# A Novel Study on Ordered Anti-Involution LA-Semihypergroups

Authors: Nabilah Abughazalah; Naveed Yaqoob
Journal: Journal of Mathematics
 (2023)

---

## Abstract

In this study, we introduce a new concept called “anti-involution” in relation to ordered LA-semihypergroups. An anti-involution is basically an involuntary automorphism, which is just a fancy term for a mathematical function that can be reversed. We looked at several fundamental results before introducing anti-involution hyperideals. We studied the anti-involution hyperideals of ordered anti-involution LA-semihypergroups using the rough set theory. In an ordered anti-involution LA-semihypergroup, theρ-upper and ρ-lower rough approximations of anti-involution hyperideals are an anti-involution hyperideals.


---

## Body

## 1. Introduction

One of the reasons for studying hyperstructures is to understand biological inheritance and physical phenomena such as nuclear fission. Chemical and redox processes are another source of inspiration for the research of hyperstructures. Classical algebraic structures can be generalized with algebraic superstructures. A composition of two elements is an element of a classical algebraic structure, and a composition of two elements is a set of algebraic superstructures. The term “hyperstructure” originally came from Marty [1], a French mathematician who pioneered hyperstructure theory. He extended the notion of classical binary operation to binary hyperoperation in his study. Several scholars directed their studies in this way, resulting in an excess of novel notions. Many applications of hyperstructures in geometry, codes, and cryptography are described in the book produced by Corsini and Leoreanu-Fotea [2] and Vougiouklis [3].Kazim and Naseeruddin [4], as the originators of the notion of almost semigroups, enlarged the notion of associative law to left invertive law in his study. Hila and Dine [5] provided the notion of left almost semihypergroups. Yaqoob et al. [6] also contributed to the idea of left almost semihypergroups. Yaqoob and Gulistan [7] introduced the notion of partially ordered left almost semihypergroups.Pawlak [8] established the notion of rough sets in 1982. Biswas and Nanda [9] applied the rough sets theory to groups. Jun [10] investigated the lower and upper approximations of gamma-subsemigroups/ideals in gamma-semigroups. Qurashi and Shabir [11, 12] generalized rough fuzzy ideals in quantales and roughness in quantale modules. Shabir and Irshad [13] studied roughness in ordered semigroups. Several writers have applied the rough set theory to various algebraic hyperstructures, such as Ameri et al. [14] applied roughness to bi-hyperideals of semihypergroups. Anvariyeh et al. [15] examined the roughness of Γ-semihypergroups. The rough set theory was applied to hyperrings by Davvaz [16], Γ-semihyperrings by Dehkordi and Davvaz [17], hyperlattices by He et al. [18], hypergroups by Leoreanu-Fotea [19], and nonassociative po-semihypergroups by Zhan et al. [20].An involutionf is a transformation [21] (or unary operation) α↦fα that satisfies the following 3 axioms:Axiom 1.
ffα=α. An involution is its own inverse.Axiom 2.
An involution is linear:fα1+α2=fα1+fα2 and λfα=fλα, where λ is a real constant.Axiom 3.
fα1α2=fα2fα1.
An anti-involution is a self-inverse transformation similar to an involution. It satisfies the Axiom1, 2, and Axiom 4:Axiom 4.
fα1α2=fα1fα2.
Foulis [22] submitted his Ph.D thesis in 1958, in which he studied unary operations in semigroups and provided some results for the theory of involution semigroups. Following that, Scheiblich and Nordahl [23] established the concept of regular ∗-semigroups. They labeled a semigroup S with a unary operation x⟶x∗ a regular ∗-semigroup if the following identities were satisfied:(1)x∗∗=x,xy∗=y∗x∗,x=xx∗x.
Following the introduction of this notion, various writers studied natural structures on semigroups with involution [24–26] and analyzed some findings on regular semigroups with involution [27]. Reilly [28] introduced a new class of ordinary ∗-semigroups. Wu [29] recently studied intraregular ordered ∗-semigroups. Aburawash [30] introduced the definition of involution group rings. Baxter [31] investigated rings with proper involution. Herstein [32, 33] added various results for involution rings.
In hyperstructures, the theory of involutions is seldom studied. Feng et al. [34] investigated regular equivalence relations on ordered ∗-semihypergroups. Yaqoob et al. [35] investigated the structures of involution Γ-semihypergroups. Tang and Yaqoob [36] introduced a fuzzy set theory to hyperideals of ordered ∗-semihypergroups.


## 2. Preliminaries and Basic Definitions

Some basic definitions are provided in this section on ordered LA-semihypergroups.Definition 1.
(see [5, 6]). A hypergroupoid Θ,∘ is said to be an LA-semihypergroup if for all ϑ,σ,ς∈Θ,(2)ϑ°σ°ς=ς°σ°ϑ.
Equation 2 is known as the left invertive law.Definition 2.
(see [7]). Let Θ be a non-empty set and “≤” be an ordered relation on Θ. Then, Θ,°,≤ is called an ordered LA-semihypergroup if(1)
Θ,° is an LA-semihypergroup;(2)
Θ,≤ is a partially ordered set;(3)
for everyϑ,σ,ς∈Θ, ϑ≤σ implies ϑ°ς≤σ°ς and ς°ϑ≤ς°σ, where ϑ°ς≤σ°ς means that for every α∈ϑ°ς there exists β∈σ°ς such that α≤β.Definition 3.
(see [7]). If ℑ is a nonempty subset of Θ,°,≤, then ℑ is the subset of Θ defined as follows:(3)ℑ=τ∈Θ:τ≤ϑ,for someϑ∈ℑ.Definition 4.
(see [7]). A non-empty subset ℑ of an ordered LA-semihypergroupΘ,°,≤ is called an LA-subsemihypergroup of Θ if ℑ°ℑ⊆ℑ.An anti-involution is an automorphism that does not change the order of the elements. The distinction between involutions and anti-involutions can exist only in algebra. The involution property does not work in left invertive structures where associativity and commutativity both do not hold.


## 3. Ordered Anti-InvolutionLA-Semihypergroups

In this section, we define the concept of an ordered anti-involution LA-semihypergroup and provide some results.Definition 5.
An ordered LA-semihypergroupΘ,°,≤ with an unary operation ⋇:Θ⟶Θ is called an ordered anti-involution LA-semihypergroup if it satisfies(1)
∀α∈Θα⋇⋇=α(2)
∀α,β∈Θα°β⋇=α⋇°β⋇Ifα⋇,β⋇∈Θ, then α⋇°β⋇=γ⋇:γ∈α°β. For any nonempty subset ℑ of an ordered anti-involution LA-semihypergroup Θ,(4)ℑ⋇=ϑ⋇∈Θ:ϑ∈ℑ.If for anyα,β∈Θ with α≤β, we have α⋇≤β⋇, then ⋇ is called an order preserving anti-involution.Example 1.
Consider a setΘ=ϑ,σ,ς with the following hyperoperation “°” and the order “≤” (Table1):
We give the covering relation “≺” as ≺=ϑ,σ,ϑ,ς. The figure of Θ is shown in Figure 1.
Then,Θ,°,≤ is an ordered LA-semihypergroup. Now, we define the anti-involution ⋇ by ϑ⋇=ϑ, σ⋇=ς, and ς⋇=σ. Then, it is easy to check that Θ,°,≤ is an ordered anti-involution LA-semihypergroup with order preserving anti-involution ⋇.Table 1
Hyperoperation “°”defined onΘ=ϑ,σ,ς.
°ϑσςϑϑϑ,σϑ,ςσϑ,ςσ,ςςςϑ,σσσ,ς(5)≤≔ϑ,ϑ,ϑ,σ,ϑ,ς,σ,σ,ς,ς.Figure 1
Figure ofΘ for Example 1.Lemma 1.
SupposeΘ is an ordered anti-involution LA-semihypergroup. Then, we have the following:(i)
ℑ⊆ℑ for any ℑ⊆Θ(ii)
ℑ⊆ℜ for any ℑ,ℜ with ℑ⊆ℜ⊆Θ(iii)
ℑ°ℜ⊆ℑ°ℜ for all ℑ,ℜ⊆Θ(iv)
ℑ⊆ℑ for all ℑ⊆Θ(v)
For any right (left, two-sided) hyperidealI of Θ, I=I(vi)
IfI and J are hyperideals of Θ, then I°J and I∩J are also hyperideals of ΘProposition 1.
Ifℑ and ℜ are nonempty subsets of an ordered anti-involution LA-semihypergroupΘ, then the following hold:(1)
ℑ⊆ℜif and only ifℑ⋇⊆ℜ⋇;(2)
ℑ∪ℜ⋇=ℑ⋇∪ℜ⋇;(3)
ℑ∩ℜ⋇=ℑ⋇∩ℜ⋇.Proof.
(i)
Considerℑ⊆ℜ. Also, consider(6)τ∈ℑ⋇⟹τ⋇∈ℑ⊆ℜ⟹τ⋇∈ℜ⟹τ∈ℜ⋇.Therefore,ℑ⋇⊆ℜ⋇. Conversely, consider ℑ⋇⊆ℜ⋇. Also, consider(7)u∈ℑ⟹u⋇∈ℑ⋇⊆ℜ⋇⟹u⋇∈ℜ⋇⟹u∈ℜ.Therefore,ℑ⊆ℜ.(ii)
Consider(8)τ∈ℑ∪ℜ⋇⇔τ⋇∈ℑ∪ℜ⇔τ⋇∈ℑorτ⋇∈ℜ⇔τ∈ℑ⋇orτ∈ℜ⋇.Hence, we obtainℑ∪ℜ⋇=ℑ⋇∪ℜ⋇.(iii)
The proof is similar to (ii).Proposition 2.
LetΘ be an ordered anti-involution LA-semihypergroup. Then,(1)
α°Θ°β⋇=α⋇°Θ°β⋇, for anyα,β∈Θ,(2)
Θ°α°Θ⋇=Θ°α⋇°Θ, for anyα∈Θ.Proof.
(1)
Letτ∈α°Θ°β⋇. By definition τ⋇∈α°Θ°β, τ⋇≤α°δ°β for some δ∈Θ. This implies that(9)τ≤α°δ°β⋇=α°δ⋇°β⋇=α⋇°δ⋇°β⋇⊆α⋇°Θ°β⋇.Therefore,τ≤α⋇°Θ°β⋇, that is, τ∈α⋇°Θ°β⋇ and we get that α°Θ°β⋇⊆α⋇°Θ°β⋇. On the other hand, if τ∈α⋇°Θ°β⋇, then for some δ∈Θ, we have τ≤α⋇°δ°β⋇. This implies that(10)τ⋇≤α⋇°δ°β⋇⋇=α⋇°δ⋇°β⋇⋇=α⋇⋇°δ⋇°β⋇⋇=α°δ⋇°β⊆α°Θ°β.So,τ≤α°Θ°β⋇, that is, τ∈α°Θ°β⋇. Therefore, α⋇°Θ°β⋇⊆α°Θ°β⋇. Consequently, α⋇°Θ°β⋇=α°Θ°β⋇.(2)
The proof is similar to (1).Definition 6.
A nonempty subsetℑ of an ordered anti-involution LA-semihypergroup Θ is called an ordered sub anti-involution LA-semihypergroup of Θ if ℑ°ℑ⊆ℑ and ℑ⋇=ℑ.Example 2.
Consider a setΘ=ϑ,σ,ς,κ,λ with the following hyperoperation “°” and the order “≤” (Table 2):
The figure ofΘ is shown in Figure 2.
Then,Θ,°,≤ is an ordered LA-semihypergroup. Now, we define the anti-involution ⋇ by ϑ⋇=ϑ, σ⋇=ς, ς⋇=σ, κ⋇=κ, and λ⋇=λ. Then, it is easy to check that Θ,°,≤ is an ordered anti-involution LA-semihypergroup with order preserving anti-involution ⋇. Here, ϑ and ϑ,σ,ς are sub anti-involution LA-semihypergroups of Θ. One can see that ϑ,σ°ϑ,σ⊆ϑ,σ but ϑ,σ⋇≠ϑ,σ, so ϑ,σ is not a sub anti-involution LA-semihypergroup of Θ.Table 2
Left almost semihypergroup (Θ=ϑ,σ,ς,κ,λ,°).
°ϑσςκλϑϑϑϑκ,λλσϑϑϑ,cκ,λλςϑϑ,σϑκ,λλκκ,λκ,λκ,λκ,λκ,λλλλλκ,λλ(11)≤≔ϑ,ϑ,ϑ,σ,ϑ,ς,σ,σ,ς,ς,κ,κ,λ,λ.Figure 2
Figure ofΘ for Example 2.Definition 7.
A nonempty subsetℑ of an ordered anti-involution LA-semihypergroup Θ is called a right (resp., left) anti-involution hyperideal of Θ if(1)
ℑ°Θ⊆ℑ (resp., Θ°ℑ⊆ℑ)(2)
Ifϑ∈ℑ and σ≤ϑ, then σ∈ℑ for every σ∈Θ(3)
ℑ⋇=ℑDefinition 8.
A nonempty subsetℑ of an ordered anti-involution LA-semihypergroup Θ is called an anti-involution hyperideal of Θ if it is both a right and a left anti-involution hyperideal of Θ.Definition 9.
A nonempty subsetℜ of an ordered anti-involution LA-semihypergroup Θ is called an anti-involution bi-hyperideal of Θ if(1)
ℜ°ℜ⊆ℜ,(2)
ℜ°Θ°ℜ⊆ℜ,(3)
Ifϑ∈ℜ and σ≤ϑ, then σ∈ℜ for every σ∈Θ,(4)
ℜ⋇=ℜ.Example 3.
Consider a setΘ=ϑ,σ,ς,κ,λ with the following hyperoperation “°” and the order “≤” (Table 3):
The figure ofΘ is shown in Figure 3.
Then,Θ,°,≤ is an ordered LA-semihypergroup. Now, we define the anti-involution ⋇ by ϑ⋇=ϑ, σ⋇=σ, ς⋇=κ, κ⋇=ς, and λ⋇=λ. Then, it is easy to check that Θ,°,≤ is an ordered anti-involution LA-semihypergroup with order preserving anti-involution ⋇. Here, λ, ϑ,σ,λ, and ς,κ,λ are anti-involution bi-hyperideals of Θ.Table 3
Hyperoperation “°”defined onΘ=ϑ,σ,ς,κ,λ.
°ϑσςκλϑσσλλλσϑ,σϑ,σλλλςλλς,κ,λκλκλλςς,κ,λλλλλλλλ(12)≤≔ϑ,ϑ,σ,σ,ς,ς,κ,κ,λ,ϑ,λ,σ,λ,ς,λ,κ,λ,λ.Figure 3
Figure ofΘ for Example 3.Definition 10.
A nonempty subsetQ of an ordered anti-involution LA-semihypergroup Θ is called an anti-involution quasi-hyperideal of Θ if(1)
Q°Θ∩Θ°Q⊆Q,(2)
Q⊆Q,(3)
Q⋇=Q.Proposition 3.
LetΘ be an ordered LA-semihypergroup with order preserving anti-involution ⋇. Then,(1)
ℑ⋇ is a left (resp., right) anti-involution hyperideal for any left (resp., right) hyperideal ℑ of Θ,(2)
ℜ⋇ is an anti-involution bi-hyperideal for any bi-hyperidealℜ of Θ,(3)
Q⋇ is an anti-involution quasi-hyperideal for any quasi-hyperidealQ of Θ.Proof.
Assumeℑ is a left hyperideal of Θ. Since Θ°ℑ⊆ℑ and Θ⋇=Θ, we have(13)Θ°ℑ⋇=Θ⋇°ℑ⋇=Θ°ℑ⋇⊆ℑ⋇.
Now, letϑ∈ℑ⋇ and σ≤ϑ, then σ⋇≤ϑ⋇∈ℑ. Thus, σ⋇∈ℑ, hence, σ∈ℑ⋇. Therefore, ℑ⋇ is a left anti-involution hyperideal of Θ. Similar is the case for right hyperideal ℑ of Θ.(2)
Similar to equation (1).(3)
LetQ be a quasi-hyperideal of Θ. Since Q°Θ∩Θ°Q⊆Q, and Θ⋇=Θ. By Proposition 1, we have(14)Q⋇°Θ∩Θ°Q⋇=Q⋇°Θ⋇∩Θ⋇°Q⋇=Q°Θ⋇∩Θ°Q⋇=Q°Θ∩Θ°Q⋇⊆Q⋇.
Now, letϑ∈Q⋇ and σ≤ϑ, then σ⋇≤ϑ⋇∈Q. Thus, σ⋇∈Q, hence, σ∈Q⋇. This shows that Q⋇ is an anti-involution quasi-hyperideal of Θ.Theorem 1.
LetΘ be an ordered LA-semihypergroup with order preserving anti-involution ⋇. Let ℑi⋇:i∈I be a family of(1)
Left (resp., right) anti-involution hyperideals ofΘ. Then, the intersection ∩i∈Iℑi⋇≠∅ is a left (resp., right) anti-involution hyperideal of Θ.(2)
Anti-involution bi-hyperideals ofΘ. Then, the intersection ∩i∈Iℑi⋇≠∅ is an anti-involution bi-hyperideal of Θ.(3)
Anti-involution quasi-hyperideals ofΘ. Then, the intersection ∩i∈Iℑi⋇≠∅ is an anti-involution quasi-hyperideal of Θ.Proof.
Straightforward.Proposition 4.
LetΘ be an ordered LA-semihypergroup with order preserving anti-involution ⋇ and ℑ be any anti-involution hyperideal of Θ. For any τ∈Θ, if τ∈Θ°τ⋇°Θ°τ⋇°Θ, then ℑ=ℑ⋇°ℑ⋇.Proof.
Letϑ∈ℑ⋇°ℑ⋇. Then, ϑ∈a1°a2 for some a1,a2∈ℑ⋇. Let a1∈Θ°a1⋇°Θ°a1⋇°Θ. Then, a1∈p1°a1⋇°p2°a1⋇°p3 for some p1,p2,p3∈Θ. Similarly, a2∈q1°a2⋇°q2°a2⋇°q3 for some q1,q2,q3∈Θ. Consequently, we have(15)ϑ∈a1°a2⊆p1°a1⋇°p2°a1⋇°p3°q1°a2⋇°q2°a2⋇°q3=a1⋇°p2°p1°a1⋇°p3°q1°a2⋇°q2°a2⋇°q3=p3°p1°a1⋇°a1⋇°p2°q1°a2⋇°q2°a2⋇°q3=p3°a1⋇°p1°a1⋇°p2°q1°a2⋇°q2°a2⋇°q3⊆Θ°ℑ°Θ°Θ⊆ℑ.
Hence,ℑ⋇°ℑ⋇⊆ℑ. On the other hand, let ϑ∈ℑ. Then, we have ϑ∈r1°ϑ⋇°r2°ϑ⋇°r3 for some r1,r2,r3∈Θ, because ϑ∈Θ°ϑ⋇°Θ°ϑ⋇°Θ. Clearly,(16)ϑ∈r1°ϑ⋇°r2°ϑ⋇°r3⊆Θ°ℑ⋇°Θ°ℑ⋇°Θ=ℑ⋇°ℑ⋇°Θ=Θ°ℑ⋇°ℑ⋇⊆ℑ⋇°ℑ⋇.
Thus,ℑ⊆ℑ⋇°ℑ⋇. Hence, ℑ=ℑ⋇°ℑ⋇.Definition 11.
An ordered anti-involution LA-semihypergroupΘ is called a regular ordered anti-involution LA-semihypergroup, if α∈α°α⋇°α, for all α∈Θ.Proposition 5.
LetΘ be an ordered LA-semihypergroup with order preserving anti-involution ⋇. If Θ is regular and has left identity, then(17)Θ°α°α⋇°Θ=Θ°α°α⋇°Θ,for any α∈Θ.Proof.
Letα∈Θ. Since Θ is regular, we have α∈α°α⋇°α. Then,(18)Θ°α°α⋇°Θ=Θ°α°α⋇°α°α⋇°Θ=α°α⋇°Θ°α°α⋇°Θ⊆α°α⋇°Θ°α⋇°Θ=α⋇°Θ°α°α⋇°Θ⊆Θ°α°α⋇°Θ.
Thus,Θ°α°α⋇°Θ⊆Θ°α°α⋇°Θ. On the other hand, we have(19)Θ°α°α⋇°Θ=Θ°α°α⋇°α°α⋇°Θ=Θ°α⋇°α°α°α⋇°Θ=α⋇°α°Θ°α°α⋇°Θ=α⋇°Θ°α°α°α⋇°Θ=α°α°α⋇°Θ°α⋇°Θ=Θ°α°α⋇°α°α⋇°Θ⊆Θ°α°α⋇°Θ.
Thus,Θ°α°α⋇°Θ⊆Θ°α°α⋇°Θ. Hence, Θ°α°α⋇°Θ=Θ°α°α⋇°Θ.


## 4. Rough Anti-Involution Hyperideals

In this section, we applied rough set theory to anti-involution hyperideals of ordered anti-involution LA-semihypergroups.Definition 12.
A relationρ on an ordered anti-involution LA-semihypergroupΘ is called a pseudohyperorder if(1)
≤⊆ρ(2)
ρ is transitive, that is, ϑ,σ,σ,ς∈ρ implies ϑ,ς∈ρ for all ϑ,σ,ς∈Θ.(3)
ρ is compatible, that is, if ϑ,σ∈ρ then, ϑ°α,σ°α∈ρ and α°ϑ,α°σ∈ρ for all ϑ,σ,α∈Θ.(4)
for allϑ,σ∈ρ, we have ϑ,σ⋇=ϑ⋇,σ⋇∈ρ.Definition 13.
A pseudohyperorder relationρ on an ordered anti-involution LA-semihypergroupΘ is said to be complete if ρNα°β=ρNα°ρNβ.Definition 14.
[20] Let α be a nonempty set and ρ be a binary relation on X. By ℘X, we mean the power set of X. For all ℑ⊆X, we define ρ− and ρ+:℘X⟶℘X by(20)ρ−ℑ=α∈X:ρNα⊆ℑ,ρ+ℑ=α∈X:ρNα∩ℑ≠∅,where ρNα=β∈X:αρβ. ρ−ℑ and ρ+ℑ are called the lower approximation and the upper approximation operations, respectively.Lemma 2.
Letρ be a pseudohyperorder on an ordered anti-involution LA-semihypergroupΘ. Then, for any α∈Θ, ρNα⋇=ρNα⋇.Proof.
Considerτ∈ρNα⋇. Then, τ⋇∈ρNα, this implies that α,τ⋇∈ρ. By definition of pseudohyperorder relation α,τ⋇⋇∈ρ, this implies that α⋇,τ∈ρ. So, τ∈ρNα⋇. Thus, we get ρNα⋇⊆ρNα⋇. Conversely, consider τ∈ρNα⋇, then α⋇,τ∈ρ. By definition of pseudohyperorder relation α⋇,τ⋇∈ρ, this implies that α,τ⋇∈ρ. So, τ⋇∈ρNα, this implies that τ∈ρNα⋇. Thus, we obtain ρNα⋇⊆ρNα⋇. Hence, ρNα⋇=ρNα⋇.Theorem 2.
Letρ be a pseudohyperorder on an ordered LA-semihypergroupΘ. If ℑ and ℜ are nonempty subsets of Θ, then(21)ρ+ℑ°ρ+ℜ⊆ρ+ℑ°ℜ.Proof.
The proof is straightforward.Theorem 3.
Letρ be a complete pseudohyperorder on an ordered LA-semihypergroupΘ. If ℑ and ℜ are nonempty subsets of Θ, then(22)ρ−ℑ°ρ−ℜ⊆ρ−ℑ°ℜ.Proof.
The proof is straightforward.Definition 15.
Assumeρ is a pseudohyperorder on an ordered anti-involution LA-semihypergroupΘ. Then, a nonempty subset ℑ of Θ is called a ρ-upper rough LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ if ρ+ℑ is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ.Theorem 4.
Letρ be a pseudohyperorder on an ordered anti-involution LA-semihypergroupΘ and ℑ an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ. Then, ρ+ℑ is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ.Proof.
Letℑ be an anti-involution bi-hyperideal of Θ.(1)
By Theorem2, we have(23)ρ+ℑ°ρ+ℑ⊆ρ+ℑ°ℑ⊆ρ+ℑ.(2)
By Theorem2, we have(24)ρ+ℑ°Θ°ρ+ℑ=ρ+ℑ°ρ+Θ°ρ+ℑ⊆ρ+ℑ°Θ°ℑ⊆ρ+ℑ.(3)
Letϑ∈ρ+ℑ and σ∈Θ such that σ≤ϑ. Then, there exists β∈ℑ, such that ϑρβ and σρϑ. Since ρ is transitive, so σρβ implies σ∈ρ+ℑ.(4)
Consider(25)τ∈ρ+ℑ⋇⟺τ⋇∈ρ+ℑ⟺ρNτ⋇∩ℑ≠∅⟺ρNτ⋇∩ℑ≠∅using Lemma2⟺ρNτ⋇∩ℑ⋇≠∅becauseℑ⋇=ℑ⟺ρNτ∩ℑ⋇≠∅using Proposition13⟺ρNτ∩ℑ≠∅⟺τ∈ρ+ℑ.
Thus,ρ+ℑ⋇=ρ+ℑ. Hence, ρ+ℑ is an anti-involution bi-hyperideal of Θ. The other cases can be seen in a similar way.Definition 16.
Assumeρ is a complete pseudohyperorder on an ordered anti-involution LA-semihypergroupΘ. Then, a nonempty subset ℑ of Θ is called a ρ-lower rough LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ if ρ−ℑ is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ.Theorem 5.
Assumeρ is a complete pseudohyperorder on an ordered anti-involution LA-semihypergroupΘ and ℑ an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ. Then, ρ−ℑ is an LA-subsemihypergroup (resp., left anti-involution hyperideal, right anti-involution hyperideal, anti-involution hyperideal, and anti-involution bi-hyperideal) of Θ.Proof.
Letℑ be an anti-involution bi-hyperideal of Θ.(1)
By Theorem3, we have(26)ρ−ℑ°ρ−ℑ⊆ρ−ℑ°ℑ⊆ρ−ℑ.(2)
By Theorem3, we have(27)ρ−ℑ°Θ°ρ−ℑ=ρ−ℑ°ρ−Θ°ρ−ℑ⊆ρ−ℑ°Θ°ℑ⊆ρ−ℑ.(3)
Letϑ∈ρ−ℑ and σ∈Θ such that σ≤ϑ. Then, there exists β∈ℑ, such that ϑρβ and σρϑ. Since ρ is transitive, so σρβ implies σ∈ρ−ℑ.(4)
Consider(28)τ∈ρ−ℑ⋇⟺τ⋇∈ρ−ℑ⟺ρNτ⋇⊆ℑ⟺ρNτ⋇⊆ℑusing Lemma2⟺ρNτ⋇⊆ℑ⋇becauseℑ⋇=ℑ⟺ρNτ⊆ℑ⟺τ∈ρ−ℑ.
Thus,ρ−ℑ⋇=ρ−ℑ. Hence, ρ−ℑ is an anti-involution bi-hyperideal of Θ. The other cases can be seen in a similar way.The following example shows that the converse of Theorems4 and 5, for the case of anti-involution bi-hyperideal, but does not hold in general.Example 4.
LetΘ=ϑ,σ,ς,κ,λ be an ordered LA-semihypergroup with the following multiplication table and the order “≤.” (Table 4)
The figure ofΘ is shown in Figure 4.
Clearly,Θ is an ordered LA-semihypergroup. We define the anti-involution ⋇ by ϑ⋇=ϑ, σ⋇=σ, ς⋇=ς, κ⋇=λ, and λ⋇=κ. Then, it is easy to check that Θ,°,≤ is an ordered anti-involutionLA-semihypergroup with order preserving anti-involution ⋇. Now, let(29)ρ=ϑ,ϑ,ϑ,σ,σ,σ,ς,ς,ς,κ,ς,λ,κ,ς,κ,κ,κ,λ,λ,ς,λ,κ,λ,λ,be a complete pseudohyperorder on Θ, such that(30)ρNϑ=ϑ,ρNσ=ϑ,σ,ρNς=ρNκ=ρNλ=ς,κ,λ.
Now, forℑ=ϑ,σ,λ⊆Θ,(31)ρ−ϑ,σ,λ=ϑ,σ,ρ+ϑ,σ,λ=ϑ,σ,ς,κ,λ.
It is clear thatρ−ϑ,σ,λ and ρ+ϑ,σ,λ are both anti-involution bi-hyperideals of Θ but ϑ,σ,λ is not an anti-involution bi-hyperideal of Θ.Table 4
LA-semihypergroup (Θ=ϑ,σ,ς,κ,λ,°).
°ϑσςκλϑϑϑϑϑϑσϑσσσσςϑσςκλκϑσλκ,λκ,λλϑσκκ,λκ,λ(29)≤≔=ϑ,ϑ,ϑ,σ,σ,σ,ς,ς,ς,κ,ς,λ,κ,κ,λ,λ.Figure 4
Figure ofΘ for Example 4.


## 5. Conclusion

Involutions can be applied to associative algebraic structures. This theory fails for noncommutative and nonassociative structures (left invertive structures). Here, we chose a noncommutative and nonassociative structure (ordered left almost semihypergroup) and then applied the concept of involutions. However, the involution theory failed on the ordered LA-semihypergroup because of its noncommutative and nonassociative nature. Then, we applied anti-involutions to an ordered LA-semihypergroup and provided some results and examples. We constructed some results on roughness in an ordered anti-involution LA-semihypergroup.In the future, the following topics might be considered for further studies:(i)
Regular and intraregular ordered anti-involution LA-semihypergroups(ii)
Prime and weakly prime hyperideals in ordered anti-involution LA-semihypergroups(iii)
Fuzzy ordered anti-involution LA-semihypergroups(iv)
Soft ordered anti-involution LA-semihypergroups



---

*Source: 6972483-2023-03-22.xml*