# Approximations of∈,∈∨qk-Fuzzy Hyperideals in Ordered LA-Semihypergroups

Authors: Nabilah Abughazalah; Naveed Yaqoob; Samavia Ashraf
Journal: Mathematical Problems in Engineering
 (2021)

---

## Abstract

This paper concerns the relationship between rough sets,∈,∈∨qk-fuzzy sets, and the LA-semihypergroups. We proved that the lower approximation (l-approx) and the upper approximation (u-approx) of different hyperideals of LA-semihypergroups become again hyperideal and also provided some examples. We also provided a result which shows that, if ϑ is not a hyperideal of an LA-semihypergroup P, then its lower approximation is also not a hyperideal.


---

## Body

## 1. Introduction

Hyperstructures represent a natural extension of classical algebraic structures. Algebraic hyperstructure theory depends on hyperoperation and their properties. One of these theories is the semihypergroup theory. Semihypergroup theory was first introduced by French Mathematician Marty in 1934 [1]. The concept of semihypergroup is the generalization of the concept of semigroup. In semigroup, the composition of two elements is an element, while in a semihypergroup the composition of two elements is a nonempty set. In semihypergroup, closure and associative law hold. Many authors studied different aspects of semihypergroup (see [2–4]). Recently, Hila and Dine [5] introduced the notion of LA-sm-hyp-group (LA-semihypergroup), and then, Yaqoob, Corsini, and Yousafzai [6] extended the work of Hila and Dine and characterized intraregular left almost semihypergroups by their HP-ideals (hyperideals) using pure left identity. The concept of ordered semihypergroup was studied by Heidari and Davvaz [7] where they used a binary relation ≤ on semihypergroup ϒ,⊛ such that the binary relation is a partial order and the structure ϒ,⊛,≤ is known as ordered semihypergroup [8].The concept of fuzzy sets was first introduced by Zadeh [9] in 1964. Fuzzy set is a set having degrees of membership between 1 and 0. Fuzzy set theory is a classical application of set theory. Applications of fuzzy sets in some algebraic structures and other structures can be seen in [10–13]. Zdzisław Pawlak [14] introduced the notion of rough sets, in the year 1982. Rough set theory constitutes a sound basis for knowledge discovery databases. It offers mathematical tools to discover pattern hidden in data. It can be used for feature selection, feature extraction, decision rule generation, and paper extraction. Applications of rough sets in some algebraic structures and algebraic hyperstructures can be seen in [15–36].


## 2. Preliminaries and Notations

Letϒ be a nonempty set. Then, the map ⊛:ϒ×ϒ⟶ρ∗ϒ is called HP-operation (hyperoperation) or join operation on the set ϒ, where ρ∗ϒ=ρϒ/∅ denotes the set of all nonempty subsets of ϒ. A hypergroupoid is a set ϒ together with a (binary) HP-operation. For any nonempty subsets of Ω1,Ω2 of ϒ, we denote(1)Ω1⊛Ω2=∪a∈Ω1,b∈Ω2a⊛b.Instead ofa⊛Ω1 and Ω2⊛a, we write a⊛Ω1 and Ω2⊛a, respectively.Definition 1.
(see [5, 6]). A hypergroupoid ϒ,⊛ is called an LA-sm-hyp-group if for every p,q,r∈ϒ, we have p⊛q⊛r=r⊛q⊛p.
The lawp⊛q⊛r=r⊛q⊛p is called a left invertive law.Definition 2.
(see [8]). Let ϒ be a nonempty set and ≤ be a partially ordered relation on ϒ. The triplet ϒ,⊛,≤ is called a partially ordered LA-sm-hyp-group if the following conditions are satisfied:(i)
ϒ,⊛is an LA-sm-hyp-group(ii)
ϒ,≤ is a partially ordered set(iii)
For everya,b,c∈ϒ, a≤b implies a⊛c≤b⊛c and c⊛a≤c⊛b, where a⊛c≤b⊛c means that for p∈a⊛c, there exist q∈b⊛c such that p≤qDefinition 3.
(see [8]). If ϒ,⊛,≤ is a partially ordered LA-sm-hyp-group and Ω⊆ϒ, then Ω is the subset of ϒ defined as follows:(2)Ω=t∈ϒ:t≤a,for somea∈Ω.
Ifp∈ϒ and Ω is a nonempty subset of ϒ, then Ωp=q,r∈ϒ×ϒ|p≤q⊛r.Definition 4.
(see [8]). A nonempty subset Ω of a partially ordered LA-sm-hyp-group is called a LAS-sm-hyp-group (LA-subsemihypergroup) of ϒ if Ω⊛Ω⊆Ω.Definition 5.
(see [8]). A nonempty subset Ω of a partially ordered LA-sm-hyp-group ϒ,⊛,≤ is called a right (resp., left) HP-ideal (hyperideal) of ϒ if(i)
Ω⊛ϒ⊆Ω,resp.,ϒ⊛Ω⊆Ω(ii)
For everya∈ϒ and b∈Ω, a≤b implies a∈Ω
IfΩ is both right HP-ideal and left HP-ideal of LA-sm-hyp-group ϒ, then Ω is called a HP-ideal (or two-sided HP-ideal) of LA-sm-hyp-group ϒ.Definition 6.
(see [8]). A nonempty subset Ω of a partially ordered LA-sm-hyp-group ϒ,⊛,≤ is called an interior HP-ideal of ϒ, if(i)
ϒ⊛Ω⊛ϒ⊆Ω(ii)
For everya∈ϒand b∈Ω, a≤b implies a∈ΩDefinition 7.
(see [8]). A nonempty subset Ω of a partially ordered LA-sm-hyp-group ϒ,⊛,≤ is called a generalized bi-HP-ideal of ϒ, if(i)
Ω⊛ϒ⊛Ω⊆Ω(ii)
For everya∈ϒ and b∈Ω, a≤b implies a∈Ω
Definitions onϵ,ϵ∨qk-fuzzy HP-ideals in LA-sm-hyp-group can be found in [10].


## 3. Rough∈,∈∨qk-Fuzzy LAS-sm-hyp-Groups

Definition 8.
Letϒ be a nonempty set and δ be a binary relation on ϒ. By ℘ϒ, we mean the power set of ϒ. For all Ω⊆ϒ, we define δupper¯and δlower¯:℘ϒ⟶℘ϒ by(3)δupper¯Ω=p∈ϒ:∃q∈Ω,such thatpδq=p∈ϒ:δCp∩Ω≠∅,δlower¯Ω=p∈ϒ:∨q,pδq⟶q∈Ω=p∈ϒ:δCp⊆Ω,where δCp=q∈ϒ:pδq.δupper¯Ω and δlower¯Ω are called the u-approx and the l-approx operations, respectively.Definition 9.
Letδ be a PH-order relation (pseudohyperorder relation [36]) on a partially ordered LA-sm-hyp-group ϒ. Let ϑ be a fuzzy subset of LA-sm-hyp-group ϒ. For every p in ϒ, we define δ-u-approx and the δ-l-approx of ϑ by(4)δupper¯ϑp=⋎p′∈δCpϑp′,δlower¯ϑp=⋏p′∈δCpϑp′.Theorem 1.
Letδ be a PH-order relation on a partially ordered LA-sm-hyp-group ϒ. Let ϑ be an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ. Then, δupper¯ϑ is an LAS-sm-hyp-group of LA-sm-hyp-group ϒ.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ and δ be a PH-order relation on ϒ. We have to show that δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ.(i)
For eachp,q in ϒ and p≤q, we have δCq⊆δCp. Now(5)δupper¯ϑp=⋎p′∈δCpϑp′≥⋎q′∈δCqminϑq′,1−k2=min⋎q′∈δCqϑq′,1−k2=minδupper¯ϑq,1−k2.Thus, we getδupper¯ϑp≥minδupper¯ϑq,1−k/2.(ii)
Now, letp,q,r in ϒ. Let(6)⋏r∈p⊛qδupper¯ϑr=⋏r∈p⊛q⋎t∈δCrϑt≥⋎t∈δCp⊛qϑt≥⋎t∈δCp⊛δCqϑt≥⋏t∈p′⊛q′⋎p′∈δCp,q′∈δCqϑt=⋎p′∈δCp,q′∈δCq⋏t∈p′⊛q′ϑt≥⋎p′∈δCp,q′∈δCqminϑp′,ϑq′,1−k2≥min⋎p′∈δCpϑp′,⋎q′∈δCqϑq′,1−k2=minδupper¯ϑp,δupper¯ϑq,1−k2.Thus, we get⋏r∈p⊛qδupper¯ϑr≥minδupper¯ϑp,δupper¯ϑq,1−k/2. Hence δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ.Theorem 2.
Letδ be a complete and symmetric PH-order relation on a partially ordered LA-sm-hyp-group ϒ. Let ϑ be an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ. Then, δlower¯ϑ is an LAS-sm-hyp-group of LA-sm-hyp-group ϒ.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ and δ be a complete and symmetric PH-order relation on ϒ. We have to show that δlower¯ϑ is an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ.(i)
For eachp,q in ϒ and p≤q, we have δCp=δCq. Now(7)δlower¯ϑp=⋏p′∈δCpϑp′=⋏q′∈δCqminϑq′,1−k2=min⋏q′∈δCqϑq′,1−k2=minδlower¯ϑq,1−k2.Thus, we getδlower¯ϑp=minδlower¯ϑq,1−k/2.(ii)
Now letp,q,r in ϒ. Let(8)⋏r∈p⊛qδlower¯ϑr=⋏r∈p⊛q⋏t∈δCrϑt=⋏t∈δCp⊛qϑt=⋏t∈δCp⊛δCqϑt≥⋏t∈p′⊛q′,⋏p′∈δCp,q′∈δCqϑt=⋏p′∈δCp,q′∈δCq,⋏t∈p′⊛q′ϑt≥⋏p′∈δCp,q′∈δCqminϑp′,ϑq′,1−k2≥min⋏p′∈δCpϑp′,⋏q′∈δCqϑq′,1−k2=minδlower¯ϑp,δlower¯ϑq,1−k2,Thus, we get⋏r∈p⊛qδlower¯ϑp≥minδlower¯ϑp,δlower¯ϑq,1−k/2. Hence δlower¯ϑ is an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ.


## 4. Rough∈,∈∨qk-Fuzzy HP-Ideals

Theorem 3.
Letδ be a PH-order on a partially ordered LA-sm-hyp-group ϒ. Let ϑ be an ϵ,ϵ∨qk-fuzzy left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ. Then, δupper¯ϑ is a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy bi-HP-ideal of LA-sm-hyp-group ϒ and δ be a PH-order relation on ϒ. We have to show that δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy bi-HP-ideal of LA-sm-hyp-group ϒ.(i)
For eachp,q in ϒ and p≤q, we have δCq⊆δCp. Now(9)δupper¯ϑp=⋎p′∈δCpϑp′≥⋎q′∈δCqminϑq′,1−k2=min⋎q′∈δCqϑq′,1−k2=minδupper¯ϑq,1−k2.Thus, we getδupper¯ϑp≥minδupper¯ϑq,1−k/2.(ii)
Now, letp,q,r in ϒ. Let(10)⋏r∈p⊛qδupper¯ϑr=⋏r∈p⊛q,⋎t∈δCrϑt≥⋎t∈δCp⊛qϑt≥⋎t∈δCp⊛δCqϑt≥⋏t∈p′⊛q′,⋎p′∈δCp,q′∈δCqϑt=⋎p′∈δCp,q′∈δCq,⋏t∈p′⊛q′ϑt≥⋎p′∈δCp,q′∈δCqminϑp′,ϑq′,1−k2≥min⋎p′∈δCpϑp′,⋎q′∈δCqϑq′,1−k2=minδupper¯ϑp,δupper¯ϑq,1−k2.Thus, we get⋏r∈p⊛qδupper¯ϑr≥minδupper¯ϑp,δupper¯ϑq,1−k/2.(iii)
Now, letp,q,r,w in ϒ. Let(11)⋏w∈p⊛q⊛rδupper¯ϑw=⋏w∈p⊛q⊛r,⋎t∈δCwϑt≥⋎t∈δCp⊛q⊛rϑt≥⋎t∈δCp⊛δCq⊛δCrϑt≥⋏t∈p′⊛q′⊛r′,⋎p′∈δCp,q′∈δCq,r′∈δCrϑt=⋎p′∈δCp,q′∈δCq,r′∈δCr,⋏t∈p′⊛q′⊛r′ϑt≥⋎p′∈δCp,q′∈δCq,r′∈δCrminϑp′,ϑr′,1−k2≥min⋎p′∈δCpϑp′,⋎r′∈δCrϑr′,1−k2=minδupper¯ϑp,δupper¯ϑr,1−k2.Thus, we get⋏w∈p⊛q⊛rδupper¯ϑw≥minδupper¯ϑp,δupper¯ϑr,1−k/2. Hence, δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy bi-HP-ideal of LA-sm-hyp-group ϒ. The other cases can be proved in a similar way.
The following example shows that the converse of Theorem3 is not true in general.Example 1.
Consider a setϒ=a,b,c,d with the following hyperoperation“⊛” defined in Table 1.
The order “≤” is ≤:=a,a,b,b,c,c,d,d,d,a. Then, ϒ,⊛,≤ becomes a partially ordered LA-sm-hyp-group. Let us define a fuzzy subset ϑ of LA-sm-hyp-group ϒ by ϑ=0.5/a,0.7/b,0.9/c,0.2/d and take t=0.4 and n=0.3. Then, it is clear that ϑ is not an ϵ,ϵ∨qk-fuzzy HP-ideal of LA-sm-hyp-group ϒ because for d≤a, at in ϑ but dt∉ϑ and ϑd+t+n=0.9<1. Now define a PH-order relation δ on ϒ by(12)δ=a,a,a,d,b,b,b,c,c,b,c,c,d,a,d,d.
By the routine calculations, we get(13)δupper¯ϑa=⋎t∈δCaϑt=0.5,δupper¯ϑb=⋎t∈δCbϑt=0.9,δupper¯ϑc=⋎t∈δCcϑt=0.9,δupper¯ϑd=⋎t∈δCdϑt=0.5.
Thus, we getδupper¯ϑ=0.5/a,0.9/b,0.9/c,0.5/d. One can see that δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy HP-ideal of LA-sm-hyp-group ϒ. Similarly, we can find the example for other cases.Table 1
Tabular form of the hyperoperation “⊛” defined in Example 1.
⊛abcdaaa,da,ddba,db,cb,cdca,dbbddddddTheorem 4.
Letδ be a complete and symmetric PH-order relation on a partially ordered LA-sm-hyp-group ϒ. Let ϑ be an ϵ,ϵ∨qk-fuzzy left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ. Then, δlower¯ϑ is a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy bi-HP-ideal of LA-sm-hyp-group ϒ and δ be a complete and symmetric PH-order relation on ϒ. We have to show that δlower¯ϑ is an ϵ,ϵ∨qk-fuzzy bi-HP-ideal of LA-sm-hyp-group ϒ.(i)
For eachp,q in ϒ and p≤q, we have δCp=δCq. Now(14)δlower¯ϑp=⋏p′∈δCpϑp′=⋏q′∈δCqminϑq′,1−k2=min⋏q′∈δCqϑq′,1−k2minδlower¯ϑq,1−k2.Thus, we getδlower¯ϑp=minδlower¯ϑq,1−k/2.(ii)
Now, letp,q,r in ϒ. Let(15)⋏r∈p⊛qδlower¯ϑr=⋏r∈p⊛q,⋏t∈δCrϑt=⋏t∈δCp⊛qϑt=⋏t∈δCp⊛δCqϑt≥⋏t∈p′⊛q′,⋏p′∈δCp,q′∈δCqϑt=⋏p′∈δCp,q′∈δCq,⋏t∈p′⊛q′ϑt≥⋏p′∈δCp,q′∈δCqminϑp′,ϑq′,1−k2≥min⋏p′∈δCpϑp′,⋏q′∈δCqϑq′,1−k2=minδlower¯ϑp,δlower¯ϑq,1−k2.Thus, we get⋏r∈p⊛qδlower¯ϑr≥minδlower¯ϑp,δlower¯ϑq,1−k/2.(iii)
Now, letp,q,r,w in ϒ. Let(16)⋏w∈p⊛q⊛rδlower¯ϑw=⋏w∈p⊛q⊛r,⋏t∈δCwϑt=⋏t∈δCp⊛q⊛rϑt=⋏t∈δCp⊛δCq⊛δCrϑt≥⋏t∈p′⊛q′⊛r′,⋏p′∈δCp,q′∈δCq,r′∈δCrϑt=⋏p′∈δCp,q′∈δCq,r′∈δCr,⋏t∈p′⊛q′⊛r′ϑt≥⋏p′∈δCp,q′∈δCq,r′∈δCrminϑp′,ϑr′,1−k2≥min⋏p′∈δCpϑp′,⋏r′∈δCrϑr′,1−k2=minδlower¯ϑp,δlower¯ϑr,1−k2,Thus, we get⋏w∈p⊛q⊛rδlower¯ϑw≥minδlower¯ϑp,δlower¯ϑr,1−k/2. Hence δlower¯ϑ is an ϵ,ϵ∨qk-fuzzy bi-HP-ideal of LA-sm-hyp-group ϒ. The other cases can be proved in a similar way.
The following theorem shows that, ifϑ is not a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ, then the l-approx is also not a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ.Theorem 5.
Letδ be a complete and symmetric PH-order relation on a partially ordered LA-sm-hyp-group ϒ and ϑ is not an ϵ,ϵ∨qk-fuzzy left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ. Then, δlower¯ϑ is also not a left HP-ideal (resp., right HP-ideal, HP-ideal, generalized bi-HP-ideal, bi-HP-ideal, and interior HP-ideal) of LA-sm-hyp-group ϒ.Proof.
Letδ be a complete and symmetric PH-order relation on ϒ. Suppose that ϑ is not an ϵ,ϵ∨qk-fuzzy interior HP-ideal of LA-sm-hyp-group ϒ, that is,(i)
Ifp≤q implies that ϑp≱minϑq,1−k/2(ii)
⋏w∈p⊛q⊛rϑw≱minϑq,1−k/2.(iii)
Letp≤q, then δCp=δCq. Now,(17)δlower¯ϑp=⋏p′∈δCpϑp′≱⋏q′∈δNqminϑq′,1−k2=min⋏q′∈δCqϑq′,1−k2=minδlower¯ϑq,1−k2.Thus, we getδlower¯ϑp≱minδlower¯ϑq,1−k/2.(iv)
Now, letp,q,r,w in ϒ. Let(18)⋏w∈p⊛q⊛rδlower¯ϑw=⋏w∈p⊛q⊛r,⋏t∈δCwϑt=⋏t∈δCp⊛q⊛rϑt=⋏t∈δCp⊛δCq⊛δCrϑt≥⋏t∈p′⊛q′⊛r′,⋏p′∈δCp,q′∈δCq,r′∈δCrϑt=⋏p′∈δCp,q′∈δCq,r′∈δCr,⋏t∈p′⊛q′⊛r′ϑt≱⋏p′∈δCp,q′∈δCq,r′∈δCrminϑq′,1−k2≥min⋏q′∈δCqϑq′,1−k2=minδlower¯ϑq,1−k2.Thus, we get⋏w∈p⊛q⊛rδlower¯ϑw≱minδlower¯ϑq,1−k/2. Hence δlower¯ϑ is not an interior HP-ideal of LA-sm-hyp-group ϒ. The other cases can be proved in a similar way.


## 5. Rough∈,∈∨qk-Fuzzy Quasi-HP-Ideals

In this section, we first provide some results onϵ,ϵ∨qk-fuzzy quasi-HP-ideals and rough ϵ,ϵ∨qk-fuzzy quasi-HP-ideals. For a partially ordered LA-sm-hyp-group ϒ, the fuzzy subset ϒ of LA-sm-hyp-group ϒ is defined as follows:(19)ϒ:ϒ⟶0,1|p⟶ϒp≔1.Definition 10.
Letϑ,η be the fuzzy subsets of a set ϒ. The product of the fuzzy subset ϑ and η is defined as ϑ⊠ηp=⋎p∈q⊛rminϑq,ηr.Definition 11.
LetΩ be a nonempty subset of an ordered LA-sm-hyp-group ϒ. Then, the characteristic function ϑΩ of Ω is a fuzzy subset of LA-sm-hyp-group ϒ, defined as follows:(20)ϑΩp=1,ifp∈Ω,0,ifp∉Ω.Definition 12.
Letϑ be a fuzzy subset of LA-sm-hyp-group ϒ. Then, ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ if and only if(i)
p≤q⇒ϑp≥minϑq,1−k/2, for all p,q in ϒ(ii)
ϑp≥minϒ⊠ϑp,ϑ⊠ϒp,1−k/2, for all p in ϒExample 2.
Consider a setϒ=a,b,c with the following hyperoperation“⊛” defined in Table 2.
The order “≤” is ≤:a,a,b,a,b,b,b,c,c,c. Then, ϒ,⊛,≤ becomes a partially ordered LA-sm-hyp-group. Now define a fuzzy subset of LA-sm-hyp-group ϒ as ϑa=0.5 and ϑb=ϑc=0.7. If we take t=0.4 and k∈0.3,1, then ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.Table 2
Tabular form of the hyperoperation “⊛” defined in Example 2.
⊛abcaa,ccb,cbb,ccccb,cb,cb,cTheorem 6.
Letϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group ϒ. Then the set ϑt=p∈Υ|ϑp≥t is a quasi-HP-ideal of LA-sm-hyp-group ϒ, where t∈0,1−k/2.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.(i)
Letp in ϒ, p≤q and q in ϑt. Then, ϑq≥t. As ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ and p≤q, so(21)ϑp≥minϑq,1−k2≥mint,1−k2=t.This implies thatp in ϑt.(ii)
Now, we have to show thatϒ⊠ϑt∩ϑt⊠ϒ⊆ϑt . Let a in ϒ⊠ϑt∩ϑt⊠ϒ. Thus, a in ϒ⊠ϑt and a in ϑt⊠ϒ. So a in s⊛p and a in q⊛t for some s,t in ϒ and p,q in ϑt. Thus, ϑp≥t and ϑq≥t. Since(22)ϒ⊠ϑa=⋎a∈s⊛pϒs∧ϑp≥ϒs∧ϑp=ϒ∧ϑp=ϑp.
Similarly,ϑ⊠ϒa≥ϑq. Thus,(23)ϑa≥minϒ⊠ϑa,ϑ⊠ϒa,1−k2≥minϑp,ϑq,1−k2≥mint,t,1−k2=t.
Thus,a in ϑt. Hence, ϑt is a quasi-HP-ideal of LA-sm-hyp-group ϒ.Corollary 1.
Letϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group ϒ. Then the set ϑ0=p∈ϒ|ϑp>0 is a quasi-HP-ideal of LA-sm-hyp-group ϒ.Corollary 2.
Letϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group ϒ. Then, the set ϑtS=p∈ϒ|ϑp>t is a quasi-HP-ideal of LA-sm-hyp-group ϒ.Lemma 1.
A nonempty subsetQ of a partially ordered LA-sm-hyp-group ϒ is a quasi-HP-ideal of LA-sm-hyp-group ϒ if and only if the characteristic function ϑQ of Q is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.Theorem 7.
Everyϵ,ϵ∨qk-fuzzy left (right) HP-ideal ϑ of LA-sm-hyp-group ϒ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy left HP-ideal of LA-sm-hyp-group ϒ and p in ϒ. Then(24)ϒ⊠ϑp=⋎p∈q⊛rϒq∧ϑr=⋎p∈q⊛rϑr.
This implies that(25)ϒ⊠ϑp∧1−k2=⋎p∈q⊛rϑp∧1−k2=⋎p∈q⊛rϑp∧1−k2≤⋎p∈q⊛r,⋏p∈q⊛rϑp=ϑp.
Hence,ϑp≥ϒ⊠ϑp∧1−k/2≥minϒ⊠ϑp,ϑ⊠ϒp,1−k/2. Thus, ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ. The other case can be proved in a similar way.Theorem 8.
Everyϵ,ϵ∨qk-fuzzy quasi-HP-ideal of a partially ordered LA-sm-hyp-group ϒ is an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ.Proof.
Supposeϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ. Let p,q in ϒ. Now for every p in p⊛q, we have(26)ϑp≥minϒ⊠ϑp,ϑ⊠ϒp,1−k2⋎p∈p⊛qϒp∧ϑq∧⋎a∈p⊛qϑp∧ϒq∧1−k2≥ϒp∧ϑq∧ϑp∧ϒq∧1−k2≥1∧ϑq∧ϑp∧1∧1−k2=ϑq∧ϑp∧1−k2.
So⋏p∈p⊛qϑp≥minϑp,ϑq,1−k/2, for all p,q in ϒ. Thus, ϑ is an ϵ,ϵ∨qk-fuzzy LAS-sm-hyp-group of LA-sm-hyp-group ϒ.Theorem 9.
Letδ be a PH-order on a partially ordered LA-sm-hyp-group ϒ. Let ϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ. Then, δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ and δ be a PH-order relation on ϒ. We have to show that δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.(i)
For eachp,q in ϒ and p≤q, we have δCq⊆δCp. Now(27)δupper¯ϑp=⋎p′∈δCpϑp′≥⋎q′∈δCqminϑq′,1−k2=min⋎q′∈δCqϑq′,1−k2=minδupper¯ϑq,1−k2.Thus, we getδupper¯ϑp≥minδupper¯ϑq,1−k/2.(ii)
Now letw in ϒ. Let(28)δupper¯ϑw=⋎t∈δCwϑt≥⋎t∈δCwminϒ⊠ϑt,ϑ⊠ϒt,1−k2=min⋎t∈δCwϒ⊠ϑt,⋎t∈δCwϑ⊠ϒt,1−k2=minϒ⊠δupper¯ϑp,δupper¯ϑ⊠ϒp,1−k2.Thus ,δupper¯ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.Theorem 10.
Letδ be a complete and symmetric PH-order on a partially ordered LA-sm-hyp-group ϒ. Let ϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ. Then, δlower¯ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.Proof.
Letϑ be an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ and δ be a complete and symmetric PH-order relation on ϒ. We have to show that δlower¯ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.(i)
For eachp,q in ϒ and p≤q, we have δCp=δCq. Now(29)δlower¯ϑp=⋏p′∈δCpϑp′=⋏q′∈δCqminϑq′,1−k2=min⋏q′∈δCqϑq′,1−k2=minδlower¯ϑq,1−k2.Thus, we getδlower¯ϑp≥minδlower¯ϑq,1−k/2.(ii)
Now letw in ϒ. Let(30)δlower¯ϑw=⋏t∈δCwϑt≥⋏t∈δCwminϒ⊠ϑt,ϑ⊠ϒt,1−k2=min⋏t∈δCwϒ⊠ϑt,⋏t∈δCwϑ⊠ϒt,1−k2=minϒ⊠δlower¯ϑp,δlower¯ϑ⊠ϒp,1−k2.Thus,δlower¯ϑ is an ϵ,ϵ∨qk-fuzzy quasi-HP-ideal of LA-sm-hyp-group ϒ.



---

*Source: 9917947-2021-05-07.xml*