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.