Bipolar Fuzzy Shift UP-Filters

Thiti Gaketem; Pannawit Khamrot; Pongpun Julatha; Rukchart Prasertpong; Aiyared Iampan

doi:10.5391/IJFIS.2023.23.3.294

Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 294-302

Published online September 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.3.294

Bipolar Fuzzy Shift UP-Filters

Thiti Gaketem¹, Pannawit Khamrot², Pongpun Julatha³, Rukchart Prasertpong⁴ , and Aiyared Iampan¹

¹Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Phayao, Thailand
²Faculty of Science and Agricultural Technology, Rajamangala University of Technology, Lanna Phitsanulok, Thailand
³Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand
⁴Division of Mathematics and Statistics, Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Nakhon Sawan, Thailand

Correspondence to :
Aiyared Iampan (aiyared.ia@up.ac.th)

Received: March 8, 2023; Revised: April 22, 2023; Accepted: June 2, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

In this paper, we present the concept of bipolar fuzzy shift UP-filters of UP-algebras by applying the concept of shift UP-filters to bipolar fuzzy sets and investigating their essential properties. In UP-algebras, we found that every bipolar fuzzy strong UP-ideal is a bipolar fuzzy shift UP-filter, and every bipolar fuzzy shift UP-filter is a bipolar fuzzy UP-filter. An important relationship between bipolar fuzzy shift UP-filters and their bipolar fuzzy characteristic functions is presented. The relations between bipolar fuzzy shift UP-filters and other shift UP-filters (fuzzy and neutrosophic shift UP-filters) are given. Finally, characterizations of bipolar fuzzy shift UP-filters are investigated in terms of level, fuzzy, and neutrosophic sets.

Keywords: UP-algebra, Shift UP-filter, Fuzzy shift UP-filter, Bipolar fuzzy shift UP-filter, Neutrosophic shift UP-filter

1. Introduction and Preliminaries

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Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

BCK-algebras [1], BCI-algebras [2], BCH-algebras [3], KU-algebras [4], PSRU-algebras [5], UP-algebras [6], and other algebraic structures have been the subject of several academic projects. They are inextricably linked to logic. For example, BCI-algebras were introduced by Iséki [1] in 1966 and have linkages with BCI-logic, which is the BCI-system in combinatory logic and has applications in functional programming. BCK- and BCI-algebras are two classes of logical algebras. Imai and Iséki [2] introduced them in 1966, and many scholars have examined them. BCK-algebras are a proper subclass of BCI-algebras, as is well known. Prabpayak and Leerawat [4] established the notion of KU-algebras in 2009. KU-algebras were extended to UP-algebras by Iampan [6] in 2017.

The concept of a fuzzy set in a nonempty set was first considered by Zadeh [7]. The fuzzy set theories developed by Zadeh and others have shown many mathematical structures to solve uncertainties. The concept of fuzzy set was developed in fuzzy set theory, for example, intuitionistic fuzzy sets, Pythagorean fuzzy sets, interval-valued fuzzy sets, neutrosophic sets, and vague sets. The fuzzy set was extended by the bipolar-valued fuzzy set whose membership degree range is [−1, 0] ∪ [0, 1], which was of interest to Zhang in 1994 [8]. In 2008, Jun and Song [9] used the concept of bipolar fuzzy sets in BCH-algebras. Many researchers have studied bipolar fuzzy sets in algebraic structures; in 2011, Lee and Jun [10] studied bipolar fuzzy a-ideals of BCI-algebras, and in 2012, Jun et al. [11] studied bipolar fuzzy CI-algebras. In 2021, Muhiuddin and Al-Kadi [12] studied bipolar fuzzy implicative ideals of BCK-algebras. Gaketem and Khamrot [13] introduced the concept of bipolar fuzzy weakly interior ideals of semigroups. The relationships between bipolar fuzzy weakly interior ideals and bipolar fuzzy left (right) ideals and between bipolar fuzzy weakly interior ideals and bipolar fuzzy interior ideals have also been discussed. Gaketem et al. [14] introduced the concept of bipolar fuzzy implicative UP-filters (BFIUPFs) in UP-algebras.

To extend the concept of shift UP-filters of UP-algebras introduced by Jun and Iampan [15] in 2019 and bipolar fuzzy set theory applied to UP-algebras by Kawila et al. [16] in 2018, in this paper, we present the concept of bipolar fuzzy shift UP-filters of UP-algebras by applying the concept of shift UP-filters to bipolar fuzzy sets and investigate its essential properties. The relations between bipolar fuzzy shift UP-filters and other shift UP-filters (fuzzy and neutrosophic shift UP-filters) are given. Finally, characterizations of bipolar fuzzy shift UP-filters are investigated in terms of level, fuzzy, and neutrosophic sets.

2. Preliminaries

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Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

First, we start with the definition of UP-algebras as follows:

Definition 2.1 ([6])

A UP-algebra is defined as ( ) of type (2, 0), where is a nonempty set, 0 is a fixed element of , and ⊠ is a binary operation on if it satisfies the following four conditions:

(1)(for all t,v,x,∈T) ((v⌧x)⌧((t⌧v)⌧(t⌧x))=0),

(2)(for all t∈T)(0⌧t=t),

(3)(for all t∈T)(t⌧0=0),

(4)(for all t,v∈T) (t⌧v=0,v⌧t=0⇒t=v).

According to [6], UP-algebras are a generalization of the concept of KU-algebras (see [4]). Unless otherwise indicated, let denote the UP-algebra ( ). The binary relation ≤ on is defined as follows:

(5)(for all t,v∈T)(t≤v⇔t⌧v=0)

and the following statements are true (see [6, 17]):

(6)(for all t∈T)(t≤t),

(7)(for all t,v,x∈T)((t≤v,v≤x)⇒t≤x),

(8)(for all t,v,x∈T)(t≤v⇒x⌧t≤x⌧v,)

(9)(for all t,v,x∈T)(t≤v⇒v⌧x≤t⌧x),(for all t,v,x∈T)

(10)(t≤v⌧t, in particular,v⌧x≤t⌧(v⌧x)),

(11)(for all t,v∈T)(v⌧t≤t⇔t=v⌧t),

(12)(for all t,v∈T)(t≤v⌧v),

(13)(for all a,t,v,x∈T)(t⌧(v⌧x)≤t⌧((a⌧v)⌧(a⌧x))),

(14)(for all a,t,v,x∈T)(((a⌧t)⌧(a⌧v))⌧x≤(t⌧v)⌧x),

(15)(for all t,v,x∈T)((t⌧v)⌧x≤v⌧x),

(16)(for all t,v,x∈T)(t≤v⇒t≤x⌧v),

(17)(for all t,v,x∈T)((t⌧v)⌧x≤t⌧(v⌧x)),

(18)(for all a,t,v,x∈T)((t⌧v)⌧x≤v⌧(a⌧x)).

Definition 2.2 ([15])

A nonempty subset of is called a shift UP-filter (SUPF) of if

(19)0∈G,

(20)(for all t,v,x∈T)(t⌧(v⌧x)∈G,t∈G⇒((x⌧v)⌧v)⌧x∈G).

Theorem 2.3 ([15])

If is a family of SUPFs of , then is an SUPF of .

A fuzzy set (FS) α in a nonempty set is a function from to the closed unit interval [0, 1] of real numbers, i.e., . A bipolar fuzzy set (BFS) ϖ [8] in a nonempty set is an object of the form

ϖ:={(t,ϖN(t),ϖP(t))∣t∈G},

where and . We use the symbol ϖ = (ϖ_N,ϖ_P ) for the for simplicity. The concept of BFSs is an extension of that of FSs. In 2018, Kawila et al. [16] introduced the concepts of a bipolar fuzzy UP-filter and bipolar fuzzy strong UP-ideal of as follows.

Definition 2.4 ([16])

A BFS ϖ = (ϖ_N,ϖ_P ) in is called

(1) a bipolar fuzzy UP-filter (BFUPF) of if the following four conditions hold:

(21)(for all t∈T)(ϖN(0)≤ϖN(t)),(22)(for all t∈T)(ϖP(0)≥ϖP(t)),(23)(for all t,v∈T)(ϖN(v)≤max{ϖN(t⌧v),ϖN(t)}),(24)(for all t,v∈T)(ϖP(v)≥min{ϖP(t⌧v),ϖP(t)}),
(2) a bipolar fuzzy strong UP-ideal (BFSUPI) of if the conditions (21), (22), and the following hold :

(25)(for all t,v,x∈T)(ϖN(t)≤max{ϖN((x⌧v)⌧(x⌧t)),ϖN(v)}),

(26)(for all t,v,x∈T)(ϖP(t)≥min{ϖP((x⌧v)⌧(x⌧t)),ϖP(v)}).

Theorem 2.5 ([16])

A BFS ϖ = (ϖ_N,ϖ_P ) in is a BFSUPI of if and only if ϖ is constant.

Example 2.6

Consider an UP-algebra with the following Cayley table:

⊠	q	t	v	x
0	w	t	v	x
q	0	t	v	x
t	0	0	v	x
v	0	0	0	x
x	0	0	w	0

(1) Define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

0 q t v x
ϖN −0.8 −0.6 −0.6 −0.1 −0.1
ϖP 0.7 0.5 0.5 0.3 0.3

Then, ϖ is a BFUPF of but not a BFSUPI.
(2) Define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

0 q t v x
ϖN −0.8 −0.8 −0.8 −0.8 −0.8
ϖP 0.6 0.6 0.6 0.6 0.6

Then, ϖ is a BFSUPI.

3. Bipolar Fuzzy Shift UP-Filters

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Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

In this section, we introduce the concept of bipolar fuzzy shift UP-filters of UP-algebras and discuss its properties. Moreover, one characterization of a shift UP-filter is investigated using the bipolar fuzzy characteristic function.

Definition 3.1

A BFS ϖ = (ϖ_N,ϖ_P ) in is called a bipolar fuzzy shift UP-filter (BFSUPF) of if the conditions (21), (22), and the following hold:

(27)(for all t,v,x∈T)(ϖN(((x⌧v)⌧v)⌧x)≤max{ϖN(t⌧(v⌧x)),ϖN(t)}),

(28)(for all t,v,x∈T)(ϖP(((x⌧v)⌧v)⌧x)≥min{ϖP(t⌧(v⌧x)),ϖP(t)}).

Example 3.2

Consider a UP-algebra with the following Cayley table:

⊠	q	t	v	x
0	q	t	v	x
q	0	t	v	x
t	0	0	t	x
v	0	0	0	x
x	q	t	v	0

Define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

	0	q	t	v	x
ϖN	−0.4	−0.4	−0.3	−0.3	−0.3
ϖP	0.7	0.7	0.5	0.5	0.5

Then, ϖ is a BFSUPF of .

Theorem 3.3

Every BFSUPI of is a BFSUPF of .

Proof

Suppose that ϖ = (ϖ_N,ϖ_P ) is a BFSUPI of . By Theorem 2.5, we get that ϖ is constant. Then, ϖ_N(0) = ϖ_N(t),ϖ_P (0) = ϖ_P (t),ϖ_N(((x⊠v)⊠v)⊠x) = max{ϖ_N(t⊠(v⊠x)),ϖ_N(t)}, and ϖ_P (((x⊠v)⊠v)⊠x) = min{ϖ_P (t⊠(v ⊠x)),ϖ_P (t)} for all . Hence, ϖ is a BFSUPF of .

In general, the converse of Theorem 3.3 is not true, as shown below.

Example 3.4

From Example 3.2, we have ϖ is a BFSUPF but not a BFSUPI of .

Theorem 3.5

Each BFSUPF of is a BFUPF of .

Proof

Let ϖ = (ϖ_N,ϖ_P ) be a BFSUPF of . Then, ϖ satisfies the conditions (21) and (22). Next, let . Thus,

ϖN(v)=ϖN(((v⌧0)⌧0)⌧v)≤max{ϖN(t⌧(0⌧v)),ϖN(t)}=max{ϖN(t⌧v),ϖN(t)},

and

ϖP(v)=ϖP(((v⌧0)⌧0)⌧v)≥min{ϖP(t⌧(0⌧v)),ϖP(t)}=min{ϖP(t⌧v),ϖP(t)}.

Hence, ϖ is a BFUPF of .

In general, the converse of Theorem 3.5 is not true, as shown below.

Example 3.6

Consider the UP-algebra in Example 3.2, and define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

	0	q	t	v	x
ϖ_N	−0.8	−0.6	−0.5	−0.5	0
ϖ_P	0.7	0.6	0.3	0.3	0.5

Then, the BFS ϖ = (ϖ_N,ϖ_P ) is a BFUPF but not a BFSUPF of . Indeed,

ϖN(((q⌧t)⌧t)⌧q)=-0.6>-0.8=max{ϖN(0⌧(t⌧q)),ϖN(0)},

and

ϖP(((q⌧t)⌧t)⌧q)=0.6<0.7=min{ϖP(0⌧(t⌧q)),ϖP(0))}.

From Theorems 3.3 and 3.5 and Examples 3.4 and 3.6, we find that a BFUPF is a general concept of a BFSUPF, and a BFSUPF is a general concept of a BFSUPI.

Theorem 3.7

If ϖ = (ϖ_N,ϖ_P ) is a BFSUPF of , then the set and ϖ_P (t) = ϖ_P (0)} is an SUPF of .

Proof

Clearly, 0 ∈ ϖ₀. Let t, v, x ∈ be such that t ⊠ (v ⊠ x) ∈ ϖ₀ and t ∈ ϖ₀. Then, ϖ_N(t) = ϖ_N(t ⊠ (v ⊠ x)) = ϖ_N(0) and ϖ_P (t) = ϖ_P (t ⊠ (v ⊠ x)) = ϖ_P (0). Thus,

ϖN(0)≤ϖN(((x⌧v)⌧v)⌧x) by (21) ≤max{ϖN(t⌧(v⌧x)),ϖN(t)} by (27) =max{ϖN(0),ϖN(0)} =ϖN(0),

and

ϖP(0)≥ϖP(((x⌧v)⌧v)⌧x) by (22) ≥min{ϖP(t⌧(v⌧x)),ϖP(t)} by (28) =min{ϖP(0),ϖP(0)} =ϖP(0).

That is, ϖ_N(((x⊠v)⊠v)⊠x) = ϖ_N(0) and ϖ_P (((x⊠v)⊠ v) ⊠ x) = ϖ_P (0), and thus, ((x ⊠ v) ⊠ v) ⊠ x ∈ ϖ₀. Hence, ϖ₀ is an SUPF of .

If is a subset of , then the bipolar fuzzy characteristic function of is denoted and defined as follows: ϖG=(ϖNG,ϖPG), where for all ,

ϖPG(t)={1,if t∈G,0,otherwise,

and

ϖNG(t)={-1,if t∈G,0,otherwise.

Theorem 3.8

A nonempty subset of is an SUPF of if and only if ϖG=(ϖNG,ϖPG) is a BFSUPF of .

Proof. (⇒)

Assume that is an SUPF of . Then, , which implies that ϖNG(0)=-1≤ϖNG(t) and ϖPG(0)=1≥ϖPG(t) for all . Thus, the conditions (21) and (22) hold. To show that the conditions (27) and (28) hold, let . If and , then ϖNG(t⌧(v⌧x))=ϖNG(t)=-1 and ϖPG(t⌧(v⌧x))=ϖPG(t)=1. By the assumption, we get Thus,

ϖNG(((x⌧v)⌧v)⌧x)=-1=max{ϖNG(t⌧(v⌧x)),ϖNG(t)},

and

ϖPG(((x⌧v)⌧v)⌧x)=1=min{ϖPG(t⌧(v⌧x)),ϖPG(t)}.

On the contrary, suppose that or . Thus,

max{ϖNG(t⌧(v⌧x)),ϖNG(t)}=0≥ϖNG(((x⌧v)⌧v)⌧x)

and

min{ϖPG(t⌧(v⌧x)),ϖPG(t)}=0≤ϖPG(((x⌧v)⌧v)⌧x).

Hence, we have the conditions (27) and (28). Therefore, we obtain that is a BFSUPF of .

(⇐) Assume that is a BFSUPF of . Because is nonempty, we obtain ϖNG(0)≤ϖNG(t)=-1 when , which implies that . Let and . Then, ϖNG(t)=ϖNG(t⌧(v⌧x))=-1. By this assumption, we have

ϖNG(((x⌧v)⌧v)⌧x)≤max{ϖNG(t⌧(v⌧x)),ϖNG(t)}=-1.

Thus, . Hence, is an SUPF of .

4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets

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Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

In this section, we characterize bipolar fuzzy shift UP-filters of UP-algebras in terms of level, fuzzy, and neutrosophic sets and find their relationships.

Definition 4.1

Let ϖ = (ϖ_N,ϖ_P ) be a BFS in a nonempty set and (r⁻, r⁺) ∈ [−1, 0] × [0, 1]. The subsets LN(ϖ;r-),UN(ϖ;r-),LP(ϖ;r+), and UP(ϖ;r+) of , defined by

LN(ϖ;r-)={t∈G∣ϖN(t)≤r-},UN(ϖ;r-)={t∈G∣ϖN(t)≥r-},LP(ϖ;r+)={t∈G∣ϖP(t)≤r+},UP(ϖ;r+)={t∈G∣ϖP(t)≥r+},

are called the negative lower r⁻-level subset, negative upper r⁻-level subset, positive lower r⁺-level subset, and positive upper r⁺-level subset of ϖ of , respectively.

Theorem 4.2

Let ϖ = (ϖ_N,ϖ_P ) be a BFS in . Then, ϖ is a BFSUPF of if and only if for all (r⁻, r⁺) ∈ [−1, 0] × [0, 1], we have the following:

(i) the subset LN(ϖ;r-) is a SUPF of if LN(ϖ;r-)≠∅ and
(ii) the subset UP(ϖ;r+) is a SUPF of if UP(ϖ;r+)≠∅.

Proof. (⇒)

Assume that ϖ is a BFSUPF of . Let r⁻ ∈ [−1, 0] be such that LN(ϖ;r-)≠∅, and let p∈LN(ϖ;r-). By using Eq. (21), we obtain ϖ_N(0) ≤ ϖ_N(p) ≤ r⁻. Then 0∈LN(ϖ;r-). It is shown that condition (19) holds. To show that condition (20) holds, let be such that t⊠ (v⊠x), t∈LN(ϖ;r-). Then, ϖ_N(t ⊠ (v ⊠ x)) ≤ r⁻ and ϖ_N(t) ≤ r⁻. By using Eq. (27), we obtain

ϖN(((x⌧v)⌧v)⌧x)≤max{ϖN(t⌧(v⌧x)),ϖN(t)}≤r-.

Thus, ((x⌧v)⌧v)⌧x∈LN(ϖ;r-). Hence, condition (20) holds. Therefore, we find that LN(ϖ;r-) is an SUPF of .

Let r⁺ ∈ [0, 1] be such that UP(ϖ;r+)≠∅. If p∈UP(ϖ;r+), then by using Eq. (22), we obtain ϖ_P (0) ≥ ϖ_P (p) ≥ r⁺, which implies that 0∈UP(ϖ;r+). Thus, condition (19) is true. Next, we show that condition (20) is true. Suppose that and t ⊠ (v ⊠ x), t∈UP(ϖ;r+). Then, ϖ_P (t ⊠ (v ⊠ x)) ≥ r⁺ and ϖ_P (t) ≥ r⁺. By using Eq. (28), we have

ϖP(((x⌧v)⌧v)⌧x)≥min{ϖP(t⌧(v⌧x)),ϖP(t)}≥r+.

Thus, ((x⌧v)⌧v)⌧x∈UP(ϖ;r+). Hence, condition (20) holds. Therefore, UP(ϖ;r+) is an SUPF of .

(⇐) Suppose that for all r⁻ ∈ [−1, 0], LN(ϖ;r-) is an SUPF of if LN(ϖ;r-) is nonempty and for all r⁺ ∈ [0, 1], UP(ϖ;r+) is an SUPF of if UP(ϖ;r+) is nonempty. Let . Then, t∈LN(ϖ;ϖN(t))≠∅ and t∈UP(ϖ;ϖP(t))≠∅. Thus, LN(ϖ;ϖN(t)) and UP(ϖ;ϖP(t)) are SUPFs of . Using Eq. (19), we have 0∈LN(ϖ;ϖN(t)) and 0∈UP(ϖ;ϖP(t)). Thus, ϖ_N(0) ≤ ϖ_N(t) and ϖ_P (0) ≥ ϖ_P (t). Hence, conditions (21) and (22) hold. Next, we show that condition (27) holds. Let and choose r⁻ = max{ϖ_N(t⊠(v⊠x)),ϖ_N(t)}. Then,ϖ_N(t⊠ (v⊠x)) ≤ r⁻ and ϖ_N(t) ≤ r⁻. Thus, t ⊠ (v ⊠x), t∈LN(ϖ;r-)≠∅. Using Eq. (20), we have ((x⌧v)⌧v)⌧x∈LN(ϖ;r-). Thus,

ϖN(((x⌧v)⌧v)⌧x)≤r-=max{ϖN(t⌧(v⌧x)),ϖN(t)}.

We have shown that condition (27) holds. Finally, to show that condition (28) holds, suppose that and choose r⁺ = min{ϖ_P (⊠(v⊠x)),ϖ_P (t)}. Then, ϖ_P (t⊠ (v⊠x)) ≥ r⁺ and ϖ_P (t) ≥ r⁺. Thus, t ⊠ (v ⊠ x), t∈UP(ϖ;r+)≠∅. Therefore, UP(ϖ;r+) is an SUPF of . By using Eq. (20) again, we obtain ((x⌧v)⌧v)⌧x∈UP(ϖ;r+). Thus,

ϖP(((x⌧v)⌧v)⌧x)≥r+=min{ϖP(t⌧(v⌧x)),ϖP(t)}.

Hence, condition (28) holds. It follows from Definition 3.1 that ϖ is a BFSUPF of .

For any r ∈ [0, 1], the set

C(ϖ;r)=LN(ϖ;-r)∩UP(ϖ;r)

is called the r-level subset of .

Corollary 4.3

If ϖ = (ϖ_N,ϖ_P ) is a BFSUPF of , then C(ϖ; r) is an SUPF of for all r ∈ [0, 1] when C(ϖ; r) is nonempty.

Proof

This follows from Theorems 2.3 and 4.2.

Remark 4.4

If is a nonempty subset of , then G=C(ϖG;1)=LN(ϖG;-1)=UP(ϖG;1).

Corollary 4.5

A nonempty subset of is an SUPF of if and only if there exist a BFSUPF ϖ = (ϖ_N,ϖ_P ) of and r ∈ [0, 1] such that

G=C(ϖ;ρ)=LN(ϖ;-r)=UP(ϖ;r).

Proof

This follows from Theorem 3.8, Corollary 4.3, and Remark 4.4.

For a BFS ϖ = (ϖ_N,ϖ_P ) in a nonempty set , the BFS ϖ¯=(ϖN¯,ϖP¯) defined by

ϖN¯(t)=-1-ϖN(t) and ϖP¯(t)=1-ϖP(t)

for all , and ϖ̄ is called the complement of .

Lemma 4.6 ([14])

For each BFS ϖ = (ϖ_N,ϖ_P ) in and element (r⁻, r⁺) of [−1, 0] × [0, 1], the following hold:

(i) LN(ϖ¯;r-)=UN(ϖ;-1-r-),
(ii) LP(ϖ¯;r+)=UP(ϖ;1-r+),
(iii) UN(ϖ¯;r-)=LN(ϖ;-1-r-), and
(iv) UP(ϖ¯;r+)=LP(ϖ;1-r+).

Theorem 4.7

Let ϖ = (ϖ_N,ϖ_P ) be a BFS in . Then, the BFS ϖ¯=(ϖN¯,ϖP¯) in is a BFSUPF of if and only if for all (r⁻, r⁺) ∈ [−1, 0] × [0, 1], we have

(i) the subset UN(ϖ;r-) is a SUPF of if UN(ϖ;r-)=∅ and
(ii) the subset LP(ϖ;r+) is a SUPF of if LP(ϖ;r+)≠∅.

Proof

This follows from Theorem 4.2 and Lemma 4.6.

In 1999, Smarandache [18] introduced a neutrosophic set (NS) N in a nonempty set as a structure of the form , where , and are called the truth, indeterminate, and false membership functions, respectively. For simplicity, we denote the by ⟨α, β, γ⟩. Songsaeng et al. [19, 20] applied neutrosophic set theory to UP-algebras, introduced many concepts of neutrosophic UP-substructures, and investigated their properties. A neutrosophic shift of the UP-filter of a UP-algebra is an interesting and important concept of neutrosophic UP-substructures and is defined as follows:

Definition 4.8 ([20])

An NS ⟨α, β, γ⟩ in is called a neutrosophic shift UP-filter (NSUPF) of if the following six conditions hold:

(29)(for all t∈T)(α(0)≥α(t)),

(30)(for all t∈T)(β(0)≤β(t)),

(31)(for all t∈T)(γ(0)≥γ(t)),

(32)(for all t,v,x∈T)(α(((x⌧v)⌧v)⌧v)≥min{α(t⌧(v⌧x)),α(t)}),

(33)(for all t,v,x∈T)(β(((x⌧v)⌧v)⌧x)≤max{β(t⌧(v⌧x)),β(t)}),

(34)(for all t,v,x∈T)(γ(((x⌧v)⌧v)⌧x)≥min{γ(t⌧(v⌧x)),γ(t)}).

Definition 4.9

An FS α in is called a fuzzy shift UP-filter (FSUPF) of if it satisfies conditions (29) and (32).

For a function α from a nonempty set into the set of all real numbers R, the functions −α, 1+α, and α −1 are defined as follows:

(35)-α:G→R,t↦-α(t),

(36)1+α:G→R,t↦1+α(t)

(37)α-1:G→R,t↦α(t)-1.

Theorem 4.10

Let α be an FS and ϖ = (ϖ_N,ϖ_P ) be the BFS in such that ϖ_N = −α and ϖ_P = α. Then, α is an FSUPF of if and only if ϖ is a BFSUPF of .

Proof

Assume that α is an FSUPF of . Then, ϖ_P (0) = α(0) ≥ α(t) = ϖ_P (t) and ϖ_N(0) = −α(0) ≤ −α(t) = ϖ_N(t) for all , which implies that ϖ satisfies conditions (21) and (22). To show that ϖ satisfies conditions (27 ) and (28), let . Then, α(((x⊠v)⊠v)⊠x) ≥ min{α(t⊠ (v⊠x)), α(t)} and −α(((x ⊠ v) ⊠ v) ⊠ x) ≤ max{−α(t ⊠ (v ⊠ x)), −α(t)}. Thus,

ϖP(((x⌧v)⌧v)⌧x)≥min{ϖP(t⌧(v⌧x)),ϖP(t)},ϖN(((x⌧v)⌧v)⌧x)≤max{ϖN(t⌧(v⌧x)),ϖN(t)}.

Thus, ϖ satisfies conditions (27) and (28). Therefore, ϖ is a BFSUPF of .

The converse of this theorem is clear.

Theorem 4.11

A BFS ϖ = (ϖ_N,ϖ_P ) in is a BFSUPF of if and only if ϖ_P and −ϖ_N are FSUPFs of .

Proof

Assume that ϖ is a BFSUPF of . Clearly, ϖ_P is an FSUPF of and −ϖ_N is an FS in . If , then ϖ_N(t) ≥ ϖ_N(0), which implies that −ϖ_N(t) ≤ −ϖ_N(0). Thus, −ϖ_N satisfies condition (29). To show that −ϖ_N satisfies condition (32), let . Then, ϖ_N(((x ⊠ v) ⊠ v) ⊠ x) ≤ max{ϖ_N(t⊠(v⊠x)),ϖ_N(t)} and −ϖ_N(((x⊠v)⊠v)⊠x) ≥ min{−ϖ_N(t ⊠ (v ⊠ x)), −ϖ_N(t)}. Thus, −ϖ_N satisfies condition (32). Hence, −ϖ_N is an FSUPF of .

Conversely, assume that ϖ_P and −ϖ_N are FSUPFs of . Then, conditions (22) and (28) hold because ϖ_P is an FSUPF of . Because −ϖ_N is an FSUPF of and ϖ_N = −(−ϖ_N), conditions (21) and (27) are satisfied. Hence, ϖ is a BFSUPF of .

For two FSs α and β in a nonempty set , we denote the BFS as (α – 1, β). In the following theorem, we provide the condition for an NS to be an NSUPF by BFSs.

Theorem 4.12

An NS ⟨α, β, γ⟩ in is an NSUPF of if and only if the BFSs (β – 1, α) and (β – 1, γ) are BFSUPFs of .

Proof

Assume that ⟨α, β, γ⟩ is an NSUPF of . Let . Then, α(0) ≥ α(t), α(((x ⊠ v) ⊠ v) ⊠ x) ≥ min{α(t ⊠ (v⊠x)), α(t)}, (β–1)(t) = β(t)–1 ≥ β(0)–1 = (β–1)(0) and

(β-1)(((x⌧v)⌧v)⌧x)=β(((x⌧v)⌧v)⌧x)-1≤max{β(t⌧(v⌧x)),β(t)}-1=max{β(t⌧(v⌧x))-1,β(t)-1}=max{(β-1)(t⌧(v⌧x)),(β-1)(t)}.

Hence, (β – 1, α) is a BFSUPF of . In the same manner, we can prove that (β – 1, γ) is a BFSUPF of .

Conversely, assume that the BFSs (β – 1, α) and (β – 1, γ) are BFSUPFs of . It is clear that ⟨α, β, γ⟩ satisfies conditions (29), (31), (32), and (34). Because (β – 1, α) is a BFSUPF of , we have

β(0)=β(0)-1+1=(β-1)(0)+1≤(β-1)(t)+1=β(t)-1+1=β(t),

and

β(((x⌧v)⌧v)⌧x)=β(((x⌧v)⌧v)⌧x)-1+1=(β-1)(((x⌧v)⌧v)⌧x)+1≤max{(β-1)(t⌧(v⌧x)),(β-1)(t)}+1=max{β(t⌧(v⌧x))-1,β(t)-1}+1=max{β(t⌧(v⌧x)),β(t)}-1+1=max{β(t⌧(v⌧x)),β(t)}

for all . Thus, ⟨α, β, γ⟩ satisfies conditions (30) and (33). Therefore, we find that ⟨α, β, γ⟩ is an NSUPF of .

Theorem 4.13

Let ϖ = (ϖ_N,ϖ_P ) be a BFS of . Then, ϖ is a BFSUPF of if and only if the NS ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ is an NSUPF of .

Proof

Assume that ϖ is a BFSUPF of . By Theorem 4.11, we obtain that ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ satisfies conditions (29), (31), (32), and (34). If , then

(1+ϖN)(t)=1+ϖN(t)≥1+ϖN(0)=(1+ϖN)(0),

and

(1+ϖN)(((x⌧v)⌧v)⌧x)=1+ϖN(((x⌧v)⌧v)⌧x)≤1+max{ϖN(t⌧(v⌧x)),ϖN(t)}=max{1+ϖN(t⌧(v⌧x)),1+ϖN(t)}=max{(1+ϖN)(t⌧(v⌧x)),(1+ϖN)(t)}.

Thus, ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ satisfies conditions (30) and (33). Hence, ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ is an NSUPF of .

Conversely, assume that ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ is an NSUPF of . Then, ϖ_P and −ϖ_N are FSUPFs of . It follows from Theorem 4.11 that ϖ is a BFSUPF of .

5. Conclusion

Go to

Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

In this paper, we introduced the concept of bipolar fuzzy shift UP-filters of UP-algebras and investigated its important properties. Relations between bipolar fuzzy shift UP-filters and other shift UP-filters (fuzzy and neutrosophic shift UP-filters) have been given. Finally, we characterized bipolar fuzzy shift UP-filters in terms of level, fuzzy, and neutrosophic sets.

In the future, we will extend bipolar fuzzy soft shift UP-filters over UP-algebras, provide some properties, and apply them to decision-making problems.

Acknowledgments

Go to

Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

This research project was supported by the Thailand Science Research and Innovation Fund and the University of Phayao (Grant No. FF66-UoE017).

Conflict of Interest

Go to

Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

No potential conflict of interest relevant to this article was reported.

References

Go to

Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

Iseki, K (1966). An algebra related with a propositional calculus. Proceedings of the Japan Academy. 42, 26-29. https://doi.org/10.3792/pja/1195522171
Imai, Y, and Iseki, K (1966). On axiom systems of propositional calculi, XIV. Proceedings of the Japan Academy. 42, 19-22. https://doi.org/10.3792/pja/1195522169
Hu, QP, and Li, X (1983). On BCH-algebras. Mathematics Seminar Notes (Kobe University). 11, 313-320.
Prabpayak, C, and Leerawat, U (2009). On ideals and congruences in KU-algebras. Scientia Magna. 5, 54-57.
Yiarayong, P, and Wachirawongsakorn, P (2018). A new generalization of BE-algebras. Heliyon. 4. article no. e00863
Iampan, A (2017). A new branch of the logical algebra: UPalgebras. Journal of Algebra and Related Topics. 5, 35-54. https://doi.org/10.22124/jart.2017.2403
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
Zhang, WR . Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis., Proceedings of the 1st International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference, 1994, San Antonio, TX, Array, pp.305-309. https://doi.org/10.1109/IJCF.1994.375115
Jun, YB, and Song, SZ. (2008) . Subalgebras and closed ideals of BCH-algebras based on bipolar-valued fuzzy sets. Scientiae Mathematicae Japonicae Online. 68, 287-297. https://www.jams.or.jp/scm/contents/e-2008-4/2008-37.pdf
Lee, KJ, and Jun, YB (2011). Bipolar fuzzy a-ideals of BCI-algebras. Communications of the Korean Mathematical Society. 26, 531-542. https://doi.org/10.4134/CKMS.2011.26.4.531
Jun, YB, Lee, KJ, and Roh, EH. (2012) . Ideals and filters in CI-algebras based on bipolar-valued fuzzy sets. Annals of Fuzzy Mathematics and Informatics. 4, 109-121. http://www.afmi.or.kr/papers/2012/Vol-04_No-01/AFMI-4-1(109-121)-J-111102.pdf
Muhiuddin, G, and Al-Kadi, D (2021). Bipolar fuzzy implicative ideals of BCK-algebras. Journal of Mathematics. 2021. article no 6623907
Gaketem, T, and Khamrot, P (2021). On some semigroups characterized in terms of bipolar fuzzy weakly interior ideals. IAENG International Journal of Computer Science. 48, 250-256.
Gaketem, T, Khamrot, P, Julatha, P, Prasertpong, R, and Iampan, A (). A novel concept of bipolar fuzzy sets in UPalgebras: bipolar fuzzy implicative UP-filters. submitted
Jun, YB, and Iampan, A (2019). Shift UP-filters and decompositions of UP-filters in UP-algebras. Missouri Journal of Mathematical Sciences. 31, 36-45. https://doi.org/10.35834/mjms/1559181624
Kawila, K, Udomsetchai, C, and Iampan, A (2018). Bipolar fuzzy UP-algebras. Mathematical and Computational Applications. 23. article no 69
Iampan, A (2018). Introducing fully UP-semigroups. Discussiones Mathematicae - General Algebra and Applications. 38, 297-306. https://doi.org/10.7151/dmgaa.1290
Smarandache, F (1999). A Unifying Field in Logics: Neutrosophic Logic. Rehoboth, NM: American Research Press
Songsaeng, M, and Iampan, A (2019). Neutrosophic set theory applied to UP-algebras. European Journal of Pure and Applied Mathematics. 12, 1382-1409. https://doi.org/10.29020/nybg.ejpam.v12i4.3543
Songsaeng, M, Shum, KP, Chinram, P, and Iampan, A (2021). Neutrosophic implicative UP-filters, neutrosophic comparative UP-filters, and neutrosophic shift UP-filters of UP-algebras. Neutrosophic Sets and Systems. 47, 620-643.

Biographies

Go to

Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion
Acknowledgments
Conflict of Interest
References
Biographies

Thiti Gaketem is a lecturer at the School of Science, University of Phayao, Thailand. He received his B.S. and M.S. degrees and Ph.D. in Mathematics from Naresuan University, Thailand. His areas of interest include the algebraic theory of semigroups and fuzzy algebraic structures. E-mail: thiti.ga@up.ac.th

Pannawit Khamrot received the B.Ed. in Mathematics from Nakhon Sawan Rajabhat University and M.S. in Applied Statistics and Ph.D. in Mathematics from Naresuan University, Thailand. He is a lecturer at the Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Phitsanulok, Phitsanulok, Thailand. His areas of interest include the algebraic theory of semigroups and bipolar fuzzy algebraic structures. E-mail: pk g@rmutl.ac.th

Pongpun Julatha is a faculty member of the Faculty of Science and Technology, Pibulsongkram Rajabhat University, Thailand. He received his B.S., M.S., and Ph.D. degrees in Mathematics from Naresuan University, Thailand. His areas of interest include the algebraic theory of semigroups, ternary semigroups, Γ-semigroups and fuzzy algebraic structures. E-mail: pongpun.j@psru.ac.th

Rukchart Prasertpong is a faculty member of the Faculty of Science and Technology, Nakhon Sawan Rajabhat University, Thailand. He received his B.S., M.S., and Ph.D. degrees in Mathematics from Naresuan University, Thailand. His areas of interest include the algebraic theory of semigroups and Γ-semigroups, fuzzy algebraic structures, and rough algebraic structures. E-mail: rukchart.p@nsru.ac.th

Aiyared Iampan was born in Nakhon Sawan, Thailand, in 1979. He is an Associate Professor at the Department of Mathematics, School of Science, University of Phayao, Phayao, Thailand. He received his B.S., M.S., and Ph.D. degrees in Mathematics from Naresuan University, Phitsanulok, Thailand, under thesis advisor Professor Dr. Manoj Siripitukdet. His areas of interest include the algebraic theory of semigroups, ternary semigroups, Γ-semigroups, lattices and ordered algebraic structures, fuzzy algebraic structures, and logical algebras. He was the founder of the Group for Young Algebraists at the University of Phayao in 2012 and one of the co-founders of the Fuzzy Algebras and Decision-Making Problems Research Unit at the University of Phayao in 2021. E-mail:aiyared.ia@up.ac.th

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(3): 294-302

Published online September 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.3.294

Bipolar Fuzzy Shift UP-Filters

Thiti Gaketem¹, Pannawit Khamrot², Pongpun Julatha³, Rukchart Prasertpong⁴ , and Aiyared Iampan¹

Correspondence to:Aiyared Iampan (aiyared.ia@up.ac.th)

Received: March 8, 2023; Revised: April 22, 2023; Accepted: June 2, 2023

Abstract

Keywords: UP-algebra, Shift UP-filter, Fuzzy shift UP-filter, Bipolar fuzzy shift UP-filter, Neutrosophic shift UP-filter

1. Introduction and Preliminaries

2. Preliminaries

First, we start with the definition of UP-algebras as follows:

Definition 2.1 ([6])

A UP-algebra is defined as ( ) of type (2, 0), where is a nonempty set, 0 is a fixed element of , and ⊠ is a binary operation on if it satisfies the following four conditions:

(1)

(for all t, v, x, \in T) ((v ⌧ x) ⌧ ((t ⌧ v) ⌧ (t ⌧ x)) = 0),

(2)

(for all t \in T) (0 ⌧ t = t),

(3)

(for all t \in T) (t ⌧ 0 = 0),

(4)

(for all t, v \in T) (t ⌧ v = 0, v ⌧ t = 0 \Rightarrow t = v) .

(5)

(for all t, v \in T) (t \leq v \Leftrightarrow t ⌧ v = 0)

and the following statements are true (see [6, 17]):

(6)

(for all t \in T) (t \leq t),

(7)

(for all t, v, x \in T) ((t \leq v, v \leq x) \Rightarrow t \leq x),

(8)

(for all t, v, x \in T) (t \leq v \Rightarrow x ⌧ t \leq x ⌧ v,)

(9)

(for all t, v, x \in T) (t \leq v \Rightarrow v ⌧ x \leq t ⌧ x), (for all t, v, x \in T)

(10)

(t \leq v ⌧ t, in particular, v ⌧ x \leq t ⌧ (v ⌧ x)),

(11)

(for all t, v \in T) (v ⌧ t \leq t \Leftrightarrow t = v ⌧ t),

(12)

(for all t, v \in T) (t \leq v ⌧ v),

(13)

(for all a, t, v, x \in T) (t ⌧ (v ⌧ x) \leq t ⌧ ((a ⌧ v) ⌧ (a ⌧ x))),

(14)

(for all a, t, v, x \in T) (((a ⌧ t) ⌧ (a ⌧ v)) ⌧ x \leq (t ⌧ v) ⌧ x),

(15)

(for all t, v, x \in T) ((t ⌧ v) ⌧ x \leq v ⌧ x),

(16)

(for all t, v, x \in T) (t \leq v \Rightarrow t \leq x ⌧ v),

(17)

(for all t, v, x \in T) ((t ⌧ v) ⌧ x \leq t ⌧ (v ⌧ x)),

(18)

(for all a, t, v, x \in T) ((t ⌧ v) ⌧ x \leq v ⌧ (a ⌧ x)) .

Definition 2.2 ([15])

A nonempty subset of is called a shift UP-filter (SUPF) of if

(19)

0 \in G,

(20)

(for all t, v, x \in T) (t ⌧ (v ⌧ x) \in G, t \in G \Rightarrow ((x ⌧ v) ⌧ v) ⌧ x \in G) .

Theorem 2.3 ([15])

If is a family of SUPFs of , then is an SUPF of .

A fuzzy set (FS) α in a nonempty set is a function from to the closed unit interval [0, 1] of real numbers, i.e., . A bipolar fuzzy set (BFS) ϖ [8] in a nonempty set is an object of the form

ϖ : = {(t, ϖ_{N} (t), ϖ_{P} (t)) ∣ t \in G},

Definition 2.4 ([16])

A BFS ϖ = (ϖ_N,ϖ_P ) in is called

(1) a bipolar fuzzy UP-filter (BFUPF) of if the following four conditions hold:

(21) $(for all t \in T) (ϖ_{N} (0) \leq ϖ_{N} (t)),$ (22) $(for all t \in T) (ϖ_{P} (0) \geq ϖ_{P} (t)),$ (23) $(for all t, v \in T) (ϖ_{N} (v) \leq max {ϖ_{N} (t ⌧ v), ϖ_{N} (t)}),$ (24) $(for all t, v \in T) (ϖ_{P} (v) \geq min {ϖ_{P} (t ⌧ v), ϖ_{P} (t)}),$
(2) a bipolar fuzzy strong UP-ideal (BFSUPI) of if the conditions (21), (22), and the following hold :

(25) $(for all t, v, x \in T) (ϖ_{N} (t) \leq max {ϖ_{N} ((x ⌧ v) ⌧ (x ⌧ t)), ϖ_{N} (v)}),$

(26) $(for all t, v, x \in T) (ϖ_{P} (t) \geq min {ϖ_{P} ((x ⌧ v) ⌧ (x ⌧ t)), ϖ_{P} (v)}) .$

Theorem 2.5 ([16])

A BFS ϖ = (ϖ_N,ϖ_P ) in is a BFSUPI of if and only if ϖ is constant.

Example 2.6

Consider an UP-algebra with the following Cayley table:

⊠	q	t	v	x
0	w	t	v	x
q	0	t	v	x
t	0	0	v	x
v	0	0	0	x
x	0	0	w	0

(1) Define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

0 q t v x
ϖN −0.8 −0.6 −0.6 −0.1 −0.1
ϖP 0.7 0.5 0.5 0.3 0.3

Then, ϖ is a BFUPF of but not a BFSUPI.
(2) Define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

0 q t v x
ϖN −0.8 −0.8 −0.8 −0.8 −0.8
ϖP 0.6 0.6 0.6 0.6 0.6

Then, ϖ is a BFSUPI.

3. Bipolar Fuzzy Shift UP-Filters

Definition 3.1

A BFS ϖ = (ϖ_N,ϖ_P ) in is called a bipolar fuzzy shift UP-filter (BFSUPF) of if the conditions (21), (22), and the following hold:

(27)

(for all t, v, x \in T) (ϖ_{N} (((x ⌧ v) ⌧ v) ⌧ x) \leq max {ϖ_{N} (t ⌧ (v ⌧ x)), ϖ_{N} (t)}),

(28)

(for all t, v, x \in T) (ϖ_{P} (((x ⌧ v) ⌧ v) ⌧ x) \geq min {ϖ_{P} (t ⌧ (v ⌧ x)), ϖ_{P} (t)}) .

Example 3.2

Consider a UP-algebra with the following Cayley table:

⊠	q	t	v	x
0	q	t	v	x
q	0	t	v	x
t	0	0	t	x
v	0	0	0	x
x	q	t	v	0

Define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

	0	q	t	v	x
ϖN	−0.4	−0.4	−0.3	−0.3	−0.3
ϖP	0.7	0.7	0.5	0.5	0.5

Then, ϖ is a BFSUPF of .

Theorem 3.3

Every BFSUPI of is a BFSUPF of .

Proof

In general, the converse of Theorem 3.3 is not true, as shown below.

Example 3.4

From Example 3.2, we have ϖ is a BFSUPF but not a BFSUPI of .

Theorem 3.5

Each BFSUPF of is a BFUPF of .

Proof

Let ϖ = (ϖ_N,ϖ_P ) be a BFSUPF of . Then, ϖ satisfies the conditions (21) and (22). Next, let . Thus,

\begin{array}{l} ϖ_{N} (v) = ϖ_{N} (((v ⌧ 0) ⌧ 0) ⌧ v) \\ \leq max {ϖ_{N} (t ⌧ (0 ⌧ v)), ϖ_{N} (t)} \\ = max {ϖ_{N} (t ⌧ v), ϖ_{N} (t)}, \end{array}

and

\begin{array}{l} ϖ_{P} (v) = ϖ_{P} (((v ⌧ 0) ⌧ 0) ⌧ v) \\ \geq min {ϖ_{P} (t ⌧ (0 ⌧ v)), ϖ_{P} (t)} \\ = min {ϖ_{P} (t ⌧ v), ϖ_{P} (t)} . \end{array}

Hence, ϖ is a BFUPF of .

In general, the converse of Theorem 3.5 is not true, as shown below.

Example 3.6

Consider the UP-algebra in Example 3.2, and define a BFS ϖ = (ϖ_N,ϖ_P ) in as follows:

	0	q	t	v	x
ϖ_N	−0.8	−0.6	−0.5	−0.5	0
ϖ_P	0.7	0.6	0.3	0.3	0.5

Then, the BFS ϖ = (ϖ_N,ϖ_P ) is a BFUPF but not a BFSUPF of . Indeed,

\begin{array}{l} ϖ_{N} (((q ⌧ t) ⌧ t) ⌧ q) = - 0.6 > - 0.8 \\ = max {ϖ_{N} (0 ⌧ (t ⌧ q)), ϖ_{N} (0)}, \end{array}

and

\begin{array}{l} ϖ_{P} (((q ⌧ t) ⌧ t) ⌧ q) = 0.6 < 0.7 \\ = min {ϖ_{P} (0 ⌧ (t ⌧ q)), ϖ_{P} (0))} . \end{array}

From Theorems 3.3 and 3.5 and Examples 3.4 and 3.6, we find that a BFUPF is a general concept of a BFSUPF, and a BFSUPF is a general concept of a BFSUPI.

Theorem 3.7

If ϖ = (ϖ_N,ϖ_P ) is a BFSUPF of , then the set and ϖ_P (t) = ϖ_P (0)} is an SUPF of .

Proof

Clearly, 0 ∈ ϖ₀. Let t, v, x ∈ be such that t ⊠ (v ⊠ x) ∈ ϖ₀ and t ∈ ϖ₀. Then, ϖ_N(t) = ϖ_N(t ⊠ (v ⊠ x)) = ϖ_N(0) and ϖ_P (t) = ϖ_P (t ⊠ (v ⊠ x)) = ϖ_P (0). Thus,

\begin{array}{l} ϖ_{N} (0) & \leq ϖ_{N} (((x ⌧ v) ⌧ v) ⌧ x) & by (21) \\ \leq max {ϖ_{N} (t ⌧ (v ⌧ x)), ϖ_{N} (t)} & by (27) \\ = max {ϖ_{N} (0), ϖ_{N} (0)} \\ = ϖ_{N} (0), \end{array}

and

\begin{array}{l} ϖ_{P} (0) & \geq ϖ_{P} (((x ⌧ v) ⌧ v) ⌧ x) & by (22) \\ \geq min {ϖ_{P} (t ⌧ (v ⌧ x)), ϖ_{P} (t)} & by (28) \\ = min {ϖ_{P} (0), ϖ_{P} (0)} \\ = ϖ_{P} (0) . \end{array}

That is, ϖ_N(((x⊠v)⊠v)⊠x) = ϖ_N(0) and ϖ_P (((x⊠v)⊠ v) ⊠ x) = ϖ_P (0), and thus, ((x ⊠ v) ⊠ v) ⊠ x ∈ ϖ₀. Hence, ϖ₀ is an SUPF of .

If is a subset of , then the bipolar fuzzy characteristic function of is denoted and defined as follows: $ϖ^{G} = (ϖ_{N}^{G}, ϖ_{P}^{G})$ , where for all ,

ϖ_{P}^{G} (t) = {\begin{array}{l} 1, & if t \in G, \\ 0, & otherwise, \end{array}

and

ϖ_{N}^{G} (t) = {\begin{array}{l} - 1, & if t \in G, \\ 0, & otherwise. \end{array}

Theorem 3.8

A nonempty subset of is an SUPF of if and only if $ϖ^{G} = (ϖ_{N}^{G}, ϖ_{P}^{G})$ is a BFSUPF of .

Proof. (⇒)

Assume that is an SUPF of . Then, , which implies that $ϖ_{N}^{G} (0) = - 1 \leq ϖ_{N}^{G} (t)$ and $ϖ_{P}^{G} (0) = 1 \geq ϖ_{P}^{G} (t)$ for all . Thus, the conditions (21) and (22) hold. To show that the conditions (27) and (28) hold, let . If and , then $ϖ_{N}^{G} (t ⌧ (v ⌧ x)) = ϖ_{N}^{G} (t) = - 1$ and $ϖ_{P}^{G} (t ⌧ (v ⌧ x)) = ϖ_{P}^{G} (t) = 1$ . By the assumption, we get Thus,

\begin{array}{l} ϖ_{N}^{G} (((x ⌧ v) ⌧ v) ⌧ x) = - 1 \\ = max {ϖ_{N}^{G} (t ⌧ (v ⌧ x)), ϖ_{N}^{G} (t)}, \end{array}

and

\begin{array}{l} ϖ_{P}^{G} (((x ⌧ v) ⌧ v) ⌧ x) = 1 \\ = min {ϖ_{P}^{G} (t ⌧ (v ⌧ x)), ϖ_{P}^{G} (t)} . \end{array}

On the contrary, suppose that or . Thus,

max {ϖ_{N}^{G} (t ⌧ (v ⌧ x)), ϖ_{N}^{G} (t)} = 0 \geq ϖ_{N}^{G} (((x ⌧ v) ⌧ v) ⌧ x)

and

min {ϖ_{P}^{G} (t ⌧ (v ⌧ x)), ϖ_{P}^{G} (t)} = 0 \leq ϖ_{P}^{G} (((x ⌧ v) ⌧ v) ⌧ x) .

Hence, we have the conditions (27) and (28). Therefore, we obtain that is a BFSUPF of .

(⇐) Assume that is a BFSUPF of . Because is nonempty, we obtain $ϖ_{N}^{G} (0) \leq ϖ_{N}^{G} (t) = - 1$ when , which implies that . Let and . Then, $ϖ_{N}^{G} (t) = ϖ_{N}^{G} (t ⌧ (v ⌧ x)) = - 1$ . By this assumption, we have

\begin{array}{l} ϖ_{N}^{G} (((x ⌧ v) ⌧ v) ⌧ x) \leq max {ϖ_{N}^{G} (t ⌧ (v ⌧ x)), ϖ_{N}^{G} (t)} \\ = - 1. \end{array}

Thus, . Hence, is an SUPF of .

4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets

In this section, we characterize bipolar fuzzy shift UP-filters of UP-algebras in terms of level, fuzzy, and neutrosophic sets and find their relationships.

Definition 4.1

Let ϖ = (ϖ_N,ϖ_P ) be a BFS in a nonempty set and (r⁻, r⁺) ∈ [−1, 0] × [0, 1]. The subsets $L_{N}^{(ϖ; r^{-})}, U_{N}^{(ϖ; r^{-})}, L_{P}^{(ϖ; r^{+})}$ , and $U_{P}^{(ϖ; r^{+})}$ of , defined by

\begin{array}{l} L_{N}^{(ϖ; r^{-})} = {t \in G ∣ ϖ_{N} (t) \leq r^{-}}, \\ U_{N}^{(ϖ; r^{-})} = {t \in G ∣ ϖ_{N} (t) \geq r^{-}}, \\ L_{P}^{(ϖ; r^{+})} = {t \in G ∣ ϖ_{P} (t) \leq r^{+}}, \\ U_{P}^{(ϖ; r^{+})} = {t \in G ∣ ϖ_{P} (t) \geq r^{+}}, \end{array}

are called the negative lower r⁻-level subset, negative upper r⁻-level subset, positive lower r⁺-level subset, and positive upper r⁺-level subset of ϖ of , respectively.

Theorem 4.2

Let ϖ = (ϖ_N,ϖ_P ) be a BFS in . Then, ϖ is a BFSUPF of if and only if for all (r⁻, r⁺) ∈ [−1, 0] × [0, 1], we have the following:

(i) the subset $L_{N}^{(ϖ; r^{-})}$ is a SUPF of if $L_{N}^{(ϖ; r^{-})} \neq \emptyset$ and
(ii) the subset $U_{P}^{(ϖ; r^{+})}$ is a SUPF of if $U_{P}^{(ϖ; r^{+})} \neq \emptyset$ .

Proof. (⇒)

Assume that ϖ is a BFSUPF of . Let r⁻ ∈ [−1, 0] be such that $L_{N}^{(ϖ; r^{-})} \neq \emptyset$ , and let $p \in L_{N}^{(ϖ; r^{-})}$ . By using Eq. (21), we obtain ϖ_N(0) ≤ ϖ_N(p) ≤ r⁻. Then $0 \in L_{N}^{(ϖ; r^{-})}$ . It is shown that condition (19) holds. To show that condition (20) holds, let be such that t⊠ (v⊠x), $t \in L_{N}^{(ϖ; r^{-})}$ . Then, ϖ_N(t ⊠ (v ⊠ x)) ≤ r⁻ and ϖ_N(t) ≤ r⁻. By using Eq. (27), we obtain

\begin{array}{l} ϖ_{N} (((x ⌧ v) ⌧ v) ⌧ x) \leq max {ϖ_{N} (t ⌧ (v ⌧ x)), ϖ_{N} (t)} \\ \leq r^{-} . \end{array}

Thus, $((x ⌧ v) ⌧ v) ⌧ x \in L_{N}^{(ϖ; r^{-})}$ . Hence, condition (20) holds. Therefore, we find that $L_{N}^{(ϖ; r^{-})}$ is an SUPF of .

Let r⁺ ∈ [0, 1] be such that $U_{P}^{(ϖ; r^{+})} \neq \emptyset$ . If $p \in U_{P}^{(ϖ; r^{+})}$ , then by using Eq. (22), we obtain ϖ_P (0) ≥ ϖ_P (p) ≥ r⁺, which implies that $0 \in U_{P}^{(ϖ; r^{+})}$ . Thus, condition (19) is true. Next, we show that condition (20) is true. Suppose that and t ⊠ (v ⊠ x), $t \in U_{P}^{(ϖ; r^{+})}$ . Then, ϖ_P (t ⊠ (v ⊠ x)) ≥ r⁺ and ϖ_P (t) ≥ r⁺. By using Eq. (28), we have

\begin{array}{l} ϖ_{P} (((x ⌧ v) ⌧ v) ⌧ x) \geq min {ϖ_{P} (t ⌧ (v ⌧ x)), ϖ_{P} (t)} \\ \geq r^{+} . \end{array}

Thus, $((x ⌧ v) ⌧ v) ⌧ x \in U_{P}^{(ϖ; r^{+})}$ . Hence, condition (20) holds. Therefore, $U_{P}^{(ϖ; r^{+})}$ is an SUPF of .

(⇐) Suppose that for all r⁻ ∈ [−1, 0], $L_{N}^{(ϖ; r^{-})}$ is an SUPF of if $L_{N}^{(ϖ; r^{-})}$ is nonempty and for all r⁺ ∈ [0, 1], $U_{P}^{(ϖ; r^{+})}$ is an SUPF of if $U_{P}^{(ϖ; r^{+})}$ is nonempty. Let . Then, $t \in L_{N}^{(ϖ; ϖ_{N} (t))} \neq \emptyset$ and $t \in U_{P}^{(ϖ; ϖ_{P} (t))} \neq \emptyset$ . Thus, $L_{N}^{(ϖ; ϖ_{N} (t))}$ and $U_{P}^{(ϖ; ϖ_{P} (t))}$ are SUPFs of . Using Eq. (19), we have $0 \in L_{N}^{(ϖ; ϖ_{N} (t))}$ and $0 \in U_{P}^{(ϖ; ϖ_{P} (t))}$ . Thus, ϖ_N(0) ≤ ϖ_N(t) and ϖ_P (0) ≥ ϖ_P (t). Hence, conditions (21) and (22) hold. Next, we show that condition (27) holds. Let and choose r⁻ = max{ϖ_N(t⊠(v⊠x)),ϖ_N(t)}. Then,ϖ_N(t⊠ (v⊠x)) ≤ r⁻ and ϖ_N(t) ≤ r⁻. Thus, t ⊠ (v ⊠x), $t \in L_{N}^{(ϖ; r^{-})} \neq \emptyset$ . Using Eq. (20), we have $((x ⌧ v) ⌧ v) ⌧ x \in L_{N}^{(ϖ; r^{-})}$ . Thus,

\begin{array}{l} ϖ_{N} (((x ⌧ v) ⌧ v) ⌧ x) \leq r^{-} \\ = max {ϖ_{N} (t ⌧ (v ⌧ x)), ϖ_{N} (t)} . \end{array}

We have shown that condition (27) holds. Finally, to show that condition (28) holds, suppose that and choose r⁺ = min{ϖ_P (⊠(v⊠x)),ϖ_P (t)}. Then, ϖ_P (t⊠ (v⊠x)) ≥ r⁺ and ϖ_P (t) ≥ r⁺. Thus, t ⊠ (v ⊠ x), $t \in U_{P}^{(ϖ; r^{+})} \neq \emptyset$ . Therefore, $U_{P}^{(ϖ; r^{+})}$ is an SUPF of . By using Eq. (20) again, we obtain $((x ⌧ v) ⌧ v) ⌧ x \in U_{P}^{(ϖ; r^{+})}$ . Thus,

\begin{array}{l} ϖ_{P} (((x ⌧ v) ⌧ v) ⌧ x) \geq r^{+} \\ = min {ϖ_{P} (t ⌧ (v ⌧ x)), ϖ_{P} (t)} . \end{array}

Hence, condition (28) holds. It follows from Definition 3.1 that ϖ is a BFSUPF of .

For any r ∈ [0, 1], the set

C (ϖ; r) = L_{N}^{(ϖ; - r)} \cap U_{P}^{(ϖ; r)}

is called the r-level subset of .

Corollary 4.3

If ϖ = (ϖ_N,ϖ_P ) is a BFSUPF of , then C(ϖ; r) is an SUPF of for all r ∈ [0, 1] when C(ϖ; r) is nonempty.

Proof

This follows from Theorems 2.3 and 4.2.

Remark 4.4

If is a nonempty subset of , then $G = C (ϖ^{G}; 1) = L_{N}^{(ϖ^{G}; - 1)} = U_{P}^{(ϖ^{G}; 1)}$ .

Corollary 4.5

A nonempty subset of is an SUPF of if and only if there exist a BFSUPF ϖ = (ϖ_N,ϖ_P ) of and r ∈ [0, 1] such that

G = C (ϖ; ρ) = L_{N}^{(ϖ; - r)} = U_{P}^{(ϖ; r)} .

Proof

This follows from Theorem 3.8, Corollary 4.3, and Remark 4.4.

For a BFS ϖ = (ϖ_N,ϖ_P ) in a nonempty set , the BFS $\bar{ϖ} = (\bar{ϖ_{N}}, \bar{ϖ_{P}})$ defined by

\bar{ϖ_{N}} (t) = - 1 - ϖ_{N} (t) and \bar{ϖ_{P}} (t) = 1 - ϖ_{P} (t)

for all , and ϖ̄ is called the complement of .

Lemma 4.6 ([14])

For each BFS ϖ = (ϖ_N,ϖ_P ) in and element (r⁻, r⁺) of [−1, 0] × [0, 1], the following hold:

(i) $L_{N}^{(\bar{ϖ}; r^{-})} = U_{N}^{(ϖ; - 1 - r^{-})}$ ,
(ii) $L_{P}^{(\bar{ϖ}; r^{+})} = U_{P}^{(ϖ; 1 - r^{+})}$ ,
(iii) $U_{N}^{(\bar{ϖ}; r^{-})} = L_{N}^{(ϖ; - 1 - r^{-})}$ , and
(iv) $U_{P}^{(\bar{ϖ}; r^{+})} = L_{P}^{(ϖ; 1 - r^{+})}$ .

Theorem 4.7

Let ϖ = (ϖ_N,ϖ_P ) be a BFS in . Then, the BFS $\bar{ϖ} = (\bar{ϖ_{N}}, \bar{ϖ_{P}})$ in is a BFSUPF of if and only if for all (r⁻, r⁺) ∈ [−1, 0] × [0, 1], we have

(i) the subset $U_{N}^{(ϖ; r^{-})}$ is a SUPF of if $U_{N}^{(ϖ; r^{-})} = \emptyset$ and
(ii) the subset $L_{P}^{(ϖ; r^{+})}$ is a SUPF of if $L_{P}^{(ϖ; r^{+})} \neq \emptyset$ .

Proof

This follows from Theorem 4.2 and Lemma 4.6.

Definition 4.8 ([20])

An NS ⟨α, β, γ⟩ in is called a neutrosophic shift UP-filter (NSUPF) of if the following six conditions hold:

(29)

(for all t \in T) (α (0) \geq α (t)),

(30)

(for all t \in T) (β (0) \leq β (t)),

(31)

(for all t \in T) (γ (0) \geq γ (t)),

(32)

(for all t, v, x \in T) (α (((x ⌧ v) ⌧ v) ⌧ v) \geq min {α (t ⌧ (v ⌧ x)), α (t)}),

(33)

(for all t, v, x \in T) (β (((x ⌧ v) ⌧ v) ⌧ x) \leq max {β (t ⌧ (v ⌧ x)), β (t)}),

(34)

(for all t, v, x \in T) (γ (((x ⌧ v) ⌧ v) ⌧ x) \geq min {γ (t ⌧ (v ⌧ x)), γ (t)}) .

Definition 4.9

An FS α in is called a fuzzy shift UP-filter (FSUPF) of if it satisfies conditions (29) and (32).

For a function α from a nonempty set into the set of all real numbers R, the functions −α, 1+α, and α −1 are defined as follows:

(35)

- α : G \to R, t \mapsto - α (t),

(36)

1 + α : G \to R, t \mapsto 1 + α (t)

(37)

α - 1 : G \to R, t \mapsto α (t) - 1.

Theorem 4.10

Let α be an FS and ϖ = (ϖ_N,ϖ_P ) be the BFS in such that ϖ_N = −α and ϖ_P = α. Then, α is an FSUPF of if and only if ϖ is a BFSUPF of .

Proof

\begin{array}{l} ϖ_{P} (((x ⌧ v) ⌧ v) ⌧ x) \\ \geq min {ϖ_{P} (t ⌧ (v ⌧ x)), ϖ_{P} (t)}, \\ ϖ_{N} (((x ⌧ v) ⌧ v) ⌧ x) \\ \leq max {ϖ_{N} (t ⌧ (v ⌧ x)), ϖ_{N} (t)} . \end{array}

Thus, ϖ satisfies conditions (27) and (28). Therefore, ϖ is a BFSUPF of .

The converse of this theorem is clear.

Theorem 4.11

A BFS ϖ = (ϖ_N,ϖ_P ) in is a BFSUPF of if and only if ϖ_P and −ϖ_N are FSUPFs of .

Proof

For two FSs α and β in a nonempty set , we denote the BFS as (α – 1, β). In the following theorem, we provide the condition for an NS to be an NSUPF by BFSs.

Theorem 4.12

An NS ⟨α, β, γ⟩ in is an NSUPF of if and only if the BFSs (β – 1, α) and (β – 1, γ) are BFSUPFs of .

Proof

Assume that ⟨α, β, γ⟩ is an NSUPF of . Let . Then, α(0) ≥ α(t), α(((x ⊠ v) ⊠ v) ⊠ x) ≥ min{α(t ⊠ (v⊠x)), α(t)}, (β–1)(t) = β(t)–1 ≥ β(0)–1 = (β–1)(0) and

\begin{array}{l} (β - 1) (((x ⌧ v) ⌧ v) ⌧ x) = β (((x ⌧ v) ⌧ v) ⌧ x) - 1 \\ \leq max {β (t ⌧ (v ⌧ x)), β (t)} - 1 \\ = max {β (t ⌧ (v ⌧ x)) - 1, β (t) - 1} \\ = max {(β - 1) (t ⌧ (v ⌧ x)), (β - 1) (t)} . \end{array}

Hence, (β – 1, α) is a BFSUPF of . In the same manner, we can prove that (β – 1, γ) is a BFSUPF of .

\begin{array}{l} β (0) = β (0) - 1 + 1 = (β - 1) (0) + 1 \\ \leq (β - 1) (t) + 1 = β (t) - 1 + 1 \\ = β (t), \end{array}

and

\begin{array}{l} β (((x ⌧ v) ⌧ v) ⌧ x) \\ = β (((x ⌧ v) ⌧ v) ⌧ x) - 1 + 1 \\ = (β - 1) (((x ⌧ v) ⌧ v) ⌧ x) + 1 \\ \leq max {(β - 1) (t ⌧ (v ⌧ x)), (β - 1) (t)} + 1 \\ = max {β (t ⌧ (v ⌧ x)) - 1, β (t) - 1} + 1 \\ = max {β (t ⌧ (v ⌧ x)), β (t)} - 1 + 1 \\ = max {β (t ⌧ (v ⌧ x)), β (t)} \end{array}

for all . Thus, ⟨α, β, γ⟩ satisfies conditions (30) and (33). Therefore, we find that ⟨α, β, γ⟩ is an NSUPF of .

Theorem 4.13

Let ϖ = (ϖ_N,ϖ_P ) be a BFS of . Then, ϖ is a BFSUPF of if and only if the NS ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ is an NSUPF of .

Proof

Assume that ϖ is a BFSUPF of . By Theorem 4.11, we obtain that ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ satisfies conditions (29), (31), (32), and (34). If , then

\begin{array}{l} (1 + ϖ_{N}) (t) = 1 + ϖ_{N} (t) \\ \geq 1 + ϖ_{N} (0) = (1 + ϖ_{N}) (0), \end{array}

and

\begin{array}{l} (1 + ϖ_{N}) (((x ⌧ v) ⌧ v) ⌧ x) \\ = 1 + ϖ_{N} (((x ⌧ v) ⌧ v) ⌧ x) \\ \leq 1 + max {ϖ_{N} (t ⌧ (v ⌧ x)), ϖ_{N} (t)} \\ = max {1 + ϖ_{N} (t ⌧ (v ⌧ x)), 1 + ϖ_{N} (t)} \\ = max {(1 + ϖ_{N}) (t ⌧ (v ⌧ x)), (1 + ϖ_{N}) (t)} . \end{array}

Thus, ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ satisfies conditions (30) and (33). Hence, ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ is an NSUPF of .

Conversely, assume that ⟨ϖ_P,1 + ϖ_N,−ϖ_N⟩ is an NSUPF of . Then, ϖ_P and −ϖ_N are FSUPFs of . It follows from Theorem 4.11 that ϖ is a BFSUPF of .

5. Conclusion

In the future, we will extend bipolar fuzzy soft shift UP-filters over UP-algebras, provide some properties, and apply them to decision-making problems.

Abstract
1. Introduction and Preliminaries
2. Preliminaries
3. Bipolar Fuzzy Shift UP-Filters
4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets
5. Conclusion

⊠	q	t	v	x
0	w	t	v	x
q	0	t	v	x
t	0	0	v	x
v	0	0	0	x
x	0	0	w	0

	0	q	t	v	x
ϖN	−0.8	−0.6	−0.6	−0.1	−0.1
ϖP	0.7	0.5	0.5	0.3	0.3

	0	q	t	v	x
ϖN	−0.8	−0.8	−0.8	−0.8	−0.8
ϖP	0.6	0.6	0.6	0.6	0.6

⊠	q	t	v	x
0	q	t	v	x
q	0	t	v	x
t	0	0	t	x
v	0	0	0	x
x	q	t	v	0

	0	q	t	v	x
ϖN	−0.4	−0.4	−0.3	−0.3	−0.3
ϖP	0.7	0.7	0.5	0.5	0.5

	0	q	t	v	x
ϖ_N	−0.8	−0.6	−0.5	−0.5	0
ϖ_P	0.7	0.6	0.3	0.3	0.5

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Article Tools

International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Bipolar Fuzzy Shift UP-Filters

Abstract

1. Introduction and Preliminaries

2. Preliminaries

Definition 2.1 ([6])

Definition 2.2 ([15])

Theorem 2.3 ([15])

Definition 2.4 ([16])

Theorem 2.5 ([16])

Example 2.6

3. Bipolar Fuzzy Shift UP-Filters

Definition 3.1

Example 3.2

Theorem 3.3

Example 3.4

Theorem 3.5

Example 3.6

Theorem 3.7

Theorem 3.8

4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets

Definition 4.1

Theorem 4.2

Corollary 4.3

Remark 4.4

Corollary 4.5

Lemma 4.6 ([14])

Theorem 4.7

Definition 4.8 ([20])

Definition 4.9

Theorem 4.10

Theorem 4.11

Theorem 4.12

Theorem 4.13

5. Conclusion

Acknowledgments

Conflict of Interest

References

Biographies

Article

Original Article

Bipolar Fuzzy Shift UP-Filters

Abstract

1. Introduction and Preliminaries

2. Preliminaries

Definition 2.1 ([6])

Definition 2.2 ([15])

Theorem 2.3 ([15])

Definition 2.4 ([16])

Theorem 2.5 ([16])

Example 2.6

3. Bipolar Fuzzy Shift UP-Filters

Definition 3.1

Example 3.2

Theorem 3.3

Example 3.4

Theorem 3.5

Example 3.6

Theorem 3.7

Theorem 3.8

4. Characterizing Bipolar Fuzzy Shift UP-Filters in Terms of Level, Fuzzy, and Neutrosophic Sets

Definition 4.1

Theorem 4.2

Corollary 4.3

Remark 4.4

Corollary 4.5

Lemma 4.6 ([14])

Theorem 4.7

Definition 4.8 ([20])

Definition 4.9

Theorem 4.10

Theorem 4.11

Theorem 4.12

Theorem 4.13

5. Conclusion

References

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