Characterizations of intra-regular LA-semihyperrings in terms of their hyperideals

1.   Introduction

The algebraic structure of left almost semigroups (for short, $LA$-semigroups), which is a generalization of commutative semigroups, was first introduced by Kazim and Naseeruddin ^[20] in 1972. An Abel-Grassmann groupoid (for short, $AG$-groupoid) is another name for it ^[33]. A non-associative and a non-commutative algebraic structure that lies midway between a groupoid and a commutative semigroup is known as an $LA$-semigroup. Regularities are interesting and important properties to examine in $LA$-semigroups. In 2010, Khan and Asif ^[21] characterized intra-regular $LA$-semigroups by the properties of their fuzzy ideals. Later, Abdullah et al. ^[3] discussed characterizations of regular $LA$-semigroups using interval valued $(\alpha, \beta)$-fuzzy ideals. Also, Khan et al. ^[22] characterized right regular $LA$-semigroups using their fuzzy left ideals and fuzzy right ideals. In 2016, Khan et al. ^[25] characterized the class of $(m, n)$-regular $LA$-semigroups by their $(m, n)$-ideals. Some characterizations of weakly regular $LA$-semigroups by using the smallest ideals and fuzzy ideals of $LA$-semigroups are investigated by Yousafzai et al. ^[40]. In addition, Sezer ^[36] have used the concept of soft sets to characterize regular, intra-regular, completely regular, weakly regular and quasi-regular $LA$-semigroups. Now, many mathematicians have investigated various characterizations of $LA$-semigroups (see, e.g., ^[2,9,41]). Furthermore, some mathematicians have considered the notion of left almost semirings (for short, $LA$-semirings), that is a generalization of left almost rings (for short, $LA$-rings) ^[37], to have different features. In 2021, the left almost structures are now widely studied such as Elmoasry ^[13] studied the concepts of rough prime and rough fuzzy prime ideals in $LA$-semigroups, Massouros and Yaqoob ^[26] investigated the theory of left and right almost groups and focused on more general structures, and Rehman et al. ^[34] introduced the notion of neutrosophic $LA$-rings and discussed various types of ideals and establish several results to better understand the characteristic behavior of neutrosophic $LA$-rings. In addition, the concept of left almost has been investigated in various algebraic structures (for example, in ordered $LA$-semigroups ^[4,18,46], in ordered $LA$-$\Gamma$-semigroups ^[8], in gamma $LA$-rings and gamma $LA$-semigroups ^[24], in $LA$-polygroups ^[7,42,44]).

Marty ^[28] introduced the concept of hyperstructures, as a generalization of ordinary algebraic structures. The composition of two elements in an ordinary algebraic structure is an element, but in an algebraic hyperstructure, the composition of two elements is a nonempty set. Many authors have developed on the concept of hyperstructures (see, e.g., ^[1,12,38]). Rehman et al. ^[35] introduced the concept of left almost hypergroups (for short, $LA$-hypergroups) and gave the examples of $LA$-hypergroups. Moreover, they introduced the concept of $LA$-hyperrings and characterized $LA$-hyperrings by their hyperideals and hypersystems. Next, the concept of weak $LA$-hypergroups was investigated by Nawaz et al. ^[30]. In 2020, Hu et al. ^[17] extended the notion of neutrosophic to $LA$-hypergroups and strong pure $LA$-semihypergroups. The concept of left almost semihypergroups (for short, $LA$-semihypergroups) is a generalization of $LA$-semigroups and commutative semihypergroups developed by Hila and Dine ^[16]. An $LA$-semihypergroups is a non-associative and non-commutative hyperstructure midway between a hypergroupoid and a commutative semihypergroup. Yaqoob et al. ^[43] have characterized intra-regular $LA$-semihypergroups by using the properties of their left and right hyperideals. Then, Gulistan et al. ^[14] defined the class of regular $LA$-semihypergroups 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. Furthermore, Khan et al. ^[19] investigated some properties of fuzzy left hyperideals and fuzzy right hyperideals in regular and intra-regular $LA$-semihypergroups. Meanwhile, the notion of ordered $LA$-semihypergroups which is a generalization of $LA$-semihypergroups was introduced by Yaqoob and Gulistan ^[45]. Also, Azhar et al. discussed some results related with fuzzy hyperideals and generalized fuzzy hyperideals of ordered $LA$-semihypergroups ^[5,15].

It is known that every semiring can be considered to be a semihyperring. This implies that some results in intra-regular semihyperrings generalized the results in intra-regular semirings. The class of intra-regular semihyperrings was investigated by Nakkhasen and Pibaljommee ^[32] in 2019. Afterward, Nawaz et al. ^[31] introduced the notion of left almost semihyperrings (for short, $LA$-semihyperrings), which is a generalization of $LA$-semirings. Recently, Nakkhasen ^[29] characterized some classes of regularities in $LA$-semihyperrings, that is, weakly regular $LA$-semihyperrings and regular $LA$-semihyperrings by the properties of their hyperideals. In this paper, we are interested in the class of intra-regular $LA$-semihyperrings. Then, we give some characterizations of intra-regular $LA$-semihyperrings by means of their hyperideals. In addition, we show how ordered $LA$-semirings can be used to create $LA$-semihyperrings.

2.   Preliminaries

First, we will review some fundamental notions and properties that are needed for this study. Let $H$ be a nonempty set. Then, the mapping $\circ:H\times H\rightarrow \mathcal{P}^*(H)$ is called a hyperoperation (see, e.g., ^[10,11,39]) on $H$ where $\mathcal{P}^*(H) = \mathcal{P}(H)\setminus\{\emptyset\}$ denotes the set of all nonempty subsets of $H$. A hypergroupoid is a nonempty set $H$ together with a hyperopartion $\circ$ on $H$. 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 ^[16] if for all $x, y, z\in H, (x\circ y)\circ z = (z\circ y)\circ x$. 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 $(A\circ B)\circ C = (C\circ B)\circ A$.

A hyperstructure $(S, +, \cdot)$ is called an $LA$-semihyperring ^[31] 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. Let $\mathbb{Z}$ be the set of all integers. The hyperoperations $\ominus$ and $\odot$ on $\mathbb{Z}$ are defined by $x\ominus y = \{y-x\}$ and $x\odot y = \{xy\}$ for all $x, y\in \mathbb{Z}$, respectively. We have that $(\mathbb{Z}, \ominus, \odot)$ is an $LA$-semihyperrings.

Example 2.2. ^[35] 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.

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

The concepts listed below will be considered in this research, as they occurred in ^[31]. For any $LA$-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., $x = ex$). We have that $S^2 = S$, for any $LA$-semihyperring $S$ with a left identity $e$. 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 also a left identity, but the converse is not true in general, see in ^[29].

Lemma 2.1. ^[31] If $S$ is an $LA$-semihyperring with a pure left identity $e$, then $x(yz) = y(xz)$for all $x, y, z\in S$.

Let $S$ be an $LA$-semihyperring. Then, 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 $S$.

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 ^[31] of $S$ if $SA\subseteq A$;

$(ii)$ $A$ is called a right hyperideal ^[31] of $S$ if $AS\subseteq A$;

$(iii)$ $A$ is called a hyperideal ^[31] of $S$ if it is both a left and a right hyperideal of $S$;

$(iv)$ $A$ is called a quasi-hyperideal ^[31] of $S$ if $SA\cap AS\subseteq A$;

$(v)$ $A$ is called a bi-hyperideal ^[31] of $S$ if $AA\subseteq A$ and $(AS)A\subseteq A$.

Example 2.3. Let $S = \{a, b, c, d\}$. Define hyperoperations $+$ and $\cdot$ on $S$ by the following tables:

We can see that $(S, +, \cdot)$ is an $LA$-semihyperring. Consider $A = \{a, b, c\}$ and $B = \{a, c\}$. It is easy to see that $A$ is a quasi-hyperideal of $S$. In addition, $B$ is a bi-hyperideal of $S$, but it is not a quasi-hyperideal of $S$ because $SB\cap BS = \{a, b\} \nsubseteq B$.

A nonempty subset $G$ of an $LA$-semihyperring $S$ is called a generalized bi-hyperideal of $S$ if $G+G\subseteq G$ and $(GS)G\subseteq G$. Obviously, every bi-hyperideal of an $LA$-semihyperring $S$ is a generalized bi-hyperideal, but the converse is not true in general. We can show this with the following example.

Example 2.4. From Example 2.3, consider $G = \{a, c, d\}$. It is not difficult to show that $G$ is a generalized bi-hyperideal of $S$. But $G$ is not a bi-hyperideal of $S$, because $c\cdot d = \{a, b\}\nsubseteq G$.

An ordered $LA$-semiring is a system $(S, +, \cdot, \leq)$ consisting of a nonempty set $S$ such that $(S, +, \cdot)$ is an $LA$-semiring, $(S, \leq)$ is a partially ordered set, and for every $a, b, x\in S$ the following conditions are satisfied: $(i)$ if $a\leq b$, then $a+x\leq b+x$ and $x+a\leq x+b$; $(ii)$ if $a\leq b$, then $a\cdot x\leq b\cdot x$ and $x\cdot a\leq x\cdot b$. For an ordered $LA$-semiring $(S, +, \cdot, \leq)$ and $x\in S$, we denote $(x] = \{s\in S\mid s\leq x\}$.

In 2014, Amjad and Yousafzai ^[6] have shown that every ordered $LA$-semigroup $(S, \cdot, \leq)$ can be considered as an $LA$-semihypergroup $(S, \circ)$ where a hyperoperation $\circ$ on $S$ defined by

Now, we apply this idea to construct an $LA$-semihyperring from an ordered $LA$-semiring as the following lemma.

Lemma 2.2. Let $(S, +, \cdot, \leq)$ be an ordered $LA$-semiring. Then $(S, \oplus, \odot)$ is an $LA$-semihyperring where the hyperoperations $\oplus$ and $\odot$ on $S$ are defined by letting $a, b\in S$,

Proof. By the Example in ^[6], it follows that $(S, \oplus)$ and $(S, \odot)$ are $LA$-semihypergroups. Next, we will show that the hyperoperation $\odot$ is distributive with respect to the hyperoperation $\oplus$ on $S$. First, we claim that $a\odot(b\oplus c) = (a\cdot (b+c)]$. Let $t\in a\odot(b\oplus c)$. Then, $t\in a\odot x$ for some $x\in b\oplus c$. So, $t\leq a\cdot x\leq a\cdot(b+c)$, then $t\in (a\cdot(b+c)]$. Hence, $a\odot(b\oplus c)\subseteq(a\cdot (b+c)]$. Let $s\in (a\cdot (b+c)]$. Then, $s\leq a\cdot (b+c)$, and so

That is, $(a\cdot (b+c)]\subseteq a\odot(b\oplus c)$. It follows that $a\odot(b\oplus c) = (a\cdot (b+c)]$. Next, we show that $(a\odot b)\oplus(a\odot c) = (a\cdot b +a\cdot c]$. Let $t \in (a\odot b)\oplus(a\odot c)$. Then $t\in x\oplus y$ for some $x\in a\odot b$ and $y\in a\odot c$. This implies that $t\leq x+y\leq a\cdot b+a\cdot c$. Thus, $t\in (a\cdot b+a\cdot c]$. Hence, $(a\odot b)\oplus(a\odot c)\subseteq(a\cdot b+a\cdot c]$. Let $s\in (a\cdot b+a\cdot c]$. Then

Hence, $(a\cdot b+a\cdot c]\subseteq (a\odot b)\oplus(a\odot c)$. Therefore, $(a\odot b)\oplus(a\odot c) = (a\cdot b+a\cdot c]$. Since $(a\cdot(b+c)] = (a\cdot b+a\cdot c]$, we obtain that $a\odot(b\oplus c) = (a\odot b)\oplus(a\odot c)$. Similarly, we can show that $(b\oplus c)\odot a = (b\odot a)\oplus(c\odot a)$. Consequently, $(S, \oplus, \odot)$ is an $LA$-semihyperring.

Example 2.5. Let $S = \{a, b, c\}$ be a set with two binary operations $+$ and $\cdot$ on $S$ defined as follows:

Then, $(S, +, \cdot)$ is an $LA$-semiring ^[27]. We define an order relation $\leq$ on $S$ by

The figure of $\leq$ on $S$ is given by

It is a routine matter to check that $(S, +, \cdot, \leq)$ is an ordered $LA$-semiring. We obtain that its associated $LA$-semihyperring $(S, \oplus, \odot)$ where $\oplus$ and $\odot$ are defined by Lemma 2.2 as follows:

Now, we can see that $A = \{a, b\}$ is a left hyperideal of $S$, but it is not a right hyperideal of $S$ because $b\odot c = \{a, c\}\nsubseteq A$.

Lemma 2.3. ^[29] Let $S$ be an $LA$-semihyperring with a pure left identity $e$. Then every right hyperideal of $S$is a hyperideal of $S$.

Lemma 2.4. ^[29] Every left (resp., right) hyperideal of an $LA$-semihyperring $S$ is a quasi-hyperideal of $S$.

Lemma 2.5. Every left (resp., right) hyperideal of an $LA$-semihyperring $S$ is a bi-hyperideal of $S$.

Proof. Let $B$ be a left hyperideal of an $LA$-semihyperring $S$. Then, $BB\subseteq SB\subseteq B$, and so $(BS)B\subseteq SB\subseteq B$. Thus, $B$ is a bi-hyperideal of $S$. For the case right hyperideals, we can prove similarly.

Lemma 2.6. ^[29] 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 $S$.

Lemma 2.7. ^[29] 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 $a^2\subseteq a^2S$.

Lemma 2.8. If $S$ is an $LA$-semihyperring with a left identity $e$, then for every $a\in S$, $Sa$ is a left hyperideal of $S$ such that $a\in Sa$.

Proof. Assume that $S$ is an $LA$-semihyperring with a left identity $e$. Let $a\in S$. Then, $a\in ea\subseteq Sa$ and $Sa+Sa = (S+S)a\subseteq Sa$. Now, by using paramedial law and left invertive law, we have

It follows that $Sa$ is a left hyperideal of $S$.

Let $J$ be a finite nonempty subset of $\mathbb{N}$ such that $J = \{j_1, j_2, j_3, \ldots, j_n\}$, where $j_1, j_2, j_3, \ldots, j_n\in\mathbb{N}$. For any $a\in S$, we denote

For any nonempty subsets $A$ and $B$ of $LA$-semihyperring $S$ and $a\in S$, we denote

Remark 2.1. Let $A$ and $B$ be any nonempty subsets of an $LA$-semihyperring $S$. Then the following statements hold:

$(i)$ $A\subseteq \Sigma A$;

$(ii)$ $A(\Sigma B)\subseteq \Sigma AB$ and $(\Sigma A)B\subseteq \Sigma AB$.

Lemma 2.9. Let $A$ be any nonempty subset of an $LA$-semihyperring $S$. If $A+A\subseteq A$, then $\Sigma aA = aA$ and $\Sigma Aa = Aa$ for all $a\in S$.

3.   Characterizations of intra-regular LA-semihyperrings

In this section, we apply the concept of intra-regular $LA$-rings, defined in ^[23], to define the notion of intra-regular $LA$-semihyperrings and study some of its properties. Finally, we give some characterizations of intra-regular $LA$-semihyperrings by the properties of many types of hyperideals of $LA$-semihyperrings.

Definition 3.1. An $LA$-semihyperring $S$ is said to be intra-regular if for every $a\in S$, $a\in \Sigma (Sa^2)S$.

Example 3.1. 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 ^[31]. Now, we can see that $S$ is intra-regular.

However, the set $S = \{a, b, c, d, e\}$ with two hyperoperations $\oplus$ and $\odot$ on $S$ as defined in Example 2.5 is not intra-regular, because $b\not\in\{a\} = \Sigma(S\odot b^2)\odot S$.

Proposition 3.1. Every left (resp., right) hyperideal of an intra-regular $LA$-semihyperring $S$ is a hyperideal of $S$.

Proof. Let $S$ be an intra-regular $LA$-semihyperring and $x\in S$. Assume that $L$ is a left hyperideal of $S$ and $a\in L$. Then, $a\in \Sigma(Sa^2)S$. Now, by using Remark 2.1 and left invertive law, we have

Thus, $L$ is a right hyperideal of $S$, and so $L$ is a hyperideal of $S$. Suppose that $R$ is a right hyperideal of $S$ and $r\in R$. Then,

Hence, $R$ is a left hyperideal of $S$. It follows that $R$ is a hyperideal of $S$.

Proposition 3.2. If $S$ is an intra-regular $LA$-semihyperring with a pure left identity $e$, then $\Sigma I^2 = I$ for every left hyperideal $I$ of $S$.

Proof. Assume that $S$ is an intra-regular $LA$-semihyperring with a pure left identity $e$. Let $I$ be a left hyperideal of $S$. Then, $\Sigma I^2\subseteq I$. Let $a\in I$. By using left invertive law, medial law and Lemma 2.1, we have

Thus, $I \subseteq \Sigma I^2$. Therefore, $\Sigma I^2 = I$.

A (resp., left, right) hyperideal $P$ of an $LA$-semihyperring $S$ is called semiprime if for any $a\in S$, $a^2\subseteq P$ implies $a\in P$.

Proposition 3.3. Every hyperideal of an intra-regular $LA$-semihyperring is semiprime.

Proof. Assume that $S$ is an intra-regular $LA$-semihyperring. Let $I$ be a hyperideal of $S$ and $a\in S$ such that $a^2\subseteq I$. Then, $a\in \Sigma(Sa^2)S \subseteq \Sigma(SI)S \subseteq \Sigma IS \subseteq \Sigma I = I$. Hence, $I$ is semiprime.

Proposition 3.4. Let $S$ be an $LA$-semihyperring $S$ with a pure left identity $e$. If $S$ satisfies $L\cup R = \Sigma LR$, for every left hyperideal $L$ and every right hyperideal $R$ of $S$ such that $R$ is semiprime, then $S$ is intra-regular.

Proof. Let $a\in S$. By Lemma 2.8 and Lemma 2.7, we have that $Sa$ is a left hyperideal and $a^2S$ is a right hyperideal of $S$ such that $a\in Sa$ and $a^2\subseteq a^2S$, respectively. Thus, by the given assumption, $a\in a^2S$. Now, by using left invertive law, medial law and Lemma 2.1, we have

This shows that $S$ is intra-regular.

Next, we give characterizations of intra-regular $LA$-semihyperrings by means of (resp., left, right) hyperideals, quasi-hyperideals, bi-hyperideals and generalized bi-hyperideals of $LA$-semihyperrings as show by the following theorems.

Theorem 3.1. Let $S$ be an $LA$-semihyperring with a pure left identity $e$. Then $S$ is intra-regular if and only if $L = L^3$, for every left hyperideal $L$ of $S$.

Proof. Assume that $S$ is intra-regular. Let $L$ be any left hyperideal of $S$. Then, $L^3 = (LL)L \subseteq (SL)L \subseteq LL \subseteq L$. Now, let $a\in L$. By Lemma 2.7, $a^2S$ is a hyperideal of $S$ such that $a^2\subseteq a^2S$. Thus, by given assumption and Proposition 3.3, we have that $a^2S$ is semiprime, and so $a\in a^2S$. Thus, by using left invertive law and Lemma 2.1, we have

Hence, $L\subseteq L^3$. Therefore, $L = L^3$.

Conversely, assume that $L = L^3$, for every left hyperideal $L$ of $S$. Let $a\in S$. By Lemma 2.8, $Sa$ is a left hyperideal of $S$ such that $a\in Sa$. Then, by the given assumption and using medial law, we have

This shows that $S$ is intra-regular.

Theorem 3.2. Let $S$ be a pure left invertible $LA$-semihyperring with a pure left identity $e$. Then the following conditions are equivalent:

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

$(ii)$ $L\cap R\subseteq \Sigma LR$, where $L$ and $R$ are any left and right hyperideals of $S$, respectively.

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $L$ be a left hyperideal and $R$ be a right hyperideal of $S$, and let $a\in L\cap R$. Then, by using left invertive law and Lemma 2.1, we have

Hence, $L\cap R\subseteq \Sigma LR$.

$(ii)\Rightarrow (i)$ Assume that $(ii)$ holds. Let $a\in S$. Since $S$ is a pure left invertible, there exists $x\in S$ such that $e = xa$. By Lemma 2.7, $a^2S$ is both a left and a right hyperideal of $S$ such that $a^2\subseteq a^2S$. Then, by using left interive law, Lemma 2.1 and given assumption, we have

Now, by using left invertive law and Remark 2.1, we have

Therefore, $S$ is intra-regular.

Theorem 3.3. Let $S$ be a pure left invertible $LA$-semihyperring with a pure left identity $e$. Then the following statements are equivalent:

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

$(ii)$ $L\cap R = \Sigma RL$, for every left hyperideal $L$ and every right hyperideal $R$ of $S$.

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $L$ and $R$ be a left hyperideal and a right hyperideal of $S$, respectively. It is easy to see that $\Sigma RL\subseteq L\cap R$. On the other hand, let $a\in L\cap R$. Then, $a\in \Sigma (Sa^2)S$. By using left invertive law, paramedial law and Lemma 2.1, we have

Hence, $L\cap R\subseteq \Sigma RL$. Therefore, $L\cap R = \Sigma RL$.

$(ii)\Rightarrow (i)$ This proof is similar to the proof of $(ii)\Rightarrow (i)$ in Theorem 3.2, because $a^2S$ is both a left hyperideal and a right hyperideal of $S$.

Theorem 3.4. 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 intra-regular;

$(ii)$ $G\cap I = (GI)G$, for every generalized bi-hyperideal $G$ and every hyperideal $I$ of $S$;

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

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

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $G$ be a generalized bi-hyperideal and $I$ be a hyperideal of $S$, and let $a\in G\cap I$. Then, $a\in\Sigma (Sa^2)S$. Now, by using left invertive law and Lemma 2.1, we have

Consider,

Then, by using (3.1), medial law, Lemma 2.1 and Lemma 2.9, we have

It follows that $S(Sa) = a(Sa)$. Thus, $a\in \Sigma(S(Sa))a = \Sigma(a(Sa))a = (a(Sa))a \subseteq (G(SI))G \subseteq (GI)G.$ Hence, $G\cap I\subseteq (GI)G$. On the other hand, $(GI)G \subseteq (SI)S \subseteq I$ and $(GI)G \subseteq (GS)G \subseteq G$, that is, $(GI)G \subseteq G\cap I$. Therefore, $G\cap I = (GI)G$.

$(ii)\Rightarrow (iii)$ Since every bi-hyperideal is a generalized bi-hyperideal of $S$, it follows that $(iii)$ holds.

$(iii)\Rightarrow (iv)$ By Lemma 2.6, we have that every quasi-hyperideal of $S$ is a bi-hyperideal. Hence, $(iv)$ holds.

$(iv)\Rightarrow (i)$ Let $L$ be a left hyperideal and $R$ be a right hyperideal of $S$. By Lemma 2.3 and Lemma 2.4, we have that $R$ is a hyperideal and $L$ is a quasi-hyperideal of $S$, respectively. By assumption, $L\cap R = (LR)L \subseteq (SR)L\subseteq RL\subseteq \Sigma RL$. On the other hand, $\Sigma RL \subseteq L\cap R$. Therefore, $L\cap R = \Sigma RL$. By Theorem 3.3, we have that $S$ is intra-regular.

Theorem 3.5. 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 intra-regular;

$(ii)$ $R\cap G\subseteq \Sigma GR$, for every generalized bi-hyperideal $G$ and every right hyperideal $R$ of $S$;

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

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

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $R$ be a right hyperideal and $G$ be a generalized bi-hyperideal of $S$, and let $a\in R\cap G$. Then, $a\in \Sigma (Sa^2)S$. Since $S(Sa)\subseteq Sa$, left invertive law, medial law and Lemma 2.1, we obtain that

Hence, $R\cap G\subseteq \Sigma GR$.

$(ii)\Rightarrow (iii)$ Since every bi-hyperideal is a generalized bi-hyperideal of $S$, it follows that $(iii)$ holds.

$(iii)\Rightarrow (iv)$ By Lemma 2.6, we have that every quasi-hyperideal of $S$ is a bi-hyperideal. Hence, $(iv)$ holds.

$(iv)\Rightarrow (i)$ Let $L$ be a left hyperideal and $R$ be a right hyperideal of $S$. By Lemma 2.4, $L$ is a quasi-hyperideal of $S$. By assumption, $L\cap R\subseteq \Sigma LR$. Therefore, $S$ is intra-regular by Theorem 3.2.

Theorem 3.6. 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 conditions are equivalent:

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

$(ii)$ $R\cap G\subseteq \Sigma RG$, for every generalized bi-hyperideal $G$ and every right hyperideal $R$ of $S$;

$(iii)$ $R\cap B\subseteq \Sigma RB$, for every bi-hyperideal $B$ and every right hyperideal $R$ of $S$;

$(iv)$ $R\cap Q\subseteq \Sigma RQ$, for every quasi-hyperideal $Q$ and every right hyperideal $R$ of $S$.

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $G$ be a generalized bi-hyperideal and $R$ be a right hyperideal of $S$. Let $a\in R\cap G$. Then, $a\in \Sigma (Sa^2)S$. Thus, by using left invertive law and Lemma 2.1, we have $a\in \Sigma (Sa^2)S = \Sigma (S(aa))S = \Sigma (a(Sa))S = \Sigma (S(Sa))a$. Since $S(Sa) = a(Sa)$, we have

This implies that $R\cap G \subseteq \Sigma RG$.

$(ii)\Rightarrow (iii)$ Since every bi-hyperideal is a generalized bi-hyperideal of $S$, it turns out that $(iii)$ holds.

$(iii)\Rightarrow (iv)$ By Lemma 2.6, we have that every quasi-hyperideal of $S$ is a bi-hyperideal. So, $(iv)$ holds.

$(iv)\Rightarrow (v)$ Let $L$ and $R$ be a left hyperideal and a right hyperideal of $S$, respectively. By Lemma 2.4, $L$ is also a quasi-hyperideal of $S$. By hypothesis, $L\cap R \subseteq \Sigma RL$. Otherwise, $\Sigma RL \subseteq L\cap R$. Hence, $L\cap R = \Sigma RL$. Therefore, $S$ is intra-regular by Theorem 3.3.

Theorem 3.7. 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 statements are equivalent:

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

$(ii)$ $L\cap G\subseteq \Sigma LG$, for every generalized bi-hyperideal $G$ and every left hyperideal $L$ of $S$;

$(iii)$ $L\cap B\subseteq \Sigma LB$, for every bi-hyperideal $B$ and every left hyperideal $L$ of $S$;

$(iv)$ $L\cap Q\subseteq \Sigma LQ$, for every quasi-hyperideal $Q$ and every left hyperideal $L$ of $S$.

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $L$ be a left hyperideal and $G$ be a generalized bi-hyperideal of $S$, and let $a\in L\cap G$. Then, $a\in \Sigma(Sa^2)S$. Now, by using left invertive law and Lemma 2.1, we have

This implies that $L\cap G\subseteq \Sigma LG$.

$(ii)\Rightarrow (iii)$ Since every bi-hyperideal is a generalized bi-hyperideal of $S$, it follows that $(iii)$ holds.

$(iii)\Rightarrow (iv)$ By Lemma 2.6, we have that every quasi-hyperideal of $S$ is a bi-hyperideal. Hence, $(iv)$ holds.

$(iv)\Rightarrow (i)$ Let $L$ be a left hyperideal and $R$ be a right hyperideal of $S$. By Lemma 2.4, $R$ is also a quasi-hyperideal of $S$. By assumption, $L\cap R\subseteq \Sigma LR$. Therefore, $S$ is intra-regular by Theorem 3.2.

Theorem 3.8. 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 statements are equivalent:

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

$(ii)$ $L\cap G\subseteq \Sigma GL$, for every generalized bi-hyperideal $G$ and every left hyperideal $L$ of $S$;

$(iii)$ $L\cap B\subseteq \Sigma BL$, for every bi-hyperideal $B$ and every left hyperideal $L$ of $S$;

$(iv)$ $L\cap Q\subseteq \Sigma QL$, for every quasi-hyperideal $Q$ and every left hyperideal $L$ of $S$.

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $G$ be a generalized bi-hyperideal and $L$ be a left hyperideal of $S$ and let $a\in L\cap G$. Then, $a\in \Sigma (Sa^2)S$. Thus, by using $S(Sa)\subseteq Sa$, left invertive law, medial law and Lemma 2.1, we have

Hence, $L\cap G \subseteq \Sigma GL$.

$(ii)\Rightarrow (iii)$ Since every bi-hyperideal of $S$ is a generalized bi-hyperideal, it follows that $(iii)$ holds.

$(iii)\Rightarrow (iv)$ The implication holds from Lemma 2.6.

$(iv)\Rightarrow (i)$ Let $L$ and $R$ be a left hyperideal and a right hyperideal of $S$, respectively. By Lemma 2.4, $R$ is also a quasi-hyperideal of $S$. By the given assumption, we have $L\cap R\subseteq \Sigma RL$. On the other hand, $\Sigma RL \subseteq L\cap R$. Therefore, $L\cap R = \Sigma RL$. By Theorem 3.3, we obtain that $S$ is intra-regular.

Theorem 3.9. 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 intra-regular;

$(ii)$ $L\cap G\cap R\subseteq \Sigma (LG)R$, for every generalized bi-hyperideal $G$, every left hyperideal $L$ and every right hyperideal $R$ of $S$;

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

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

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $G$ be a generalized bi-hyperideal, $L$ be a left hyperideal and $R$ be a right hyperideal of $S$, and let $a\in L\cap G\cap R$. Then, $a\in \Sigma (Sa^2)S$. We note that $S(Sa) = a(Sa)$. Then, by using left invertive law, medial law, paramedial law and Lemma 2.1, we have

Hence, $L\cap G\cap R\subseteq \Sigma (LG)R$.

$(ii)\Rightarrow (iii)$ Since every bi-hyperideal is a generalized bi-hyperideal of $S$, it follows that $(iii)$ holds.

$(iii)\Rightarrow (iv)$ By Lemma 2.6, we have that every quasi-hyperideal of $S$ is a bi-hyperideal. Hence, $(iv)$ holds.

$(iv)\Rightarrow (i)$ Let $L$ be a left hyperideal and $R$ be a right hyperideal of $S$. By Lemma 2.4, $L$ is a quasi-hyperideal of $S$. By assumption, $L\cap R = L\cap L\cap R\subseteq \Sigma (LL)R\subseteq \Sigma (SL)R\subseteq \Sigma LR$. By Theorem 3.2, we obtain that $S$ is intra-regular.

Theorem 3.10. 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 intra-regular;

$(ii)$ $L\cap G\cap R\subseteq \Sigma (RG)L$, for every generalized bi-hyperideal $G$, every left hyperideal $L$ and every right hyperideal $R$ of $S$;

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

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

Proof. $(i)\Rightarrow (ii)$ Assume that $S$ is intra-regular. Let $G$ be a generalized bi-hyperideal, $L$ be a left hyperideal and $R$ be a right hyperideal of $S$. Let $a\in L\cap G\cap R$. Then, $a\in \Sigma (Sa^2)S$. Since $S(Sa) \subseteq (\Sigma S(Sa))a \subseteq Sa$ and by Lemma 2.9, we have $S(Sa) \subseteq (\Sigma S(Sa))a \subseteq (\Sigma Sa)a = (Sa)a$. By the given assumption, left invertive law, medial law, paramedial law and Lemma 2.1, we have

This shows that, $L\cap G\cap R\subseteq \Sigma (RG)L$.

$(ii)\Rightarrow (iii)$ Since every bi-hyperideal of $S$ is a generalized bi-hyperideal, which implies that $(iii)$ holds.

$(iii)\Rightarrow (iv)$ The proof follows from Lemma 2.6.

$(iv)\Rightarrow (v)$ Let $L$ be a left hyperideal and $R$ be a right hyperideal of $S$. Also, $L$ is a quasi-hyperideal of $S$ by Lemma 2.4. By assumption, we have that $L\cap R = L\cap L \cap R \subseteq \Sigma (RL)L \subseteq \Sigma (RS)L \subseteq \Sigma RL$. Otherwise, $\Sigma RL \subseteq L\cap R$. Hence, $L\cap R = \Sigma RL$. Therefore, $S$ is intra-regular by Theorem 3.3.

The following theorem, we can prove similarly.

Theorem 3.11. 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 intra-regular;

$(ii)$ $R\cap G \subseteq \Sigma (RG)R$, for every generalized bi-hyperideal $G$ and every right hyperideal $R$ of $S$;

$(iii)$ $R\cap B \subseteq \Sigma (RB)R$, for every bi-hyperideal $B$ every right hyperideal $R$ of $S$;

$(iv)$ $R\cap Q \subseteq \Sigma (RQ)R$, for every quasi-hyperideal $Q$ and every right hyperideal $R$ of $S$.

4.   Conclusions

In 2018, the concept of $LA$-semihyperrings was introduced by Nawaz et al. ^[31] as a generalization of $LA$-semirings. In Section 2, we have shown that some $LA$-semihyperring can be constructed from an ordered $LA$-semiring as shown in Lemma 2.2. This means that the $LA$-semihyperring is also a generalization of an ordered $LA$-semiring. In Section 3, we applied the concept of intra-regular $LA$-rings, appeared in ^[23], to define the concept of intra-regular $LA$-semihyperrings and discussed some of its properties. Finally, we characterized the class of intra-regular $LA$-semihyperrings by using (resp., left, right) hyperideals, quasi-hyperideals, bi-hyperideals and generalized bi-hyperideals of $LA$-semihyperrings were shown in Theorem 3.1 - Theorem 3.11. In our future study, we can consider the characterizations of the class of both regular and intra-regular $LA$-semihyperrings based on different types of hyperideals of $LA$-semihyperrings.

Acknowledgments

This research was financially supported by Faculty of Science, Mahasarakham University (Grant year 2020).

Conflict of interest

The author declares no conflict of interest.