now, we recall the concepts of different types of hyperideals of $ LA $-semihyperrings which occurred in ^[30] as follows. Let $ S $ be an $ LA $-semihyperring and a nonempty subset $ A $ of $ S $ such that $ A+A\subseteq A $. Then:

$(i) $ $ A $ is called a left hyperideal of $ S $ if $ SA\subseteq A $;

$(ii)$ $ A $ is called a right hyperideal of $ S $ if $ AS\subseteq A $;

$(iii)$ $ A $ is called a hyperideal of $ S $ if it is both a left and a right hyperideal of $ S $;

$(iv)$ $ A $ is called a quasi-hyperideal of $ S $ if $ SA\cap AS\subseteq A $;

$(v)$ $ A $ is called a bi-hyperideal of $ S $ if $ AA\subseteq A $ and $ (AS)A\subseteq A $.

Example 2.4. Let $ S = \{a, b, c, d, e\} $ be a set with the hyperoperations $ + $ and $ \cdot $ on $ S $ defined as follows:

Then, $ (S, +, \cdot) $ is an $ LA $-semihyperring. Now, we can see that $ A = \{a, b, e\} $ is a left hyperideal of $ S $, but it is not a right hyperideal, because $ b\cdot d = \{a, c\}\nsubseteq\{a, b, e\} $.

Proposition 2.5. Let $ S $ be an $ LA $-semihyperring such that $ S = S^2 $. Then every right hyperideal of $ S $ is a hyperideal.

Proof. Let $ R $ be a right hyperideal of $ S $. Let $ a\in SR $. Then $ a\in sr $ for some $ r\in R $ and $ s\in S $. Since $ S = S^2 $, $ s\in xy $ for some $ x, y\in S $. By using the left invertive law, we have

Thus, $ SR\subseteq R $. This shows that $ R $ is a left hyperideal of $ S $. Therefore, $ R $ is a hyperideal of $ S $.

For any $ LA $-semihyperring $ S $ with a pure left identity $ e $, we have that $ S = S^2 $. Then, we have the following lemma.

Lemma 2.6. Let $ S $ be an $ LA $-semihyperring with a pure left identity $ e $. Then every right hyperideal of $ S $ is a hyperideal of $ S $.

Lemma 2.7. Every left (resp., right) hyperideal of an $ LA $-semihyperring $ S $ is a quasi-hyperideal of $ S $.

Proof. Let $ Q $ be a left hyperideal of an $ LA $-semihyperring $ S $. Then, $ Q+Q\subseteq Q $ and $ SQ\cap QS\subseteq SQ\subseteq Q $. Hence, $ Q $ is a quasi-hyperideal of $ S $. For the case of the right hyperideal, we can prove similarly.

Lemma 2.8. The intersection of a left hyperideal $ L $ and a right hyperideal $ R $ of an $ LA $-semihyperring $ S $ is a quasi-hyperideal of $ S $.

Proof. It is easy to show that $ L\cap R+L\cap R\subseteq L\cap R $. Next, consider

Hence, $ L\cap R $ is a quasi-hyperideal of $ S $.

Lemma 2.9. Let $ S $ be an $ LA $-semihyperring with a left identity $ e $ such that $ (xe)S\subseteq xS $ for all $ x\in S $. Then every quasi-hyperideal of $ S $ is a bi-hyperideal of $ S $.

Proof. Let $ B $ be a quasi-hyperideal of $ S $. Then, $ B+B\subseteq B $ and $ SB\cap BS\subseteq B $. Clearly, $ BB\subseteq B $. Now, we have that $ (BS)B\subseteq SB $. Next, let $ x\in (BS)B $. So, $ x\in (as)b $ for some $ a, b\in B $ and $ s\in S $. By assumption and using the medial law, we have

Thus, $ (BS)B\subseteq BS $. It follows that $ (BS)B\subseteq SB\cap BS\subseteq B $. Therefore, $ B $ is a bi-hyperideal of $ S $.

Lemma 2.10. If $ S $ is an $ LA $-semihyperring with a pure left identity $ e $, then for every $ a\in S $, $ a^2S $ is a hyperideal of $ S $ such that $ a^2\subseteq a^2S $.

Proof. Assume that $ S $ is an $ LA $-semihyperring with a pure left identity $ e $. Let $ a\in S $. By using the left invertive law, we have

Then, using Lemma 2.3, the left invertive law and the paramedial law, we have

and

Hence, $ a^2S $ is a hyperideal of $ S $. Now, using the left invertive law, we have

This completes the proof.

3.   Weakly regular $ LA $-semihyperrings

In this section, the class of weakly regular $ LA $-semihyperrings has been studied, we give some characterizations of weakly regular $ LA $-semihyperrings by using the concepts of left hyperideals and right hyperideals of $ LA $-semihyperrings.

Definition 3.1. An element $ a $ of an $ LA $-semihyperrnig $ S $ is said to be weakly regular if there exist $ x, y\in S $ such that $ a\in (ax)(ay) $. The $ LA $-semihyperring $ S $ is called weakly regular if every element of $ S $ is weakly regular.

Example 3.2. In Example 2.1, we can show that there exist $ x, y\in S $ such that $ a\in (ax)(ay) $ for all $ a\in S $. Therefore, $ S $ is weakly regular.

Theorem 3.3. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $. Then $ S $ is weakly regular if and only if $ R_1\cap R_2\subseteq R_1R_2 $, where both $ R_1 $ and $ R_2 $ are right hyperideals of $ S $.

Proof. Assume that $ S $ is weakly regular. Let $ R_1 $ and $ R_2 $ be right hyperideals of $ S $ and $ a\in R_1\cap R_2 $. Then, there exist $ x, y\in S $ such that $ a\in (ax)(ay)\subseteq (R_1S)(R_2S)\subseteq R_1R_2 $. Hence, $ R_1\cap R_2\subseteq R_1R_2 $.

Conversely, let $ a\in S $. Since $ S $ is a pure left invertible $ LA $-semihyperring, there exists $ x\in S $ such that $ e = xa $. By Lemma 2.10, we have that $ a^2S $ is a right hyperideal of $ S $ and $ a^2\subseteq a^2S $. Then, by using assumption, the left invertive law and Lemma 2.3, we have

Next, using the left invertive law and Lemma 2.3, we have

This implies that $ a\in (ax)(ay) $ for some $ x, y\in S $. Therefore, $ S $ is weakly regular.

The proof of the following theorem is similar to Theorem 3.3.

Theorem 3.4. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $. Then $ S $ is weakly regular if and only if $ L_1\cap L_2\subseteq L_1L_2 $, where both $ L_1 $ and $ L_2 $ are left hyperideals of $ S $.

Theorem 3.5. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $. Then $ S $ is weakly regular if and only if $ R\cap L\subseteq L^2R^2 $, for every right hyperideal $ R $ and left hyperideal $ L $ of $ S $.

Proof. Assume that $ S $ is weakly regular. Let $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $ and $ a\in R\cap L $. Then, there exist $ x, y\in S $ such that $ a\in (ax)(ay) $. By using the left invertive law, the medial law, the paramedial law and Lemma 2.3, we have

Therefore, $ R\cap L\subseteq L^2R^2 $.

Conversely, let $ R_1 $ and $ R_2 $ be right hyperideals of $ S $. By Lemma 2.6, we have that $ R_1 $ is also a left hyperideal of $ S $. By assumption, $ R_1\cap R_2\subseteq R_1^2R_2^2\subseteq R_1R_2 $. Consequently, $ S $ is weakly regular by Theorem 3.3.

Theorem 3.6. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $. Then $ S $ is weakly regular if and only if $ R\cap L\subseteq L^3R $, for every right hyperideal $ R $ and left hyperideal $ L $ of $ S $.

Proof. Let $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $ and $ a\in R\cap L $. By assumption, there exist $ x, y\in S $ such that $ a\in (ax)(ay) $. Then, by using the left invertive law, the medial law, the paramedial law and Lemma 2.3, we have

Hence, $ R\cap L\subseteq L^3R $.

Conversely, let $ R_1 $ and $ R_2 $ be right hyperideals of $ S $. By Lemma 2.6, we have that $ R_1 $ also a left hyperideal of $ S $. By assumption, $ R_1\cap R_2\subseteq R_1^3R_2 = ((R_1R_1)R_1)R_2\subseteq R_1R_2 $. By Theorem 3.1, $ S $ is weakly regular.

4.   Regular $ LA $-semihyperrings

In this section, we characterize the class of regular $ LA $-semihyperrings in terms of (resp., left, right) hyperideals, quasi-hyperideals and bi-hyperideals of $ LA $-semihyperrings.

Definition 4.1. An element $ a $ of an $ LA $-semihyperrnig $ S $ is said to be regular if there exists an element $ x\in S $ such that $ a\in (ax)a $. The $ LA $-semihyperring $ S $ is called regular if every element of $ S $ is regular.

Example 4.2. In Example 2.2, we have that there exists $ x\in S $ such that $ a\in (ax)a $ for all $ a\in S $. Hence, $ S $ is regular.

Lemma 4.3. Let $ S $ be an $ LA $-semihyperring. Then the following conditions are equivalent:

$ $(i)$ $ $ S $ is regular;

$ $(ii)$ $ $ a\in (aS)a $, for every $ a\in S $;

$ $(iii)$ $ $ A\subseteq (AS)A $, for all $ \emptyset\neq A\subseteq S $.

Theorem 4.4. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $. Then $ S $ is regular if and only if $ R\cap L = RL $, for every right hyperideal $ R $ and left hyperideal $ L $ of $ S $.

Proof. Assume that $ S $ is regular. Let $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $ and let $ a\in R\cap L $. Then, $ a\in (aS)a\subseteq (RS)L\subseteq RL $. It follows that $ R\cap L\subseteq RL $. Since $ RL\subseteq R $ and $ RL\subseteq L $, we have $ RL\subseteq R\cap L $. Thus, $ R\cap L = RL $.

Conversely, let $ a\in S $. Since $ S $ is a pure left invertible $ LA $-semihyperring, there exists $ x\in S $ such that $ e = xa $. By Lemma 2.10, $ a^2S $ is both a right hyperideal and a left hyperideal of $ S $. Moreover, $ a^2\subseteq a^2S $. Then, by using the given assumption, Lemma 2.3 and the left invertive law, we have

Hence, using the invertive law, we have

Therefore, $ S $ is regular.

Theorem 4.5. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $ such that $ (xe)S\subseteq xS $ for all $ x\in S $. Then the following statements are equivalent:

$ $(i)$ $ $ S $ is regular;

$ $(ii)$ $ $ (BS)B = B $, for every bi-hyperideal $ B $ of $ S $;

$ $(iii)$ $ $ (QS)Q = Q $, for every quasi-hyperideal $ Q $ of $ S $.

Proof. $ (i)\Rightarrow (ii) $ Assume that $ S $ is regular. Let $ B $ be a bi-hyperideal of $ S $ and $ a\in B $. Then, $ a\in (aS)a\subseteq (BS)B $. Thus, $ B\subseteq (BS)B $. On the other hand $ (BS)B\subseteq B $. Hence, $ (BS)B = B $.

$ (ii)\Rightarrow (iii) $ It follows from Lemma 2.9.

$ (iii)\Rightarrow (i) $ Let $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $. By Lemma 2.8, $ R\cap L $ is a quasi-hyperideal of $ S $. By assumption, we have that $ R\cap L = ((R\cap L)S)(R\cap L)\subseteq (RS)L\subseteq RL $. Any other way, $ RL\subseteq R\cap L $. Thus, $ R\cap L = RL $. Therefore, $ S $ is regular by Theorem 4.4.

Theorem 4.6. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $ such that $ (xe)S\subseteq xS $ for all $ x\in S $. Then the following statements are equivalent:

$(i)$ $ S $ is regular;

$(ii)$ $ B\cap I\subseteq (BI)B $, for every bi-hyperideal $ B $ and hyperideal $ I $ of $ S $;

$(iii)$ $ Q\cap I\subseteq (QI)Q $, for every quasi-hyperideal $ Q $ and hyperideal $ I $ of $ S $.

Proof. $ (i)\Rightarrow (ii) $ Assume that $ S $ is regular. Let $ B $ be a bi-hyperideal and $ I $ be a hyperideal of $ S $.

Now, let $ a\in B\cap I $. It turns out that $ a\in (aS)a $. Thus, by left invertive law and Lemma 2.3, we have

Hence, $ B\cap I\subseteq (BI)B $.

$ (ii)\Rightarrow (iii) $ By Lemma 2.9, we have that every quasi-hyperideal of $ S $ is a bi-hyperideal. Hence, $ (iii) $ holds.

$ (iii)\Rightarrow (i) $ Let $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $. Then, $ R\cap L $ is a quasi-hyperideal of $ S $ by Lemma 2.8. Since $ (iii) $ holds, we get that $ R\cap L = (R\cap L)\cap S\subseteq ((R\cap L)S)(R\cap L) \subseteq (RS)L\subseteq RL $. Also, $ R\cap L = RL $. By Theorem 4.4, $ S $ is regular.

Theorem 4.7. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $ such that $ (xe)S\subseteq xS $ for all $ x\in S $. Then the following conditions are equivalent:

$ $(i)$ $ $ S $ is regular;

$ $(ii)$ $ $ B\cap L\subseteq (BS)L $, for every bi-hyperideal $ B $ and left hyperideal $ L $ of $ S $;

$ $(iii)$ $ $ Q\cap L\subseteq (QS)L $, for every quasi-hyperideal $ Q $ and left hyperideal $ L $ of $ S $.

Proof. $ (i)\Rightarrow (ii) $ Assume that $ S $ is regular. Let $ B $ be a bi-hyperideal and $ L $ be a left hyperideal of $ S $ and $ a\in B\cap L $. Then, $ a\in (aS)a $. By using the left invertive law, we have

Hence, $ B\cap L\subseteq (BS)L $.

$ (ii)\Rightarrow (iii) $ Since every quasi-hyperideal is a bi-hyperideal of $ S $, $ (iii) $ holds.

$ (iii)\Rightarrow (i) $ Let $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $. By Lemma 2.7, $ R $ is also a quasi-hyperideal of $ S $. By assumption, $ R\cap L\subseteq (RS)L\subseteq RL $. So, $ R\cap L = RL $. Therefore, $ S $ is regular by Theorem 4.4.

The proof of the following theorem is similar to Theorem 4.7.

Theorem 4.8. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $ such that $ (xe)S\subseteq xS $ for all $ x\in S $. Then the following conditions are equivalent:

$(i) $ $ S $ is regular;

$(ii) $ $ B\cap R\subseteq (RS)B $, for every bi-hyperideal $ B $ and right hyperideal $ R $ of $ S $;

$ (iii) $ $ Q\cap R\subseteq (RS)Q $, for every quasi-hyperideal $ Q $ and right hyperideal $ R $ of $ S $.

Theorem 4.9. Let $ S $ be a pure left invertible $ LA $-semihyperring with a pure left identity $ e $ such that $ (xe)S\subseteq xS $ for all $ x\in S $. Then the following conditions are equivalent:

$ (i) $ $ S $ is regular;

$ (ii) $ $ B\cap R\cap L\subseteq (BR)L $, for every bi-hyperideal $ B $, right hyperideal $ R $ and left hyperideal $ L $ of $ S $;

$ (iii) $ $ Q\cap R\cap L\subseteq (QR)L $, for every quasi-hyperideal $ Q $, right hyperideal $ R $ and left hyperideal $ L $ of $ S $.

Proof. $ (i)\Rightarrow (ii) $ Assume that $ S $ is regular. Let $ B $ be a bi-hyperideal, $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $ and $ a\in B\cap R\cap L $. Then, $ a\in (aS)a $. By using the medial law, we have

This implies that $ B\cap R\cap L\subseteq (BR)L $.

$ (ii)\Rightarrow (iii) $ The implication follows by Lemma 2.9.

$ (iii)\Rightarrow (i) $ Let $ R $ be a right hyperideal and $ L $ be a left hyperideal of $ S $. By Lemma 2.7, $ R $ is also a quasi-hyperideal of $ S $. By the hypothesis, we have that $ R\cap L = R\cap R\cap L\subseteq (RR)L\subseteq RL $. Since $ RL\subseteq R\cap L $, it follows that $ R\cap L = RL $. By Theorem 4.4, $ S $ is regular.

5.   Conclusions

In this paper, the classes of weakly regular $ LA $-semihyperrings and regular $ LA $-semihyperrings have been considered. In Section 3, the characterizations of weakly regular $ LA $-semihyperrings by the properties of their left hyperideals and right hyperideals were shown in Theorem 3.3–Theorem 3.6. In Section 4, the fundamental characterization of regular $ LA $-semihyperrings by using their left hyperideals and right hyperideals has been given in Theorem 4.4. Finally, we characterized regular $ LA $-semihyperrings in terms of (resp., left, right) hyperideals, quasi-hyperideals and bi-hyperideals of $ LA $-semihyperrings were shown in Theorem 4.5–Theorem 4.9. In our future work, we will characterize the class of intra-regular $ LA $-semihyperrings by using the concept of their hyperideals.

Acknowledgments

This research project was financially supported by Mahasarakham University.

Conflict of interest

The author declares no conflict of interest.