Cubic hyperideals in LA-semihypergroups

Abstract

The notion of cubic set was introduced in [23] as a generalization of the notion of fuzzy sets and intuitionistic fuzzy sets. In this paper, we initiate a study of cubic sets in left almost semihypergroups. By using the concept of cubic sets, we introduce the notion of cubic sub LA-semihypergroups (hyperideals and bi-hyperideals) and discuss some basic results on cubic sets in LA-semihypergroups.

Keywords

LA-semihypergroups cubic sets cubic hyperideals

1 Introduction

Marty [1] introduced the notion of a hyperstructure in 1934 and then, several authors continued their researches in this direction. Nowadays hyperstructures are widely studied from theoretical point of view and for their applications in many subjects of pure and applied mathematics. 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 basic definitions and theorems about hyperstructures can be found in [2, 3]. The concept of a semihypergroup is a generalization of the concept of a semigroup. Many authors studied different aspects of semihypergroups. Some principal notions about semihypergroups theory can be found in [4]. Recently, Hila and Dine [5] Hila introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semigroups. Yaqoob, Corsini and Yousafzai [6] extended the work of Hila and Dine and characterized intra-regular left almost semihypergroups by their hyperideals using pure left identities. Other results on LA-semihypergroups can be found in [7, 17, 18].

In 1965, Zadeh [8] introduced the notion of a fuzzy subset of a non-empty set X, as a function from X to [0, 1]. After the introduction of fuzzy sets, several researchers conducted the researches on generalizations of fuzzy sets with huge applications in computer, logics, automata and many branches of pure and applied mathematics. The notion of i-v fuzzy sets was first introduced by Zadeh [9] as an extension of fuzzy sets. Interval-valued fuzzy sets have been actively used in real-life applications. In [10], Narayanan and Manikantan introduced the notion of i-v fuzzy ideals generated by an i-v fuzzy subsets in semigroup. In 2008, Shabir and Khan introduced the concept of i-v fuzzy ideals generated by an i-v fuzzy subset in an ordered semigroup [11]. Recently, fuzzy set theory has been developed in the context of hyperalgebraic structure theory. In [12], Davvaz introduced the concept of fuzzy hyperideals in a semihypergroup. More on fuzzy hyperstructures one can find in [13]. But in fuzzy sets theory, there is no means to incorporate the hesitation or uncertainty in the membership degrees. Atanassov introduced the idea of defining a fuzzy set by describing a membership degree and a non-membership degree separately in such a way that the sum of the two degrees must not exceed one. Such pairs were called Intuitionistic Fuzzy Sets [19, 20]. This concept has been applied to various algebraic structures. The relations between intuitionistic fuzzy sets and algebraic structures have been already considered by many mathematicians. In [21], Davvaz established the intuitionistic fuzzification of the concept of hyperideals in a semihypergroup and investigated some of their properties. In [14], Ersoy and Davvaz and in [15, 16], Hila and Abdullah studied the structure of intuitionistic fuzzy sets in Γ-semihypergroups. Atanassov and Gargov [22] initiated the notion of i-v intuitionistic fuzzy sets which is a generalization of both intuitionistic fuzzy sets and interval valued fuzzy sets.

Several applications of fuzzy, intuitionistic and i-v (intuitionistic) fuzzy sets in different fields of real life and decision making, motivated the importance of cubic sets introduced by Jun, Kim and Yang [23] as a new type of fuzzy sets. Cubic sets are a generalization of fuzzy sets, interval-valued fuzzy sets and intuitionistic fuzzy sets. The theory of cubic sets attracted several mathematicians. Jun and others [24 –27] studied the theory of cubic sets in different algebraic structures. Yaqoob and others investigated some properties of cubic hyperideals in left almost Γ-semihypergroups [28] and cubic KU-ideals of KU-algebras [29].

In this paper, we initiate a study of cubic sets in left almost semihypergroups. We define a cubic left (right, two sided) hyperideal, cubic bi-hyperideal and cubic interior hyperideal in a left almost semihypergroup. We use cubic left (right, two-sided) and cubic bi-hyperideals to characterize some classes of left almost semihypergroups.

2 Preliminaries

In this section, we present some preliminary results that will be used throughout the paper. Recall first the basic terms and definitions from the hyperstructure theory.

Definition 2.1. A map $\circ : H \times H \to P^{*} (H)$ is called a hyperoperation or a join operation on a set H, where H is a non-empty set and $P^{*} (H) = P (H) ∖ {\emptyset}$ denotes the set of all non-empty subsets of H.

Definition 2.2. A hypergroupoid is a pair (H, ∘) , where ∘ is a hyperoperation on H.

Definition 2.3. [5, 6] A hypergroupoid (H, ∘) is called an LA-semihypergroup if for all x, y, z ∈ H, $(x \circ y) \circ z = (z \circ y) \circ x .$

The law (x ∘ y) ∘ z = (z ∘ y) ∘ x is called a left invertive law. In an LA-semihypergroup, the medial law (x ∘ y) ∘ (z ∘ w) = (x ∘ z) ∘ (y ∘ w) holds for all x, y, z, w ∈ H .

Let A and B be two non-empty subsets of H. Then we define $\begin{matrix} A \circ B & = & ⋃_{a \in A, b \in B} a \circ b, a \circ A = {a} \circ A, \\ a \circ B = {a} \circ B . \end{matrix}$

Example 2.4. Consider an LA-semigroup (H, ·) with a left identity and let A be a non-empty subset of H . Define a hyperoperation ∘ on H by x ∘ y = x · (A · y) , where x, y ∈ H . Then (H, ∘) becomes an LA-semihypergroup. Indeed, $\begin{matrix} (x \circ y) \circ z & = & (x \cdot (A \cdot y)) \circ z = (x \cdot (A \cdot y)) \cdot (A \cdot z) \\ = & (x \cdot A) \cdot ((A \cdot y) \cdot z) \\ = & (z \cdot (A \cdot y)) \cdot (A \cdot x) = (z \circ y) \circ x . \end{matrix}$

This implies that (x ∘ y) ∘ z = (z ∘ y) ∘ x holds for all x, y, z ∈ H . Hence (H, ∘) is an LA-semihypergroup.

Example 2.5. Let $H = {(\begin{matrix} a & 0 \\ 0 & b \end{matrix}) : a, b \in ℝ ∖ {0}}$ be the set of 2 × 2 commutative matrices. For every x, y ∈ H, we define a binary hyperoperation ∘ on H by x ∘ y = yKx^-1, where $K = {(\begin{matrix} a & 0 \\ 0 & b \end{matrix}) : a, b \in ℚ ∖ {0}} \subseteq H .$ Then, $\begin{matrix} (x \circ y) \circ z & = & zK ({yKx}^{- 1})^{- 1} = {zKxK}^{- 1} y^{- 1} \\ = & {xKzK}^{- 1} y^{- 1} = xK ({yKz}^{- 1})^{- 1} \\ = & (z \circ y) \circ x . \end{matrix}$

This implies that (x ∘ y) ∘ z = (z ∘ y) ∘ x holds for all x, y, z ∈ H . Notice that (x ∘ y) ∘ z = x ∘ (y ∘ z) does not hold. Hence (H, ∘) is an LA-semihypergroup.

Example 2.6. Let $ℂ$ be the field of complex numbers, endowed with a binary hyperoperation ∘ defined by $x \circ y = y ℤ \bar{x},$ where $\bar{x}$ is conjugate of complex number x . Then $(ℂ, \circ)$ becomes an LA-semihypergroup.

An element e ∈ H is called a left identity (resp., pure left identity) if for all x ∈ H, x ∈ e ∘ x (resp., x = e ∘ x). An LA-semihypergroup may or may not contains a left identity and a pure left identity.

Example 2.7. [6] Let H = {x, y, z, w} with the binary hyperoperation defined below:

∘	x	y	z	w
x	x	y	z	w
y	z	{y, z}	{y, z}	w
z	y	{y, z}	{y, z}	w
w	w	w	w	H

Clearly H is not a semihypergroup because {z} = (z ∘ x) ∘ x ¬ = z ∘ (x ∘ x) = {y} . Thus H is an LA-semihypergroup because the elements of H satisfies the left invertive law. Here x is a pure left identity because for all a ∈ H, a = x ∘ a .

Lemma 2.8. [6] Let H be an LA-semihypergroup with a pure left identity e . Then x ∘ (y ∘ z) = y ∘ (x ∘ z) holds for all x, y, z ∈ H .

Lemma 2.9. [6] Let H be an LA-semihypergroup with a pure left identity e . Then the paramedial law (x ∘ y) ∘ (z ∘ w) = (w ∘ z) ∘ (y ∘ x) holds for all x, y, z, w ∈ H .

Definition 2.10. [5] Let H be an LA-semihypergroup. A non-empty subset A of H is called a sub LA-semihypergroup of H if x ∘ y ⊆ A for every x, y ∈ A .

Definition 2.11. [5] A subset I of an LA-semihypergroup H is called a right (resp., left) hyperideal of H if I ∘ H ⊆ I (resp., H ∘ I ⊆ I) and is called a hyperideal if it is two-sided hyperideal.

Definition 2.12. [6] By a bi-hyperideal (resp., generalized bi-hyperideal) of an LA-semihypergroup H, we mean a sub LA-semihypergroup (resp., non-empty subset) B of H such that (B ∘ H) ∘ B ⊆ B .

Definition 2.13. [6] By an interior hyperideal of an LA-semihypergroup H, we mean a subset A of H such that (H ∘ A) ∘ H ⊆ A .

An LA-semihypergroup H is called a regular LA-semihypergroup if for every x ∈ H, x ∈ (x ∘ y) ∘ x, for some y ∈ H. An element a of H is called a left regular element of H if there exists x ∈ H such that a ∈ x ∘ (a ∘ a) and H is called left regular if every element of H is left regular.

Now we recall the concept of i-v fuzzy sets.

An interval number is $\tilde{a} = [a^{-}, a^{+}]$ , where 0 ≤ a^- ≤ a⁺ ≤ 1. Let D [0, 1] denotes the family of all closed subintervals of [0, 1], i.e., $\begin{matrix} D [0, 1] & = & {\tilde{a} = [a^{-}, a^{+}] : a^{-} \leq a^{+}, for \\ a^{-}, a^{+} \in I} . \end{matrix}$

We define the operations “⪰”, “⪯”, “=”, “rmin” and “rmax” for two elements in D [0, 1] . Let $\tilde{a} = [a^{-}, a^{+}]$ and $\tilde{b} = [b^{-}, b^{+}]$ in D [0, 1] . Then

$\tilde{a} ⪰ \tilde{b}$ if and only if a^- ≥ b^- and a⁺ ≥ b⁺,

$\tilde{a} ⪯ \tilde{b}$ if and only if a^- ≤ b^- and a⁺ ≤ b⁺,

$\tilde{a} = \tilde{b}$ if and only if a^- = b^- and a⁺ = b⁺,

$rmin {\tilde{a}, \tilde{b}} = [min {a^{-}, b^{-}}, min {a^{+}, b^{+}}]$ ,

$rmax {\tilde{a}, \tilde{b}} = [max {a^{-}, b^{-}}, max {a^{+}, b^{+}}] .$

It is obvious that (D [0, 1] , ⪯ , ∨ , ∧) is a complete lattice with $\tilde{0} = [0, 0]$ as its least element and $\tilde{1} = [1, 1]$ as its greatest element. Let $\tilde{a_{i}} \in D [0, 1]$ where i ∈ Λ . We define $\begin{matrix} \underset{i \in Λ}{rinf} {\tilde{a}}_{i} = [inf_{i \in Λ} a_{i}^{-}, inf_{i \in Λ} a_{i}^{+}], \\ \underset{i \in Λ}{rsup} {\tilde{a}}_{i} = [sup_{i \in Λ} a_{i}^{-}, sup_{i \in Λ} a_{i}^{+}] . \end{matrix}$ An interval valued fuzzy set (briefly, IVF-set) ${\tilde{η}}_{A}$ on X is defined by: ${\tilde{η}}_{A} = {〈 x, [η_{A}^{-} (x), η_{A}^{+} (x)] 〉 : x \in X}$ , where $η_{A}^{-} (x) \leq η_{A}^{+} (x)$ , for all x ∈ X. Then the ordinary fuzzy sets $η_{A}^{-} : X \to [0, 1]$ and $η_{A}^{+} : X \to [0, 1]$ are called a lower fuzzy set and an upper fuzzy set of $\tilde{η}$ , respectively. If ${\tilde{η}}_{A} (x) = [η_{A}^{-} (x), η_{A}^{+} (x)],$ then $A = {〈 x, {\tilde{η}}_{A} (x) 〉 : x \in X},$ where ${\tilde{η}}_{A} : X ⟶ D [0, 1] .$

3 Cubic hyperideals in LA-semihypergroups

Jun and others [23], introduced the idea of cubic set defined on a non-empty set X as follows: $ℑ = {〈 x, {\tilde{η}}_{ℑ} (x), ϑ_{ℑ} (x) 〉 : x \in X},$ which is briefly denoted by $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉,$ where ${\tilde{η}}_{ℑ} : X ⟶ D [0, 1]$ and ϑ_ℑ : X → [0, 1].

Let $ℑ_{1} = 〈 {\tilde{η}}_{ℑ_{1}}, ϑ_{ℑ_{1}} 〉$ and $ℑ_{2} = 〈 {\tilde{η}}_{ℑ_{2}}, ϑ_{ℑ_{2}} 〉$ be two cubic sets in an LA-semihypergroup H . Then $\begin{array}{l} ℑ_{1} \cap ℑ_{2} = {〈 x, {\tilde{η}}_{ℑ_{1} \cap ℑ_{2}} (x), ϑ_{ℑ_{1} \cap ℑ_{2}} (x) 〉 : x \in H} \\ = {〈 x, r m i n {{\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{2}} (x)}, \max {ϑ_{ℑ_{1}} (x), \\ ϑ_{ℑ_{2}} (x)} 〉 : x \in H}, \\ ℑ_{1} \cup ℑ_{2} = {〈 x, {\tilde{η}}_{ℑ_{1} \cup ℑ_{2}} (x), ϑ_{ℑ_{1} \cup ℑ_{2}} (x) 〉 : x \in H} \\ = {〈 x, r m a x {{\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{2}} (x)}, \min {ϑ_{ℑ_{1}} (x), \\ ϑ_{ℑ_{2}} (x)} 〉 : x \in H}, \end{array}$ and $ℑ_{1} * ℑ_{2} = {〈 x, {\tilde{η}}_{ℑ_{1} * ℑ_{2}} (x), ϑ_{ℑ_{1} * ℑ_{2}} (x) 〉 : x \in H},$ where $\begin{matrix} {\tilde{η}}_{ℑ_{1} * ℑ_{2}} (x) \\ = {\begin{matrix} \underset{x \in y \circ z}{rsup} {rmin {{\tilde{η}}_{ℑ_{1}} (y), {\tilde{η}}_{ℑ_{2}} (z)}} & if x \in y \circ z \\ [0, 0] & otherwise \end{matrix} \\ ϑ_{ℑ_{1} * ℑ_{2}} (x) \\ = {\begin{matrix} inf_{x \in y \circ z} {max {ϑ_{ℑ_{1}} (y), ϑ_{ℑ_{2}} (z)}} & if x \in y \circ z \\ 1 & otherwise . \end{matrix} \end{matrix}$

Denote by $C (H)$ the family of all cubic sets in H . For any ℑ₁, $ℑ_{2} \in C (H),$ $ℑ_{1} \subseteq ℑ_{2} \Leftrightarrow {\tilde{η}}_{ℑ_{1}} ⪯ {\tilde{η}}_{ℑ_{2}},$ ϑ_ℑ₁ ≥ ϑ_ℑ₂.

Proposition 3.1. If H is an LA-semihypergroup, then the set $(C (H), *)$ is an LA-semihypergroup.

Proof. The proof is straightforward. □

Proposition 3.2. If H is an LA-semihypergroup, then the medial law $(ℑ_{1} * ℑ_{2}) * (ℑ_{3} * ℑ_{4}) = (ℑ_{1} * ℑ_{3}) * (ℑ_{2} * ℑ_{4})$ holds in $C (H),$ for all ℑ₁, ℑ ₂, ℑ ₃ and ℑ₄ in $C (H) .$

Proof. The proof is straightforward. □

Proposition 3.3. Let H be an LA-semihypergroup with a pure left identity. Then the following properties hold in $C (H) :$

ℑ₁∗ (ℑ ₂ ∗ ℑ ₃) = ℑ ₂ ∗ (ℑ ₁ ∗ ℑ ₃) ;

(ℑ ₁ ∗ ℑ ₂) ∗ (ℑ ₃ ∗ ℑ ₄) = (ℑ ₄ ∗ ℑ ₃) ∗ (ℑ ₂ ∗ ℑ ₁) ;

for all ℑ₁, ℑ ₂, ℑ ₃ and ℑ₄ in $C (H) .$

Proof. (i) Let $ℑ_{1} = 〈 {\tilde{η}}_{ℑ_{1}}, ϑ_{ℑ_{1}} 〉,$ $ℑ_{2} = 〈 {\tilde{η}}_{ℑ_{2}}, ϑ_{ℑ_{2}} 〉$ and $ℑ_{3} = 〈 {\tilde{η}}_{ℑ_{3}}, ϑ_{ℑ_{3}} 〉$ be in $C (H)$ . There exist x and y in H such that z ∈ x ∘ y . Then $\begin{matrix} {\tilde{η}}_{ℑ_{1} * (ℑ_{2} * ℑ_{3})} (z) \\ = \underset{z \in x \circ y}{rsup} {rmin {{\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{2} * ℑ_{3}} (y)}} \\ = \underset{z \in x \circ y}{rsup} {rmin {{\tilde{η}}_{ℑ_{1}} (x), \\ \underset{y \in t \circ w}{rsup} {rmin {{\tilde{η}}_{ℑ_{2}} (t), {\tilde{η}}_{ℑ_{3}} (w)}}}} \\ = \underset{z \in x \circ y}{rsup} \underset{y \in t \circ w}{rsup} {rmin {{\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{2}} (t), {\tilde{η}}_{ℑ_{3}} (w)}} \\ = \underset{z \in x \circ (t \circ w)}{rsup} {rmin {{\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{2}} (t), {\tilde{η}}_{ℑ_{3}} (w)}} \\ = \underset{z \in t \circ (x \circ w)}{rsup} {rmin {{\tilde{η}}_{ℑ_{2}} (t), {\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{3}} (w)}}, \\ (by Lemma 2.8) \\ = \underset{z \in t \circ m}{rsup} \underset{m \in x \circ w}{rsup} {rmin {{\tilde{η}}_{ℑ_{2}} (t), {\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{3}} (w)}} \\ = \underset{z \in t \circ m}{rsup} {rmin {{\tilde{η}}_{ℑ_{2}} (t), \\ \underset{m \in x \circ w}{rsup} {rmin {{\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{3}} (w)}}}} \\ = \underset{z \in t \circ m}{rsup} {rmin {{\tilde{η}}_{ℑ_{2}} (t), {\tilde{η}}_{ℑ_{1} * ℑ_{3}} (m)}} \\ = {\tilde{η}}_{ℑ_{2} * (ℑ_{1} * ℑ_{3})} (z), \end{matrix}$ and $\begin{matrix} ϑ_{ℑ_{1} * (ℑ_{2} * ℑ_{3})} (z) \\ = inf_{z \in x \circ y} {max {ϑ_{ℑ_{1}} (x), ϑ_{ℑ_{2} * ℑ_{3}} (y)}} \\ = inf_{z \in x \circ y} {max {ϑ_{ℑ_{1}} (x), \\ inf_{y \in t \circ w} {max {ϑ_{ℑ_{2}} (t), ϑ_{ℑ_{3}} (w)}}}} \\ = inf_{z \in x \circ y} inf_{y \in t \circ w} {max {ϑ_{ℑ_{1}} (x), ϑ_{ℑ_{2}} (t), ϑ_{ℑ_{3}} (w)}} \\ = inf_{z \in x \circ (t \circ w)} {max {ϑ_{ℑ_{1}} (x), ϑ_{ℑ_{2}} (t), ϑ_{ℑ_{3}} (w)}} \\ = inf_{z \in t \circ (x \circ w)} {max {ϑ_{ℑ_{2}} (t), ϑ_{ℑ_{1}} (x), ϑ_{ℑ_{3}} (w)}}, \\ (by Lemma 2.8) \\ = inf_{z \in t \circ m} inf_{m \in x \circ w} {max {ϑ_{ℑ_{2}} (t), ϑ_{ℑ_{1}} (x), ϑ_{ℑ_{3}} (w)}} \\ = inf_{z \in t \circ m} {max {ϑ_{ℑ_{2}} (t), \\ inf_{m \in x \circ w} {max {ϑ_{ℑ_{1}} (x), ϑ_{ℑ_{3}} (w)}}}} \\ = inf_{z \in t \circ m} {max {ϑ_{ℑ_{2}} (t), ϑ_{ℑ_{1} * ℑ_{3}} (m)}} \\ = ϑ_{ℑ_{2} * (ℑ_{1} * ℑ_{3})} (z) . \end{matrix}$

Let z be an element of H such that z ∉ x ∘ y, for some x, y ∈ H. Then we have ${\tilde{η}}_{ℑ_{1} * (ℑ_{2} * ℑ_{3})} (z) = [0, 0] = {\tilde{η}}_{ℑ_{2} * (ℑ_{1} * ℑ_{3})} (z),$ and $ϑ_{ℑ_{1} * (ℑ_{2} * ℑ_{3})} (z) = 1 = ϑ_{ℑ_{2} * (ℑ_{1} * ℑ_{3})} (z) .$

Hence this shows that ℑ₁ ∗ (ℑ ₂ ∗ ℑ ₃) = ℑ ₂ ∗ (ℑ ₁ ∗ ℑ ₃) holds in $C (H)$ .

(2) The proof is similar to (1). □

Lemma 3.4. Let H be an LA-semihypergroup with a pure left identity. Then for all x ∈ H, $H * H = H,$ where $H = 〈 \tilde{1}, 0 〉,$ $\tilde{1} (x) = \tilde{1}$ and 0 (x) =0 .

Proof. The proof is straightforward. □

Definition 3.5. Let H be an LA-semihypergroup. A cubic set $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ in H is called a cubic sub LA-semihypergroup of H if for all x, y ∈ H,

$\underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (z)} ⪰ rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (y)}$

$sup_{z \in x \circ y} {ϑ_{ℑ} (z)} \leq max {ϑ_{ℑ} (x), ϑ_{ℑ} (y)} .$

Example 3.6. Let H = {e, x, y, z, w} be a set with the binary hyperoperation defined as follow:

∘	e	x	y	z	w
e	e	e	e	e	e
x	e	w	w	{y, w}	w
y	e	w	w	{x, w}	w
z	e	{x, w}	{y, w}	{z, w}	w
w	e	w	w	w	w

Clearly H is not a semihypergroup because {x, w} = (x ∘ z) ∘ z ¬ = x ∘ (z ∘ z) = {y, w} . Thus H is an LA-semihypergroup because the elements of H satisfies the left invertive law. Let

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

be a cubic subset of H, which is defined by

H	${\tilde{η}}_{ℑ}$	ϑ _ℑ
e	[0.7, 0.8]	0.2
{x, y}	[0.4, 0.47]	0.6
z	[0.2, 0.3]	0.9
w	[0.5, 0.6]	0.4

It can be seen that

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

is a cubic sub LA-semihypergroup of H .

Definition 3.7. Let H be an LA-semihypergroup. A cubic set $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ in H is called a cubic left (resp., right) hyperideal of H if for all x, y ∈ H,

$\underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (z)} ⪰ {\tilde{η}}_{ℑ} (y)$ (resp., $\underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (z)} ⪰ {\tilde{η}}_{ℑ} (x)$ ),

$sup_{z \in x \circ y} {ϑ_{ℑ} (y)} \leq ϑ_{ℑ} (y)$ (resp., $sup_{z \in x \circ y} {ϑ_{ℑ} (z)} \leq ϑ_{ℑ} (x)$ ).

Example 3.8. Let H = {x, y, z} be a set with the binary hyperoperation defined as follow:

∘	x	y	z
x	z	z	z
y	{x, z}	z	z
z	z	z	z

Clearly H is not a semihypergroup because {z} = (y ∘ y) ∘ x ¬ = y ∘ (y ∘ x) = {x, z} . Thus H is an LA-semihypergroup because the elements of H satisfies the left invertive law. Let

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

be a cubic subset of H, which is defined by

H	${\tilde{η}}_{ℑ}$	ϑ _ℑ
x	[0.2, 0.3]	0.8
y	[0.4, 0.5]	0.5
z	[0.7, 0.9]	0.2

It can be seen that

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

is a cubic left hyperideal of H . But

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

is not a cubic right hyperideal of H, because

\begin{matrix} [0.2, 0.3] & = & \underset{t \in y \circ x}{rinf} {{\tilde{η}}_{ℑ} (t)} ≽ {\tilde{η}}_{ℑ} (y) = [0.4, 0.5] \\ 0.8 & = & sup_{t \in y \circ x} {ϑ_{ℑ} (t)} ≰ ϑ_{ℑ} (y) = 0.5 . \end{matrix}

A cubic set ℑ in H is called a two-sided cubic hyperideal of H if it is both a cubic left and a cubic right hyperideal of H.

Example 3.9. Let H = {e, x, y, z, w} be a set with the binary hyperoperation defined as follow:

∘	e	x	y	z	w
e	{e, y}	y	{x, y}	{z, w}	w
x	{x, y}	y	y	{z, w}	w
y	{x, y}	{x, y}	{x, y}	{z, w}	w
z	{z, w}	{z, w}	{z, w}	z	w
w	w	w	w	w	w

Clearly H is not a semihypergroup because {x, y} = (x ∘ x) ∘ y ¬ = x ∘ (x ∘ y) = {y} . Thus H is an LA-semihypergroup because the elements of H satisfies the left invertive law. Let

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

be a cubic subset of H, which is defined by

H	${\tilde{η}}_{ℑ}$	ϑ _ℑ
e	[0.1, 0.21]	0.7
{x, y}	[0.24, 0.3]	0.5
z	[0.35, 0.4]	0.2
w	[0.47, 0.5]	0.1

It can be seen that

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

is a cubic hyperideal of H .

Definition 3.10. Let H be an LA-semihypergroup. A cubic subset $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ of H is called a cubic interior hyperideal of H if for all x, y, z ∈ H,

$\underset{a \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ {\tilde{η}}_{ℑ} (y)$ ,

$sup_{a \in (x \circ y) \circ z} {ϑ_{ℑ} (a)} \leq ϑ_{ℑ} (y)$ .

Example 3.11. Consider the LA-semihypergroup H = {e, x, y, z, w} defined in the Example 3.6. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic subset of H, which is defined by ${\tilde{η}}_{ℑ} (a) = {\begin{matrix} [0.6, 0.9] & if a \in {e, w} \\ [0.1, 0.3] & if a \in {x, y, z} \end{matrix}$ and $ϑ_{ℑ} (a) = {\begin{matrix} 0.4 & if a \in {e, w} \\ 0.7 & if a \in {x, y, z} . \end{matrix}$

It can be seen that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic interior hyperideal of H .

Definition 3.12. Let H be an LA-semihypergroup. A cubic subset $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ of H is called a cubic generalized bi-hyperideal of H if for all x, y, z ∈ H,

$\underset{a \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (z)}$ ,

$sup_{a \in (x \circ y) \circ z} {ϑ_{ℑ} (a)} \leq max {ϑ_{ℑ} (x), ϑ_{ℑ} (z)}$ .

Definition 3.13. Let H be an LA-semihypergroup. A cubic sub LA-semihypergroup $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ of H is called a cubic bi-hyperideal of H if for all x, y, z ∈ H,

$\underset{a \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (z)}$ ,

$sup_{a \in (x \circ y) \circ z} {ϑ_{ℑ} (a)} \leq max {ϑ_{ℑ} (x), ϑ_{ℑ} (z)}$ .

Example 3.14. Let us consider $\begin{matrix} H & = & {(\begin{matrix} a & 0 \\ 0 & b \end{matrix}) : a, b \in ℤ^{+}}, \\ K & = & {(\begin{matrix} a & 0 \\ 0 & 1 \end{matrix}) : a \in ℤ^{+}} \end{matrix}$ be sets of 2 × 2 commutative matrices. Then H is an LA-semihypergroup under the hyperoperation ∘ on H, defined by x ∘ y = y · K · adjx, for all x, y ∈ H . Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic subset of H, which is defined by ${\tilde{η}}_{ℑ} (x) = {\begin{matrix} [0.4, 0.5] & if a, b \in 2 ℤ^{+} \\ [0.1, 0.3] & otherwise \end{matrix}$ and $ϑ_{ℑ} (x) = {\begin{matrix} 0.2 & if a, b \in 2 ℤ^{+} \\ 0.8 & otherwise . \end{matrix}$

It can be seen that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic bi-hyperideal of H .

Let H be an LA-semihypergroup. Let $\tilde{t} \in D [0, 1]$ and s ∈ [0, 1] . Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic set in H. Then the set $U (ℑ; \tilde{t}, s) = {x \in H : {\tilde{η}}_{ℑ} (x) ⪰ \tilde{t}, ϑ_{ℑ} (x) \leq s}$ is called a cubic level set of $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉 .$

Theorem 3.15. Let H be an LA-semihypergroup and $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic set in H. Then $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a left (resp., right) cubic hyperideal of H if and only if for all $\tilde{t} \in D [0, 1]$ and s ∈ [0, 1], the set $U (ℑ; \tilde{t}, s)$ is either empty or a left (resp., right) cubic hyperideal of H.

Proof. The proof is straightforward. □

Let H be an LA-semihypergroup. Then the cubic characteristic function $χ_{ℑ} = 〈 {\tilde{η}}_{χ_{ℑ}}, ϑ_{χ_{ℑ}} 〉$ of $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is defined by $\begin{matrix} {\tilde{η}}_{χ_{ℑ}} = {\begin{matrix} [1, 1] & if x \in ℑ \\ [0, 0] & if x \notin ℑ \end{matrix} and \\ ϑ_{χ_{ℑ}} = {\begin{matrix} 0 & if x \in ℑ \\ 1 & if x \notin ℑ . \end{matrix} \end{matrix}$

Theorem 3.16. Let H be an LA-semihypergroup and $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ a cubic set in H. The following statements are equivalent:

$ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic sub LA-semihypergroup (resp., left hyperideal, right hyperideal, generalized bi-hyperideal, bi-hyperideal, interior hyperideal) of H.

ℑ∗ ℑ ⊆ ℑ (resp., $H * ℑ \subseteq ℑ,$ $ℑ * H \subseteq ℑ,$ $(ℑ * H) * ℑ \subseteq ℑ,$ ℑ∗ ℑ ⊆ ℑ and $(ℑ * H) * ℑ \subseteq ℑ,$ $(H * ℑ) * H \subseteq ℑ)$ , where $H = 〈 \tilde{1}, 0 〉,$ $\tilde{1} (x) = \tilde{1}$ and 0 (x) =0, for all x ∈ H.

Proof. Let us suppose that ℑ is a cubic generalized bi-hyperideal of H and let a ∈ H. If $((ℑ * H) * ℑ) (a) = 〈 [0, 0], 1 〉$ , then $(ℑ * H) * ℑ \subseteq ℑ$ .

Otherwise, there exist x, y, z, t ∈ H such that a ∈ x ∘ y and x ∈ z ∘ t. From the definition of cubic generalized bi-hyperideal, we have $\underset{a \in (x \circ y) \subseteq ((z \circ t) \circ y)}{rinf} {\tilde{η}}_{ℑ} (a) ⪰ rmin {{\tilde{η}}_{ℑ} (z), {\tilde{η}}_{ℑ} (y)}$ and $sup_{a \in (x \circ y) \subseteq ((z \circ t) \circ y)} {ϑ_{ℑ} (a)} \leq max {ϑ_{ℑ} (z), ϑ_{ℑ} (y)} .$

Therefore, $\begin{matrix} {\tilde{η}}_{(ℑ * H) * ℑ} (a) \\ = \underset{a \in x \circ y}{rsup} {rmin {{\tilde{η}}_{ℑ * H} (x), {\tilde{η}}_{ℑ} (y)}} \\ ⪯ \underset{a \in x \circ y}{rsup} {rmin {\underset{x \in z \circ t}{rsup} {rmin {{\tilde{η}}_{ℑ} (z), {\tilde{1}}_{H} (t)}, {\tilde{η}}_{ℑ} (y)}}} \\ ⪯ \underset{a \in x \circ y}{rsup} {rmin {\underset{x \in z \circ t}{rsup} {rmin {{\tilde{η}}_{ℑ} (z), [1, 1]}, {\tilde{η}}_{ℑ} (y)}}} \\ ⪯ \underset{a \in (z \circ t) \circ y}{rsup} {rmin {{\tilde{η}}_{ℑ} (z), {\tilde{η}}_{ℑ} (y)}} ⪯ \underset{a \in (z \circ t) \circ y}{rsup} {\tilde{η}}_{ℑ} (a), \end{matrix}$

and $\begin{array}{l} ϑ_{(ℑ * ℋ) * ℑ} (a) = \inf_{a \in x ° y} {\max {ϑ_{ℑ * ℋ} (x), ϑ_{ℑ} (y)}} \\ \geq \inf_{a \in x ° y} {\max {\inf_{x \in z ° t} {\max {ϑ_{ℑ} (z), {\tilde{1}}_{ℋ} (t)}, ϑ_{ℑ} (y)}}} \\ \geq \inf_{a \in x ° y} {\max {\inf_{x \in z ° t} {\max {ϑ_{ℑ} (z), 0}, ϑ_{ℑ} (y)}}}} \\ \geq \inf_{a \in (z ° t) ° y} {\max {ϑ_{ℑ} (z), ϑ_{ℑ} (y)}} \geq \inf_{a \in (z ° t) ° y} ϑ_{ℑ} (a) . \end{array}$

Thus $(ℑ * H) * ℑ \subseteq ℑ$ .

Conversely, let us suppose that $(ℑ * H) * ℑ \subseteq ℑ$ . Let x, y, z ∈ H and take a ∈ (x ∘ y) ∘ z. Then we have $\begin{matrix} \underset{a \in (x \circ y) \circ z}{rinf} {\tilde{η}}_{ℑ} (a) ⪰ {\tilde{η}}_{ℑ} (a) ⪰ {\tilde{η}}_{(ℑ * H) * ℑ} (a) \\ ⪰ rmin {\underset{t \in x \circ y}{rinf} {\tilde{η}}_{ℑ * H} (t), {\tilde{η}}_{ℑ} (z)} \\ ⪰ rmin {\underset{x \circ y = u \circ v}{rsup} {rmin {{\tilde{η}}_{ℑ} (u), {\tilde{1}}_{H} (v)}, {\tilde{η}}_{ℑ} (z)}} \\ ⪰ rmin {rmin {{\tilde{η}}_{ℑ} (x), {\tilde{1}}_{H} (y)}, {\tilde{η}}_{ℑ} (z)} \\ = rmin {rmin {{\tilde{η}}_{ℑ} (x), [1, 1]}, {\tilde{η}}_{ℑ} (z)} \\ = rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (z)}, \end{matrix}$ and $\begin{matrix} sup_{a \in (x \circ y) \circ z} ϑ_{ℑ} (a) \leq ϑ_{ℑ} (a) \leq ϑ_{(ℑ * H) * ℑ} (a) \\ \leq max {sup_{t \in x \circ y} ϑ_{ℑ * H} (t), ϑ_{ℑ} (z)} \\ \leq max {inf_{x \circ y = u \circ v} {max {ϑ_{ℑ} (u), 0_{H} (v)}, ϑ_{ℑ} (z)}} \\ \leq max {max {{\tilde{η}}_{ℑ} (x), 0_{H} (y)}, {\tilde{η}}_{ℑ} (z)} \\ = max {max {{\tilde{η}}_{ℑ} (x), 0}, {\tilde{η}}_{ℑ} (z)} \\ = max {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (z)} . \end{matrix}$

Therefore, $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic generalized bi-hyperideal of H. The other cases can be proved in a similar way. □

Theorem 3.17. If {ℑ _i} _i∈Λ is a family of cubic left hyperideals (resp., right hyperideals, bi-hyperideals, interior hyperideals) of an LA-semihypergroup H, then ⋂_i∈Λ ℑ _i is a cubic left hyperideal (resp., right hyperideal, bi-hyperideal, interior hyperideal) of H, where $⋂_{i \in Λ} ℑ_{i} = (⋀_{i \in Λ} {\tilde{η}}_{ℑ_{i}}, ⋁_{i \in Λ} ϑ_{ℑ})$ and $\begin{matrix} ⋀_{i \in Λ} {\tilde{η}}_{ℑ_{i}} (x) = rinf {{\tilde{η}}_{ℑ_{i}} (x) : i \in Λ, x \in H} \\ ⋁_{i \in Λ} ϑ_{ℑ_{i}} (x) = sup {ϑ_{ℑ_{i}} (x) : i \in Λ, x \in H} . \end{matrix}$

Proof. The proof is straightforward. □

Proposition 3.18. Let $ℑ_{1} = 〈 {\tilde{η}}_{ℑ_{1}}, ϑ_{ℑ_{1}} 〉$ be a cubic right hyperideal of H and $ℑ_{2} = 〈 {\tilde{η}}_{ℑ_{2}}, ϑ_{ℑ_{2}} 〉$ be a cubic left hyperideal of H. Then ℑ₁ ∗ ℑ ₂ ⊆ ℑ ₁ ∩ ℑ ₂.

Proof. Let $ℑ_{1} = 〈 {\tilde{η}}_{ℑ_{1}}, ϑ_{ℑ_{1}} 〉$ be a cubic right hyperideal of H and $ℑ_{2} = 〈 {\tilde{η}}_{ℑ_{2}},$ ϑ_ℑ₂〉 be a cubic left hyperideal of H. Let z ∈ H and suppose that there exist x, y ∈ H such that z ∈ x ∘ y. Then $\begin{matrix} {\tilde{η}}_{ℑ_{1} * ℑ_{2}} (z) = \underset{z \in x \circ y}{rsup} {rmin {{\tilde{η}}_{ℑ_{1}} (x), {\tilde{η}}_{ℑ_{2}} (y)}} \\ ⪯ \underset{z \in x \circ y}{rsup} {rmin {\underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ_{1}} (z)}, \underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ_{2}} (z)}}} \\ = rmin {{\tilde{η}}_{ℑ_{1}} (z), {\tilde{η}}_{ℑ_{2}} (z)} = {\tilde{η}}_{ℑ_{1} \cap ℑ_{2}} (z), \end{matrix}$ and $\begin{matrix} ϑ_{ℑ_{1} * ℑ_{2}} (z) = inf_{z \in x \circ y} {max {ϑ_{ℑ_{1}} (x), ϑ_{ℑ_{2}} (y)}} \\ \geq inf_{z \in x \circ y} {max {sup_{z \in x \circ y} {ϑ_{ℑ_{1}} (z)}, sup_{z \in x \circ y} {ϑ_{ℑ_{2}} (z)}}} \\ = max {ϑ_{ℑ_{1}} (z), ϑ_{ℑ_{2}} (z)} = ϑ_{ℑ_{1} \cap ℑ_{2}} (z) . \end{matrix}$

Let us suppose that there do not exist x, y ∈ H such that z ∈ x ∘ y. Then, ${\tilde{η}}_{ℑ_{1} * ℑ_{2}} (z) = [0, 0] ⪯ {\tilde{η}}_{ℑ_{1} \cap ℑ_{2}} (z)$ and ϑ_{ℑ₁∗ℑ₂} (z) =1 ≥ ϑ_{ℑ₁∩ℑ₂} (z). This completes the proof. □

Lemma 3.19. Let H be an LA-semihypergroup with a pure left identity. Then every cubic right hyperideal is a cubic hyperideal.

Proof. The proof is straightforward. □

Theorem 3.20. If $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic left hyperideal of H with a pure left identity, then $ℑ \cup (ℑ * H)$ and ℑ ∪ (ℑ ∗ ℑ) are cubic hyperideals of H .

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic left hyperideal of H . We have $\begin{matrix} (ℑ \cup (ℑ * H)) * H & = & (ℑ * H) \cup ((ℑ * H) * H) \\ = & (ℑ * H) \cup ((H * H) * ℑ) \\ = & (ℑ * H) \cup (H * ℑ) \\ (by Lemma 3.4) \\ \subseteq & (ℑ * H) \cup ℑ . \end{matrix}$

Hence, $ℑ \cup (ℑ * H)$ is a cubic right hyperideal of H. By Lemma 3.19, $ℑ \cup (ℑ * H)$ is a cubic hyperideal of H. In a similar way we can prove that ℑ ∪ (ℑ ∗ ℑ) is a cubic hyperideal. □

Proposition 3.21. Let H be an LA-semihypergroup with a pure left identity. Then for any cubic left hyperideal $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ which is idempotent in $C (H),$ the following properties hold:

ℑ is a cubic interior hyperideal;

ℑ is a cubic bi-hyperideal.

Proof. Let ℑ be a cubic left hyperideal of H .

(i) We have $(H * ℑ) * H \subseteq ℑ * H = (ℑ * ℑ) * H = (H * ℑ) * ℑ \subseteq ℑ * ℑ = ℑ .$ This implies that ℑ is a cubic interior hyperideal H .

(ii) We have $(ℑ * H) * ℑ \subseteq (ℑ * H) * (ℑ * ℑ) = (ℑ * ℑ) * (H * ℑ) \subseteq ℑ * ℑ = ℑ .$ This implies that ℑ is a cubic bi-hyperideal of H . □

Theorem 3.22. Let H be a regular LA-semihypergroup. Then $(ℑ * H) * ℑ = ℑ$ for every cubic generalized bi-hyperideal $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ of H.

Proof. Let H be a regular LA-semihypergroup and $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic generalized bi-hyperideal of H. Let a ∈ H and so a ∈ (a ∘ x) ∘ a for some x ∈ H. Thus we have $\begin{matrix} {\tilde{η}}_{(ℑ * H) * ℑ} (a) \\ = \underset{a \in y \circ z}{rsup} {rmin {{\tilde{η}}_{ℑ * H} (y), {\tilde{η}}_{ℑ} (z)}} \\ ⪰ rmin {\underset{t \in a \circ x}{rsup} {\tilde{η}}_{ℑ * H} (t), {\tilde{η}}_{ℑ} (a)} \\ ⪰ rmin {\underset{a \circ x = p \circ q}{rsup} {rmin {{\tilde{η}}_{ℑ} (p), {\tilde{1}}_{H} (q)}, {\tilde{η}}_{ℑ} (a)}} \\ ⪰ rmin {{\tilde{η}}_{ℑ} (a), {\tilde{1}}_{H} (x), {\tilde{η}}_{ℑ} (a)} \\ ⪰ rmin {{\tilde{η}}_{ℑ} (a), [1, 1], {\tilde{η}}_{ℑ} (a)} = {\tilde{η}}_{ℑ} (a) . \end{matrix}$ and $\begin{matrix} ϑ_{(ℑ * H) * ℑ} (a) \\ = inf_{a \in y \circ z} max {ϑ_{ℑ * H} (y), ϑ_{ℑ} (z)} \\ \leq max {inf_{t \in a \circ x} ϑ_{ℑ * H} (t), ϑ_{ℑ} (a)} \\ \leq max {inf_{a \circ x = p \circ q} max {ϑ_{ℑ} (p), 0_{H} (q)}, ϑ_{ℑ} (a)} \\ \leq max {ϑ_{ℑ} (a), 0_{H} (x), ϑ_{ℑ} (a)} \\ = max {ϑ_{ℑ} (a), 0, ϑ_{ℑ} (a)} = ϑ_{ℑ} (a) . \end{matrix}$

Therefore, $ℑ \subseteq (ℑ * H) * ℑ$ . Since $(ℑ * H) * ℑ \subseteq ℑ$ , then by Theorem 3.16, we have $ℑ = (ℑ * H) * ℑ$ for every cubic generalized bi-hyperideal $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ of H. □

Proposition 3.23. If H is an LA-semihypergroup, then ℑ₁ ∩ ℑ ₂ ⊆ ℑ ₁ ∗ ℑ ₂ for every cubic generalized bi-hyperideal $ℑ_{1} = 〈 {\tilde{η}}_{ℑ_{1}}, ϑ_{ℑ_{1}} 〉$ and every cubic left hyperideal $ℑ_{2} = 〈 {\tilde{η}}_{ℑ_{2}}, ϑ_{ℑ_{2}} 〉$ of H.

Proof. The proof is straightforward. □

Proposition 3.24. If H is a regular LA-semi-hypergroup, then ℑ₁ ∩ ℑ ₂ ∩ ℑ ₃ ⊆ (ℑ ₁ ∗ ℑ ₂) ∗ ℑ ₃ for every cubic right hyperideal $ℑ_{1} = 〈 {\tilde{η}}_{ℑ_{1}}, ϑ_{ℑ_{1}} 〉$ , every cubic generalized bi-hyperideal $ℑ_{2} = 〈 {\tilde{η}}_{ℑ_{2}}, ϑ_{ℑ_{2}} 〉$ and every cubic left hyperideal $ℑ_{3} = 〈 {\tilde{η}}_{ℑ_{3}}, ϑ_{ℑ_{3}} 〉$ of H.

Proof. The proof is straightforward. □

Proposition 3.25. Let H be a regular LA-semihypergroup H with a pure left identity e. If $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic right hyperideal of H, then ℑ (x ∘ y) = ℑ (y ∘ x) holds for all x, y ∈ H.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic right hyperideal of a regular LA-semihypergroup H with a pure left identity e . Let x, y ∈ H . Since H is regular, x ∈ (x ∘ a) ∘ x and y ∈ (y ∘ b) ∘ y for some a, b ∈ H. Now by using the medial and paramedial laws, we get $\begin{matrix} x \circ y \subseteq ((x \circ a) \circ x) \circ ((y \circ b) \circ y) \\ = (y \circ x) \circ ((y \circ b) \circ (x \circ a)) . \end{matrix}$

Since $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic right hyperideal, then for every w ∈ x ∘ y ⊆ (y ∘ x) ∘ ((y ∘ b) ∘ (x ∘ a)) , we have $\underset{w \in x \circ y \subseteq (y \circ x) \circ ((y \circ b) \circ (x \circ a))}{rinf} {{\tilde{η}}_{ℑ} (w)} ⪰ \underset{s \in y \circ x}{rinf} {{\tilde{η}}_{ℑ} (s)},$ and $sup_{w \in x \circ y \subseteq (y \circ x) \circ ((y \circ b) \circ (x \circ a))} {ϑ_{ℑ} (w)} \leq sup_{s \in y \circ x} {ϑ_{ℑ} (s)} .$

Again by using the medial and paramedial laws, we get $\begin{matrix} y \circ x \subseteq ((y \circ b) \circ y) \circ ((x \circ a) \circ x) \\ = (x \circ y) \circ ((x \circ a) \circ (y \circ b)) . \end{matrix}$

Since $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic right hyperideal, for every t ∈ y ∘ x ⊆ (x ∘ y) ∘ ((x ∘ a) ∘ (y ∘ b)) , we have $\underset{t \in y \circ x \subseteq (x \circ y) \circ ((x \circ a) \circ (y \circ b))}{rinf} {{\tilde{η}}_{ℑ} (t)} ⪰ \underset{p \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (p)},$ and $sup_{t \in y \circ x \subseteq (x \circ y) \circ ((x \circ a) \circ (y \circ b))} {ϑ_{ℑ} (t)} \leq sup_{p \in x \circ y} {ϑ_{ℑ} (p)},$

This shows that ℑ (x ∘ y) = ℑ (y ∘ x) holds for all x, y ∈ H. □

Proposition 3.26. Let H be an LA-semihypergroup with a pure left identity e. Then $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic right hyperideal of H if and only if it is a cubic interior hyperideal of H.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic right hyperideal of H. For x, y, z ∈ H, we have $\begin{matrix} \underset{a \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ \underset{w \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (w)} \\ = \underset{w \in (e \circ x) \circ y}{rinf} {{\tilde{η}}_{ℑ} (w)} = \underset{w \in (y \circ x) \circ e}{rinf} {{\tilde{η}}_{ℑ} (w)} \\ ⪰ \underset{s \in y \circ x}{rinf} {{\tilde{η}}_{ℑ} (s)} ⪰ {\tilde{η}}_{ℑ} (y), \end{matrix}$ and $\begin{matrix} sup_{a \in (x \circ y) \circ z} {ϑ_{ℑ} (a)} \leq sup_{w \in x \circ y} {ϑ_{ℑ} (w)} \\ = sup_{w \in (e \circ x) \circ y} {ϑ_{ℑ} (w)} = sup_{w \in (y \circ x) \circ e} {ϑ_{ℑ} (w)} \\ \leq sup_{s \in y \circ x} {ϑ_{ℑ} (s)} \leq ϑ_{ℑ} (y), \end{matrix}$ which implies that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic interior hyperideal. Conversely, for any x and y in H we have $\underset{a \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (a)} = \underset{a \in (e \circ x) \circ y}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ {\tilde{η}}_{ℑ} (x),$ and $sup_{a \in x \circ y} {ϑ_{ℑ} (a)} = sup_{a \in (e \circ x) \circ y} {ϑ_{ℑ} (a)} \leq ϑ_{ℑ} (x),$ This shows that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic right hyperideal of H. This completes the proof. □

Proposition 3.27. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic left hyperideal of an LA-semihypergroup H with a pure left identity e. Then every cubic interior hyperideal ℑ is a cubic bi-hyperideal.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic left hyperideal of H . Then for every z ∈ H, there exist x, y ∈ H such that $\underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (z)} ⪰ {\tilde{η}}_{ℑ} (y) and sup_{z \in x \circ y} {ϑ_{ℑ} (z)} \leq ϑ_{ℑ} (y) .$

Let e be a pure left identity in H . We have, $\underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (z)} = \underset{z \in (e \circ x) \circ y}{rinf} {{\tilde{η}}_{ℑ} (z)} ⪰ {\tilde{η}}_{ℑ} (x)$ and $sup_{z \in x \circ y} {ϑ_{ℑ} (z)} = sup_{z \in (e \circ x) \circ y} {ϑ_{ℑ} (z)} \leq ϑ_{ℑ} (x) .$

This implies that $\underset{z \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (z)} ⪰ rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (y)}$ and $sup_{z \in x \circ y} {ϑ_{ℑ} (z)} \leq max {ϑ_{ℑ} (x), ϑ_{ℑ} (y)} .$

Thus ℑ is a cubic sub LA-semihypergroup of H . Now for any x, y, z ∈ H, $\begin{matrix} \underset{a \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} = \underset{a \in (x \circ (e \circ y)) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} \\ = \underset{a \in (e \circ (x \circ y)) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ \underset{t \in x \circ y}{rinf} {{\tilde{η}}_{ℑ} (t)} \\ = \underset{t \in (e \circ x) \circ y}{rinf} {{\tilde{η}}_{ℑ} (t)} ⪰ {\tilde{η}}_{ℑ} (x), \end{matrix}$ and $\begin{matrix} sup_{a \in (x \circ y) \circ z} {ϑ_{ℑ} (a)} = sup_{a \in (x \circ (e \circ y)) \circ z} {ϑ_{ℑ} (a)} \\ = sup_{a \in (e \circ (x \circ y)) \circ z} {ϑ_{ℑ} (a)} \leq sup_{t \in x \circ y} {ϑ_{ℑ} (t)} \\ = sup_{t \in (e \circ x) \circ y} {ϑ_{ℑ} (t)} \leq ϑ_{ℑ} (x) . \end{matrix}$

Also $\begin{matrix} \underset{a \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} = \underset{a \in (z \circ y) \circ x}{rinf} {{\tilde{η}}_{ℑ} (a)} \\ = \underset{a \in (e \circ (z \circ y)) \circ x}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ \underset{t \in z \circ y}{rinf} {{\tilde{η}}_{ℑ} (t)} \\ = \underset{t \in (e \circ z) \circ y}{rinf} {{\tilde{η}}_{ℑ} (t)} ⪰ {\tilde{η}}_{ℑ} (z), \end{matrix}$

and $\begin{matrix} sup_{a \in (x \circ y) \circ z} {ϑ_{ℑ} (a)} = sup_{a \in (z \circ y) \circ x} {ϑ_{ℑ} (a)} \\ = sup_{a \in (e \circ (z \circ y)) \circ x} {ϑ_{ℑ} (a)} \leq sup_{t \in z \circ y} {ϑ_{ℑ} (t)} \\ = sup_{t \in (e \circ z) \circ y} {ϑ_{ℑ} (t)} \leq ϑ_{ℑ} (z) . \end{matrix}$

Hence we get $\underset{a \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (a)} ⪰ rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (z)}$ and $sup_{a \in (x \circ y) \circ z} {ϑ_{ℑ} (a)} \leq max {ϑ_{ℑ} (x), ϑ_{ℑ} (z)} .$

This shows that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic bi-hyperideal of H . □

Theorem 3.28. Let H be a regular LA-semi-hypergroup with a pure left identity e. Then $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic left hyperideal of H if and only if $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic bi-hyperideal of H .

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic left hyperideal of H. Let x, y, z ∈ H. We have $\begin{matrix} \underset{w \in (x \circ y) \circ z}{rinf} {{\tilde{η}}_{ℑ} (w)} = \underset{w \in (z \circ y) \circ x}{rinf} {{\tilde{η}}_{ℑ} (w)} \\ ⪰ {\tilde{η}}_{ℑ} (x) ⪰ rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (z)}, \end{matrix}$ and $\begin{matrix} sup_{w \in (x \circ y) \circ z} {ϑ_{ℑ} (w)} = sup_{w \in (z \circ y) \circ x} {ϑ_{ℑ} (w)} \leq ϑ_{ℑ} (x) \\ \leq max {ϑ_{ℑ} (x), ϑ_{ℑ} (z)} . \end{matrix}$

This shows that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic generalized bi-hyperideal of H, and clearly $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic sub LA-semihypergroup. Therefore $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic bi-hyperideal of H .

Conversely, assume that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic bi-hyperideal of H . Let x, y ∈ H . Since H is regular, then by Lemma 2.8, the left invertive law and medial law, we have $\begin{matrix} x \circ y \\ \subseteq x \circ ((y \circ z) \circ y) = (y \circ z) \circ (x \circ y) \\ = ((x \circ y) \circ z) \circ y = ((x \circ y) \circ (e \circ z)) \circ y \\ = ((x \circ e) \circ (y \circ z)) \circ y = (y \circ ((x \circ e) \circ z)) \circ y . \end{matrix}$ Thus for every w ∈ x ∘ y ⊆ (y ∘ ((x ∘ e) ∘ z)) ∘ y, we have $\begin{matrix} \underset{w \in x \circ y \subseteq (y \circ ((x \circ e) \circ z)) \circ y}{rinf} {{\tilde{η}}_{ℑ} (w)} & ⪰ & rmin {{\tilde{η}}_{ℑ} (y), {\tilde{η}}_{ℑ} (y)} \\ = & {\tilde{η}}_{ℑ} (y), \end{matrix}$ and $\begin{matrix} sup_{w \in x \circ y \subseteq (y \circ ((x \circ e) \circ z)) \circ y} {ϑ_{ℑ} (w)} & \leq & max {ϑ_{ℑ} (y), ϑ_{ℑ} (y)} \\ = & ϑ_{ℑ} (y) . \end{matrix}$ Hence $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic left hyperideal of H . □

Lemma 3.29. If H is a left regular LA-semi-hypergroup, then $H * H = H$ .

Proof. The proof is straightforward. □

Lemma 3.30. For every cubic interior hyperideal $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ of a left regular LA-semihypergroup H with a pure left identity, $H * ℑ \subseteq ℑ * H$ .

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be any cubic interior hyperideal of H with a pure left identity. By using the left invertive law, paramedial law, Lemma 3.29 and Theorem 3.16, we have $\begin{matrix} H * ℑ & = & (H * H) * ℑ = (ℑ * H) * H \\ = & (ℑ * H) * (H * H) = (H * H) * (H * ℑ) \\ = & ((H * ℑ) * H) * H \subseteq ℑ * H . \end{matrix}$

This shows that $H * ℑ \subseteq ℑ * H .$ □

Theorem 3.31. If H is a left regular LA-semi-hypergroup H with a pure left identity, then every cubic generalized bi-hyperideal is a cubic sub LA-semihypergroup.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic generalized bi-hyperideal of a left regular LA-semihypergroup H. Let x, y ∈ H. Then there exists z ∈ H, such that x ∈ z ∘ (x ∘ x). From the definition of a cubic generalized bi-hyperideal and Lemma 2.8, it follows that: $\begin{matrix} \underset{a \in x \circ y \subseteq (z \circ (x \circ x)) \circ y = (x \circ (z \circ x)) \circ y}{rinf} {{\tilde{η}}_{ℑ} (a)} \\ ⪰ rmin {{\tilde{η}}_{ℑ} (x), {\tilde{η}}_{ℑ} (y)} \end{matrix}$

and $\begin{matrix} sup_{a \in x \circ y \subseteq (z \circ (x \circ x)) \circ y = (x \circ (z \circ x)) \circ y} {ϑ_{ℑ} (a)} \\ \leq max {ϑ_{ℑ} (x), ϑ_{ℑ} (y)} . \end{matrix}$

Therefore, $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic sub LA-semihypergroup. □

Lemma 3.32. If H is a left regular (resp. regular) LA-semihypergroup, then every cubic left (resp. right) hyperideal $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ of H is cubic idempotent.

Proof. Let ℑ be any cubic left hyperideal of a left regular LA-semihypergroup H with a pure left identity. Then $ℑ * ℑ \subseteq H * ℑ \subseteq ℑ$ . Since H is left regular, for every a ∈ H, there exists x ∈ H such that a ∈ x ∘ (a ∘ a) . By Lemma 2.8, we have $a \in x \circ (a \circ a) = a \circ (x \circ a) .$

Therefore $\begin{matrix} {\tilde{η}}_{ℑ * ℑ} (a) & = & \underset{a \in a \circ (x \circ a)}{rsup} {rmin {{\tilde{η}}_{ℑ} (a), \underset{s \in x \circ a}{rinf} {\tilde{η}}_{ℑ} (s)}} \\ ⪰ & rmin {{\tilde{η}}_{ℑ} (a), {\tilde{η}}_{ℑ} (a)} = {\tilde{η}}_{ℑ} (a), \end{matrix}$ and $\begin{matrix} ϑ_{ℑ * ℑ} (a) & = & inf_{a \in a \circ (x \circ a)} {max {ϑ_{ℑ} (a), sup_{s \in x \circ a} ϑ_{ℑ} (s)}} \\ \leq & max {ϑ_{ℑ} (a), ϑ_{ℑ} (a)} = ϑ_{ℑ} (a) . \end{matrix}$

Thus we get ℑ ⊆ ℑ ∗ ℑ . Hence this shows that $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is cubic idempotent. The other case can be proved in a similar way. □

4 Homomorphism problems

In this section, we discuss some properties concerning the image and inverse image of cubic hyperideals.

Definition 4.1. Let (H₁, ∘ ₁) and (H₂, ∘ ₂) be two LA-semihypergroups. If there exists a mapping $F : H_{1} \to H_{2}$ such that $F (x \circ_{1} y) = F (x) \circ_{2} F (y)$ , for all x, y ∈ H₁, then we say that $F$ is a homomorphism from H₁ to H₂. Moreover, if $F$ is a bijection, then $F$ is called an isomorphism, and H₁ and H₂ are isomorphic.

Example 4.2. Let H₁ = {x, y, z} and H₂ = {a, b, c} be two LA-semihypergroups with the hyperoperations defined in the following tables:

∘₁	x	y	z
x	z	{x, z}	z
y	z	{y, z}	{x, z}
z	{x, z}	{x, z}	{x, z}

∘₂	a	b	c
a	{a, c}	{a, c}	{a, c}
b	{a, c}	{a, b}	a
c	a	{a, c}	a

and let $F : H_{1} ⟶ H_{2}$ be defined by f (x) = c, f (y) = b, f (z) = a . Then, clearly, $F$ is an isomorphism.

Definition 4.3. Let $F$ be a mapping from an LA-semihypergroup H₁ to an LA-semihypergroup H₂ and $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic subset of H₁ . The image $F (ℑ) = 〈 {\tilde{η}}_{F (ℑ)}, ϑ_{F (ℑ)} 〉$ of ℑ is a cubic subset of H₂ and is defined by $\begin{matrix} {\tilde{η}}_{F (ℑ)} (y) = {\begin{matrix} \underset{t \in F^{- 1} (y)}{rsup} {{\tilde{η}}_{ℑ} (t)} & if F^{- 1} (y) \neg = \emptyset \\ [0, 0] & otherwise, \end{matrix} \\ ϑ_{F (ℑ)} (y) = {\begin{matrix} inf_{t \in F^{- 1} (y)} {ϑ_{ℑ} (t)} & if F^{- 1} (y) \neg = \emptyset \\ 1 & otherwise . \end{matrix} \end{matrix}$

Definition 4.4. Let $F$ be a mapping from an LA-semihypergroup H₁ to an LA-semihypergroup H₂ and $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic subset in H₂. Then the inverse image of ℑ under $F$ is denoted by $F^{- 1} (ℑ) = 〈 {\tilde{η}}_{F^{- 1} (ℑ)}, ϑ_{F^{- 1} (ℑ)} 〉$ and is defined by ${\tilde{η}}_{F^{- 1} (ℑ)} (x) = {\tilde{η}}_{ℑ} (F (x)) and ϑ_{F^{- 1} (ℑ)} (x) = ϑ_{ℑ} (F (x)) .$

Theorem 4.5. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a surjective homomorphism from an LA-semi-hypergroup H₁ to an LA-semihypergroup H₂. If $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic sub LA-semihypergroup (resp., hyperideal, left hyperideal, right hyperideal, bi-hyperideal, interior hyperideal) of H₁, then the image of ℑ under $F$ is a cubic sub LA-semihypergroup (resp., hyperideal, left hyperideal, right hyperideal, bi-hyperideal, interior hyperideal) of H₂ .

Proof. The proof is straightforward. □

Theorem 4.6. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a surjective homomorphism from an LA-semihypergroup H₁ to an LA-semihypergroup H₂. If H₁ contains a pure left identity e, then the image of cubic interior hyperideal of H₁ is a cubic right hyperideal of H₂ .

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic interior hyperideal of H₁ and let x₂, y₂ ∈ H₂. Then there exist x₁, y₁, z₁ ∈ H₁ such that $F (x_{1}) = x_{2},$ $F (y_{1}) = y_{2}$ and $F (z_{1}) = z_{2} .$ Now, we have $\begin{matrix} \underset{z_{2} \in x_{2} \circ_{2} y_{2}}{rinf} {{\tilde{η}}_{F (ℑ)} (z_{2})} \\ = \underset{z_{2} \in x_{2} \circ_{2} y_{2}}{rinf} {\underset{t \in F^{- 1} (z_{2})}{rsup} {\tilde{η}}_{ℑ} (t)} \\ ⪰ \underset{z_{2} \in x_{2} \circ_{2} y_{2}}{rinf} {{\tilde{η}}_{ℑ} (z_{1})} \\ = \underset{F (z_{1}) \in F (x_{1}) \circ_{2} F (y_{1})}{rinf} {{\tilde{η}}_{ℑ} (z_{1})} \\ = \underset{F (z_{1}) \in F (x_{1} \circ_{1} y_{1})}{rinf} {{\tilde{η}}_{ℑ} (z_{1})} \\ = \underset{z_{1} \in x_{1} \circ_{1} y_{1}}{rinf} {{\tilde{η}}_{ℑ} (z_{1})} = \underset{z_{1} \in (e \circ_{1} x_{1}) \circ_{1} y_{1}}{rinf} {{\tilde{η}}_{ℑ} (z_{1})} \\ ⪰ {\tilde{η}}_{ℑ} (x_{1}) ⪰ \underset{x_{1} \in F^{- 1} (x_{2})}{rsup} {\tilde{η}}_{ℑ} (x_{1}) ⪰ {\tilde{η}}_{F (ℑ)} (x_{2}), \end{matrix}$ and $\begin{matrix} sup_{z_{2} \in x_{2} \circ_{2} y_{2}} {ϑ_{F (ℑ)} (z_{2})} \\ = sup_{z_{2} \in x_{2} \circ_{2} y_{2}} {inf_{t \in F^{- 1} (z_{2})} ϑ_{ℑ} (t)} \\ \leq sup_{z_{2} \in x_{2} \circ_{2} y_{2}} {ϑ_{ℑ} (z_{1})} = sup_{F (z_{1}) \in F (x_{1}) \circ_{2} F (y_{1})} {ϑ_{ℑ} (z_{1})} \\ = sup_{F (z_{1}) \in F (x_{1} \circ_{1} y_{1})} {ϑ_{ℑ} (z_{1})} \\ = sup_{z_{1} \in x_{1} \circ_{1} y_{1}} {ϑ_{ℑ} (z_{1})} = sup_{z_{1} \in (e \circ_{1} x_{1}) \circ_{1} y_{1}} {ϑ_{ℑ} (z_{1})} \\ \leq ϑ_{ℑ} (x_{1}) \leq sup_{x_{1} \in F^{- 1} (x_{2})} ϑ_{ℑ} (x_{1}) \leq ϑ_{F (ℑ)} (x_{2}) . \end{matrix}$

This shows that the image of a cubic interior hyperideal of H₁ is a cubic right hyperideal of H₂ . □

In the following example we show that in an LA-semihypergroup H₁ without pure left identities, the image of a cubic subset ℑ under $F$ can be or cannot be a cubic right hyperideal of H₂.

Example 4.7. Let H₁ = {x, y, z, w, t} and H₂ = {a, b, c, d} be two LA-semihypergroups defined by the following Cayley tables:

∘₁	x	y	z	w	t
x	x	x	x	x	x
y	x	{y, t}	t	{x, w}	{z, t}
z	x	{z, t}	t	{x, w}	t
w	x	{x, w}	{x, w}	w	{x, w}
t	x	{z, t}	{z, t}	{x, w}	{z, t}

∘₂	a	b	c	d
a	c	{a, c}	c	d
b	c	{b, c}	{a, c}	d
c	{a, c}	{a, c}	{a, c}	d
d	d	d	d	d

We define a homomorphism $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ by: $F : H_{1} ⟶ H_{2} | r ⟶ F (r) : = {\begin{matrix} d & if r \in {x, w} \\ b & if r = y \\ a & if r = z \\ c & if r = t \end{matrix}$

We define a cubic subset

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

of H₁ by:

H ₁	${\tilde{η}}_{ℑ}$	ϑ _ℑ
x	[0.93, 0.97]	0.3
y	[0.22, 0.27]	0.9
w	[0.79, 0.81]	0.4
{z, t}	[0.63, 0.71]	0.6

Here

ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉

is a cubic subset of H₁ . It can be easily verified that the image of ℑ under

F

is a cubic right hyperideal of H₂ .

Theorem 4.8. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a homomorphism from an LA-semihypergroup H₁ to an LA-semihypergroup H₂. If $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ is a cubic sub LA-semihypergroup (resp. left hyperideal, right hyperideal, bi-hyperideal, interior hyperideal) of H₂, then the preimage $F^{- 1} (ℑ)$ of ℑ under $F$ is a cubic sub LA-semihypergroup (resp. hyperideal, left hyperideal, right hyperideal, bi-hyperideal, interior hyperideal) of H₁ .

Proof. The proof is straightforward. □

Theorem 4.9. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a homomorphism from an LA-semihypergroup H₁ to an LA-semihypergroup H₂. If H₁ contains a pure left identity e, then the preimage of every cubic right hyperideal of H₂ is a cubic interior hyperideal of H₁.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic right hyperideal of H₂ and x, a, y ∈ H₁ . Then by the left invertive law, $\begin{matrix} \underset{z \in (x \circ_{1} a) \circ_{1} y}{rinf} {{\tilde{η}}_{F^{- 1} (ℑ)} (z)} = \underset{z \in (x \circ_{1} a) \circ_{1} y}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ \underset{F (z) \in (F (x) \circ_{2} F (a)) \circ_{2} F (y)}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ \underset{F (s) \in F (x) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ = \underset{F (s) \in F (e \circ_{1} x) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ = \underset{F (s) \in (F (e) \circ_{2} F (x)) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ = \underset{F (s) \in (F (a) \circ_{2} F (x)) \circ_{2} F (e)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ ⪰ \underset{F (w) \in F (a) \circ_{2} F (x)}{rinf} {{\tilde{η}}_{ℑ} (F (w))} ⪰ {\tilde{η}}_{ℑ} (F (a)) \\ = {\tilde{η}}_{F^{- 1} (ℑ)} (a), \end{matrix}$ and $\begin{matrix} sup_{z \in (x \circ_{1} a) \circ_{1} y} {ϑ_{F^{- 1} (ℑ)} (z)} \\ = sup_{z \in (x \circ_{1} a) \circ_{1} y} {ϑ_{ℑ} (F (z))} \\ \leq sup_{F (z) \in (F (x) \circ_{2} F (a)) \circ_{2} F (y)} {ϑ_{ℑ} (F (z))} \\ \leq sup_{F (s) \in F (x) \circ_{2} F (a)} {ϑ_{ℑ} (F (s))} \\ = sup_{F (s) \in F (e \circ_{1} x) \circ_{2} F (a)} {ϑ_{ℑ} (F (s))} \\ = sup_{F (s) \in (F (e) \circ_{2} F (x)) \circ_{2} F (a)} {ϑ_{ℑ} (F (s))} \\ = sup_{F (s) \in (F (a) \circ_{2} F (x)) \circ_{2} F (e)} {ϑ_{ℑ} (F (s))} \\ \leq sup_{F (w) \in F (a) \circ_{2} F (x)} {ϑ_{ℑ} (F (w))} \leq ϑ_{ℑ} (F (a)) \\ = ϑ_{F^{- 1} (ℑ)} (a) . \end{matrix}$

This shows that the preimage of every cubic right hyperideal of H₂ is a cubic interior hyperideal of H₁. □

Theorem 4.10. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a homomorphism from an LA-semihypergroup H₁ to an LA-semihypergroup H₂. If H₁ contains a pure left identity e, then the preimage of every cubic interior hyperideal of H₂ is a cubic generalized bi-hyperideal of H₁.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic interior hyperideal of H₂ and x, a, y ∈ H₁ . Then by the left invertive law, $\begin{matrix} \underset{z \in (x \circ_{1} a) \circ_{1} y}{rinf} {{\tilde{η}}_{F^{- 1} (ℑ)} (z)} \\ = \underset{z \in (e \circ_{1} (x \circ_{1} a)) \circ_{1} y}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ \underset{F (z) \in (F (e) \circ_{2} (F (x) \circ_{2} F (a))) \circ_{2} F (y)}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ \underset{F (s) \in F (x) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ = \underset{F (s) \in F (e \circ_{1} x) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ = \underset{F (s) \in (F (e) \circ_{2} F (x)) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ ⪰ {\tilde{η}}_{ℑ} (F (x)) = {\tilde{η}}_{F^{- 1} (ℑ)} (x) . \end{matrix}$

Also $\begin{matrix} \underset{z \in (x \circ_{1} a) \circ_{1} y}{rinf} {{\tilde{η}}_{F^{- 1} (ℑ)} (z)} = \underset{z \in (y \circ_{1} a) \circ_{1} x}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ = \underset{z \in (e \circ_{1} (y \circ_{1} a)) \circ_{1} x}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ \underset{F (z) \in (F (e) \circ_{2} (F (y) \circ_{2} F (a))) \circ_{2} F (x)}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ \underset{F (s) \in F (y) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ = \underset{F (s) \in F (e \circ_{1} y) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ = \underset{F (s) \in (F (e) \circ_{2} F (y)) \circ_{2} F (a)}{rinf} {{\tilde{η}}_{ℑ} (F (s))} \\ ⪰ {\tilde{η}}_{ℑ} (F (y)) = {\tilde{η}}_{F^{- 1} (ℑ)} (y) . \end{matrix}$

From equations (1) and (2), we get $\begin{matrix} \underset{z \in (x \circ_{1} a) \circ_{1} y}{rinf} {{\tilde{η}}_{F^{- 1} (ℑ)} (z)} \\ ⪰ rmin {{\tilde{η}}_{F^{- 1} (ℑ)} (x), {\tilde{η}}_{F^{- 1} (ℑ)} (y)} . \end{matrix}$

Similarly we get $\begin{matrix} sup_{z \in (x \circ_{1} a) \circ_{1} y} {ϑ_{F^{- 1} (ℑ)} (z)} \\ \leq max {ϑ_{F^{- 1} (ℑ)} (x), ϑ_{F^{- 1} (ℑ)} (y)} . \end{matrix}$

This shows that the preimage of every cubic interior hyperideal of H₂ is a generalized bi-hyperideal of H₁. □

Theorem 4.11. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a homomorphism from an LA-semihypergroup H₁ to an LA-semihypergroup H₂. If H₁ is regular, then the preimage of every cubic interior hyperideal of H₂ is a cubic right hyperideal of H₁.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic interior hyperideal of H₂ and x, y ∈ H₁ . Since H₁ is regular, then there exists w ∈ H₁ such that x ∈ (x ∘ ₁w) ∘ ₁x . Then by the left invertive law, $\begin{matrix} \underset{z \in x \circ_{1} y \subseteq ((x \circ_{1} w) \circ_{1} x) \circ_{1} y = (y \circ_{1} x) \circ_{1} (x \circ_{1} w)}{rinf} {{\tilde{η}}_{F^{- 1} (ℑ)} (z)} \\ ⪰ \underset{F (z) \in (F (y) \circ_{2} F (x)) \circ_{2} (F (x) \circ_{2} F (w))}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ {\tilde{η}}_{ℑ} (F (x)) = {\tilde{η}}_{F^{- 1} (ℑ)} (x), \end{matrix}$ and $\begin{matrix} sup_{z \in x \circ_{1} y \subseteq ((x \circ_{1} w) \circ_{1} x) \circ_{1} y = (y \circ_{1} x) \circ_{1} (x \circ_{1} w)} {ϑ_{F^{- 1} (ℑ)} (z)} \\ \leq sup_{F (z) \in (F (y) \circ_{2} F (x)) \circ_{2} (F (x) \circ_{2} F (w))} {ϑ_{ℑ} (F (z))} \\ \leq ϑ_{ℑ} (F (x)) = ϑ_{F^{- 1} (ℑ)} (x) . \end{matrix}$

This shows that the preimage of every cubic interior hyperideal of H₂ is a cubic right hyperideal of H₁. □

Theorem 4.12. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a homomorphism from an LA-semihypergroup H₁ to an LA-semihypergroup H₂. If H₁ is regular with a pure left identity, then the preimage of every cubic right hyperideal of H₂ is a cubic left hyperideal of H₁.

Proof. Let $ℑ = 〈 {\tilde{η}}_{ℑ}, ϑ_{ℑ} 〉$ be a cubic right hyperideal of H₂ and x, y ∈ H₁ . Since H₁ is regular, then there exists w ∈ H₁ such that y ∈ (y ∘ ₁w) ∘ ₁y . Then by Lemma 2.8, $\begin{matrix} \underset{z \in x \circ_{1} y \subseteq x \circ_{1} ((y \circ_{1} w) \circ_{1} y) = (y \circ_{1} w) \circ_{1} (x \circ_{1} w)}{rinf} {{\tilde{η}}_{F^{- 1} (ℑ)} (z)} \\ ⪰ \underset{F (z) \in (F (y) \circ_{2} F (w)) \circ_{2} (F (x) \circ_{2} F (w))}{rinf} {{\tilde{η}}_{ℑ} (F (z))} \\ ⪰ \underset{F (s) \in F (y) \circ_{2} F (w)}{rinf} {\tilde{η}}_{ℑ} (F (s)) ⪰ {\tilde{η}}_{ℑ} (F (y)) \\ = {\tilde{η}}_{F^{- 1} (ℑ)} (y), \end{matrix}$ and $\begin{matrix} sup_{z \in x \circ_{1} y \subseteq x \circ_{1} ((y \circ_{1} w) \circ_{1} y) = (y \circ_{1} w) \circ_{1} (x \circ_{1} w)} {ϑ_{F^{- 1} (ℑ)} (z)} \\ \leq sup_{F (z) \in (F (y) \circ_{2} F (w)) \circ_{2} (F (x) \circ_{2} F (w))} {ϑ_{ℑ} (F (z))} \\ \leq sup_{F (s) \in F (y) \circ_{2} F (w)} ϑ_{ℑ} (F (s)) \leq ϑ_{ℑ} (F (y)) \\ = ϑ_{F^{- 1} (ℑ)} (y) . \end{matrix}$

This shows that the preimage of every cubic right hyperideal of H₂ is a cubic left hyperideal of H₁. □

Theorem 4.13. Let $F : (H_{1}, \circ_{1}) ⟶ (H_{2}, \circ_{2})$ be a homomorphism from an LA-semihypergroup H₁ to an LA-semihypergroup H₂. If H₁ is left regular with a pure left identity, then the preimage of every cubic generalized bi-hyperideal of H₂ is a cubic sub LA-semihypergroup of H₁.

Proof. The proof is straightforward. □

5 Conclusions

In this paper, we studied the structure of cubic sets in left almost semihypergroups. We defined cubic left (right, two sided) hyperideal, cubic bi-hyperideal and cubic interior hyperideal in a left almost semihypergroup and used them to characterize some classes of left almost semihypergroups. These results will serve as a base for future work, combining cubic sets, soft sets, rough sets and applying them in real-life applications inspired by [30 –35].

Footnotes

Acknowledgments

Sincere gratitude is expressed to the authorities of Deanship of Scientific Research, Majmaah University, Saudi Arabia. The financial support, continuous help and fruitful cooperation provided to the First Author are gratefully acknowledged. This research work was provided through Project No. 37/97.

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