# Certain Concepts ofQ-Hesitant Fuzzy Ideals

Authors: Lubna Abdul Aziz Alleheb; Kholood Mohammad Alsager
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The hesitant fuzzy set model has attracted the interest of scholars in various fields. The striking framework of hesitant fuzzy sets is keen to provide a larger domain of preference for fuzzy information modeling of deployment membership. Starting from the hybrid properties of hesitant fuzzy ideals (HFI), this paper constructs a new generalized hybrid structureQ-HFI. The concept of Q-hesitant fuzzy exchange ideal in BCK-algebra is considered. Lastly, Q-hesitant fuzzy exchange ideal features are described.


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## Body

## 1. Introduction

When dealing with information on all aspects of uncertainty, nonclassical logic always makes use of classical logic. Nonclassical logic is a useful tool in computer science because it deals with fuzzy information and uncertainty. In the literature, the study ofBCK/BCI-algebras was first proposed by Imai and Iséki [1] in 1966 and such algebras can be regarded as a generalization of propositional logic. The study BCK/BCI-algebras have been developed by many people and have been extended to the fuzzy setting. After the introduction of fuzzy sets introduced by Zadeh [2], there have been many generalizations of this fundamental concept. In 2010, Torra [3] considered hesitant fuzzy sets. The hesitant fuzzy set model is useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision-makers.Algebraic structures provide sufficient motivation for researchers to examine various concepts and stem from the broader field of abstract algebra blur set frame. In 2011, Xia and Xu [4] described hesitant fuzzy information aggregation techniques, and this concept was applied to BCK/BCI-algebras, EQ-algebras, residuated lattices, MTL-algebras, and K-algebras [5–9]. Jun and Ahn [6] investigated the concept of hesitant fuzzy subalgebras and HFIs of BCK/BCI-algebras. In 2018, Alshehri et al. [10] put forward the concept of new types of HFIs in BCK-algebras. As a continuation of this study, we describe certain concepts, including Q-HFIs andQ-hesitant fuzzy commutative ideals in BCK-algebras.


## 2. Basic Notions

A setU with a constant element 0 and a binary operation ∗ is said to be a BCK-algebra [1] if it satisfies the axioms:For all⊓,⊑,⇕∈U,
(1)BCK−1⊑∗⊓∗⊓∗m∗⇕∗⊑=0,BCK−2⊓∗⊓∗⊓∗⊑=0,BCK−3⊓∗⊓=0,BCK−40∗⊓=0,BCK−5⊓∗⊑=0,⊑∗⊓=0implythat⊓=⊑.In aBCK-algebra U, we can define the relation ≤ by ⊓≤⊑ if and only if ⊓∗⊑=0.Then,U;≤ is a partially ordered set with the least element 0. In any BCK-algebra U, the following properties hold:
(2)⊓∗⊑∗⇕=⊓∗⇕m∗⊑,⊓∗⊑≤⊓,⊓∗0=⊓,⊓∗⇕∗⊑∗⇕≤⊓∗⊑,⊓∗⊓∗⊓∗⊑=⊓∗⊑.⊓≤⊑ implies ⊓∗⇕≤⊑∗⇕ and ⇕∗⊑≤⇕∗⊓ for all ⊓,⊑,⇕∈U.LetU be a BCK-algebra and let A be a nonempty subset of U. Then, A is called an ideal of U [11] if it satisfies the following:
(1)
I∞0∈A(2)
Iϵ⊑∈A and ⊓∗⊑∈A imply that ⊓∈A for all ⊓,⊑∈UA subsetA of a BCK-algebra U is called a commutative ideal [12] of U if it satisfies the following:
(1)
I∞0∈A(2)
C⊓∗⊑∗⇕∈A and ⇕∈A imply that ⊓∗⊑∗⊑∗⊓∈A for all ⊓,⊑,⇕∈UA fuzzy set inU is said to be a fuzzy ideal of U if it satisfies the following:
(1)
F∞μ0≥μ⊓ for all ⊓∈U(2)
Fϵμ⊓≥minμ⊓∗⊑,μ⊑ for all ⊓,⊑∈ULetU be a reference set and A be a nonempty subset of U, a hesitant fuzzy set.HU=⊓,⊓U:⊓∈U on U [3] satisfying the following condition:
(3)U⊓=ϕforall⊓∉A,is called a hesitant fuzzy set related to U (briefly, A-hesitant fuzzy set) on U and is represented by HA=⊓,A⊓:⊓∈U, where A is a mapping from U to ✓0,1 with A⊓=ϕ for all ⊓∉A.LetU be a reference set and A be a nonempty subset of U, an A-hesitant fuzzy set HA=⊓,⊓A:⊓∈U of U is called a HFI [6] of U related to A (briefly, A-HFI of U if it satisfies the following:
(1)
H∞A0≥A⊓ for all ⊓∈U(2)
HϵA⊓≥minA⊓∗⊑,A⊑ for all u,v∈UGiven a nonempty subsetA of U, an A-hesitant fuzzy setHA=⊓,A:⊓∈U of U is called a hesitant fuzzy commutative ideal [10] of U related to A (briefly, A-hesitant fuzzy commutative ideal of U) if it satisfiesH∞A0≥A⊓ for all ⊓∈UHA⊓∗⊑∗⊑∗⊓≥minA⊓∗⊑∗⇕,A⇕ for all ⊓,⊑,⇕∈ULetU be a nonempty finite universe and Q be a nonempty set. A Q-hesitant fuzzy set AQ is a set given by
(4)AQ=⊓,∐,AQ⊓,∐:⊓∈U,∐∈Q,where AQ:U×Q⟶0,1. The function AQ⊓,∐ is called the membership function of Q-hesitant fuzzy set, and the set of all Q-hesitant fuzzy set over U×Q will be denoted by QHFU×Q.LetHU be a hesitant fuzzy set of a BCK-algebra U. The set
(5)HU∫=⊓∈UHU⊓≥∫,where ∫;∈✓0,1 is called a hesitant fuzzy ∫-level set of HU.Theorem 1 (see [6]).
For a subalgebra,A of a BCK-algebra U, every A-HFI is an A-hesitant fuzzy subalgebra.Proposition 2 (see [13]).
InBCK-algebra U the following conditions hold, for all ⊓,⊑,⇕∈U,
(6)⊓∗⇕∗⇕∗⊑∗⇕≤⊓∗⊑∗⇕,⊓∗⇕∗⊓∗⊓∗⇕=⊓∗⇕∗⇕,⊓∗⊑∗⊑∗⊓∗⊑∗⊓∗⊑∗⊑∗⊓≤⊓∗⊑.


## 3.Q-Hesitant Fuzzy Ideals

Definition 3.
LetU be a nonempty finite universe, Q be a nonempty set and A be the subset of U, a Q-hesitant fuzzy ideal HAQofU×Q (briefly: QHF-ideal) if it satisfies the following assertion:
(1)
Q∞HAQ′,∐≥AQ⊓,∐∀⊓∈U,∐∈Q(2)
QϵHAQ⊓,∐≥minAQ⊓∗⊑,∐,AQ⊑,∐,∀⊓,⊑∈U,∐∈QExample 1 DenoteU=0,a,b,c.
The binary operation∗ on U is given by Cayley (Table 1).
For a subsetA=0,a,b. Let HAQ=⊓,∐,AQ⊓,∐⊓∈U,∐∈Q be a QHFS of U×Q defined by
(7)AQ:U×Q↦0,12,if⊓,∐=′,∐,0,13,if⊓,∐=a,b,c,∐.
Then,HAQ is a QHF-ideal of U×Q.Table 1∗0abc00000aa0a0bbb00ccba0Proposition 4.
LetA be a subset of U and HAQ be a Q-HFI of U×Q. Then, the following assertions are valid:
(1)
⊓≤⊑⟹AQ⊓,∐≥AQ⊑,∐ for all ⊓,⊑∈U,∐∈Q(2)
⊓∗⊑≤⇕⟹AQ⊓,∐≥minAQ⊑,∐,AQ⇕,∐ for all ⊓,⊑,⇕∈U,∐∈QProof.
(1)
Suppose⊓≤⊑ implies ⊓∗⊑=0∈A (for all ⊓,⊑∈A) and so
(8)AQ⊑,∐=minAQ′,∐,AQ⊑,∐=minAQ⊓∗⊑,∐,AQ⊑,∐≤AQ⊓,∐,by QϵH(2)
Suppose⊓∗⊑≤⇕ implies ⊓∗⊑∗⇕=0∈A (for all ⊓,⊑,⇕∈U) so
(9)AQ⇕,∐=minAQ′,∐,AQ⇕,∐=minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐≤AQ⊓∗⊑,∐
It follows that(10)minAQ⊑,∐,AQ⇕,∐≤minAQ⊓∗⊑,∐,AQ⊑,∐≤AQ⊓,∐.Proposition 5.
Every AQHF-ideal of U×Q satisfies the following condition:
(1)
UQ⊑,∐≤UQ⊓,∐ with ⊓≤⊑ for all ⊓,⊑∈U,∐∈Q(2)
minUQ⊑,∐,UQ⇕,∐≤UQ⊓,∐ with ⊓∗⊑≤⇕ for all ⊓,⊑,⇕∈U,∐∈QTheorem 6.
IfHAQ a Q-HFI of U×Q, then for any ⊓,⊣∞,⊣ϵ,⋯,⊣\∈U, and
(11)⋯⊓∗⊣∞∗⊣ϵ∗⋯∗⊣\=0⟹AQ⊓,∐≥minAQ⊣∞,∐,AQ⊣ϵ,∐,⋯,AQ⊣\,∐.Theorem 7.
LetHAQ be a Q-HFI of U×Q. Then, the following are equivalent:
(i)
AQ⊓∗⊑∗⊑,∐≤AQ⊓∗⊑,∐ for all ⊓,⊑∈U,∐∈Q(ii)
AQ⊓∗⊑∗⇕,∐≤AQ⊓∗⇕∗⊑∗⇕,∐ for all ⊓,⊑,⇕∈U,∐∈QProof.
i⟹ii Suppose condition  is valid. Since
(12)⊓∗⊑∗⇕∗⇕∗⇕=⊓∗⇕∗⊑∗⇕∗⇕≤⊓∗⊑∗⇕.
Applying, by Proposition2 and ⊣, we have
(13)AQ⊓∗⊑∗⇕,∐≤AQ⊓∗⊑∗⇕∗⇕∗⇕,∐≤AQ⊓∗⊑∗⇕∗⇕,∐=AQ⊓∗⇕∗⊑∗⇕,∐.
Hence, conditionii holds
ii⟹i Suppose condition ii is valid. If we put ⇕=⊑ in ii then
(14)AQ⊓∗⊑∗⊑,∐≤AQ⊓∗⊑∗⊑∗⊑,∐=AQ⊓∗⊑∗′,∐=AQ⊓∗⊑,∐,
hence, the conditioni holds.
The proof is complete.Theorem 8.
LetHUQ be a Q-HFI of U×Q, then the set
(15)H∫=⊓∈U,∐∈QUQ∫,∐≤UQ⊓,∐,
is an ideal ofU×Q for all ∫∈U.Proof.
Let⊓,⊑∈U,∐∈Q be such that ⊓∗⊑,∐∈H∫ and ⊑,∐∈H∫. Then,
(16)UQ∫,∐≤UQ⊓∗⊑,∐,UQ∫,∐≤UQ⊑,∐.
It follows fromQ∞H,QϵH that
(17)UQ∫,∐≤minUQ⊓∗⊑,∐,UQ⊑,∐≤UQ⊓,∐≤UQ′,∐.So that′,∐∈H∫ and ⊓,∐∈H∫, therefore, H∫ is an ideal of U×Q for all ∫∈U.Theorem 9.
Suppose thatHAQ is a Q-hesitant fuzzy set of U×Q, where A is a nonempty subset of U. Then, the following are equivalent:
(i)
HAQ is a Q-HFI of U×Q(ii)
For any∫∈✓0,1, the set HAQ∫=⊓∈U:UQ⊓,∐≥∫ is an ideal of U×QProof.
⇒ Assume that HAQ is a QHF-ideal of U×Q. Let u,v∈U and s∈✓0,1 be such that ⊓∗⊑,∐∈HAQ∫ and ⊑,∐∈HAQ∫.Then,
(18)AQ⊓∗⊑,∐≥∫,AQ⊑,∐≥∫.
It follows that(19)AQ′,∐≥AQ⊓,∐≥minAQ⊓∗⊑,∐,AQ⊑,∐≥∫.
Hence,′,∐∈HAQ∫ and ⊓,∐∈HAQ∫
ThereforeHAQ∫ is an ideal of U×Q
⟸ Suppose that HAQ∫ is an ideal of U×Q. For any ⊓∈U,∐∈Q, let AQ⊓,∐=∫⊓,∐
Then⊓,∐∈HAQ∫⊓,∐. Since HAQ∫⊓,∐ is an ideal of U×Q. we have
(20)′,∐∈HAQ∫⊓,∐andsoAQ⊓,∐=∫⊓,∐≤AQ′,∐
LetAQ⊓∗⊑=∫⊓∗⊑,∐ and AQ⊑,∐∫⊑,∐. Then ⊓∗⊑,∐∈HAQ∫ and ⊑,∐∈HAQ∫ such that ∫=min∫⊓∗⊑,∐,∫⊑,∐, which imply that ⊓,∐∈HAQ∫. Thus,
(21)AQ⊓,∐≥∫=minAQ⊓∗⊑,∐,AQ⊑,∐.
Therefore,HAQ is a QHF-ideal of U×Q.


## 4.Q-Hesitant Fuzzy Commutative Ideals

Definition 10.
LetU be a universal set and Q be a nonempty set. A Q-hesitant fuzzy commutative ideal (Q-HFCI) of U×Q if it satisfies the following assertion:
(1)
Q∞HAQ′,∐≥AQ⊓,∐∀⊓∈U,∐∈Q(2)
QHAQ⊓∗⊑∗⊑∗⊓,∐≥minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐∀⊓,⊑,⇕∈U,∐∈QExample 1.
LetU=0,a,b,c be a set with the binary operation ∗ which is defined in Cayley (Table 2).
Let⊔′,⊔∞,⊔∈∈√0,1 such that ⊔′≻⊔∞≻⊔∈. We define a QHFS of U×Q as follows:
(22)UQ:U×Q⟶√0,1,⊓,∐↦t0,if⊓,∐=′,∐,t1,if⊓,∐=a,∐,t2,if⊓,∐=b,c,∐,where t0=0,1≻t1=0.2,0.6≽t2=0.2,0.3. By direct calculations, one can see that HQ is QHFC-ideal of U×Q.Table 2∗0abc00000aa00abba0bcccc0Theorem 11.
EveryQ-hesitant fuzzy CI is a Q-hesitant fuzzy ideal of U×Q.Proof.
Assume thatHAQ is a QHFC-ideal of a BCK-algebra U, for any ⊓,⇕∈U,∐∈Q. We have
(23)minAQ⊓∗⇕,∐,AQ⇕,∐=minAQ⊓∗′∗⇕,∐,AQ⇕,∐≤AQ⊓∗′∗′∗⊓,∐=AQ⊓,∐.
Therefore,HAQ is a QHF-ideal in U×Q.The following example shows that the converse of theorem 6 is not true.Example 2.
LetU=0,a,b,c,d be a set with the Cayley (Table 3).
Lete0,e1,e2∈✓0,1 such that e0≻e1≻e2. We define a mapping
(24)UQ~U×Q⟶✓0,1,⊓,∐↦e0,if⊓,∐=′,∐,e1,if⊓,∐=⊣,∐,e2,if⊓,∐=b,c,d,∐,
wheree0=0,1≻e1=0.4,0.8±e2=0.5,0.6. It is routine to verify that HUQ is QHF-ideal of U×Q. But it is not a QHFC-ideal of U×Q. Since
(25)UQb∗c∗c∗b,∐≱minUQb∗c∗0,∐,UQ′,∐.Table 3∗0abcd000000aa0a00bbb000cccc00ddddc0Theorem 12.
LetHUQ be a Q-HFI of a BCK-algebra U. Then, HUQ is a Q-hesitant fuzzy CI of U×Q if and only if it satisfies the following condition:
(26)UQ⊓∗⊑∗⊑∗⊓,∐≥UQ⊓∗⊑,∐∀⊓,⊑∈U,∐∈Q.Proof.
Assume thatHUQ is QHFC-ideal. Taking m=0 in QCH and using Q∞H. Also, we use ⊓∗0=⊓.
(27)UQ⊓∗⊑∗⊑∗⊓,∐≥minUQ⊓∗⊑∗′,∐,UQ′,∐=UQ⊓∗⊑.
Conversely, LetHUQ As HUQ be a QHF-ideal of U×Q satisfying condition (1).
Then,(28)UQ⊓∗⊑,∐≥minUQ⊓∗⊑∗⇕,∐,UQ⇕,∐,∀⊓,⊑,⇕∈U,∐∈Q,combining (1) and (2), then we obtain QcH.
The proof is complete.Lemma 13.
AnyQ-HFI of a BCK-algebra U satisfies
(29)⊓∗⊑≤⇕⟹UQ⊓,∐≥minUQ⊑,∐,UQ⇕,∐.Proof.
Assume that⊓∗⊑≤⇕ holds. Then,
(30)UQ⊓∗⊑,∐≥minUQ⊓∗⊑∗⇕,∐,UQ⇕,∐=minUQ′,∐,UQ⇕,∐=UQ⇕,∐.
It follows that(31)UQ⊓,∐≥minUQ⊓∗⊑,∐,UQ⊑,∐≥minUQ⊑,∐,UQ⇕,∐.
The proof is complete.Theorem 14.
For any commutative in aBCK-algebra U. Every Q-HFI is commutative.Proof.
LetHUQ be a QHF-ideal of a commutative BCK-algebra U. It is sufficient to show that HUQ satisfies condition QH .Let ⊓,⊑,⇕∈U. Then,
(32)⊓∗⊑∗⊑∗⊓∗⊓∗⊑∗⇕∗⇕=⊓∗⊑∗⊑∗⊓∗⇕∗⊓∗⊑∗⇕≤⊓∗⊑∗⊑∗⊓∗⊓∗⊑=⊓∗⊓∗⊑∗⊑∗⊑∗⊓=0.
That is,(33)⊓∗⊑∗⊑∗⊓∗⊓∗⊑∗⇕≤⇕.
By Lemma13, we have
(34)UQ⊓∗⊑∗⊑∗⊓,∐≥minUQ⊓∗⊑∗⇕,∐,UQ⇕,∐.
Thus,QcH holds. Therefore, HUQ is a Q-HFCI.Definition 15.
LetHUQ be a Q-hesitant CI of a BCK-algebra U, for ∫∈✓0,1, the set HUQ∫=⊓∈U,∐∈QHUQ′,∐≥∫ of a CI is called Q-hesitant ∫-level CI of HUQ.Theorem 16.
InBCK-algebra U, any CI of can be realized as Q-hesitant ∫-level CI of some Q-HFCI of U×Q.Proof.
LetC be a CI of BCK-algebra U and let HAQ be a Q-hesitant fuzzy set of U×Q defined by
(35)AQ⊓,∐=∫,if⊓∈C,0,ifotherwise,
where∫∈✓0,1. Let ⊓,⊑,⇕∈U.
If⊓∗⊑∗⇕∈C and ⇕∈C, then ⊓∗⊑∗⊑∗⊓∈C Thus,
(36)AQ=AQ⇕,∐=AQ⊓∗⊑∗⊑∗⊓,∐=∫,and so
(37)AQ⊓∗⊑∗⊑∗⊓,∐≥minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐,(i)
If⊓∗⊑∗⇕∉C and ⇕∉C, then AQ⊓∗⊑∗⇕,∐=AQ⇕,∐=0
Hence,(38)AQ⊓∗⊑∗⊑∗⊓,∐≥minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐.(ii)
If exactly one of⊓∗⊑∗∐ and ⇕ belongs to C, then exactly one of AQ⊓∗⊑∗⇕,∐ and AQ⇕,∐ is equal to zero. So,(39)AQ⊓∗⊑∗⊑∗⊓,∐≥minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐
The results above show(40)AQ⊓∗⊑∗⊑∗⊓,∐≥minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐forall⊓,⊑,⇕∈U,∐∈Q.
It is clear thatAQ′,∐≥AQ⊓,∐ for all ⊓∈U. Therefore, HAQ is a QHFC-ideal of U×Q. Obviously, HAQ∫=C.Theorem 17.
IfHAQ a QHFC-ideal of a BCK-algebra U. Then, two-level CI HAQ∫∞ and HAQ∫∈ where ∫∞<∫∈ of HAQ are equal if and only if there is no u∈U such that ∫∞≤AQ⊓,∐<∫∈.Proof.
LetHAQ∫∞=HAQ∫∈. If there exists ⊓∈U such that ∫∞≤AQ⊓,∐<∫∈, then HAQ∫∈⊆HAQ∫∞. This is impossible. Conversely, assume that there is no ⊓∉U such that ∫∞≤AQ⊓,∐<∫∈.
∫∞<∫∈ implies HAQ∫∈⊆HAQ∫∞ If ⊓,∐∈HAQ∫∞, then AQ⊓,∐≥∫∞ and so AQ⊓,∐≥∫∈, because AQ⊓,∐≮∫∈.
Hence,⊓,∐∈HAQ∫∈ which says that HAQ∫∞⊆HAQ∫∈.
Thus,HAQ∫∞=HAQ∫∈.
This completes the proof.LetAQ be a Q-hesitant fuzzy set in U and let ImAQ denote the image of AQ.Theorem 18.
LetU be a BCK-algebra and HAQ a Q-HFCI of U×Q. If ImAQ=∫∞,∫∈,⋯,∫\, where ∫∞⊲∫∈⊲⋯∫\, then the family of CIs HAQ∫≻≻=∞,∈⋯,\ constitutes all the level CIs of AQ.Proof.
Let∫∈✓0,1 and ∫∉ImhAQ. If ∫<∫∞, then AQ∫∞⊆AQ∫. Since AQ∫∞=U, we have AQ∫=U and AQ∫=AQ∫∞.
If∫><∫<∫>+∞1≤>≤\−∞, then there is no ⊓∈U such that ∫≤AQ⊓,∐<∫>+∞. From above theorem 10, it follows that AQ∫=AQ∫>+∞. This shows that for any ∫∈✓0,1 with ∫≤AQ′,∐, the level CI AQ∫ is in AQ∫>:1≤>≤\.Lemma 19.
Given aBCK-algebra U and HAQ a QHFC-ideal over U×Q. If ∫∞ and ∫∈ belong to ImAQ such that AQ∫∞=AQ∫∈, then ∫∞=∫∈.Proof.
Assume that∫∞≠∫∈, say ∫∞<∫∈. Then, there is ⊓∈U,∐∈Q such that AQ⊓,∐=∫∞<∫∈, and so ⊓,∐∈AQ∫∞ and ⊓,∐∉AQ∫∈. Thus, AQ∫∞≠AQ∫∈, which is a contradiction to our fact. This completes the proof.


## 5.Q-Hesitant Fuzzy Characteristic CIs

A mapping:U⟶V of a BCK-algebra is called a homomorphism if satisfying the identity ⊓∗⊑=⊓∗⊑ for all ⊓,⊑∈U. Throughout, AutU will denote the BCK-algebra of automorphisms of U.Definition 20.
Let:U⟶V be a homomorphism of BCK-algebras. For any QHFC-ideal AQ of V, we define a new QHFC-ideal AQ in U by
(41)AQ⊓,∐=AQ⊓,∐forall⊓∈U,∐∈Q.Theorem 21.
Let:U⟶V be a homomorphism of BCK-algebra U. If AQ is a QHFC-ideal of V, then AQ is a QHFC-ideal of U.Proof.
Let⊓∈U,∐∈Q(42)AQ⊓,∐=AQ⊓,∐≤AQ′,∐=AQ′,∐=AQ′,∐.
Let⊓,⊑,⇕∈U,∐∈Q(43)minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐=minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐=minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐=minAQ⊓∗⊑∗⇕,∐,AQ⇕,∐≤AQ⊓∗⊑∗⊑∗⊓,∐=AQ⊓∗⊑∗⊑∗⊓,∐.Definition 22.
A CIC of a BCK-algebra U is called a characteristic CI (CCI) of U if C=C for all ∈AutU.Definition 23.
AQ-HFCI of a BCK-algebra U is called a Q-hesitant fuzzy CCI of U×Q if
(44)AQ⊓,∐=AQ⊓,∐forall⊓∈U,∐∈Qand∀∈AutU.Theorem 24.
LetHAQ be a Q-hesitant fuzzy characteristic CI of U×Q. Then, each ∫-level CI of AQ is a characteristic commutative ideal of U×Q.Proof.
Assume∫∈ImAQ,∈AutU and ⊓,∐∈AQ∫. Since HAQ is a Q-hesitant fuzzy characteristic commutative ideal of U×Q, we have AQ⊓,∐=AQ⊓,∐≥∫.
It follows that⊓,∐∈AQ∫ and hence AQ∫⊆AQ∫.
To show the reverse inclusion, let⊓,∐∈AQ∫ and let ⊑∈U be such that ⊑=⊓. Then, AQ⊑,∐=AQ⊑,∐=AQ⊓,∐≥∫ whence ⊑,∐∈AQ∫. It follows that ⊓=⊑∈AQ∫ so that AQ∫⊆AQ∫. Thus, AQ∫,∫∈ImAQ is a CCI of U×Q.
The proof of the following lemma is obvious, and we omit the proof.Lemma 25.
LetHAQ be a Q-HFCI of U×Q and let ⊓∈U. Then, AQ⊓,∐=∫ if and only if ⊓,∐∈AQ∫∞ and ⊓,∐∉AQ∫∈, for all ∫∈≥∫∞.
Now, we consider the inverse of Theorem24Theorem 26.
LetHAQ be a Q-HFCI of U×Q. If each level CI of AQ is a CCI of U, then HAQ is a Q-hesitant fuzzy characteristic commutative ideal of U×Q.Proof.
Let⊓∈U,∐∈Q,∈AutU and AQ⊓,∐=∫∞. Then, ⊓,∐∈AQ∫∞ and ⊓,∐∉AQ∫∈ for all ∫∈≥∫∞, by Lemma 25 Since AQ∫∞=AQ∫∞ by hypothesis, we have ⊓,∐∈AQ∫∞ and hence AQ⊓,∐≥∫∞. Let ∫∈=AQ⊓,∐. If possible, let ∫∈≥∫∞.
Then,⊓,∐∈AQ∫∈=AQ∫∈. Since { is one to one, it follows that ⊓,∐∈AQ∫∈, which is a contradiction. Hence, AQ⊓,∐=∫∞=AQ⊓,∐. It follows that HAQ is a Q-hesitant fuzzy CCI of U×Q. This completes the proof.


## 6. Conclusions

A new concept of HFI is considered by applying a two-dimensional membership function, namely,Q-HFI. Several properties and theorems of Q-HFI are proved. In this regard, we propose the concept of Q-HFCI in BCK-algebra and prove some related properties. We have considered the features of Q-HFCI. We study some feature properties related to Q-HFCI. Our future research is to find ways to apply Q-HFI to a wide range of logical algebraic systems, such as pseudo-BCI-algebras [14, 15]. For other notions, the readers are suggested to see [16–28].



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*Source: 7099148-2022-07-22.xml*