Pythagorean Fuzzy Implicative/Comparative/Shift UP-Filters of UP-Algebras with Approximations

Akarachai Satirad; Ronnason Chinram; Pongpun Julath; Aiyared Iampan

doi:10.5391/IJFIS.2023.23.1.56

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International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 56-78

Published online March 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.1.56

Pythagorean Fuzzy Implicative/Comparative/Shift UP-Filters of UP-Algebras with Approximations

Akarachai Satirad¹, Ronnason Chinram², Pongpun Julath³ , and Aiyared Iampan¹

¹Fuzzy Algebras and Decision-Making Problems Research Unit, Department of Mathematics, School of Science, University of Phayao, Mae Ka, Mueang, Thailand
²Division of Computational Science, Faculty of Science, Prince of Songkla University, Hat Yai, Thailand
³Department of Mathematics, Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok, Thailand

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

Received: April 19, 2022; Revised: September 11, 2022; Accepted: March 6, 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. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

This study aims to introduce new types of Pythagorean fuzzy sets (PFSs) in University of Phayao (UP)-algebras, which we refer to as Pythagorean fuzzy implicative UP-filters (PFIUPFs), Pythagorean fuzzy comparative UP-filters (PFCUPFs), and Pythagorean fuzzy shift UP-filters (PFSUPFs). In addition, we will discuss the relationships between some assertions of PFSs and PFIUPFs (resp., PFCUPFs, PFSUPFs) in UP-algebras and find sufficient conditions for studying the generalizations of three PFSs in UP-algebras. As a result of the study, we found their generalization to be as follows: every PFCUPF and PFSUPF is a PFUPF, and every PFIUPF is a PFUPI. Moreover, we study the upper and lower approximations of PFSs.

Keywords: UP-algebra, Pythagorean fuzzy implicative UP-filter, Pythagorean fuzzy comparative UP-filter, Pythagorean fuzzy shift UP-filter

1. Introduction and Preliminaries

Go to

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

Logical algebras constitute a significant class of algebras that include many other algebraic structures. Examples of these include BCK-algebras [1], BCI algebras [2, 3], BE algebras [4]; P-algebras [5], K-algebra [6–8], K(G)-algebra [9], gK-algebra [10], and extensions of KU/UP-algebras [11]. These are strongly connected with logic. For example, BCI-algebras introduced by Iséki [2] in 1966 have connections with BCI logic, being the BCI system in combinatory logic, which has applications in functional programming languages. BCK- and BCI-algebras are two classes of logical algebra, which were introduced by Imai and Iséki [1, 2] in 1966 and have been extensively investigated by many researchers.

The concept of rough sets was first described by Pawlak [12] in 1982. After its introduction, several studies have been conducted on the generalization of the concept of rough sets and its application in many algebraic structures. For example, in 2002, Jun [13] and Dudek et al. [14] applied the rough set theory to BCK- and BCI-algebras. Between 2019–2020, Ansari et al. [15] and Klinseesook et al. [16] applied the rough set theory to UP-algebras.

Zadeh [17] pioneered the concept of fuzzy sets (FSs) in 1965. The FS theories developed by Zadeh and others have found many applications in the domain of mathematics and elsewhere. After the introduction of the concept of FSs by Zadeh [17], Atanassov [18] defined a new concept called an intuitionistic fuzzy set, which is a generalization of an FS; Yager [19] introduced a new class of non-standard fuzzy subsets, called Pythagorean fuzzy sets (PFSs), and the related idea of Pythagorean membership grades. Jun et al. [20] introduced the concept of intuitionistic fuzzy quasi-associative ideals in BCI-algebras. Akram and Dar [21] introduced the notions of T-fuzzy subalgebras and T-fuzzy H-ideals in BCI-algebras. In 2006, Akram and Zhan [22] introduced the notion of sensible fuzzy ideals of BCK-algebras with respect to a t-conorm and provided the conditions for a sensible fuzzy subalgebra to be a sensible fuzzy ideal with respect to a t-conorm.

The concept of PFSs has been applied to semigroups, ternary semigroups, and many logical algebras, including the following: The concept of rough Pythagorean fuzzy ideals in semigroups was presented by Hussain et al. [23] in 2019. The upper and lower approximations of Pythagorean fuzzy left (resp., right) ideals, bi-ideals, interior ideals, and (1, 2)-ideals in semigroups were then shown. Jansi and Mohana [24] studied the characteristics of bipolar Pythagorean fuzzy A-ideals of BCI-algebras. Jana et al. [25] created a few Pythagorean fuzzy Dombi aggregation operators and applied them to multiple-attribute decision-making. Senapati and Chen [26] applied interval-valued PFSs according to the concepts of Hamacher t-norm and t-conorm to multi-attribute decision-making problems. Furthermore, the links between bipolar Pythagorean fuzzy subalgebras, bipolar Pythagorean fuzzy ideals, and bipolar Pythagorean fuzzy A-ideals were investigated. Chinram and Panityakul [27] researched rough Pythagorean fuzzy ideals in ternary semigroups in 2020. Satirad et al. [28] extended the notion of PFSs to University of Phayao (UP)-algebras in 2021 and developed five varieties of PFSs. They also examined the upper and lower estimates of PFSs.

In this study, we introduce three types of PFSs in UP-algebras: Pythagorean fuzzy implicative UP-filters (PFIUPFs), Pythagorean fuzzy comparative UP-filters (PFCUPFs), and Pythagorean fuzzy shift UP-filters (PFSUPFs), and investigate their properties. In addition, we discover the sufficient conditions for, and the relationships between the PFIUPFs, PFCUPFs, and PFSUPFs, respectively with the PFSs defined in [28]. Finally, we describe the concept of lower and upper estimates of PFIUPFs, PFCUPFs, and PFSUPFs in UP-algebras.

Before we proceed, let us examine the definition of UP-algebras.

Definition 1.1 [5]

A UP-algebra is defined as of type (2, 0), where is a non-empty set, ∘ is a binary operation on , and 0 is a fixed element of , satisfying the following axioms:

(∀x1,x2,x3∈U) ((x2∘x3)∘(x1∘x2)∘(x1∘x3))=0),(UP-1)(∀x1∈U) (0∘x1=x1),(UP-2)(∀x1∈U) (x1∘0=0),(UP-3)(∀x1,x2∈U) (x1∘x2=0,x2∘x1=0⇒x1=x2).(UP-4)

For more examples and studies of UP-algebras, see [15, 29–36].

In a UP-algebra , the following assertions are valid (see [5, 30]).

(1)(∀x1∈U) (x1∘x1=0),

(2)(∀x1,x2,x3∈U)(x1∘x2=0,x2∘x3=0⇒x1∘x3=0),

(3)(∀x1,x2,x3∈U)(x1∘x2=0⇒(x3∘x1)∘(x3∘x2)=0),

(4)(∀x1,x2,x3∈U)(x1∘x2=0⇒(x2∘x3)∘(x1∘x3)=0),

(5)(∀x1,x2∈U) (x1∘(x2∘x1)=0),

(6)(∀x1,x2∈U) ((x2∘x1)∘x1=0)⇔x1=x2∘x1),

(7)(∀x1,x2∈U) (x1∘(x2∘x2)=0),

(8)(∀x0,x1,x2,x3∈U)((x1∘(x2∘x3))∘(x1∘((x0∘x2)∘(x0∘x3)))=0),

(9)(∀x0,x1,x2,x3∈U)((((x0∘x1)∘(x0∘x2))∘x3)∘((x1∘x2)∘x3)=0),

(10)(∀x1,x2,x3∈U)(((x1∘x2)∘x3)∘(x2∘x3)=0),

(11)(∀x1,x2,x3∈U)(x1∘x2=0⇒x1∘(x3∘x2)=0),

(12)(∀x1,x2,x3∈U)(((x1∘x2)∘x3)∘(x1∘(x2∘x3))=0),

(13)(∀x0,x1,x2,x3∈U)(((x1∘x2)∘x3)∘(x2∘(x0∘x3))=0).

From [5], the binary relation ≤ on a UP-algebra is defined as follows:

(∀x1,x2∈U) (x1≤x2⇔x1∘x2=0).

Definition 1.2 [17]

A fuzzy set (FS) F in a non-empty set is described by its membership function, f_F. To every point , this function associates a real number f_F(x₁) in the closed interval [0, 1]. The real number f_F(x₁) is interpreted for the point as a degree of membership of an object to the FS F; that is, .

Definition 1.3 [17]

Let F be an FS in a non-empty set . The complement of F, denoted by F̃, is described by its membership function, f_F̃, which is defined as follows:

(14)(∀x1∈U) (fF˜(x1)=1-fF(x1)).

Definition 1.4 [17]

Let F₁ and F₂ be FSs in a non-empty set . The relations ⊆ and = and the operations ∪ and ∩ are defined as follows:

(1) ,
(2) F₁ = F₂ ⇔ F₁ ⊆ F₂, F₁ ⊇ F₂,
(3) ,
(4) .

The following two propositions will be used in the following sections:

Proposition 1.5 [28]

Let F be an FS in a non-empty set . Then the following assertions are valid:

(1) ,
(2) .

Proposition 1.6 [28]

Let {F_i}_i_∈_I and {G_i}_i_∈_I be non-empty families of FSs in a non-empty set , where I is an arbitrary index set, F and G an FSs in , and Y be a non-empty subset of . Then the following assertions are valid:

(1) (∀x1,x2∈U) (infi∈I{min{fFi(x1),fFi(x2)}}=min{infi∈I{fFi(x1)},infi∈I{fFi(x2)}}),
(2) (∀x1,x2∈U) (supi∈I{max{fFi(x1),fFi(x2)}}=max{supi∈I{fFi(x1)},supi∈I{fFi(x2)}}),
(3) (∀x1,x2∈U) (infi∈I{max{fFi(x1),fFi(x2)}}≥max{infi∈I{fFi(x1)},infi∈I{fFi(x2)}}),
(4) (∀x1,x2∈U) (supi∈I{min{fFi(x1),fFi(x2)}}≤min{supi∈I{fFi(x1)},supi∈I{fFi(x2)}}),
(5) for all i ∈ I ⇒ sup_i_∈_I{f_{F_i} (x₁)} ≤ sup_i_∈_I{f_{G_i} (x₁)}),
(6) (∀x₁ ∈ Y )(f_F(x₁) ≤ f_G(x₁) ⇒ sup_x_₁_∈_Y {f_F(x₁)} ≤ sup_x_₁_∈_Y {f_G(x₁)}),
(7) for all i ∈ I ⇒ inf_i_∈_I{f_{F_i} (x₁)} ≤ inf_i_∈_I{f_{G_i} (x₁)}),
(8) (∀x₁ ∈ Y )(f_F(x₁) ≤ f_G(x₁) ⇒ inf_x_₁_∈_Y {f_F(x₁)} ≤ inf_x_₁_∈_Y {f_G(x₁)}).

The following definition can be considered based on the concepts expounded in [?].

Definition 1.7

An FS F in a UP-algebra is called

(1) a fuzzy implicative UP-filter (FIUPF) of if it satisfies

(15)(∀x1∈U) (fF(0)≥fF(x1)),(16)(∀x1,x2,x3∈U) (fF(x1∘x3)≥min{fF(x1∘(x2∘x3)),fF(x1∘x2)}),
(2) a fuzzy comparative UP-filter (FCUPF) of if it satisfies (15) and

(17)(∀x1,x2,x3∈U) (fF(x2)≥min{fF(x1∘((x2∘x3)∘x2)),fF(x1)}),
(3) a fuzzy shift UP-filter (FSUPF) of if it satisfies (15) and

(18)(∀x1,x2,x3∈U) (fF(((x3∘x2)∘x2)∘x3)≥min{fF(x1∘(x2∘x3)),fF(x1)}).

2. PFSs in UP-Algebras

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Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

In 2013, Yager and Abbasov [?, 19] introduced the concept of PFSs for the first time.

Definition 2.1

A Pythagorean fuzzy set (PFS) P in a non-empty set is described by their membership function μ_P and non-membership function ν_P. For every point , these functions associate real numbers μ_P(x₁) and ν_P(x₁) in the closed interval [0, 1] with the following condition:

(19)(∀x1∈U) (0≤μP(x1)2+νP(x1)2≤1).

The real numbers μ_P(x₁) and ν_P(x₁) are interpreted for the point as a degree of membership and non-membership of an object , respectively, to the PFS P, that is, . For simplicity, PFS P is denoted by P = (μ_P, ν_P). We say that a PFS P in is a constant Pythagorean fuzzy set (CPFS) if its membership function μ_P and nonmembership function ν_P are constant.

Definition 2.2

Let P = (μ_P, ν_P) and Q = (μ_Q, ν_Q) be PFSs in a non-empty set . The relations ⊆ and = and the operations ∪ and ∩ are defined as follows:

(1) ,
(2) P = Q ⇔ P ⊆ Q, P ⊇ Q,
(3) P ∪Q = (μ_P ∪ μ_Q, ν_P ∩ ν_Q),
(4) P ∩Q = (μ_P ∩ μ_Q, ν_P ∪ ν_Q).

Note that P ∪ Q and P ∩ Q are PFSs in . Indeed, if we assume , then (μ_P∪μ_Q)(x₁) = max{μ_P(x₁), μ_Q(x₁)} and (ν_P∩ν_Q)(x₁) = min{ν_P(x₁), ν_Q(x₁)}. Thus we consider

0≤(μP∪μQ) (x1))2+(νP∩νQ) (x1))2=max{πP(x1),μQ(x1)}2+min{νP(x1),νQ(x1)}2=(μP(x1))2+min{νP(x1),νQ(x1)}2(WLOG, assume that max{μP(x1),μQ(x1)}=μP(x1))≤(μP(x1))2+(νP(x1))2≤1.

This implies that P ∪ Q is a PFS in . The proof of P ∩ Q is similar to that of P ∪ Q. Hence, we can denote P ∪Q = (μ_P ∪ μ_Q, ν_P ∩ ν_Q) = (μ_P_∪_Q, ν_P_∪_Q) and P ∩ Q = (μ_P ∩ μ_Q, ν_P ∪ ν_Q) = (μ_P_∩_Q, ν_P_∩_Q).

Hereafter, we shall let be a UP-algebra unless otherwise stated.

Definition 2.3 [28]

A PFS P = (μ_P, ν_P) in is called

(1) a Pythagorean fuzzy UP-subalgebra (PFUPS) of if it satisfies

(20)(∀x1,x2∈U) (μP(x1∘x2)≥min{μP(x1),μP(x2)}),(21)(∀x1,x2∈U) (νP(x1∘x2)≤max{νP(x1),νP(x2)}),
(2) a Pythagorean fuzzy near UP-filter (PFNUPF) of if it satisfies

(22)(∀x1,x2∈U) (μP(x1∘x2)≥μP(x2)),(23)(∀x1,x2∈U) (νP(x1∘x2)≤νP(x2)),
(3) a Pythagorean fuzzy UP-filter (PFUPF) of if it satisfies

(24)(∀x1∈U) (μP(0)≥μP(x1)),(25)(∀x1∈U) (νP(0)≤νP(x1)),(26)(∀x1,x2∈U) (μP(x2)≥min{μP(x1∘x2),μP(x1)}),(27)(∀x1,x2∈U) (νP(x2)≤max{νP(x1∘x2),νP(x1)}),
(4) a Pythagorean fuzzy UP-ideal (PFUPI) of if it satisfies (24), (25), and

(28)(∀x1,x2,x3∈U) (μP(x1∘x3)≥min{μP(x1∘(x2∘x3)),μP(x2)}),(29)(∀x1,x2,x3∈U) (νP(x1∘x3)≤max{νP(x1∘(x2∘x3)),νP(x2)}),
(5) a Pythagorean fuzzy strong UP-ideal (PFSUPI) of if it satisfies (24), (25), and

(30)(∀x1,x2,x3∈U) (μP(x1)≥min{μP((x3∘x2)∘(x3∘x1)),μP(x2)}),(31)(∀x1,x2,x3∈U) (νP(x1)≤max{νP((x3∘x2)∘(x3∘x1)),νP(x2)}).

Satirad et al. [28] proved that the concept of PFUPSs is a generalization of PFNUPFs, PFNUPFs is a generalization of PFUPFs, PFUPFs is a generalization of PFUPIs, and PFUPIs is a generalization of PFSUPIs. Furthermore, they proved that PFSUPIs and CPFSs are coincident in UP-algebras .

Next, we introduce three notions of PFSs in UP-algebras.

Definition 2.4

A PFS P = (μ_P, ν_P) in is called

(1) a Pythagorean fuzzy implicative UP-filter (PFIUPF) of if it satisfies (24), (25), and

(32)(∀x1,x2,x3∈U) (μP(x1∘x3)≥min{μP(x1∘(x2∘x3)),μP(x1∘x2)}),(33)(∀x1,x2,x3∈U) (νP(x1∘x3)≤max{νP(x1∘(x2∘x3)),νP(x1∘x2)}),
(2) a Pythagorean fuzzy comparative UP-filter (PFCUPF) of if it satisfies (24), (25), and

(34)(∀x1,x2,x3∈U) (μP(x2)≥min{μP(x1∘((x2∘x3)∘x2)),μP(x1)}),(35)(∀x1,x2,x3∈U) (νP(x2)≤max{νP(x1∘((x2∘x3)∘x2)),νP(x1)}),
(3) a Pythagorean fuzzy shift UP-filter (PFSUPF) of if it satisfies (24), (25), and

(36)(∀x1,x2,x3∈U) (μP(((x3∘x2)∘x2)∘x3)≥min{μP(x1∘(x2∘x3)),μP(x1)}),(37)(∀x1,x2,x3∈U) (νP(((x3∘x2)∘x2)∘x3)≤max{νP(x1∘(x2∘x3)),νP(x1)}).

Theorem 2.5

Every PFIUPF of is a PFUPF.

Proof

Let P = (μ_P, ν_P) be a PFIUPF of . Then, for all x₁, ,

μP(x2)=μP(0∘x2)(by (UP-2))≥min{μP(0∘(x1∘x2)),μP(0∘x1)}(by (32))=min{μP(x1∘x2),μP(x1)},(by (UP-2))

and

νP(x2)=νP(0∘x2)(by (UP-2))≤max{νP(0∘(x1∘x2)),νP(0∘x1)}(by (33))=max{νP(x1∘x2),νP(x1)}.(by (UP-2))

Therefore, P is a PFUPF of .

The converse of Theorem 2.5 does not hold in general, as is shown in the following example.

Example 2.6

Consider a UP-algebra where is defined in Table 1.

If we define a PFS P = (μ_P, ν_P) as presented in Table 2, then, P is a PFUPF of . Because μ_P(3 ∘ 4) = μ_P(3) = 0.3 ≱ 0.8 = min{0.8, 0.8} = min{μ_P(0), μ_P(0)} = min{μ_P(3 ∘ (3 ∘ 4)), μ_P(3 ∘ 3)}, we have that P is not a PFIUPF of .

Theorem 2.7

Every PFCUPF of is a PFUPF.

Proof

Let P = (μ_P, ν_P) be a PFCUPF of . Then, for all x₁, ,

μP(x2)≤min{μP(x1∘((x2∘0)∘x2)),μP(x1)}(by (34))=min{μP(x1∘(0∘x2)),μP(x1)}(by (UP-3))=min{μP(x1∘x2),μP(x1)},(by (UP-2))

and

νP(x2)≤max{νP(x1∘((x2∘0)∘x2)),νP(x1)}(by (35))=max{νP(x1∘(0∘x2)),νP(x1)}(by (UP-3))=max{νP(x1∘x2),νP(x1)}.(by (UP-2))

Therefore, P is a PFUPF of .

Similarly, the converse of Theorem 2.7 does not hold in general, as is shown in the following example.

Example 2.8

Consider a UP-algebra where is defined in Table 3.

If we define a PFS P = (μ_P, ν_P) as presented in Table 4, then, P is a PFUPF of . Because μ_P(2) = 0.6 ≱ 1 = min{1, 1} = min{μ_P(0), μ_P(0)} = min{μ_P(0 ∘ ((2 ∘ 3) ∘ 2)), μ_P(0)}, we have that P is not a PFCUPF of .

Theorem 2.9

Every PFSUPF of is a PFUPF.

Proof

Let P = (μ_P, ν_P) be a PFSUPF of . Then for all x₁, ,

μP(x2)=μP(0∘x2)(by (UP-2))=μP((0∘0)∘x2)(by (1))=μP(((x2∘0)∘0)∘x2)(by (UP-3))≥min{μP(x1∘(0∘x2)),μP(x1)}(by (36))=min{μP(x1∘x2),μP(x1)},(by (UP-2))

and

νP(x2)=νP(0∘x2)(by (UP-2))=νP((0∘0)∘x2)(by (1))=νP(((x2∘0)∘0)∘x2)(by (UP-3))≤max{νP(x1∘(0∘x2)),νP(x1)}(by (37))=max{νP(x1∘x2),νP(x1)}.(by (UP-2))

Therefore, P is a PFUPF of .

The converse of Theorem 2.9 does not hold in general. This is shown in the following example.

Example 2.10

Consider a UP-algebra where is defined in Table 5.

If we define a PFS P = (μ_P, ν_P) as presented in Table 6, then, P is a PFUPF of . Because μ_P(((1 ∘ 2) ∘ 2) ∘ 1) = μ_P(1) = 0.5 ≱ 0.9 = min{0.9, 0.9} = min{μ_P(0), μ_P(0)} = min{μ_P (0 ∘ (2 ∘ 1)), μ_P(0)}, we have that P is not a PFSUPF of .

Theorem 2.11

Every PFIUPF of is a PFUPI.

Proof

Let P = (μ_P, ν_P) be a PFIUPF of . By Theorem 2.5, we have that P as a PFUPF, and thus, P is a PFNUPF. Then, for all x₁, x₂, ,

μP(x1∘x3)≥min{μP(x1∘(x2∘x3)),μP(x1∘x2)}(by (32))≥min{μP(x1∘(x2∘x3)),μP(x2)},(by (22))

and

νP(x1∘x3)≤max{νP(x1∘(x2∘x3)),νP(x1∘x2)}(by (33))≤max{νP(x1∘(x2∘x3)),νP(x2)}.(by (23))

Therefore, P is a PFUPI of .

The converse of Theorem 2.11 does not hold in general. This is shown in the following example.

Example 2.12

Consider a UP-algebra where is defined in Table 7.

If we define a PFS P = (μ_P, ν_P) as presented in Table 8, then, P is a PFUPI of . Because μ_P(3 ∘ 2) = μ_P(1) = 0.5 ≱ 0.6 = min{0.6, 0.6} = min{μ_P(0), μ_P(0)} = min{μ_P(3 ∘ (3 ∘ 2)), μ_P(3 ∘ 3)}, we have that P is not a PFIUPF of .

Theorem 2.13

Every PFSUPI of is a PFIUPF (respectively, PFCUPF, PFSUPF).

Proof

Let P = (μ_P, ν_P) be a PFSUPI of . Because P is constant, we have that P is a PFIUPF (resp., PFCUPF, PFSUPF) of .

The converse of Theorem 2.13 does not hold in general, as is shown in the following examples.

Example 2.14

Consider a UP-algebra where is defined in Table 9.

If we define a PFS P = (μ_P, ν_P) as presented in Table 10, then, P is a PFIUPF of . However, P is not a CPFS of . Therefore, P is not a PFSUPI of .

Example 2.15

Consider a UP-algebra where is defined in Table 11.

If we define a PFS P = (μ_P, ν_P) as presented in Table 12, then, P is a PFCUPF of . However, P is not a CPFS of . Therefore, P is not a PFSUPI of .

Example 2.16

Consider a UP-algebra where is defined in Table 13.

If we define a PFS P = (μ_P, ν_P) as presented in Table 14, then, P is a PFSUPF of . However, P is not a CPFS of . Therefore, P is not a PFSUPI of .

Next, we present some examples for studying the generalization of new notions of PFSs and original PFSs in UP-algebras.

Example 2.17

Consider a UP-algebra where is defined in Table 15.

If we define a PFS P = (μ_P, ν_P) as presented in Table 16, then, P is a PFSUPF of . Because μ_P(2 ∘ 3) = μ_P(2) = 0.1 ≱ 0.5 = min{0.5, 0.5} = min{μ_P(0), μ_P(0)} = min{μ_P(2 ∘ (2 ∘ 3)), μ_P(2 ∘ 2)}, we have that P is not a PFIUPF of .

Example 2.18

From Example 2.14, if we define a PFS P = (μ_P, ν_P) as presented in Table 17, then, P is a PFIUPF of . Because ν_P(((2 ∘ 1) ∘ 1) ∘ 2) = ν_P(2) = 0.6 ≰ 0.2 = max{0.1, 0.2} = max{ν_P(0), ν_P(3)} = max{ν_P(3 ∘ (1 ∘ 2)), ν_P(3)}, we have that P is not a PFSUPF of .

Example 2.19

From Example 2.14, if we define a PFS P = (μ_P, ν_P) as presented in Table 18, then, P is a PFIUPF of . Because ν_P(2) = 0.8 ≰ 0.6 = max{0.5, 0.6} = max{ν_P(0), ν_P(3)} = max{ν_P(3 ∘ ((2 ∘ 1) ∘ 2)), ν_P(3)}, we have that P is not a PFCUPF of .

Example 2.20

Consider a UP-algebra where is defined in Table 19.

If we define a PFS P = (μ_P, ν_P) as presented in Table 20, then, P is a PFUPI of . Because ν_P(4) = 0.8 ≰ 0.2 = max{0.2, 0.2} = max{ν_P(0), ν_P(0)} = max{ν_P(0 ∘ ((4 ∘ 1) ∘ 4)), ν_P(0)}, we have that P is not a PFCUPF of .

Example 2.21

Consider a UP-algebra where is defined in Table 21.

If we define a PFS P = (μ_P, ν_P) as presented in Table 22, then, P is a PFUPI of . Because ν_P(((1 ∘ 2) ∘ 2) ∘ 1) = ν_P(1) = 0.5 ≰ 0.3 = max{0.3, 0.3} = max{ν_P(0), ν_P(0)} = max{ν_P (0 ∘ (2 ∘ 1)), ν_P(0)}, we have that P is not a PFSUPF of .

Example 2.22

From Example 2.17, if we define a PFSP = (μ_P, ν_P) as presented in Table 23, then, P is a PFSUPF of . Because ν_P(2 ∘ 3) = ν_P(2) = 0.6 ≰ 0.5 = max{0.5, 0.5} = max{ν_P(0), ν_P(1)} = max{ν_P(2 ∘ (1 ∘ 3)), ν_P(1)}, we have that P is not a PFUPI of .

Example 2.23

From Example 2.17, if we define a PFS P = (μ_P, ν_P) as presented in Table 24, then, P is a PFSUPF of . Because ν_P(2) = 0.8 ≰ 0.6 = max{0.6, 0.6} = max{ν_P(0), ν_P(0)} = max{ν_P(0 ∘ ((2 ∘ 3) ∘ 2)), ν_P(0)}, we have that P is not a PFCUPF of .

We obtain a diagram of the generalization of new PFSs in UP-algebras, which is shown in Figure 1.

If F is an FS in , then (f_F, f_F̃) is a PFS in . Indeed, for all ,

0≤(fF(x1))2+(fF˜(x1))2=(fF(x1))2+(1-fF(x1))2≤fF(x1)+1-2fF(x1)+(fF(x1))2≤fF(x1)+1-2fF(x1)+fF(x1)=1.

Theorem 2.24

Let F be an FS in . Then the following statements hold:

(1) F is an FIUPF of if and only if (f_F, f_F̃) is a PFIUPF of ,
(2) F is an FCUPF of if and only if (f_F, f_F̃) is a PFCUPF of ,
(3) F is an FSUPF of if and only if (f_F, f_F̃) is a PFSUPF of .

Proof

(1) Assuming that F is an FIUPF of , then, for all x₁, ,

fF(0)≥fF(x1),(by (15))fF˜(0)≤fF˜(x1),(by Proposition 1.5(1))fF(x1∘x3)≥min{fF(x1∘(x2∘x3)),fF(x1∘x2)},(by (16))

and

fF˜(x1∘x3)=1-fF(x1∘x3) ≤1-min{fF(x1∘(x2∘x3)),fF(x1∘x2)}(by (16))=max{fF˜(x1∘(x2∘x3)),fF˜(x1∘x2)}.(by Proposition 1.5(2))

This implies that (f_F, f_F̃) is a PFIUPF of .

Conversely, assuming that (f_F, f_F̃) is a PFIUPF of , then, F satisfies conditions (24) and (32). Hence, F is an FIUPF of .

(2) Assuming that F is an FCUPF of , then, for all x₁, ,

fF(0)≥fF(x1),(by (15))fF˜(0)≤fF˜(x1),(by Proposition 1.5(1))fF(x2) ≥min{fF(x1∘((x2∘x3)∘x2)),fF(x1)}(by (17))

and

fF˜(x2)=1-fF(x2) ≤1-min{fF(x1∘((x2∘x3)∘x2),fF(x1)}(by (17))=max{fF˜(x1∘((x2∘x3)∘x2),fF˜(x1)}.(by Proposition 1.5(2))

This implies that (f_F, f_F̃) is a PFCUPF of .

Conversely, assuming that (f_F, f_F̃) is a PFCUPF of , then, F satisfies conditions (24) and (34). Hence, F is an FCUPF of .

(3) Assuming that F is an FSUPF of , then, for all x₁, ,

fF(0)≥fF(x1),(by (15))fF˜(0)≤fF˜(x1),(by Proposition 1.5(1))fF(((x3∘x2)∘x2)∘x3) ≥min{fF(x1∘(x2∘x3)),fF(x1)},(by (18))

and

fF˜(((x3∘x2)∘x2)∘x3)=1-fF(((x3∘x2)∘x2)∘x3) ≤1-min{fF(x1∘(x2∘x3)),fF(x1)}(by (18))=max{fF˜(x1∘(x2∘x3)),fF˜(x1)}.(by Proposition 1.5(2))

This implies that (f_F, f_F̃) is a PFSUPF of .

Conversely, assuming that (f_F, f_F̃) is a PFSUPF of , then, F satisfies conditions (24) and (36). Hence, F is an FSUPF of .

3. Sufficient Conditions for PFSs in UP-Algebras

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Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

In this section, we present some conditions for studying the generalizations of new PFSs and original PFSs in UP-algebras.

Theorem 3.1

If P is a PFUPI of satisfying

(38)(∀x1,x2,x3∈U) (μP(x1∘(x2∘x3))≥μP(x2)⇒μP(x2)≥μP(x1∘x2),νP(x1∘(x2∘x3))≤νP(x2)⇒νP(x2)≤νP(x1∘x2),)

then P is a PFIUPF of .

Proof

Let P = (μ_P, ν_P) be a PFUPI of satisfying (38). Then P satisfies conditions (24) and (25). In the case of μ_P(x₁∘ (x₂ ∘ x₃)) < μ_P(x₂), ν_P(x₁ ∘ (x₂ ∘ x₃)) > ν_P(x₂) can easily be verified. Next, let x₁, x₂, ,

μP(x1∘x3)≥min{μP(x1∘(x2∘x3)),μP(x2)}(by (28))≥min{μP(x1∘(x2∘x3)),μP(x1∘x2)},(by (38) for μP)

and

νP(x1∘x3)≥max{νP(x1∘(x2∘x3)),νP(x2)}(by (29))≥max{νP(x1∘(x2∘x3)),νP(x1∘x2)},(by (38) for νP)

Therefore, P is a PFIUPF of .

Theorem 3.2

If P is a PFUPI of satisfying

(39)(∀x1,x2,x3∈U) (μP(x1∘x2)≥μP(x1∘((x1∘x2)∘x3))⇒μP(x1∘((x1∘x2)∘x3))≥μP(x1∘(x2∘x3)),νP(x1∘x2)≤νP(x1∘((x1∘x2)∘x3))⇒νP(x1∘((x1∘x2)∘x3))≤νP(x1∘(x2∘x3)),)

then P is a PFIUPF of .

Proof

Let P = (μ_P, ν_P) be a PFUPI of satisfying (39). Then P satisfies conditions (24) and (25). In the case of μ_P(x₁∘ x₂) < μ_P(x₁ ∘((x₁ ∘x₂) ∘x₃)), ν_P(x₁ ∘x₂) > ν_P(x₁ ∘((x₁ ∘ x₂) ∘ x₃)) can easily be verified. Next, let x₁, x₂, ,

μP(x1∘x3)≥min{μP(x1∘((x1∘x2)∘x3)),μP(x1∘x2)}(by (28))≥min{μP(x1∘(x2∘x3)),μP(x1∘x2)},(by (39) for μP)

and

νP(x1∘x3)≤max{νP((x1∘x2)∘z)),νP(x1∘x2)}(by (29))≤max{νP(x1∘(x2∘x3)),νP(x1∘x2)}.(by (39) for νP)

Therefore, P is a PFIUPF of .

Theorem 3.3

If P is a PFUPF of satisfying

(40)(∀x1,x2,x3∈U) (μP(x1)≥μP(x1∘x2)⇒μP(x1∘x2)≥μP(x1∘((x2∘x3)∘x2)),νP(x1)≤νP(x1∘x2)⇒νP(x1∘x2)≤νP(x1∘((x2∘x3)∘x2)),)

then P is a PFCUPF of .

Proof

Let P = (μ_P, ν_P) be a PFUPF of satisfying (40). Then P satisfies conditions (24) and (25). In the case of μ_P(x₁) < μ_P(x₁ ∘ x₂), ν_P(x₁) > ν_P(x₁ ∘ x₂) can easily be verified. Next, let x₁, x₂, ,

μP(x2)≥min{μP(x1∘x2),μP(x1)}(by (26))≥min{μP(x1∘((x2∘x3)∘x2)),μP(x1)},(by (40) for μP)

and

νP(x2)≤max{νP(x1∘x2),νP(x1)}(by (27))≤max{νP(x1∘((x2∘x3)∘x2)),νP(x1)}.(by (40) for νP)

Therefore, P is a PFCUPF of .

Theorem 3.4

If P is a PFUPF of satisfying

(41)(∀x1,x2,x3∈U) (μP(x1)≥μP(x1∘(((x3∘x2)∘x2)∘x3))⇒μP(x1∘(((x3∘x2)∘x2)∘x3))≥μP(x1∘(x2∘x3)),νP(x1)≤νP(x1∘(((x3∘x2)∘x2)∘x3))⇒νP(x1∘(((x3∘x2)∘x2)∘x3))≤νP(x1∘(x2∘x3)),)

then P is a PFSUPF of .

Proof

Let P = (μ_P, ν_P) be a PFUPF of satisfying (41). Then P satisfies conditions (24) and (25). In the case of μ_P(x₁) < μ_P(x₁ ∘ (((x₃ ∘ x₂) ∘ x₂) ∘ x₃)), ν_P(x₁) > ν_P(x₁ ∘ (((x₃ ∘ x₂) ∘ x₂) ∘ x₃)) can easily be verified. Next, let x₁, x₂, ,

μP(((x3∘x2)∘x2)∘x3)≥min{μP(x1∘(((x3∘x2)∘x2)∘x3)),μP(x1)}(by (26))≥min{μP(x1∘(x2∘x3)),μP(x1)},(by (41) for μP)

and

νP(((x3∘x2)∘x2)∘x3)≤max{νP(x1∘(((x3∘x2)∘x2)∘x3)),νP(x1)}(by (27))≤max{νP(x1∘(x2∘x3)),νP(x1)}.(by (41) for νP)

Therefore, P is a PFSUPF of .

Theorem 3.5

If P is a PFS in satisfying

(∀x0,x1,x2,x3∈U) (x0≤x1∘(x2∘x3)⇒{μP(x1∘x3)≥min{μP(x0),μP(x1∘x2)},νP(x1∘x3)≤max{νP(x0),νP(x1∘x2)},)

then P is a PFIUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (by (UP-2)). Let . By (UP-3), we have that x₁ ∘ (0 ∘ (x₁ ∘ 0)) = 0; that is, x₁ ≤ 0 ∘ (x₁ ∘ 0). It follows from (by (UP-2)) that

μP(0)=μP(0∘0) ≥min{μP(x1),μP(0∘x1)} =min{μP(x1,μP(x1)} =μP(x1),(by (UP-2))νP(0)=νP(0∘0) ≤max{νP(x1),νP(0∘x1)} =max{νP(x1),νP(x1)} =νP(x1).(by (UP-2))

Next, let x₁, x₂, . By (1), we have that (x₁ ∘(x₂ ∘x₃))∘ (x₁ ∘ (x₂ ∘ x₃)) = 0, that is, x₁ ∘ (x₂ ∘ x₃) ≤ x₁ ∘ (x₂ ∘ x₃). It follows from (by (UP-2)) that

μP(x1∘x3)≥min{μP(x1∘(x2∘x3)),μP(x1∘x2)},νP(x1∘x3)≤max{νP(x1∘(x2∘x3)),νP(x1∘x2)}.

Therefore, P is a PFIUPF of .

Theorem 3.6

If P is a PFS in satisfying

(42)(∀x0,x1,x2,x3∈U) (x0≤x1∘((x2∘x3)∘x2⇒{μP(x2)≥min{μP(a),μP(x1)},νP(x2)≤max{νP(a),νP(x1)},)

then P is a PFCUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (42). Let . By (UP-3), we have that x₁ ∘ (x₁ ∘ ((0 ∘ x₁) ∘ 0)) = 0; that is, x₁ ≤ x₁ ∘ ((0 ∘ x₁) ∘ 0). It follows from (42) that

μP(0)≥min{μP(x1),μP(x1)}=μP(x1),νP(0)≤max{νP(x1),νP(x1)}=νP(x1).

Next, let x₁, x₂, . By (1), we have that (x₁ ∘((x₂ ∘x₃)∘ x₂))∘(x₁ ∘((x₂ ∘x₃)∘x₂)) = 0, that is, x₁ ∘((x₂ ∘x₃)∘x₂) ≤ x₁ ∘ ((x₂ ∘ x₃) ∘ x₂). It follows from (42) that

μP(x2)≥min{μP(x1∘((x2∘x3)∘x2)),μP(x1)},νP(x2)≤max{νP(x1∘((x2∘x3)∘x2)),νP(x1)}.

Therefore, P is a PFCUPF of .

Theorem 3.7

If P is a PFS in satisfying (26), (27), and

(43)(∀x1,x2,x3∈U) (μP(x1∘((x2∘x3)∘x2))≥μP((x1∘((x2∘x3)∘x2))∘x2)⇒μP((x1∘((x2∘x3)∘x2))∘x2)≥μP(x1),νP(x1∘((x2∘x3)∘x2))≤νP((x1∘((x2∘x3)∘x2))∘x2)⇒νP((x1∘((x2∘x3)∘x2))∘x2)≤νP(x1),)

then P is a PFCUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (26), (27), and (43). Let . By (UP-2) and (UP-3), we have that

μP(x1∘((0∘x1)∘0))=μP(0)≥μP(0)=μP((x1∘((0∘x1)∘0))∘0),νP(x1∘((0∘x1)∘0))=νP(0)≤νP(0)=νP((x1∘((0∘x1)∘0))∘0).

It follows from (43) that

μP(0)=μP((x1∘((0∘x1)∘0))∘0)≥μP(x1),νP(0)=νP((x1∘((0∘x1)∘0))∘0)≤νP(x1).

Thus, P satisfies conditions (24) and (25). In the case of μ_P(x₁∘ ((x₂ ∘ x₃) ∘ x₂)) < μ_P((x₁ ∘ ((x₂ ∘ x₃) ∘ x₂)) ∘ x₂), ν_P(x₁ ∘ ((x₂ ∘ x₃) ∘ x₂)) > ν_P((x₁ ∘ ((x₂ ∘ x₃) ∘ x₂)) ∘ x₂) can easily be verified. Next, let x₁, x₂, . Then

μP(x2)≥min{μP((x1∘((x2∘x3)∘x2))∘x2),μP(x1∘((x2∘x3)∘x2))}(by (26))=min{μP(x1∘((x2∘x3)∘x2)),μP(x1)},(by (43) for μP)νP(x2)≤max{νP((x1∘((x2∘x3)∘x2))∘x2),νP(x1∘((x2∘x3)∘x2))}(by (27))=max{νP(x1∘((x2∘x3)∘x2)),νP(x1)}.(by (43) for νP)

Therefore, P is a PFCUPF of .

Theorem 3.8

If P is a PFS in satisfying

(44)(∀x0,x1,x2,x3∈U) (x0≤x1∘(x2∘x3)⇒{μP(((x3∘x2)∘x2)∘x3≥min{μP(x0),μP(x1)},νP(((x3∘x2)∘x2)∘x3≤max{νP(x0),νP(x1)},)

then P is a PFSUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (44). Let . By (UP-3), we have that x₁ ∘ (x₁ ∘ (x₁ ∘ 0)) = 0; that is, x₁ ≤ x₁ ∘ (x₁ ∘ 0). It follows from (44) that:

μP(0)=μP(((0∘x1)∘x1)∘0 ≥min{μP(x1),μP(x1)}=μP(x1),(by (UP-2))νP(0)=νP(((0∘x1)∘x1)∘0 ≤max{νP(x1),νP(x1)}=νP(x1).(by (UP-2))

Next, let x₁, x₂, . By (1), we have that (x₁ ∘ (x₂ ∘x₃)) ∘ (x₁ ∘ (x₂ ∘ x₃)) = 0; that is, x₁ ∘ (x₂ ∘ x₃) ≤ x₁ ∘ (x₂ ∘ x₃). It follows from (44) that:

μP(((x3∘x2)∘x2)∘x3≥min{μP(x1∘(x2∘x3)),μP(x1)},νP(((x3∘x2)∘x2)∘x3≤max{νP(x1∘(x2∘x3)),νP(x1)}.

Therefore, P is a PFSUPF of .

Theorem 3.9

If P is a PFS in satisfying (26), (27), and

(45)(∀x1,x2,x3∈U) (μP(x1∘(x2∘x3))≥μP((x1∘(x2∘x3))∘(((x3∘x2)∘x2)∘x3)⇒μP((x1∘(x2∘x3))∘(((x3∘x2)∘x2)∘x3)≥μP(x1),νP(x1∘(x2∘x3))≤νP((x1∘(x2∘x3))∘(((x3∘x2)∘x2)∘x3)⇒νP((x1∘((x2∘x3))∘(((x3∘x2)∘x2)∘x3)≤νP(x1),)

then P is a PFSUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (26), (27), and (45). Let . By (UP-2) and (UP-3), we have that

μP(x1∘(x1∘0))=μP(0)≥μP(0)=μP((x1∘(x1∘0))∘(((0∘x1)∘x1)∘0)),νP(x1∘(x1∘0))=νP(0)≤νP(0)=νP((x1∘(x1∘0))∘(((0∘x1)∘x1)∘0)).

It follows from (45) that

μP(0)=μP((x1∘(x1∘0))∘(((0∘x1)∘x1)∘0)≥μP(x1),νP(0)=νP((x1∘(x1∘0))∘(((0∘x1)∘x1)∘0)≤νP(x1).

Thus, P satisfies conditions (24) and (25). In the case of μ_P(x₁∘ (x₂ ∘ x₃)) < μ_P((x₁ ∘ (x₂ ∘ x₃)) ∘ (((x₃ ∘ x₂) ∘ x₂) ∘ x₃), ν_P(x₁ ∘(x₂ ∘x₃)) > ν_P((x₁ ∘(x₂ ∘x₃))∘(((x₃ ∘x₂)∘x₂)∘x₃) can easily be verified. Next, let x₁, x₂, . Then

μP(((x3∘x2)∘x2)∘x3) ≥min{μP((x1∘(x2∘x3)) ∘(((x3∘x2)∘x2)∘x3),μP(x1∘(x2∘x3))}(by (26))=min{μP(x1∘(x2∘x3)),μP(x1)},(by (45) for μP)νP(((x3∘x2)∘x2)∘x3) ≤max{νP((x1∘(x2∘x3)) ∘(((x3∘x2)∘x2)∘x3),νP(x1∘(x2∘x3))}(by (27))=max{νP(x1∘(x2∘x3)),νP(x1)}.(by (45) for μP)

Therefore, P is a PFSUPF of .

We obtain a diagram of the sufficient conditions for new PFSs in UP-algebras, which is shown in Figure 2.

4. Approximations

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Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

Let ρ be an equivalence relation on a non-empty set . If , then the ρ-class of x₁ is the set (x₁)_ρ defined as follows:

(x1)ρ={x2∈U∣(x1,x2)∈ρ}.

An equivalence relation ρ on is called a congruence relation if:

(∀x1,x2,x3∈U) ((x1,x2)∈ρ⇒(x1∘x3,x2∘x3)∈ρ,(x3∘x1,x3∘x2)∈ρ).

For the non-empty subsets A and B of , we denote

AB=A∘B={x1∘x2∣x1∈A and x2∈B}.

If ρ is a congruence on , then

(∀x1,x2∈U) ((x1)ρ(x2)ρ⊆(x1∘x2)ρ).(see [16])

A congruence relation ρ on is said to be complete if

(∀x1,x2∈U) ((x1)ρ(x2)ρ=(x1∘x2)ρ).

Definition 4.1

Let ρ be an equivalence relation on a non-empty set and P = (μ_P, ν_P) a PFS in . The upper approximation is defined as

ρ+(P)={(x1,μ¯P(x1),ν¯P(x1))∣x1∈U},

where μ¯P(x1)=supa∈(x1)ρ{μP(a)} and ν¯P(x1)=infa∈(x1)ρ{νP(a)}.

The lower approximation is defined as

ρ-(P)={(x1,μ_P(x1),ν_P(x1))∣x1∈U},

where μ_P(x1)=infa∈(x1)ρ{μP(a)} and ν_P(x1)=supa∈(x1)ρ{νP(a)}.

It is easy to prove that ρ⁺(P) and ρ⁻(P) are PFSs in . Thus, we can denote the upper and lower approximations by ρ⁺(P) = (μ̄_P, ν̄_P) and ρ⁻(P) = (μ_P, ν_P), respectively.

Proposition 4.2

Let P = (μ_P, ν_P) and Q = (μ_Q, ν_Q) be PFSs in . If ρ is an equivalence relation on , then the following statements hold:

ρ⁻(P) ⊆ P ⊆ ρ⁺(P),
P ⊆ Q ⇒ ρ⁺(P) ⊆ ρ⁺(Q), ρ⁻(P) ⊆ ρ⁻(Q),
ρ⁺(P ∪ Q) = ρ⁺(P) ∪ ρ⁺(Q),
ρ⁺(P ∩ Q) ⊆ ρ⁺(P) ∩ ρ⁺(Q),
ρ⁻(P ∪ Q) ⊇ ρ⁻(P) ∪ ρ⁻(Q),
ρ⁻(P ∩ Q) = ρ⁻(P) ∩ ρ⁻(Q).

Proof

Let ρ be an equivalence relation of .

(1) Then for all ,

μ_P(x1)=infa∈(x1)ρ{μP(a)}≤μP(x1)≤supa∈(x1)ρ{μP(a)}=μ¯P(x1),

and

ν_P(x1)=supa∈(x1)ρ{νP(a)}≥νP(x1)≥infa∈(x1)ρ{νP(a)}=ν¯P(x1).

By Definition 2.2 (1), we have that ρ⁻(P) ⊆ P ⊆ ρ⁺(P).

(2) If P ⊆ Q, then μ_P(x₁) ≤ μ_Q(x₁) and ν_P(x₁) ≥ ν_Q(x₁) for all . We consider

μ¯P(x1)=supa∈(x1)ρ{μP(a)} ≤supa∈(x1)ρ{μQ(a)}(by Proposition 1.6 (6))=μ¯Q(x1), ν¯P(x1)=infa∈(x1)ρ{μP(a)} ≥infa∈(x1)ρ{μQ(a)}(by Proposition 1.6 (8))=ν¯Q(x1), μ_P(x1)=infa∈(x1)ρ{μP(a)} ≤infa∈(x1)ρ{μQ(a)}(by Proposition 1.6 (8))μ_Q(x1),

and

ν_P(x1)=supa∈(x1)ρ{μP(a)} ≥supa∈(x1)ρ{μQ(a)}(by Proposition 1.6 (6))ν_Q(x1).

By Definition 2.2 (1), we have that ρ⁺(P) ⊆ ρ⁺(Q) and ρ⁻(P) ⊆ ρ⁻(Q).

(3) By Definition 2.2 (3), we have that P∪Q = (μ_P_∪_Q, ν_P_∪_Q). Then

ρ+(P∪Q)=(μ¯P∪Q,ν¯P∪Q),

and

ρ+(P)∪ρ+(Q)=(μ¯P∪μ¯Q,ν¯P∩ν¯Q).

Thus, for all ,

μ¯P∪Q(x1)=supa∈(x1)ρ{μP∪Q(a)} =supa∈(x1)ρ{μP∪μQ(a)} =supa∈(x1)ρ{max{μP(a),μQ(a)}} =max{supa∈(x1)ρ{μP(a)},supa∈(x1)ρ{μQ(a)}}(by Proposition 1.6 (2))=max{μ¯P(x1),μ¯Q(x1)} =(μ¯P∪μ¯Q) (x1),

and

ν¯P∪Q(x1)=infa∈(x1)ρ{νP∪Q(a)} =infa∈(x1)ρ{νP∩νQ(a)} =infa∈(x1)ρ{min{νP(a),νQ(a)}} =min{infa∈(x1)ρ{νP(a)},infa∈(x1)ρ{νQ(a)}}(by Proposition 1.6 (1))=min{ν¯P(x1),ν¯Q(x1)} =(ν¯P∩ν¯Q) (x1).

Hence, ρ⁺(P ∪ Q) = ρ⁺(P) ∪ ρ⁺(Q).

(4) By Definition 2.2 (4), we have that P∩Q = (μ_P_∩_Q, ν_P_∩_Q). Then,

ρ+(P∩Q)=(μ¯P∩Q,ν¯P∩Q),

and

ρ+(P)∩ρ+(Q)=(μ¯P∩μ¯Q,ν¯P∪ν¯Q).

Thus, for all ,

μ¯P∩Q(x1)=supa∈(x1)ρ{μP∩Q(a)} =supa∈(x1)ρ{μP∩μQ(a)} =supa∈(x1)ρ{min{μP(a),μQ(a)}} ≤min{supa∈(x1)ρ{μP(a)},supa∈(x1)ρ{μQ(a)}}(by Proposition 1.6 (4))=min{μ¯P(x1),μ¯Q(x1)} =(μ¯P∩μ¯Q) (x1),

and

ν¯P∩Q(x1)=infa∈(x1)ρ{νP∩Q(a)} =infa∈(x1)ρ{νP∪νQ(a)} =infa∈(x1)ρ{max{νP(a),νQ(a)}} ≥max{infa∈(x1)ρ{νP(a)},infa∈(x1)ρ{νQ(a)}}(by Proposition 1.6 (3))=max{ν¯P(x1),ν¯Q(x1)} =(ν¯P∪ν¯Q) (x1).

Therefore, ρ⁺(P ∩ Q) ⊆ ρ⁺(P) ∩ ρ⁺(Q).

(5) By Definition 2.2 (3), we have that P∪Q = (μ_P_∪_Q, ν_P_∪_Q). Then

ρ-(P∪Q)=(μ_P∪Q,ν_P∪Q),

and

ρ-(P)∪ρ-(Q)=(μ_P∪μ_Q,ν_P∩ν_Q).

Thus, for all ,

μ_P∪Q(x1)=infa∈(x1)ρ{μP∪Q(a)} =infa∈(x1)ρ{(μP∪μQ) (a)} =infa∈(x1)ρ{max{μP(a),μQ(a)}} ≥max{infa∈(x1)ρ{μP(a)},infa∈(x1)ρ{μQ(a)}}(by Proposition 1.6 (3))=max{μ_P(x1),μ_Q(x1)} =(μ_P∪μ_Q) (x1),

and

ν_P∪Q(x1)=supa∈(x1)ρ{νP∪Q(a)} =supa∈(x1)ρ{(νP∩νQ) (a)} =supa∈(x1)ρ{min{νP(a),νQ(a)}} ≤min{supa∈(x1)ρ{νP(a)},supa∈(x1)ρ{νQ(a)}}(by Proposition 1.6 (4))=min{ν_P(x1),ν_Q(x1)} =(ν_P∩ν_Q) (x1).

Hence, ρ⁻(P ∪ Q) ⊇ ρ⁻(P) ∪ ρ⁻(Q).

(6) By Definition 2.2 (4), we have that P ∩ Q = (μ_P_∩_Q, ν_P_∩_Q). Then

ρ-(P∩Q)=(μ_P∩Q,ν_P∩Q),

and

ρ-(P)∩ρ-(Q)=(μ_P∩μ_Q,ν_P∪ν_Q).

Thus, for all ,

μ_P∩Q(x1)=infa∈(x1)ρ{μP∩Q(a)} =infa∈(x1)ρ{(μP∩μQ) (a)} =infa∈(x1)ρ{min{μP(a),μQ(a)}} =min{infa∈(x1)ρ{μP(a)},infa∈(x1)ρ{μQ(a)}}(by Proposition 1.6 (1))=min{μ_P(x1),μ_Q(x1)} =(μ_P∩μ_Q) (x1),

and

ν_P∩Q(x1)=supa∈(x1)ρ{νP∩Q(a)} =supa∈(x1)ρ{(νP∪νQ) (a)} =supa∈(x1)ρ{max{νP(a),νQ(a)}} =max{supa∈(x1)ρ{νP(a)},supa∈(x1)ρ{νQ(a)}}(by Proposition 1.6 (2))=max{ν_P(x1),ν_Q(x1)} =(ν_P∪ν_Q) (x1).

Hence, ρ⁻(P ∩ Q) = ρ⁻(P) ∩ ρ⁻(Q).

Theorem 4.3

Let ρ be a congruence relation on and P = (μ_P, ν_P) be a PFS in . Then, the following statements hold:

(1) if P is a PFIUPF of , (0)_ρ = {0}, and ρ is complete, then ρ⁻(P) is a PFIUPF of ,
(2) if P is a PFCUPF of and (0)_ρ = {0}, then ρ⁻(P) is a PFCUPF of ,
(3) if P is a PFSUPF of , (0)_ρ = {0}, and ρ is complete, then ρ⁻(P) is a PFSUPF of .

Proof

(1) Assuming that P is a PFIUPF of , (0)_ρ = {0}, and ρ is complete, then, for all x₁, x₂, ,

μ_P(0)=infa∈(0)ρ{μP(a)}=μP(0) ≥μP(x1)≥infb∈(x1)ρ{μP(b)} =μ_P(x1), ν_P(0)=supa∈(0)ρ{νP(a)}=νP(0) ≤νP(x1)≤supb∈(x1)ρ{νP(b)} =ν_P(x1), μ_P(x1∘x3)=infd∈(x1∘x3)ρ{μP(d)} =infd∈(x1)ρ(x3)ρ{μP(d)}(by ρ is complete)=infa∘c∈(x1)ρ(x3)ρ{μP(a∘c)} ≥infa∘(b∘c)∈(x1)ρ((x2)ρ(x3)ρ),a∘b∈(x1)ρ(x2)ρ{min{μP(a∘(b∘c)),μP(a∘b)}}(by (32))=infa∘(b∘c)∈(x1∘(x2∘x3))ρ,a∘b∈(x1∘x2)ρ{min{μP(a∘(b∘c)),μP(a∘b)}}(by ρ is complete)=min{infa∘(b∘c)∈(x1∘(x2∘x3))ρμP(a∘(b∘c)),infa∘b∈(x1∘x2)ρ{μP(a∘b)}}(by Proposition 1.6 (1))=min{μ_P(x1∘(x2∘x3)),μ_P(x1∘x2)},

and

ν_P(x1∘x3)=supd∈(x1∘x3)ρ{νP(d)} =supd∈(x1)ρ(x3)ρ{νP(d)}(by ρ is complete)=supa∘c∈(x1)ρ(x3)ρ{νP(a∘c)} ≤supa∘(b∘c)∈(x1)ρ((x2)ρ(x3)ρ),a∘b∈(x1)ρ(x2)ρ{max{νP(a∘(b∘c)),νP(a∘b)}}(by (33))=supa∘(b∘c)∈(x1∘(x2∘x3))ρ,a∘b∈(x1∘x2)ρ{max{νP(a∘(b∘c)),νP(a∘b)}}(by ρ is complete)=max{supa∘(b∘c)∈(x1∘(x2∘x3))ρνP(a∘(b∘c)),supa∘b∈(x1∘x2)ρνP(a∘b)}}(by Proposition 1.6 (2))=max{ν_P(x1∘(x2∘x3)),ν_P(x1∘x2)}.

Hence, ρ⁻(P) is a PFIUPF of .

(2) Assuming that P is a PFCUPF of and (0)_ρ = {0}, then, for all x₁, x₂, ,

μ_P(0)=infa∈(0)ρ{μP(a)}=μP(0)≥μP(x1) ≥infb∈(x1)ρ{μP(b)} =μ_P(x1), ν_P(0)=supa∈(0)ρ{νP(a)}=νP(0)≤νP(x1) ≤supb∈(x1)ρ{νP(b)} =ν_P(x1), μ_P(x2)=infb∈(x2)ρ{μP(b)} ≥infa∘((b∘c)∘b)∈(x1)ρ(((x2)ρ(x3)ρ)(x2)ρ),a∈(x1)ρ{min{μP(a∘((b∘c)∘b)),μP(a)}}(by (34))≥infa∘((b∘c)∘b)∈(x1∘((x2∘x3)∘x2))ρ,a∈(x1)ρ{min{μP(a∘((b∘c)∘b)),μP(a)}}(by ρ is congruence)=min{infa∘((b∘c)∘b)∈(x1∘((x2∘x3)∘x2))ρ{μP(a∘((b∘c)∘b))},infa∈(x1)ρ{μP(a)}}(by Proposition 1.6 (1))=min{μ_P(x1∘((x2∘x3)∘x2)),μ_P(x1)},

and

ν_P(x2)=supb∈(x2)ρ{νP(b)} ≤supa∘((b∘c)∘b)∈(x1)ρ(((x2)ρ(x3)ρ) (x2)ρ),a∈(x1)ρ{max{νP(a∘((b∘c)∘b)),νP(a)}}(by (35))≤supa∘((b∘c)∘b)∈(x1∘((x2∘x3)∘x2))ρ,a∈(x1)ρ{max{νP(a∘((b∘c)∘b)),νP(a)}}(by ρ is congruence)=max{supa∘((b∘c)∘b)∈(x1∘((x2∘x3)∘x2))ρ{νP(a∘((b∘c)∘b))},supa∈(x1)ρ{νP(a)}}(by Proposition 1.6 (2))=max{ν_P(x1∘((x2∘x3)∘x2)),ν_P(x1)}.

Hence, ρ⁻(P) is a PFCUPF of .

(3) Assuming that P is a PFSUPF of , (0)_ρ = {0}, and ρ is complete, then, for all x₁, x₂, ,

μ_P(0)=infa∈(0)ρ{μP(a)}=μP(0) ≥μP(x1)≥infb∈(x1)ρ{μP(b)} =μ_P(x1), ν_P(0)=supa∈(0)ρ{νP(a)}=νP(0) ≤νP(x1)≤supb∈(x1)ρ{νP(b)} =ν_P(x1), μ_P(((x3∘x2)∘x2)∘x3)=infd∈(((x3∘x2)∘x2)∘x3)ρ{μP(d)} =infd∈(((x3)ρ(x2)ρ) (x2)ρ) (x3)ρ{μP(d)}(by ρ is complete)=inf((c∘b)∘b)∘c∈(((x3)ρ(x2)ρ) (x2)ρ) (x3)ρ{μP(((c∘b)∘b)∘c)} ≥infa∘(b∘c)∈(x1)ρ((x2)ρ(x3)ρ),a∈(x1)ρ{min{μP(a∘(b∘c)),μP(a)}}(by (36))=infa∘(b∘c)∈(x1∘(x2∘x3))ρ,a∘b∈(x1∘x2)ρ{min{μP(a∘(b∘c)),μP(a∘b)}}(by ρ is complete)=min{infa∘(b∘c)∈(x1∘(x2∘x3))ρμP(a∘(b∘c)),infa∈(x1)ρ{μP(a)}}(by Proposition 1.6 (1))=min{μ_P(x1∘(x2∘x3)),μ_P(x1)},

and

ν_P(((x3∘x2)∘x2)∘x3=supd∈(((x3∘x2)∘x2)∘x3)ρ{νP(d)} =supd∈(((x3)ρ(x2)ρ) (x2)ρ) (x3)ρ{νP(d)}(by ρ is complete)=sup((c∘b)∘b)∘c∈(((x3)ρ(x2)ρ) (x2)ρ) (x3)ρ{νP(((c∘b)∘b)∘c)} ≤supa∘(b∘c)∈(x1)ρ((x2)ρ(x3)ρ),a∈(x1)ρ{max{νP(a∘(b∘c)),νP(a)}}(by (37))=supa∘(b∘c)∈(x1∘(x2∘x3))ρ,a∈(x1)ρ{max{νP(a∘(b∘c)),νP(a)}}(by ρ is complete)=max{supa∘(b∘c)∈(x1∘(x2∘x3))ρνP(a∘(b∘c)),supa∈(x1)ρνP(a)}}(by Proposition 1.6 (2))=max{ν_P(x1∘(x2∘x3)),ν_P(x1)}.

Hence, ρ⁻(P) is a PFSUPF of .

The following example shows that Theorem 4.3 (1) may not be true if (0)_ρ ≠ {0} and ρ is incomplete.

Example 4.4

Consider a UP-algebra where is defined in Table 25.

If we define a PFS P = (μ_P, ν_P) as presented in Table 26, then P = (μ_P, ν_P) is a PFIUPF of . Let

ρ={(0,0),(1,1),(2,2),(3,3),(0,1),(1,0),(0,3),(3,0)}.

Then ρ is a congruence relation on . Thus,

(0)ρ=(1)ρ=(3)ρ={0,1,3},(2)ρ={2}.

However, ρ is not complete because

{0}={2}{2}=(2)ρ(2)ρ≠(2∘2)ρ=(0)ρ={0,1,3}.

Given that μ_P (0) = min{μ_P(0), μ_P(1), μ_P(3)} = min{1, 0.1, 0.3} = 0.1 ≱ 0.3 = μ_P(2) = μ_P (2) and ν_P(0) = max{ν_P(0), ν_P(1), ν_P(3)} = max{0, 0.5, 0.2} = 0 ≰ 0.2 = ν_P(2) = ν_P(2), we have that ρ⁻(P) is not a PFIUPF of .

The following example shows that Theorem 4.3 (2) may not be true if (0)_ρ = {0} and ρ is not complete.

Example 4.5

Consider a UP-algebra where is defined in Table 27.

If we define a PFS P = (μ_P, ν_P) as presented in Table 28, then P = (μ_P, ν_P) is a PFCUPF of . Let

ρ={(0,0),(1,1),(2,2),(3,3),(0,2),(2,0)}.

Then ρ is a congruence relation on . Thus

(0)ρ=(2)ρ={0,2},(1ρ)={1},(3)ρ={3}.

However, ρ is not complete because

{0}={1}{1}=(1)ρ(1)ρ≠(1∘1)ρ=(0)ρ={0,2}.

Given that μ_P (0) = min{μ_P(0), μ_P(2)} = min{0.8, 0.5} = 0.5 ≱ 0.8 = μ_P(3) = μ_P (3) and ν_P(0) = max{ν_P(0), ν_P(2)} = max{0.2, 0.7} = 0.7 ≰ 0.2 = ν_P(3) = ν_P(3), we have that ρ⁻(P) is not a PFCUPF of .

The following example shows that Theorem 4.3 (3) may not be true if (0)_ρ ≠ {0} and ρ is not complete.

Example 4.6

By Example 4.5, if we define a PFS P = (μ_P, ν_P) as presented in Table 29, then P = (μ_P, ν_P) is a PFSUPF of . Let

ρ={(0,0),(1,1),(2,2),(3,3),(0,2),(2,0)}.

Then ρ is a congruence relation on . Thus,

(0)ρ=(2)ρ={0,2},(1ρ)={1},(3)ρ={3}.

However, ρ is not complete because

{0}={3}{3}=(3)ρ(3)ρ≠(3∘3)ρ=(0)ρ={0,2}.

Given that μ_P (0) = min{μ_P(0), μ_P(2)} = min{1, 0.1} = 0.1 ≱ 0.2 = μ_P(1) = μ_P (1) and ν_P(0) = max{ν_P(0), ν_P(2)} = max{0, 0.9} = 0.9 ≰ 0.4 = ν_P(3) = ν_P(3), we have that ρ⁻(P) is not a PFSUPF of .

Using Theorem 4.3, we discussed the relationship between PFSs and the lower approximations. Next, we studied the relationship between PFSs and the upper approximations. We found that their relationship cannot be proven in the same way as Theorem 4.3. Hence, we assumed that ρ is an equivalence relation on and P = (μ_P, ν_P) is a PFS in . The following three examples show that if P is a PFIUPF (resp., PFCUPF, PFSUPF) of , then the upper approximation ρ⁺(P) is not a PFIUPF (resp., PFCUPF, PFSUPF) in general.

Example 4.7

Consider a UP-algebra where is defined in Table 30.

If we define a PFS P = (μ_P, ν_P) as presented in Table 31, then P = (μ_P, ν_P) is a PFIUPF of . Let

ρ={(0,0),(1,1),(2,2),(3,3),(0,3),(3,0)}.

Then ρ is an equivalence relation on . Thus,

(0)ρ=(3)ρ={0,3},(1ρ)={1},(2)ρ={2}.

Given that μ̄_P(0 ∘ 2) = μ_P(2) = 0.3 ≱ 0.5 = min{ 0.6, 0.5} = min{max{0.6, 0.3}, 0.5} = min{μ̄_P(3), μ_P(1)} = min{μ̄_P(0 ∘ (1 ∘ 2)), μ̄_P(0 ∘ 1)}, we have that ρ⁺(P) is not a PFIUPF of .

Example 4.8

From Example 4.7, if we define a PFS P = (μ_P, ν_P) as presented in Table 32, then P = (μ_P, ν_P) is a PFCUPF of . Let

ρ={(0,0),(1,1),(2,2),(3,3),(0,3),(3,0)}.

Then ρ is an equivalence relation on . Thus,

(0)ρ=(3)ρ={0,3},(1ρ)={1},(2)ρ={2}.

Given that μ̄_P(2) = 0.1 ≱ 0.2 = min{0.8, 0.2} = min{max {0.8, 0.1}, 0.2} = min{μ̄_P(3), μ̄_P(1)} = min{μ̄_P(1 ∘ ((2 ∘ 3) ∘ 2)), μ̄_P(1)}, we find that ρ⁺(P) is not a PFCUPF of .

Example 4.9

From Example 4.7, if we define a PFS P = (μ_P, ν_P) as presented in Table 33, then P = (μ_P, ν_P) is a PFSUPF of . Let

ρ={(0,0),(1,1),(2,2),(3,3),(0,3),(3,0)}.

Then ρ is an equivalence relation on . Thus,

(0)ρ=(3)ρ={0,3},(1ρ)={1},(2)ρ={2}.

Given that μ̄_P(((2 ∘ 0) ∘ 0) ∘ 2) = μ̄_P(2) = 0.2 ≱ 0.8 = min{0.9, 0.8} = min{ max{0.9, 0.2}, 0.8} = min{ μ̄_P(3), μ̄_P(1)} = min{μ̄_P(1 ∘ (0 ∘ 2)), μ̄_P(1)}, we have that ρ⁺(P) is not a PFSUPF of .

Open Problem

Is the upper approximation ρ⁺(P) a PFIUPF (resp., PFCUPF, PFSUPF) of if P is a PFIUPF (resp., PFCUPF, PFSUPF) of and ρ is congruent?

5. Conclusion and Future Works

Go to

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

In this study, we introduced a new concept of PFSs in UP-algebras and explored three types of PFSs in UP-algebras, namely PFIUPFs, PFCUPFs, and PFSUPFs. Furthermore, we discovered the sufficient conditions for, and the relationships between some assertions of PFSs and PFIUPFs (resp., PFCUPFs, and PFSUPFs) in UP-algebras and studied the upper and lower approximations of PFSs. We proved that the concept of PFUPSs is a generalization of PFNUPFs, PFNUPFs is a generalization of PFUPFs, PFUPFs is a generalization of PFUPIs, PFUPFs is a generalization of PFCUPFs, PFUPFs is a generalization of PFSUPFs, PFUPIs is a generalization of PFIUPFs, PFIUPFs is a generalization of PFSUPIs, PFCUPFs is a generalization of PFSUPIs, and PFSUPFs is a generalization of PFSUPIs. Accordingly, we obtained a diagram of the generalization of PFSs in UP-algebras, which is shown in Figure 3.

Some important topics for consideration in our future study of UP-algebras include:

(1) to study the roughness of PFSs, as defined by Pawlak [12],
(2) to study the soft set theory of PFSs based on the concept of fuzzy soft sets defined by Maji et al. [?],
(3) to extend the study results of PFSs in UP-algebras to Fermatean fuzzy sets, which are defined by Senapati and Yager [?], and
(4) to apply the results from this study to our research on PFS operators, based on Fermatean fuzzy sets guidelines in the studies [?, ?, ?].

Acknowledgments

Go to

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
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. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

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

Figures

Go to

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

Fig. 1.

New PFSs in UP-algebras.

Fig. 2.

Sufficient conditions for new PFSs in UP-algebras.

Fig. 3.

Eight types of PFSs in UP-algebras.

Tables

Go to

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

Table. 1.

Table 1. Cayley table for Example 2.6.

∘	1	2	3	4
0	1	2	3	4
1	0	2	3	4
2	0	0	3	3
3	1	2	0	3
4	1	2	0	0

Table. 2.

Table 2. A PFS for Example 2.6.

	0	1	2	3	4
μ_P	0.8	0.7	0.5	0.3	0.3
ν_P	0	0.2	0.3	0.5	0.5

Table. 3.

Table 3. Cayley table for Example 2.8.

∘	1	2	3	4
0	1	2	3	4
1	0	2	3	4
2	0	0	3	4
3	0	0	0	4
4	1	2	3	0

Table. 4.

Table 4. A PFS for Example 2.8.

	0	1	2	3	4
μ_P	1	0.6	0.6	0.6	0.4
ν_P	0	0	0.1	0.1	0.4

Table. 5.

Table 5. Cayley table for Example 2.10.

∘	1	2	3	4
0	1	2	3	4
1	0	2	2	4
2	0	0	2	4
3	0	0	0	4
4	1	2	3	0

Table. 6.

Table 6. A PFS for Example 2.10.

	0	1	2	3	4
μ_P	0.9	0.5	0.2	0.2	0.2
ν_P	0.3	0.3	0.4	0.4	0.4

Table. 7.

Table 7. Cayley table for Example 2.12.

∘	1	2	3	4
0	1	2	3	4
1	0	2	3	4
2	0	0	3	4
3	0	1	0	4
4	0	0	0	0

Table. 8.

Table 8. A PFS for Example 2.12.

	0	1	2	3	4
μ_P	0.6	0.5	0.2	0.1	0.1
ν_P	0.3	0.4	0.5	0.6	0.8

Table. 9.

Table 9. Cayley table for Example 2.14.

∘	1	2	3	4
0	1	2	3	4
1	0	0	0	4
2	1	0	0	4
3	1	2	0	4
4	1	2	3	0

Table. 10.

Table 10. A PFS for Example 2.14.

	0	1	2	3	4
μ_P	0.5	0.4	0.4	0.4	0.3
ν_P	0.4	0.5	0.5	0.5	0.8

Table. 11.

Table 11. Cayley table for Example 2.15.

∘	1	2	3	4
0	1	2	3	4
1	0	1	2	4
2	0	0	2	4
3	0	0	0	4
4	0	0	2	0

Table. 12.

Table 12. A PFS for Example 2.15.

	0	1	2	3	4
μ_P	0.9	0.9	0.9	0.9	0.3
ν_P	0.3	0.3	0.3	0.3	0.6

Table. 13.

Table 13. Cayley table for Example 2.16.

∘	1	2	3	4
0	1	2	3	4
1	0	1	2	4
2	0	0	1	4
3	0	0	0	4
4	1	2	3	0

Table. 14.

Table 14. A PFS for Example 2.16.

	0	1	2	3	4
μ_P	0.5	0.2	0.2	0.2	0.1
ν_P	0.2	0.6	0.6	0.6	0.8

Table. 15.

Table 15. Cayley table for Example 2.17.

∘	1	2	3
0	1	2	3
1	0	2	2
2	0	0	2
3	0	0	0

Table. 16.

Table 16. A PFS for Example 2.17.

	0	1	2	3
μ_P	0.5	0.2	0.1	0.1
ν_P	0.4	0.7	0.9	0.9

Table. 17.

Table 17. A PFS for Example 2.18.

	0	1	2	3	4
μ_P	0.8	0.1	0.2	0.6	0.1
ν_P	0.1	0.7	0.6	0.2	0.7

Table. 18.

Table 18. A PFS for Example 2.19.

	0	1	2	3	4
μ_P	0.5	0.2	0.3	0.4	0.2
ν_P	0.5	0.9	0.8	0.6	0.9

Table. 19.

Table 19. Cayley table for Example 2.20.

∘	1	2	3	4
0	1	2	3	4
1	0	0	0	0
2	1	0	0	4
3	1	2	0	4
4	1	2	3	0

Table. 20.

Table 20. A PFS for Example 2.20.

	0	1	2	3	4
μ_P	0.7	0.1	0.4	0.6	0.1
ν_P	0.2	0.7	0.6	0.4	0.7

Table. 21.

Table 21. Cayley table for Example 2.21.

∘	1	2	3
0	1	2	3
1	0	2	3
2	0	0	3
3	0	0	0

Table. 22.

Table 22. A PFS for Example 2.21.

	0	1	2	3
μ_P	0.6	0.4	0.2	0.2
ν_P	0.3	0.5	0.9	0.9

Table. 23.

Table 23. A PFS for Example 2.22.

	0	1	2	3
μ_P	0.7	0.7	0.4	0.4
ν_P	0.5	0.5	0.6	0.6

Table. 24.

Table 24. A PFS for Example 2.23.

	0	1	2	3
μ_P	0.5	0.5	0.1	0.1
ν_P	0.6	0.6	0.8	0.8

Table. 25.

Table 25. Cayley table for Example 4.4.

∘	1	2	3
0	1	2	3
1	0	2	0
2	1	0	3
3	1	2	0

Table. 26.

Table 26. A PFS for Example 4.4.

	0	1	2	3
μ_P	1	0.1	0.3	0.3
ν_P	0	0.5	0.2	0.2

Table. 27.

Table 27. Cayley table for Example 4.5.

∘	1	2	3
0	1	2	3
1	0	2	3
2	1	0	3
3	1	2	0

Table. 28.

Table 28. A PFS for Example 4.5.

	0	1	2	3
μ_P	0.8	0.3	0.5	0.8
ν_P	0.2	0.9	0.7	0.2

Table. 29.

Table 29. A PFS for Example 4.6.

	0	1	2	3
μ_P	1	0.2	0.1	0.5
ν_P	0	0.6	0.9	0.4

Table. 30.

Table 30. Cayley table for Example 4.7.

∘	1	2	3
0	1	2	3
1	0	3	3
2	0	0	0
3	1	1	0

Table. 31.

Table 31. A PFS for Example 4.7.

	0	1	2	3
μ_P	0.6	0.5	0.3	0.3
ν_P	0.4	0.5	0.7	0.7

Table. 32.

Table 32. A PFS for Example 4.8.

	0	1	2	3
μ_P	0.8	0.2	0.1	0.1
ν_P	0.2	0.6	0.9	0.9

Table. 33.

Table 33. A PFS for Example 4.9.

	0	1	2	3
μ_P	0.9	0.8	0.2	0.2
ν_P	0.3	0.4	08	0.8

References

Go to

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

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Iseki, K (1980). On BCI-algebras. Mathematics Seminar Notes 8. Kyoto, Japan: Kyoto University, pp. 125-130
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Iampan, A (2017). A new branch of the logical algebra: UP-algebras. Journal of Algebra and Related Topics. 5, 35-54. https://doi.org/10.22124/jart.2017.2403
Akram, M (2021). Spherical fuzzy K-algebras. Journal of Algebraic Hyperstructures and Logical Algebras. 2, 85-98. https://doi.org/10.52547/hatef.jahla.2.3.7
Dar, KH, and Akram, M (2005). On a K-algebra built on a group. Southeast Asian Bulletin of Mathematics. 29, 41-49.
Dar, KH, and Akram, M (2010). Characterization of K-algebras by self maps II. Annals of the University of Craiova-Mathematics and Computer Science Series. 37, 96-103.
Dar, KH, and Akram, M (2004). Characterization of a K(G)-algebra by self maps. Southeast Asian Bulletin of Mathematics. 28, 601-610.
Naghibi, R, Anvariyeh, SM, and Davvaz, B (2021). gK-algebra associated to polygroups. Afrika Matematika. 32, 1409-1420. https://doi.org/10.1007/s13370-021-00908-3
Satirad, A, Chinram, R, and Iampan, A (2020). Four new concepts of extensions of KU/UP-algebras. Missouri Journal of Mathematical Sciences. 32, 138-157. https://doi.org/10.35834/2020/3202138
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Ansari, MA, Koam, ANA, and Haider, A (2019). Rough set theory applied to UP-algebras. Italian Journal of Pure and Applied Mathematics. 2019, 388-402.
Klinseesook, T, Bukok, S, and Iampan, A (2020). Rough set theory applied to UP-algebras. Journal of Information and Optimization Sciences. 41, 705-722. https://doi.org/10.1080/02522667.2018.1552511
Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
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Jun, YB, Akram, M, and Pasha, MA (2005). Intuitionistic fuzzy quasi-associative ideals in BCI-algebras. Southeast Asian Bulletin of Mathematics. 29, 903-914.
Akram, M, and Dar, KH (2005). T-fuzzy ideals in BCI-algebras. International Journal of Mathematics and Mathematical Sciences. 2005. article no 839196
Akram, M, and Zhan, J (2006). On sensible fuzzy ideals of BCK-algebras with respect to a t-conorm. International Journal of Mathematics and Mathematical Sciences. 2006. article no 035930
Hussain, A, Mahmood, T, and Ali, MI (2019). Rough Pythagorean fuzzy ideals in semigroups. Computational and Applied Mathematics. 38. article no 67
Jansi, R, and Mohana, K (2019). Bipolar Pythagorean fuzzy A-ideals of BCI-algebra. International Journal of Innovative Science, Engineering & Technology. 6, 102-109.
Jana, C, Senapati, T, and Pal, M (2019). Pythagorean fuzzy Dombi aggregation operators and its applications in multiple attribute decision-making. International Journal of Intelligent Systems. 34, 2019-2038. https://doi.org/10.1002/int.22125
Senapati, T, and Chen, G (2021). Some novel interval-valued Pythagorean fuzzy aggregation operator based on Hamacher triangular norms and their application in MADM issues. Computational and Applied Mathematics. 40. article no 109
Chinram, R, and Panityakul, T (2020). Rough Pythagorean fuzzy ideals in ternary semigroups. Journal of Mathematics and Computer Science. 20, 302-3012. https://doi.org/10.22436/jmcs.020.04.04
Satirad, A, Chinram, R, and Iampan, A (2021). Pythagorean fuzzy sets in UP-algebras and approximations. AIMS Math. 6, 6002-6032. https://doi.org/10.3934/math.2021354
Ansari, MA, Haidar, A, and Koam, AN (2018). On a graph associated to UP-algebras. Mathematical and Computational Applications. 23. article no 61
Iampan, A (2018). Introducing fully UP-semigroups. Discussiones Mathematicae-General Algebra and Applications. 38, 297-306. https://doi.org/10.7151/dmgaa.1290
Iampan, A (2021). Multipliers and near UP-filters of UP-algebras. Journal of Discrete Mathematical Sciences and Cryptography. 24, 667-680. https://doi.org/10.1080/09720529.2019.1649027
Iampan, A, Songsaeng, M, and Muhiuddin, G (2020). Fuzzy duplex UP-algebras. European Journal of Pure and Applied Mathematics. 13, 459-471. https://doi.org/10.29020/nybg.ejpam.v13i3.3752
Satirad, A, Mosrijai, P, and Iampan, A (2019). Formulas for finding UP-algebras. International Journal of Mathematics and Computer Science. 14, 403-409.
Satirad, A, Mosrijai, P, and Iampan, A (2019). Generalized power UP-algebras. International Journal of Mathematics and Computer Science. 14, 17-25.
Senapati, T, Jun, YB, and Shum, KP (2018). Cubic set structure applied in UP-algebras. Discrete Mathematics, Algorithms and Applications. 10. article no 1850049
Senapati, T, Muhiuddin, G, and Shum, KP (2017). Representation of UP-algebras in interval-valued intuitionistic fuzzy environment. Italian Journal of Pure and Applied Mathematics. 38, 497-517.
Kankaew, P, Yuphaphin, S, Lapo, N, Chinram, R, and Iampan, A (2022). Picture fuzzy set theory applied to UP-algebras. Missouri Journal of Mathematical Sciences. 34, 94-120. https://doi.org/10.35834/2022/3401094
Yager, RR, and Abbasov, AM (2013). Pythagorean membership grades, complex numbers, and decision making. International Journal of Intelligent Systems. 28, 436-452. https://doi.org/10.1002/int.21584
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Mishra, AR, Rani, P, Saha, A, Senapati, T, Hezam, IM, and Yager, RR (2022). Fermatean fuzzy copula aggregation operators and similarity measures-based complex proportional assessment approach for renewable energy source selection. Complex & Intelligent Systems. 8, 5223-5248. https://doi.org/10.1007/s40747-022-00743-4
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Senapati, T, and Yager, RR (2019). Some new operations over Fermatean fuzzy numbers and application of Fermatean fuzzy WPM in multiple criteria decision making. Informatica. 30, 391-412. https://doi.org/10.15388/informatica.2019.211

Biographies

Go to

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works
Acknowledgments
Conflict of Interest
Figures
Tables
References
Biographies

Akarachai Satirad was born in Phayao, Thailand, in 1995. He is a Ph.D. student in the Department of Mathematics, School of Science, University of Phayao, Phayao, Thailand. He obtained his B.Sc. and M.Sc. degrees in mathematics from the University of Phayao, Phayao, Thailand, with his thesis under the advisorship of associate professor Dr. Aiyared Iampan. His areas of interest include fuzzy algebraic structures and logical algebra.

E-mail: akarachai.sa@gmail.com

Ronnason Chinram was born in Ranong, Thailand, in 1975. He obtained his M.Sc. and Ph.D. degrees from Chulalongkorn University, Thailand. From 1997, he has been with the Prince of Songkla University, Thailand, and is currently an associate professor of mathematics. Moreover, he is head at both the Division of Computational Science and the Algebra and Applications Research Unit. He currently has more than 100 research publications in well-reputed international journals. His research interests include semigroup theory, algebraic systems, fuzzy mathematics, and decision-making problems.

E-mail: ronnason.c@psu.ac.th

Pongpun Julatha is a member of the Faculty of Science and Technology, Pibulsongkram Rajabhat University, Thailand. He obtained his B.Sc., M.Sc., 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

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 obtained his B.Sc., M.Sc., and Ph.D. degrees in Mathematics from Naresuan University, Phitsanulok, Thailand, with his thesis under the advisorship of Professor Dr. Manoj Siripitukdet. His areas of interest include the algebraic theory of semigroups, ternary semigroups, Γ-semigroups, lattices, ordered algebraic structures, fuzzy algebraic structures, and logical algebras. He was the founder of the Group for Young Algebraists in the University of Phayao in 2012, and one of the co-founders of the Fuzzy Algebras and Decision-Making Problems Research Unit in 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(1): 56-78

Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.56

Pythagorean Fuzzy Implicative/Comparative/Shift UP-Filters of UP-Algebras with Approximations

Akarachai Satirad¹, Ronnason Chinram², Pongpun Julath³ , and Aiyared Iampan¹

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

Received: April 19, 2022; Revised: September 11, 2022; Accepted: March 6, 2023

Abstract

Keywords: UP-algebra, Pythagorean fuzzy implicative UP-filter, Pythagorean fuzzy comparative UP-filter, Pythagorean fuzzy shift UP-filter

1. Introduction and Preliminaries

Before we proceed, let us examine the definition of UP-algebras.

Definition 1.1 [5]

A UP-algebra is defined as of type (2, 0), where is a non-empty set, ∘ is a binary operation on , and 0 is a fixed element of , satisfying the following axioms:

\begin{array}{l} (\forall x_{1}, x_{2}, x_{3} \in U) \\ ((x_{2} \circ x_{3}) \circ (x_{1} \circ x_{2}) \circ (x_{1} \circ x_{3})) = 0), & (UP- 1) \\ (\forall x_{1} \in U) (0 \circ x_{1} = x_{1}), & (UP- 2) \\ (\forall x_{1} \in U) (x_{1} \circ 0 = 0), & (UP- 3) \\ (\forall x_{1}, x_{2} \in U) \\ (x_{1} \circ x_{2} = 0, x_{2} \circ x_{1} = 0 \Rightarrow x_{1} = x_{2}) . & (UP- 4) \end{array}

For more examples and studies of UP-algebras, see [15, 29–36].

In a UP-algebra , the following assertions are valid (see [5, 30]).

(1)

(\forall x_{1} \in U) (x_{1} \circ x_{1} = 0),

(2)

(\forall x_{1}, x_{2}, x_{3} \in U) (x_{1} \circ x_{2} = 0, x_{2} \circ x_{3} = 0 \Rightarrow x_{1} \circ x_{3} = 0),

(3)

(\forall x_{1}, x_{2}, x_{3} \in U) (x_{1} \circ x_{2} = 0 \Rightarrow (x_{3} \circ x_{1}) \circ (x_{3} \circ x_{2}) = 0),

(4)

(\forall x_{1}, x_{2}, x_{3} \in U) (x_{1} \circ x_{2} = 0 \Rightarrow (x_{2} \circ x_{3}) \circ (x_{1} \circ x_{3}) = 0),

(5)

(\forall x_{1}, x_{2} \in U) (x_{1} \circ (x_{2} \circ x_{1}) = 0),

(6)

(\forall x_{1}, x_{2} \in U) ((x_{2} \circ x_{1}) \circ x_{1} = 0) \Leftrightarrow x_{1} = x_{2} \circ x_{1}),

(7)

(\forall x_{1}, x_{2} \in U) (x_{1} \circ (x_{2} \circ x_{2}) = 0),

(8)

(\forall x_{0}, x_{1}, x_{2}, x_{3} \in U) ((x_{1} \circ (x_{2} \circ x_{3})) \circ (x_{1} \circ ((x_{0} \circ x_{2}) \circ (x_{0} \circ x_{3}))) = 0),

(9)

(\forall x_{0}, x_{1}, x_{2}, x_{3} \in U) ((((x_{0} \circ x_{1}) \circ (x_{0} \circ x_{2})) \circ x_{3}) \circ ((x_{1} \circ x_{2}) \circ x_{3}) = 0),

(10)

(\forall x_{1}, x_{2}, x_{3} \in U) (((x_{1} \circ x_{2}) \circ x_{3}) \circ (x_{2} \circ x_{3}) = 0),

(11)

(\forall x_{1}, x_{2}, x_{3} \in U) (x_{1} \circ x_{2} = 0 \Rightarrow x_{1} \circ (x_{3} \circ x_{2}) = 0),

(12)

(\forall x_{1}, x_{2}, x_{3} \in U) (((x_{1} \circ x_{2}) \circ x_{3}) \circ (x_{1} \circ (x_{2} \circ x_{3})) = 0),

(13)

(\forall x_{0}, x_{1}, x_{2}, x_{3} \in U) (((x_{1} \circ x_{2}) \circ x_{3}) \circ (x_{2} \circ (x_{0} \circ x_{3})) = 0) .

From [5], the binary relation ≤ on a UP-algebra is defined as follows:

(\forall x_{1}, x_{2} \in U) (x_{1} \leq x_{2} \Leftrightarrow x_{1} \circ x_{2} = 0) .

Definition 1.2 [17]

Definition 1.3 [17]

Let F be an FS in a non-empty set . The complement of F, denoted by F̃, is described by its membership function, f_F̃, which is defined as follows:

(14)

(\forall x_{1} \in U) (f_{\tilde{F}} (x_{1}) = 1 - f_{F} (x_{1})) .

Definition 1.4 [17]

Let F₁ and F₂ be FSs in a non-empty set . The relations ⊆ and = and the operations ∪ and ∩ are defined as follows:

(1) ,
(2) F₁ = F₂ ⇔ F₁ ⊆ F₂, F₁ ⊇ F₂,
(3) ,
(4) .

The following two propositions will be used in the following sections:

Proposition 1.5 [28]

Let F be an FS in a non-empty set . Then the following assertions are valid:

(1) ,
(2) .

Proposition 1.6 [28]

(1) $(\forall x_{1}, x_{2} \in U) (inf_{i \in I} {min {f_{F i} (x_{1}), f_{F i} (x_{2})}} = min {{inf}_{i \in I} {f_{F i} (x_{1})}, {inf}_{i \in I} {f_{F i} (x_{2})}})$ ,
(2) $(\forall x_{1}, x_{2} \in U) (sup_{i \in I} {max {f_{F i} (x_{1}), f_{F i} (x_{2})}} = max {{sup}_{i \in I} {f_{F i} (x_{1})}, {sup}_{i \in I} {f_{F i} (x_{2})}})$ ,
(3) $(\forall x_{1}, x_{2} \in U) (inf_{i \in I} {max {f_{F i} (x_{1}), f_{F i} (x_{2})}} \geq max {{inf}_{i \in I} {f_{F i} (x_{1})}, {inf}_{i \in I} {f_{F i} (x_{2})}})$ ,
(4) $(\forall x_{1}, x_{2} \in U) (sup_{i \in I} {min {f_{F i} (x_{1}), f_{F i} (x_{2})}} \leq min {{sup}_{i \in I} {f_{F i} (x_{1})}, {sup}_{i \in I} {f_{F i} (x_{2})}})$ ,
(5) for all i ∈ I ⇒ sup_i_∈_I{f_{F_i} (x₁)} ≤ sup_i_∈_I{f_{G_i} (x₁)}),
(6) (∀x₁ ∈ Y )(f_F(x₁) ≤ f_G(x₁) ⇒ sup_x_₁_∈_Y {f_F(x₁)} ≤ sup_x_₁_∈_Y {f_G(x₁)}),
(7) for all i ∈ I ⇒ inf_i_∈_I{f_{F_i} (x₁)} ≤ inf_i_∈_I{f_{G_i} (x₁)}),
(8) (∀x₁ ∈ Y )(f_F(x₁) ≤ f_G(x₁) ⇒ inf_x_₁_∈_Y {f_F(x₁)} ≤ inf_x_₁_∈_Y {f_G(x₁)}).

The following definition can be considered based on the concepts expounded in [?].

Definition 1.7

An FS F in a UP-algebra is called

(1) a fuzzy implicative UP-filter (FIUPF) of if it satisfies

(15) $(\forall x_{1} \in U) (f_{F} (0) \geq f_{F} (x_{1})),$ (16) $(\forall x_{1}, x_{2}, x_{3} \in U) (f_{F} (x_{1} \circ x_{3}) \geq min {f_{F} (x_{1} \circ (x_{2} \circ x_{3})), f_{F} (x_{1} \circ x_{2})}),$
(2) a fuzzy comparative UP-filter (FCUPF) of if it satisfies (15) and

(17) $(\forall x_{1}, x_{2}, x_{3} \in U) (f_{F} (x_{2}) \geq min {f_{F} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), f_{F} (x_{1})}),$
(3) a fuzzy shift UP-filter (FSUPF) of if it satisfies (15) and

(18) $(\forall x_{1}, x_{2}, x_{3} \in U) (f_{F} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \geq min {f_{F} (x_{1} \circ (x_{2} \circ x_{3})), f_{F} (x_{1})}) .$

2. PFSs in UP-Algebras

In 2013, Yager and Abbasov [?, 19] introduced the concept of PFSs for the first time.

Definition 2.1

(19)

(\forall x_{1} \in U) (0 \leq μ_{P} {(x_{1})}^{2} + ν_{P} {(x_{1})}^{2} \leq 1) .

Definition 2.2

Let P = (μ_P, ν_P) and Q = (μ_Q, ν_Q) be PFSs in a non-empty set . The relations ⊆ and = and the operations ∪ and ∩ are defined as follows:

(1) ,
(2) P = Q ⇔ P ⊆ Q, P ⊇ Q,
(3) P ∪Q = (μ_P ∪ μ_Q, ν_P ∩ ν_Q),
(4) P ∩Q = (μ_P ∩ μ_Q, ν_P ∪ ν_Q).

Note that P ∪ Q and P ∩ Q are PFSs in . Indeed, if we assume , then (μ_P∪μ_Q)(x₁) = max{μ_P(x₁), μ_Q(x₁)} and (ν_P∩ν_Q)(x₁) = min{ν_P(x₁), ν_Q(x₁)}. Thus we consider

\begin{array}{l} 0 \leq (μ_{P} \cup μ_{Q}) {(x_{1}))}^{2} + (ν_{P} \cap ν_{Q}) {(x_{1}))}^{2} \\ = max {π_{P} (x_{1}), μ_{Q} (x_{1})}^{2} + min {ν_{P} (x_{1}), ν_{Q} (x_{1})}^{2} \\ = {(μ_{P} (x_{1}))}^{2} + min {ν_{P} (x_{1}), ν_{Q} (x_{1})}^{2} \\ (WLOG, assume that max {μ_{P} (x_{1}), μ_{Q} (x_{1})} = μ_{P} (x_{1})) \\ \leq {(μ_{P} (x_{1}))}^{2} + {(ν_{P} (x_{1}))}^{2} \\ \leq 1. \end{array}

Hereafter, we shall let be a UP-algebra unless otherwise stated.

Definition 2.3 [28]

A PFS P = (μ_P, ν_P) in is called

(1) a Pythagorean fuzzy UP-subalgebra (PFUPS) of if it satisfies

(20) $(\forall x_{1}, x_{2} \in U) (μ_{P} (x_{1} \circ x_{2}) \geq min {μ_{P} (x_{1}), μ_{P} (x_{2})}),$ (21) $(\forall x_{1}, x_{2} \in U) (ν_{P} (x_{1} \circ x_{2}) \leq max {ν_{P} (x_{1}), ν_{P} (x_{2})}),$
(2) a Pythagorean fuzzy near UP-filter (PFNUPF) of if it satisfies

(22) $(\forall x_{1}, x_{2} \in U) (μ_{P} (x_{1} \circ x_{2}) \geq μ_{P} (x_{2})),$ (23) $(\forall x_{1}, x_{2} \in U) (ν_{P} (x_{1} \circ x_{2}) \leq ν_{P} (x_{2})),$
(3) a Pythagorean fuzzy UP-filter (PFUPF) of if it satisfies

(24) $(\forall x_{1} \in U) (μ_{P} (0) \geq μ_{P} (x_{1})),$ (25) $(\forall x_{1} \in U) (ν_{P} (0) \leq ν_{P} (x_{1})),$ (26) $(\forall x_{1}, x_{2} \in U) (μ_{P} (x_{2}) \geq min {μ_{P} (x_{1} \circ x_{2}), μ_{P} (x_{1})}),$ (27) $(\forall x_{1}, x_{2} \in U) (ν_{P} (x_{2}) \leq max {ν_{P} (x_{1} \circ x_{2}), ν_{P} (x_{1})}),$
(4) a Pythagorean fuzzy UP-ideal (PFUPI) of if it satisfies (24), (25), and

(28) $(\forall x_{1}, x_{2}, x_{3} \in U) (μ_{P} (x_{1} \circ x_{3}) \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{2})}),$ (29) $(\forall x_{1}, x_{2}, x_{3} \in U) (ν_{P} (x_{1} \circ x_{3}) \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{2})}),$
(5) a Pythagorean fuzzy strong UP-ideal (PFSUPI) of if it satisfies (24), (25), and

(30) $(\forall x_{1}, x_{2}, x_{3} \in U) (μ_{P} (x_{1}) \geq min {μ_{P} ((x_{3} \circ x_{2}) \circ (x_{3} \circ x_{1})), μ_{P} (x_{2})}),$ (31) $(\forall x_{1}, x_{2}, x_{3} \in U) (ν_{P} (x_{1}) \leq max {ν_{P} ((x_{3} \circ x_{2}) \circ (x_{3} \circ x_{1})), ν_{P} (x_{2})}) .$

Next, we introduce three notions of PFSs in UP-algebras.

Definition 2.4

A PFS P = (μ_P, ν_P) in is called

(1) a Pythagorean fuzzy implicative UP-filter (PFIUPF) of if it satisfies (24), (25), and

(32) $(\forall x_{1}, x_{2}, x_{3} \in U) (μ_{P} (x_{1} \circ x_{3}) \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1} \circ x_{2})}),$ (33) $(\forall x_{1}, x_{2}, x_{3} \in U) (ν_{P} (x_{1} \circ x_{3}) \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1} \circ x_{2})}),$
(2) a Pythagorean fuzzy comparative UP-filter (PFCUPF) of if it satisfies (24), (25), and

(34) $(\forall x_{1}, x_{2}, x_{3} \in U) (μ_{P} (x_{2}) \geq min {μ_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), μ_{P} (x_{1})}),$ (35) $(\forall x_{1}, x_{2}, x_{3} \in U) (ν_{P} (x_{2}) \leq max {ν_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), ν_{P} (x_{1})}),$
(3) a Pythagorean fuzzy shift UP-filter (PFSUPF) of if it satisfies (24), (25), and

(36) $(\forall x_{1}, x_{2}, x_{3} \in U) (μ_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1})}),$ (37) $(\forall x_{1}, x_{2}, x_{3} \in U) (ν_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1})}) .$

Theorem 2.5

Every PFIUPF of is a PFUPF.

Proof

Let P = (μ_P, ν_P) be a PFIUPF of . Then, for all x₁, ,

\begin{array}{l} μ_{P} (x_{2}) & = μ_{P} (0 \circ x_{2}) & (by (UP- 2)) \\ \geq min {μ_{P} (0 \circ (x_{1} \circ x_{2})), μ_{P} (0 \circ x_{1})} & (by (32)) \\ = min {μ_{P} (x_{1} \circ x_{2}), μ_{P} (x_{1})}, & (by (UP- 2)) \end{array}

and

\begin{array}{l} ν_{P} (x_{2}) & = ν_{P} (0 \circ x_{2}) & (by (UP- 2)) \\ \leq max {ν_{P} (0 \circ (x_{1} \circ x_{2})), ν_{P} (0 \circ x_{1})} & (by (33)) \\ = max {ν_{P} (x_{1} \circ x_{2}), ν_{P} (x_{1})} . & (by (UP- 2)) \end{array}

Therefore, P is a PFUPF of .

The converse of Theorem 2.5 does not hold in general, as is shown in the following example.

Example 2.6

Consider a UP-algebra where is defined in Table 1.

Theorem 2.7

Every PFCUPF of is a PFUPF.

Proof

Let P = (μ_P, ν_P) be a PFCUPF of . Then, for all x₁, ,

\begin{array}{l} μ_{P} (x_{2}) & \leq min {μ_{P} (x_{1} \circ ((x_{2} \circ 0) \circ x_{2})), μ_{P} (x_{1})} & (by (34)) \\ = min {μ_{P} (x_{1} \circ (0 \circ x_{2})), μ_{P} (x_{1})} & (by (UP- 3)) \\ = min {μ_{P} (x_{1} \circ x_{2}), μ_{P} (x_{1})}, & (by (UP- 2)) \end{array}

and

\begin{array}{l} ν_{P} (x_{2}) & \leq max {ν_{P} (x_{1} \circ ((x_{2} \circ 0) \circ x_{2})), ν_{P} (x_{1})} & (by (35)) \\ = max {ν_{P} (x_{1} \circ (0 \circ x_{2})), ν_{P} (x_{1})} & (by (UP- 3)) \\ = max {ν_{P} (x_{1} \circ x_{2}), ν_{P} (x_{1})} . & (by (UP- 2)) \end{array}

Therefore, P is a PFUPF of .

Similarly, the converse of Theorem 2.7 does not hold in general, as is shown in the following example.

Example 2.8

Consider a UP-algebra where is defined in Table 3.

Theorem 2.9

Every PFSUPF of is a PFUPF.

Proof

Let P = (μ_P, ν_P) be a PFSUPF of . Then for all x₁, ,

\begin{array}{l} μ_{P} (x_{2}) & = μ_{P} (0 \circ x_{2}) & (by (UP- 2)) \\ = μ_{P} ((0 \circ 0) \circ x_{2}) & (by (1)) \\ = μ_{P} (((x_{2} \circ 0) \circ 0) \circ x_{2}) & (by (UP- 3)) \\ \geq min {μ_{P} (x_{1} \circ (0 \circ x_{2})), μ_{P} (x_{1})} & (by (36)) \\ = min {μ_{P} (x_{1} \circ x_{2}), μ_{P} (x_{1})}, & (by (UP- 2)) \end{array}

and

\begin{array}{l} ν_{P} (x_{2}) & = ν_{P} (0 \circ x_{2}) & (by (UP- 2)) \\ = ν_{P} ((0 \circ 0) \circ x_{2}) & (by (1)) \\ = ν_{P} (((x_{2} \circ 0) \circ 0) \circ x_{2}) & (by (UP- 3)) \\ \leq max {ν_{P} (x_{1} \circ (0 \circ x_{2})), ν_{P} (x_{1})} & (by (37)) \\ = max {ν_{P} (x_{1} \circ x_{2}), ν_{P} (x_{1})} . & (by (UP- 2)) \end{array}

Therefore, P is a PFUPF of .

The converse of Theorem 2.9 does not hold in general. This is shown in the following example.

Example 2.10

Consider a UP-algebra where is defined in Table 5.

Theorem 2.11

Every PFIUPF of is a PFUPI.

Proof

Let P = (μ_P, ν_P) be a PFIUPF of . By Theorem 2.5, we have that P as a PFUPF, and thus, P is a PFNUPF. Then, for all x₁, x₂, ,

\begin{array}{l} μ_{P} (x_{1} \circ x_{3}) & \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1} \circ x_{2})} & (by (32)) \\ \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{2})}, & (by (22)) \end{array}

and

\begin{array}{l} ν_{P} (x_{1} \circ x_{3}) & \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1} \circ x_{2})} & (by (33)) \\ \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{2})} . & (by (23)) \end{array}

Therefore, P is a PFUPI of .

The converse of Theorem 2.11 does not hold in general. This is shown in the following example.

Example 2.12

Consider a UP-algebra where is defined in Table 7.

Theorem 2.13

Every PFSUPI of is a PFIUPF (respectively, PFCUPF, PFSUPF).

Proof

Let P = (μ_P, ν_P) be a PFSUPI of . Because P is constant, we have that P is a PFIUPF (resp., PFCUPF, PFSUPF) of .

The converse of Theorem 2.13 does not hold in general, as is shown in the following examples.

Example 2.14

Consider a UP-algebra where is defined in Table 9.

If we define a PFS P = (μ_P, ν_P) as presented in Table 10, then, P is a PFIUPF of . However, P is not a CPFS of . Therefore, P is not a PFSUPI of .

Example 2.15

Consider a UP-algebra where is defined in Table 11.

If we define a PFS P = (μ_P, ν_P) as presented in Table 12, then, P is a PFCUPF of . However, P is not a CPFS of . Therefore, P is not a PFSUPI of .

Example 2.16

Consider a UP-algebra where is defined in Table 13.

If we define a PFS P = (μ_P, ν_P) as presented in Table 14, then, P is a PFSUPF of . However, P is not a CPFS of . Therefore, P is not a PFSUPI of .

Next, we present some examples for studying the generalization of new notions of PFSs and original PFSs in UP-algebras.

Example 2.17

Consider a UP-algebra where is defined in Table 15.

Example 2.18

Example 2.19

Example 2.20

Consider a UP-algebra where is defined in Table 19.

Example 2.21

Consider a UP-algebra where is defined in Table 21.

Example 2.22

Example 2.23

We obtain a diagram of the generalization of new PFSs in UP-algebras, which is shown in Figure 1.

If F is an FS in , then (f_F, f_F̃) is a PFS in . Indeed, for all ,

\begin{array}{l} 0 \leq {(f_{F} (x_{1}))}^{2} + {(f_{\tilde{F}} (x_{1}))}^{2} \\ = {(f_{F} (x_{1}))}^{2} + {(1 - f_{F} (x_{1}))}^{2} \\ \leq f_{F} (x_{1}) + 1 - 2 f_{F} (x_{1}) + {(f_{F} (x_{1}))}^{2} \\ \leq f_{F} (x_{1}) + 1 - 2 f_{F} (x_{1}) + f_{F} (x_{1}) \\ = 1. \end{array}

Theorem 2.24

Let F be an FS in . Then the following statements hold:

(1) F is an FIUPF of if and only if (f_F, f_F̃) is a PFIUPF of ,
(2) F is an FCUPF of if and only if (f_F, f_F̃) is a PFCUPF of ,
(3) F is an FSUPF of if and only if (f_F, f_F̃) is a PFSUPF of .

Proof

(1) Assuming that F is an FIUPF of , then, for all x₁, ,

\begin{array}{l} f_{F} (0) \geq f_{F} (x_{1}), & (by (15)) \\ f_{\tilde{F}} (0) \leq f_{\tilde{F}} (x_{1}), & (by Proposition 1.5 (1)) \\ f_{F} (x_{1} \circ x_{3}) \geq min {f_{F} (x_{1} \circ (x_{2} \circ x_{3})), f_{F} (x_{1} \circ x_{2})}, & (by (16)) \end{array}

and

\begin{array}{l} f_{\tilde{F}} (x_{1} \circ x_{3}) & = 1 - f_{F} (x_{1} \circ x_{3}) \\ \leq 1 - min {f_{F} (x_{1} \circ (x_{2} \circ x_{3})), f_{F} (x_{1} \circ x_{2})} & (by (16)) \\ = max {f_{\tilde{F}} (x_{1} \circ (x_{2} \circ x_{3})), f_{\tilde{F}} (x_{1} \circ x_{2})} . & (by Proposition 1.5 (2)) \end{array}

This implies that (f_F, f_F̃) is a PFIUPF of .

Conversely, assuming that (f_F, f_F̃) is a PFIUPF of , then, F satisfies conditions (24) and (32). Hence, F is an FIUPF of .

(2) Assuming that F is an FCUPF of , then, for all x₁, ,

\begin{array}{l} f_{F} (0) \geq f_{F} (x_{1}), & (by (15)) \\ f_{\tilde{F}} (0) \leq f_{\tilde{F}} (x_{1}), & (by Proposition 1.5 (1)) \\ f_{F} (x_{2}) \\ \geq min {f_{F} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), f_{F} (x_{1})} & (by (17)) \end{array}

and

\begin{array}{l} f_{\tilde{F}} (x_{2}) & = 1 - f_{F} (x_{2}) \\ \leq 1 - min {f_{F} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}), f_{F} (x_{1})} & (by (17)) \\ = max {f_{\tilde{F}} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}), f_{\tilde{F}} (x_{1})} . & (by Proposition 1.5 (2)) \end{array}

This implies that (f_F, f_F̃) is a PFCUPF of .

Conversely, assuming that (f_F, f_F̃) is a PFCUPF of , then, F satisfies conditions (24) and (34). Hence, F is an FCUPF of .

(3) Assuming that F is an FSUPF of , then, for all x₁, ,

\begin{array}{l} f_{F} (0) \geq f_{F} (x_{1}), & (by (15)) \\ f_{\tilde{F}} (0) \leq f_{\tilde{F}} (x_{1}), & (by Proposition 1.5 (1)) \\ f_{F} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \\ \geq min {f_{F} (x_{1} \circ (x_{2} \circ x_{3})), f_{F} (x_{1})}, & (by (18)) \end{array}

and

\begin{array}{l} f_{\tilde{F}} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) & = 1 - f_{F} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \\ \leq 1 - min {f_{F} (x_{1} \circ (x_{2} \circ x_{3})), f_{F} (x_{1})} & (by (18)) \\ = max {f_{\tilde{F}} (x_{1} \circ (x_{2} \circ x_{3})), f_{\tilde{F}} (x_{1})} . & (by Proposition 1.5 (2)) \end{array}

This implies that (f_F, f_F̃) is a PFSUPF of .

Conversely, assuming that (f_F, f_F̃) is a PFSUPF of , then, F satisfies conditions (24) and (36). Hence, F is an FSUPF of .

3. Sufficient Conditions for PFSs in UP-Algebras

In this section, we present some conditions for studying the generalizations of new PFSs and original PFSs in UP-algebras.

Theorem 3.1

If P is a PFUPI of satisfying

(38)

(\forall x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} μ_{P} (x_{1} \circ (x_{2} \circ x_{3})) \geq μ_{P} (x_{2}) \\ \Rightarrow μ_{P} (x_{2}) \geq μ_{P} (x_{1} \circ x_{2}), \\ ν_{P} (x_{1} \circ (x_{2} \circ x_{3})) \leq ν_{P} (x_{2}) \\ \Rightarrow ν_{P} (x_{2}) \leq ν_{P} (x_{1} \circ x_{2}), \end{array})

then P is a PFIUPF of .

Proof

\begin{array}{l} μ_{P} (x_{1} \circ x_{3}) & \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{2})} & (by (28)) \\ \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1} \circ x_{2})}, & (by (38) for μ_{P}) \end{array}

and

\begin{array}{l} ν_{P} (x_{1} \circ x_{3}) & \geq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{2})} & (by (29)) \\ \geq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1} \circ x_{2})}, & (by (38) for ν_{P}) \end{array}

Therefore, P is a PFIUPF of .

Theorem 3.2

If P is a PFUPI of satisfying

(39)

(\forall x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} μ_{P} (x_{1} \circ x_{2}) \geq μ_{P} (x_{1} \circ ((x_{1} \circ x_{2}) \circ x_{3})) \\ \Rightarrow μ_{P} (x_{1} \circ ((x_{1} \circ x_{2}) \circ x_{3})) \geq μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), \\ ν_{P} (x_{1} \circ x_{2}) \leq ν_{P} (x_{1} \circ ((x_{1} \circ x_{2}) \circ x_{3})) \\ \Rightarrow ν_{P} (x_{1} \circ ((x_{1} \circ x_{2}) \circ x_{3})) \leq ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), \end{array})

then P is a PFIUPF of .

Proof

\begin{array}{l} μ_{P} (x_{1} \circ x_{3}) & \geq min {μ_{P} (x_{1} \circ ((x_{1} \circ x_{2}) \circ x_{3})), μ_{P} (x_{1} \circ x_{2})} & (by (28)) \\ \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1} \circ x_{2})}, & (by (39) for μ_{P}) \end{array}

and

\begin{array}{l} ν_{P} (x_{1} \circ x_{3}) & \leq max {ν_{P} ((x_{1} \circ x_{2}) \circ z)), ν_{P} (x_{1} \circ x_{2})} & (by (29)) \\ \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1} \circ x_{2})} . & (by (39) for ν_{P}) \end{array}

Therefore, P is a PFIUPF of .

Theorem 3.3

If P is a PFUPF of satisfying

(40)

(\forall x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} μ_{P} (x_{1}) \geq μ_{P} (x_{1} \circ x_{2}) \\ \Rightarrow μ_{P} (x_{1} \circ x_{2}) \geq μ_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), \\ ν_{P} (x_{1}) \leq ν_{P} (x_{1} \circ x_{2}) \\ \Rightarrow ν_{P} (x_{1} \circ x_{2}) \leq ν_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), \end{array})

then P is a PFCUPF of .

Proof

\begin{array}{l} μ_{P} (x_{2}) & \geq min {μ_{P} (x_{1} \circ x_{2}), μ_{P} (x_{1})} & (by (26)) \\ \geq min {μ_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), μ_{P} (x_{1})}, & (by (40) for μ_{P}) \end{array}

and

\begin{array}{l} ν_{P} (x_{2}) & \leq max {ν_{P} (x_{1} \circ x_{2}), ν_{P} (x_{1})} & (by (27)) \\ \leq max {ν_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), ν_{P} (x_{1})} . & (by (40) for ν_{P}) \end{array}

Therefore, P is a PFCUPF of .

Theorem 3.4

If P is a PFUPF of satisfying

(41)

(\forall x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} μ_{P} (x_{1}) \geq μ_{P} (x_{1} \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})) \\ \Rightarrow μ_{P} (x_{1} \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})) \\ \geq μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), \\ ν_{P} (x_{1}) \leq ν_{P} (x_{1} \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})) \\ \Rightarrow ν_{P} (x_{1} \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})) \\ \leq ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), \end{array})

then P is a PFSUPF of .

Proof

\begin{array}{l} μ_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) & \geq min {μ_{P} (x_{1} \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})), μ_{P} (x_{1})} & (by (26)) \\ \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1})}, & (by (41) for μ_{P}) \end{array}

and

\begin{array}{l} ν_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) & \leq max {ν_{P} (x_{1} \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})), ν_{P} (x_{1})} & (by (27)) \\ \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1})} . & (by (41) for ν_{P}) \end{array}

Therefore, P is a PFSUPF of .

Theorem 3.5

If P is a PFS in satisfying

(\forall x_{0}, x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} x_{0} \leq x_{1} \circ (x_{2} \circ x_{3}) \\ \Rightarrow {\begin{matrix} μ_{P} (x_{1} \circ x_{3}) \geq min {μ_{P} (x_{0}), μ_{P} (x_{1} \circ x_{2})}, \\ ν_{P} (x_{1} \circ x_{3}) \leq max {ν_{P} (x_{0}), ν_{P} (x_{1} \circ x_{2})}, \end{matrix} \end{array})

then P is a PFIUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (by (UP-2)). Let . By (UP-3), we have that x₁ ∘ (0 ∘ (x₁ ∘ 0)) = 0; that is, x₁ ≤ 0 ∘ (x₁ ∘ 0). It follows from (by (UP-2)) that

\begin{array}{l} μ_{P} (0) & = μ_{P} (0 \circ 0) \\ \geq min {μ_{P} (x_{1}), μ_{P} (0 \circ x_{1})} \\ = min {μ_{P} (x_{1}, μ_{P} (x_{1})} \\ = μ_{P} (x_{1}), & (by (UP- 2)) \\ ν_{P} (0) & = ν_{P} (0 \circ 0) \\ \leq max {ν_{P} (x_{1}), ν_{P} (0 \circ x_{1})} \\ = max {ν_{P} (x_{1}), ν_{P} (x_{1})} \\ = ν_{P} (x_{1}) . & (by (UP- 2)) \end{array}

\begin{array}{l} μ_{P} (x_{1} \circ x_{3}) \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1} \circ x_{2})}, \\ ν_{P} (x_{1} \circ x_{3}) \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1} \circ x_{2})} . \end{array}

Therefore, P is a PFIUPF of .

Theorem 3.6

If P is a PFS in satisfying

(42)

(\forall x_{0}, x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} x_{0} \leq x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2} \\ \Rightarrow {\begin{matrix} μ_{P} (x_{2}) \geq min {μ_{P} (a), μ_{P} (x_{1})}, \\ ν_{P} (x_{2}) \leq max {ν_{P} (a), ν_{P} (x_{1})}, \end{matrix} \end{array})

then P is a PFCUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (42). Let . By (UP-3), we have that x₁ ∘ (x₁ ∘ ((0 ∘ x₁) ∘ 0)) = 0; that is, x₁ ≤ x₁ ∘ ((0 ∘ x₁) ∘ 0). It follows from (42) that

\begin{array}{l} μ_{P} (0) \geq min {μ_{P} (x_{1}), μ_{P} (x_{1})} = μ_{P} (x_{1}), \\ ν_{P} (0) \leq max {ν_{P} (x_{1}), ν_{P} (x_{1})} = ν_{P} (x_{1}) . \end{array}

\begin{array}{l} μ_{P} (x_{2}) \geq min {μ_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), μ_{P} (x_{1})}, \\ ν_{P} (x_{2}) \leq max {ν_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), ν_{P} (x_{1})} . \end{array}

Therefore, P is a PFCUPF of .

Theorem 3.7

If P is a PFS in satisfying (26), (27), and

(43)

(\forall x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} μ_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \\ \geq μ_{P} ((x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \circ x_{2}) \\ \Rightarrow μ_{P} ((x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \circ x_{2}) \geq μ_{P} (x_{1}), \\ ν_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \\ \leq ν_{P} ((x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \circ x_{2}) \\ \Rightarrow ν_{P} ((x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \circ x_{2}) \leq ν_{P} (x_{1}), \end{array})

then P is a PFCUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (26), (27), and (43). Let . By (UP-2) and (UP-3), we have that

\begin{array}{l} μ_{P} (x_{1} \circ ((0 \circ x_{1}) \circ 0)) = μ_{P} (0) \geq μ_{P} (0) = μ_{P} ((x_{1} \circ ((0 \circ x_{1}) \circ 0)) \circ 0), \\ ν_{P} (x_{1} \circ ((0 \circ x_{1}) \circ 0)) = ν_{P} (0) \leq ν_{P} (0) = ν_{P} ((x_{1} \circ ((0 \circ x_{1}) \circ 0)) \circ 0) . \end{array}

It follows from (43) that

\begin{array}{l} μ_{P} (0) = μ_{P} ((x_{1} \circ ((0 \circ x_{1}) \circ 0)) \circ 0) \geq μ_{P} (x_{1}), \\ ν_{P} (0) = ν_{P} ((x_{1} \circ ((0 \circ x_{1}) \circ 0)) \circ 0) \leq ν_{P} (x_{1}) . \end{array}

\begin{array}{l} μ_{P} (x_{2}) & \geq min {μ_{P} ((x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \circ x_{2}), μ_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}))} & (by (26)) \\ = min {μ_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), μ_{P} (x_{1})}, & (by (43) for μ_{P}) \\ ν_{P} (x_{2}) & \leq max {ν_{P} ((x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})) \circ x_{2}), ν_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}))} & (by (27)) \\ = max {ν_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), ν_{P} (x_{1})} . & (by (43) for ν_{P}) \end{array}

Therefore, P is a PFCUPF of .

Theorem 3.8

If P is a PFS in satisfying

(44)

(\forall x_{0}, x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} x_{0} \leq x_{1} \circ (x_{2} \circ x_{3}) \\ \Rightarrow {\begin{matrix} \begin{array}{l} μ_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3} \\ \geq min {μ_{P} (x_{0}), μ_{P} (x_{1})}, \end{array} \\ \begin{array}{l} ν_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3} \\ \leq max {ν_{P} (x_{0}), ν_{P} (x_{1})}, \end{array} \end{matrix} \end{array})

then P is a PFSUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (44). Let . By (UP-3), we have that x₁ ∘ (x₁ ∘ (x₁ ∘ 0)) = 0; that is, x₁ ≤ x₁ ∘ (x₁ ∘ 0). It follows from (44) that:

\begin{array}{l} μ_{P} (0) & = μ_{P} (((0 \circ x_{1}) \circ x_{1}) \circ 0 \\ \geq min {μ_{P} (x_{1}), μ_{P} (x_{1})} = μ_{P} (x_{1}), & (by (UP- 2)) \\ ν_{P} (0) & = ν_{P} (((0 \circ x_{1}) \circ x_{1}) \circ 0 \\ \leq max {ν_{P} (x_{1}), ν_{P} (x_{1})} = ν_{P} (x_{1}) . & (by (UP- 2)) \end{array}

Next, let x₁, x₂, . By (1), we have that (x₁ ∘ (x₂ ∘x₃)) ∘ (x₁ ∘ (x₂ ∘ x₃)) = 0; that is, x₁ ∘ (x₂ ∘ x₃) ≤ x₁ ∘ (x₂ ∘ x₃). It follows from (44) that:

\begin{array}{l} μ_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3} \geq min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1})}, \\ ν_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3} \leq max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1})} . \end{array}

Therefore, P is a PFSUPF of .

Theorem 3.9

If P is a PFS in satisfying (26), (27), and

(45)

(\forall x_{1}, x_{2}, x_{3} \in U) (\begin{array}{l} μ_{P} (x_{1} \circ (x_{2} \circ x_{3})) \\ \geq μ_{P} ((x_{1} \circ (x_{2} \circ x_{3})) \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \\ \Rightarrow μ_{P} ((x_{1} \circ (x_{2} \circ x_{3})) \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \geq μ_{P} (x_{1}), \\ ν_{P} (x_{1} \circ (x_{2} \circ x_{3})) \\ \leq ν_{P} ((x_{1} \circ (x_{2} \circ x_{3})) \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \\ \Rightarrow ν_{P} ((x_{1} \circ ((x_{2} \circ x_{3})) \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \leq ν_{P} (x_{1}), \end{array})

then P is a PFSUPF of .

Proof

Let P = (μ_P, ν_P) be a PFS in satisfying (26), (27), and (45). Let . By (UP-2) and (UP-3), we have that

\begin{array}{l} μ_{P} (x_{1} \circ (x_{1} \circ 0)) = μ_{P} (0) \\ \geq μ_{P} (0) \\ = μ_{P} ((x_{1} \circ (x_{1} \circ 0)) \circ (((0 \circ x_{1}) \circ x_{1}) \circ 0)), \\ ν_{P} (x_{1} \circ (x_{1} \circ 0)) = ν_{P} (0) \\ \leq ν_{P} (0) \\ = ν_{P} ((x_{1} \circ (x_{1} \circ 0)) \circ (((0 \circ x_{1}) \circ x_{1}) \circ 0)) . \end{array}

It follows from (45) that

\begin{array}{l} μ_{P} (0) = μ_{P} ((x_{1} \circ (x_{1} \circ 0)) \circ (((0 \circ x_{1}) \circ x_{1}) \circ 0) \\ \geq μ_{P} (x_{1}), \\ ν_{P} (0) = ν_{P} ((x_{1} \circ (x_{1} \circ 0)) \circ (((0 \circ x_{1}) \circ x_{1}) \circ 0) \\ \leq ν_{P} (x_{1}) . \end{array}

\begin{array}{l} μ_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \\ \geq min {μ_{P} ((x_{1} \circ (x_{2} \circ x_{3})) \\ \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}), μ_{P} (x_{1} \circ (x_{2} \circ x_{3}))} & (by (26)) \\ = min {μ_{P} (x_{1} \circ (x_{2} \circ x_{3})), μ_{P} (x_{1})}, & (by (45) for μ_{P}) \\ ν_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) \\ \leq max {ν_{P} ((x_{1} \circ (x_{2} \circ x_{3})) \\ \circ (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}), ν_{P} (x_{1} \circ (x_{2} \circ x_{3}))} & (by (27)) \\ = max {ν_{P} (x_{1} \circ (x_{2} \circ x_{3})), ν_{P} (x_{1})} . & (by (45) for μ_{P}) \end{array}

Therefore, P is a PFSUPF of .

We obtain a diagram of the sufficient conditions for new PFSs in UP-algebras, which is shown in Figure 2.

4. Approximations

Let ρ be an equivalence relation on a non-empty set . If , then the ρ-class of x₁ is the set (x₁)_ρ defined as follows:

{(x_{1})}_{ρ} = {x_{2} \in U ∣ (x_{1}, x_{2}) \in ρ} .

An equivalence relation ρ on is called a congruence relation if:

\begin{array}{l} (\forall x_{1}, x_{2}, x_{3} \in U) ((x_{1}, x_{2}) \in ρ \\ \Rightarrow (x_{1} \circ x_{3}, x_{2} \circ x_{3}) \in ρ, (x_{3} \circ x_{1}, x_{3} \circ x_{2}) \in ρ) . \end{array}

For the non-empty subsets A and B of , we denote

A B = A \circ B = {x_{1} \circ x_{2} ∣ x_{1} \in A and x_{2} \in B} .

If ρ is a congruence on , then

\begin{array}{l} (\forall x_{1}, x_{2} \in U) ({(x_{1})}_{ρ} {(x_{2})}_{ρ} \subseteq {(x_{1} \circ x_{2})}_{ρ}) . & (see [16]) \end{array}

A congruence relation ρ on is said to be complete if

(\forall x_{1}, x_{2} \in U) ({(x_{1})}_{ρ} {(x_{2})}_{ρ} = {(x_{1} \circ x_{2})}_{ρ}) .

Definition 4.1

Let ρ be an equivalence relation on a non-empty set and P = (μ_P, ν_P) a PFS in . The upper approximation is defined as

ρ^{+} (P) = {(x_{1}, {\bar{μ}}_{P} (x_{1}), {\bar{ν}}_{P} (x_{1})) ∣ x_{1} \in U},

where ${\bar{μ}}_{P} (x_{1}) = sup_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}$ and ${\bar{ν}}_{P} (x_{1}) = inf_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)}$ .

The lower approximation is defined as

ρ^{-} (P) = {(x_{1}, {\underline{μ}}_{P} (x_{1}), {\underline{ν}}_{P} (x_{1})) ∣ x_{1} \in U},

where ${\underline{μ}}_{P} (x_{1}) = inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}$ and ${\underline{ν}}_{P} (x_{1}) = sup_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)}$ .

It is easy to prove that ρ⁺(P) and ρ⁻(P) are PFSs in . Thus, we can denote the upper and lower approximations by ρ⁺(P) = (μ̄_P, ν̄_P) and ρ⁻(P) = (μ_P, ν_P), respectively.

Proposition 4.2

Let P = (μ_P, ν_P) and Q = (μ_Q, ν_Q) be PFSs in . If ρ is an equivalence relation on , then the following statements hold:

ρ⁻(P) ⊆ P ⊆ ρ⁺(P),
P ⊆ Q ⇒ ρ⁺(P) ⊆ ρ⁺(Q), ρ⁻(P) ⊆ ρ⁻(Q),
ρ⁺(P ∪ Q) = ρ⁺(P) ∪ ρ⁺(Q),
ρ⁺(P ∩ Q) ⊆ ρ⁺(P) ∩ ρ⁺(Q),
ρ⁻(P ∪ Q) ⊇ ρ⁻(P) ∪ ρ⁻(Q),
ρ⁻(P ∩ Q) = ρ⁻(P) ∩ ρ⁻(Q).

Proof

Let ρ be an equivalence relation of .

(1) Then for all ,

\begin{array}{l} {\underline{μ}}_{P} (x_{1}) = inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)} \\ \leq μ_{P} (x_{1}) \\ \leq sup_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)} \\ = {\bar{μ}}_{P} (x_{1}), \end{array}

and

\begin{array}{l} {\underline{ν}}_{P} (x_{1}) = sup_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)} \\ \geq ν_{P} (x_{1}) \\ \geq inf_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)} \\ = {\bar{ν}}_{P} (x_{1}) . \end{array}

By Definition 2.2 (1), we have that ρ⁻(P) ⊆ P ⊆ ρ⁺(P).

(2) If P ⊆ Q, then μ_P(x₁) ≤ μ_Q(x₁) and ν_P(x₁) ≥ ν_Q(x₁) for all . We consider

\begin{array}{l} {\bar{μ}}_{P} (x_{1}) & = sup_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)} \\ \leq sup_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)} & (by Proposition 1.6 (6)) \\ = {\bar{μ}}_{Q} (x_{1}), \\ {\bar{ν}}_{P} (x_{1}) & = inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)} \\ \geq inf_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)} & (by Proposition 1.6 (8)) \\ = {\bar{ν}}_{Q} (x_{1}), \\ {\underline{μ}}_{P} (x_{1}) & = inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)} \\ \leq inf_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)} & (by Proposition 1.6 (8)) \\ {\underline{μ}}_{Q} (x_{1}), \end{array}

and

\begin{array}{l} {\underline{ν}}_{P} (x_{1}) & = sup_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)} \\ \geq sup_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)} & (by Proposition 1.6 (6)) \\ {\underline{ν}}_{Q} (x_{1}) . \end{array}

By Definition 2.2 (1), we have that ρ⁺(P) ⊆ ρ⁺(Q) and ρ⁻(P) ⊆ ρ⁻(Q).

(3) By Definition 2.2 (3), we have that P∪Q = (μ_P_∪_Q, ν_P_∪_Q). Then

ρ^{+} (P \cup Q) = (\bar{μ}_{P \cup Q}, \bar{ν}_{P \cup Q}),

and

ρ^{+} (P) \cup ρ^{+} (Q) = ({\bar{μ}}_{P} \cup {\bar{μ}}_{Q}, {\bar{ν}}_{P} \cap {\bar{ν}}_{Q}) .

Thus, for all ,

\begin{array}{l} {\bar{μ}}_{P \cup Q} (x_{1}) & = sup_{a \in {(x_{1})}_{ρ}} {μ_{P \cup Q} (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {μ_{P} \cup μ_{Q} (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {max {μ_{P} (a), μ_{Q} (a)}} \\ = max {sup_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}, sup_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)}} & (by Proposition 1.6 (2)) \\ = max {{\bar{μ}}_{P} (x_{1}), {\bar{μ}}_{Q} (x_{1})} \\ = ({\bar{μ}}_{P} \cup {\bar{μ}}_{Q}) (x_{1}), \end{array}

and

\begin{array}{l} {\bar{ν}}_{P \cup Q} (x_{1}) & = inf_{a \in {(x_{1})}_{ρ}} {ν_{P \cup Q} (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {ν_{P} \cap ν_{Q} (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {min {ν_{P} (a), ν_{Q} (a)}} \\ = min {inf_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)}, inf_{a \in {(x_{1})}_{ρ}} {ν_{Q} (a)}} & (by Proposition 1.6 (1)) \\ = min {{\bar{ν}}_{P} (x_{1}), {\bar{ν}}_{Q} (x_{1})} \\ = ({\bar{ν}}_{P} \cap {\bar{ν}}_{Q}) (x_{1}) . \end{array}

Hence, ρ⁺(P ∪ Q) = ρ⁺(P) ∪ ρ⁺(Q).

(4) By Definition 2.2 (4), we have that P∩Q = (μ_P_∩_Q, ν_P_∩_Q). Then,

ρ^{+} (P \cap Q) = ({\bar{μ}}_{P \cap Q}, {\bar{ν}}_{P \cap Q}),

and

ρ^{+} (P) \cap ρ^{+} (Q) = ({\bar{μ}}_{P} \cap {\bar{μ}}_{Q}, {\bar{ν}}_{P} \cup {\bar{ν}}_{Q}) .

Thus, for all ,

\begin{array}{l} {\bar{μ}}_{P \cap Q} (x_{1}) & = sup_{a \in {(x_{1})}_{ρ}} {μ_{P \cap Q} (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {μ_{P} \cap μ_{Q} (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {min {μ_{P} (a), μ_{Q} (a)}} \\ \leq min {sup_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}, sup_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)}} & (by Proposition 1.6 (4)) \\ = min {{\bar{μ}}_{P} (x_{1}), {\bar{μ}}_{Q} (x_{1})} \\ = ({\bar{μ}}_{P} \cap {\bar{μ}}_{Q}) (x_{1}), \end{array}

and

\begin{array}{l} {\bar{ν}}_{P \cap Q} (x_{1}) & = inf_{a \in {(x_{1})}_{ρ}} {ν_{P \cap Q} (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {ν_{P} \cup ν_{Q} (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {max {ν_{P} (a), ν_{Q} (a)}} \\ \geq max {inf_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)}, inf_{a \in {(x_{1})}_{ρ}} {ν_{Q} (a)}} & (by Proposition 1.6 (3)) \\ = max {{\bar{ν}}_{P} (x_{1}), {\bar{ν}}_{Q} (x_{1})} \\ = ({\bar{ν}}_{P} \cup {\bar{ν}}_{Q}) (x_{1}) . \end{array}

Therefore, ρ⁺(P ∩ Q) ⊆ ρ⁺(P) ∩ ρ⁺(Q).

(5) By Definition 2.2 (3), we have that P∪Q = (μ_P_∪_Q, ν_P_∪_Q). Then

ρ^{-} (P \cup Q) = ({\underline{μ}}_{P \cup Q}, {\underline{ν}}_{P \cup Q}),

and

ρ^{-} (P) \cup ρ^{-} (Q) = ({\underline{μ}}_{P} \cup {\underline{μ}}_{Q}, {\underline{ν}}_{P} \cap {\underline{ν}}_{Q}) .

Thus, for all ,

\begin{array}{l} {\underline{μ}}_{P \cup Q} (x_{1}) & = inf_{a \in {(x_{1})}_{ρ}} {μ_{P \cup Q} (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {(μ_{P} \cup μ_{Q}) (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {max {μ_{P} (a), μ_{Q} (a)}} \\ \geq max {inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}, inf_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)}} & (by Proposition 1.6 (3)) \\ = max {{\underline{μ}}_{P} (x_{1}), {\underline{μ}}_{Q} (x_{1})} \\ = ({\underline{μ}}_{P} \cup {\underline{μ}}_{Q}) (x_{1}), \end{array}

and

\begin{array}{l} {\underline{ν}}_{P \cup Q} (x_{1}) & = sup_{a \in {(x_{1})}_{ρ}} {ν_{P \cup Q} (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {(ν_{P} \cap ν_{Q}) (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {min {ν_{P} (a), ν_{Q} (a)}} \\ \leq min {sup_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)}, sup_{a \in {(x_{1})}_{ρ}} {ν_{Q} (a)}} & (by Proposition 1.6 (4)) \\ = min {{\underline{ν}}_{P} (x_{1}), {\underline{ν}}_{Q} (x_{1})} \\ = ({\underline{ν}}_{P} \cap {\underline{ν}}_{Q}) (x_{1}) . \end{array}

Hence, ρ⁻(P ∪ Q) ⊇ ρ⁻(P) ∪ ρ⁻(Q).

(6) By Definition 2.2 (4), we have that P ∩ Q = (μ_P_∩_Q, ν_P_∩_Q). Then

ρ^{-} (P \cap Q) = ({\underline{μ}}_{P \cap Q}, {\underline{ν}}_{P \cap Q}),

and

ρ^{-} (P) \cap ρ^{-} (Q) = ({\underline{μ}}_{P} \cap {\underline{μ}}_{Q}, {\underline{ν}}_{P} \cup {\underline{ν}}_{Q}) .

Thus, for all ,

\begin{array}{l} {\underline{μ}}_{P \cap Q} (x_{1}) & = inf_{a \in {(x_{1})}_{ρ}} {μ_{P \cap Q} (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {(μ_{P} \cap μ_{Q}) (a)} \\ = inf_{a \in {(x_{1})}_{ρ}} {min {μ_{P} (a), μ_{Q} (a)}} \\ = min {inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}, inf_{a \in {(x_{1})}_{ρ}} {μ_{Q} (a)}} & (by Proposition 1.6 (1)) \\ = min {{\underline{μ}}_{P} (x_{1}), {\underline{μ}}_{Q} (x_{1})} \\ = ({\underline{μ}}_{P} \cap {\underline{μ}}_{Q}) (x_{1}), \end{array}

and

\begin{array}{l} {\underline{ν}}_{P \cap Q} (x_{1}) & = sup_{a \in {(x_{1})}_{ρ}} {ν_{P \cap Q} (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {(ν_{P} \cup ν_{Q}) (a)} \\ = sup_{a \in {(x_{1})}_{ρ}} {max {ν_{P} (a), ν_{Q} (a)}} \\ = max {sup_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)}, sup_{a \in {(x_{1})}_{ρ}} {ν_{Q} (a)}} & (by Proposition 1.6 (2)) \\ = max {{\underline{ν}}_{P} (x_{1}), {\underline{ν}}_{Q} (x_{1})} \\ = ({\underline{ν}}_{P} \cup {\underline{ν}}_{Q}) (x_{1}) . \end{array}

Hence, ρ⁻(P ∩ Q) = ρ⁻(P) ∩ ρ⁻(Q).

Theorem 4.3

Let ρ be a congruence relation on and P = (μ_P, ν_P) be a PFS in . Then, the following statements hold:

(1) if P is a PFIUPF of , (0)_ρ = {0}, and ρ is complete, then ρ⁻(P) is a PFIUPF of ,
(2) if P is a PFCUPF of and (0)_ρ = {0}, then ρ⁻(P) is a PFCUPF of ,
(3) if P is a PFSUPF of , (0)_ρ = {0}, and ρ is complete, then ρ⁻(P) is a PFSUPF of .

Proof

(1) Assuming that P is a PFIUPF of , (0)_ρ = {0}, and ρ is complete, then, for all x₁, x₂, ,

\begin{array}{l} {\underline{μ}}_{P} (0) & = inf_{a \in {(0)}_{ρ}} {μ_{P} (a)} = μ_{P} (0) \\ \geq μ_{P} (x_{1}) \geq inf_{b \in {(x_{1})}_{ρ}} {μ_{P} (b)} \\ = {\underline{μ}}_{P} (x_{1}), \\ {\underline{ν}}_{P} (0) & = sup_{a \in {(0)}_{ρ}} {ν_{P} (a)} = ν_{P} (0) \\ \leq ν_{P} (x_{1}) \leq sup_{b \in {(x_{1})}_{ρ}} {ν_{P} (b)} \\ = {\underline{ν}}_{P} (x_{1}), \\ {\underline{μ}}_{P} (x_{1} \circ x_{3}) & = inf_{d \in {(x_{1} \circ x_{3})}_{ρ}} {μ_{P} (d)} \\ = inf_{d \in {(x_{1})}_{ρ} {(x_{3})}_{ρ}} {μ_{P} (d)} & (by ρ is complete) \\ = inf_{a \circ c \in {(x_{1})}_{ρ} {(x_{3})}_{ρ}} {μ_{P} (a \circ c)} \\ \geq inf_{\begin{array}{l} a \circ (b \circ c) \in {(x_{1})}_{ρ} ({(x_{2})}_{ρ} {(x_{3})}_{ρ}), \\ a \circ b \in {(x_{1})}_{ρ} {(x_{2})}_{ρ} \end{array}} {min {μ_{P} (a \circ (b \circ c)), μ_{P} (a \circ b)}} & (by (32)) \\ = inf_{\begin{array}{l} a \circ (b \circ c) \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ}, \\ a \circ b \in {(x_{1} \circ x_{2})}_{ρ} \end{array}} {min {μ_{P} (a \circ (b \circ c)), μ_{P} (a \circ b)}} & (by ρ is complete) \\ = min {inf_{a \circ (b \circ c) \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ}} μ_{P} (a \circ (b \circ c)), inf_{a \circ b \in {(x_{1} \circ x_{2})}_{ρ}} {μ_{P} (a \circ b)}} & (by Proposition 1.6 (1)) \\ = min {{\underline{μ}}_{P} (x_{1} \circ (x_{2} \circ x_{3})), {\underline{μ}}_{P} (x_{1} \circ x_{2})}, \end{array}

and

\begin{array}{l} {\underline{ν}}_{P} (x_{1} \circ x_{3}) & = sup_{d \in {(x_{1} \circ x_{3})}_{ρ}} {ν_{P} (d)} \\ = sup_{d \in {(x_{1})}_{ρ} {(x_{3})}_{ρ}} {ν_{P} (d)} & (by ρ is complete) \\ = sup_{a \circ c \in {(x_{1})}_{ρ} {(x_{3})}_{ρ}} {ν_{P} (a \circ c)} \\ \leq sup_{\begin{array}{l} a \circ (b \circ c) \in {(x_{1})}_{ρ} ({(x_{2})}_{ρ} {(x_{3})}_{ρ}), \\ a \circ b \in {(x_{1})}_{ρ} {(x_{2})}_{ρ} \end{array}} {max {ν_{P} (a \circ (b \circ c)), ν_{P} (a \circ b)}} & (by (33)) \\ = sup_{\begin{array}{l} a \circ (b \circ c) \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ}, \\ a \circ b \in {(x_{1} \circ x_{2})}_{ρ} \end{array}} {max {ν_{P} (a \circ (b \circ c)), ν_{P} (a \circ b)}} & (by ρ is complete) \\ = max {sup_{a \circ (b \circ c) \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ}} ν_{P} (a \circ (b \circ c)), sup_{a \circ b \in {(x_{1} \circ x_{2})}_{ρ}} ν_{P} (a \circ b)}} & (by Proposition 1.6 (2)) \\ = max {{\underline{ν}}_{P} (x_{1} \circ (x_{2} \circ x_{3})), {\underline{ν}}_{P} (x_{1} \circ x_{2})} . \end{array}

Hence, ρ⁻(P) is a PFIUPF of .

(2) Assuming that P is a PFCUPF of and (0)_ρ = {0}, then, for all x₁, x₂, ,

\begin{array}{l} {\underline{μ}}_{P} (0) & = inf_{a \in {(0)}_{ρ}} {μ_{P} (a)} = μ_{P} (0) \geq μ_{P} (x_{1}) \\ \geq inf_{b \in {(x_{1})}_{ρ}} {μ_{P} (b)} \\ = {\underline{μ}}_{P} (x_{1}), \\ {\underline{ν}}_{P} (0) & = sup_{a \in {(0)}_{ρ}} {ν_{P} (a)} = ν_{P} (0) \leq ν_{P} (x_{1}) \\ \leq sup_{b \in {(x_{1})}_{ρ}} {ν_{P} (b)} \\ = {\underline{ν}}_{P} (x_{1}), \\ {\underline{μ}}_{P} (x_{2}) & = inf_{b \in {(x_{2})}_{ρ}} {μ_{P} (b)} \\ \geq inf_{\begin{array}{l} a \circ ((b \circ c) \circ b) \\ \in {(x_{1})}_{ρ} (({(x_{2})}_{ρ} {(x_{3})}_{ρ}) {(x_{2})}_{ρ}), \\ a \in {(x_{1})}_{ρ} \end{array}} {min {μ_{P} (a \circ ((b \circ c) \circ b)), μ_{P} (a)}} & (by (34)) \\ \geq inf_{\begin{array}{l} a \circ ((b \circ c) \circ b) \\ \in {(x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}))}_{ρ}, \\ a \in {(x_{1})}_{ρ} \end{array}} {min {μ_{P} (a \circ ((b \circ c) \circ b)), μ_{P} (a)}} & (by ρ is congruence) \\ = min {inf_{\begin{array}{l} a \circ ((b \circ c) \circ b) \\ \in {(x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}))}_{ρ} \end{array}} {μ_{P} (a \circ ((b \circ c) \circ b))}, inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}} & (by Proposition 1.6 (1)) \\ = min {{\underline{μ}}_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), {\underline{μ}}_{P} (x_{1})}, \end{array}

and

\begin{array}{l} {\underline{ν}}_{P} (x_{2}) & = sup_{b \in {(x_{2})}_{ρ}} {ν_{P} (b)} \\ \leq sup_{\begin{array}{l} a \circ ((b \circ c) \circ b) \\ \in {(x_{1})}_{ρ} (({(x_{2})}_{ρ} {(x_{3})}_{ρ}) {(x_{2})}_{ρ}), \\ a \in {(x_{1})}_{ρ} \end{array}} {max {ν_{P} (a \circ ((b \circ c) \circ b)), ν_{P} (a)}} & (by (35)) \\ \leq sup_{\begin{array}{l} a \circ ((b \circ c) \circ b) \\ \in {(x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}))}_{ρ}, \\ a \in {(x_{1})}_{ρ} \end{array}} {max {ν_{P} (a \circ ((b \circ c) \circ b)), ν_{P} (a)}} & (by ρ is congruence) \\ = max {sup_{\begin{array}{l} a \circ ((b \circ c) \circ b) \\ \in {(x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2}))}_{ρ} \end{array}} {ν_{P} (a \circ ((b \circ c) \circ b))}, sup_{a \in {(x_{1})}_{ρ}} {ν_{P} (a)}} & (by Proposition 1.6 (2)) \\ = max {{\underline{ν}}_{P} (x_{1} \circ ((x_{2} \circ x_{3}) \circ x_{2})), {\underline{ν}}_{P} (x_{1})} . \end{array}

Hence, ρ⁻(P) is a PFCUPF of .

(3) Assuming that P is a PFSUPF of , (0)_ρ = {0}, and ρ is complete, then, for all x₁, x₂, ,

\begin{array}{l} {\underline{μ}}_{P} (0) & = inf_{a \in {(0)}_{ρ}} {μ_{P} (a)} = μ_{P} (0) \\ \geq μ_{P} (x_{1}) \geq inf_{b \in {(x_{1})}_{ρ}} {μ_{P} (b)} \\ = {\underline{μ}}_{P} (x_{1}), \\ {\underline{ν}}_{P} (0) & = sup_{a \in {(0)}_{ρ}} {ν_{P} (a)} = ν_{P} (0) \\ \leq ν_{P} (x_{1}) \leq sup_{b \in {(x_{1})}_{ρ}} {ν_{P} (b)} \\ = {\underline{ν}}_{P} (x_{1}), \\ {\underline{μ}}_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3}) & = inf_{d \in {(((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})}_{ρ}} {μ_{P} (d)} \\ = inf_{d \in (({(x_{3})}_{ρ} {(x_{2})}_{ρ}) {(x_{2})}_{ρ}) {(x_{3})}_{ρ}} {μ_{P} (d)} & (by ρ is complete) \\ = inf_{((c \circ b) \circ b) \circ c \in (({(x_{3})}_{ρ} {(x_{2})}_{ρ}) {(x_{2})}_{ρ}) {(x_{3})}_{ρ}} {μ_{P} (((c \circ b) \circ b) \circ c)} \\ \geq inf_{\begin{array}{l} a \circ (b \circ c) \\ \in {(x_{1})}_{ρ} ({(x_{2})}_{ρ} {(x_{3})}_{ρ}), \\ a \in {(x_{1})}_{ρ} \end{array}} {min {μ_{P} (a \circ (b \circ c)), μ_{P} (a)}} & (by (36)) \\ = inf_{\begin{array}{l} a \circ (b \circ c) \\ \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ}, \\ a \circ b \in {(x_{1} \circ x_{2})}_{ρ} \end{array}} {min {μ_{P} (a \circ (b \circ c)), μ_{P} (a \circ b)}} & (by ρ is complete) \\ = min {inf_{\begin{array}{l} a \circ (b \circ c) \\ \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ} \end{array}} μ_{P} (a \circ (b \circ c)), inf_{a \in {(x_{1})}_{ρ}} {μ_{P} (a)}} & (by Proposition 1.6 (1)) \\ = min {{\underline{μ}}_{P} (x_{1} \circ (x_{2} \circ x_{3})), {\underline{μ}}_{P} (x_{1})}, \end{array}

and

\begin{array}{l} {\underline{ν}}_{P} (((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3} & = sup_{d \in {(((x_{3} \circ x_{2}) \circ x_{2}) \circ x_{3})}_{ρ}} {ν_{P} (d)} \\ = sup_{d \in (({(x_{3})}_{ρ} {(x_{2})}_{ρ}) {(x_{2})}_{ρ}) {(x_{3})}_{ρ}} {ν_{P} (d)} & (by ρ is complete) \\ = sup_{((c \circ b) \circ b) \circ c \in (({(x_{3})}_{ρ} {(x_{2})}_{ρ}) {(x_{2})}_{ρ}) {(x_{3})}_{ρ}} {ν_{P} (((c \circ b) \circ b) \circ c)} \\ \leq sup_{\begin{array}{l} a \circ (b \circ c) \\ \in {(x_{1})}_{ρ} ({(x_{2})}_{ρ} {(x_{3})}_{ρ}), \\ a \in {(x_{1})}_{ρ} \end{array}} {max {ν_{P} (a \circ (b \circ c)), ν_{P} (a)}} & (by (37)) \\ = sup_{\begin{array}{l} a \circ (b \circ c) \\ \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ}, \\ a \in {(x_{1})}_{ρ} \end{array}} {max {ν_{P} (a \circ (b \circ c)), ν_{P} (a)}} & (by ρ is complete) \\ = max {sup_{\begin{array}{l} a \circ (b \circ c) \\ \in {(x_{1} \circ (x_{2} \circ x_{3}))}_{ρ} \end{array}} ν_{P} (a \circ (b \circ c)), sup_{a \in {(x_{1})}_{ρ}} ν_{P} (a)}} & (by Proposition 1.6 (2)) \\ = max {{\underline{ν}}_{P} (x_{1} \circ (x_{2} \circ x_{3})), {\underline{ν}}_{P} (x_{1})} . \end{array}

Hence, ρ⁻(P) is a PFSUPF of .

The following example shows that Theorem 4.3 (1) may not be true if (0)_ρ ≠ {0} and ρ is incomplete.

Example 4.4

Consider a UP-algebra where is defined in Table 25.

If we define a PFS P = (μ_P, ν_P) as presented in Table 26, then P = (μ_P, ν_P) is a PFIUPF of . Let

ρ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 1), (1, 0), (0, 3), (3, 0)} .

Then ρ is a congruence relation on . Thus,

{(0)}_{ρ} = {(1)}_{ρ} = {(3)}_{ρ} = {0, 1, 3}, {(2)}_{ρ} = {2} .

However, ρ is not complete because

{0} = {2} {2} = {(2)}_{ρ} {(2)}_{ρ} \neq {(2 \circ 2)}_{ρ} = {(0)}_{ρ} = {0, 1, 3} .

The following example shows that Theorem 4.3 (2) may not be true if (0)_ρ = {0} and ρ is not complete.

Example 4.5

Consider a UP-algebra where is defined in Table 27.

If we define a PFS P = (μ_P, ν_P) as presented in Table 28, then P = (μ_P, ν_P) is a PFCUPF of . Let

ρ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 2), (2, 0)} .

Then ρ is a congruence relation on . Thus

{(0)}_{ρ} = {(2)}_{ρ} = {0, 2}, (1_{ρ}) = {1}, {(3)}_{ρ} = {3} .

However, ρ is not complete because

{0} = {1} {1} = {(1)}_{ρ} {(1)}_{ρ} \neq {(1 \circ 1)}_{ρ} = {(0)}_{ρ} = {0, 2} .

The following example shows that Theorem 4.3 (3) may not be true if (0)_ρ ≠ {0} and ρ is not complete.

Example 4.6

By Example 4.5, if we define a PFS P = (μ_P, ν_P) as presented in Table 29, then P = (μ_P, ν_P) is a PFSUPF of . Let

ρ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 2), (2, 0)} .

Then ρ is a congruence relation on . Thus,

{(0)}_{ρ} = {(2)}_{ρ} = {0, 2}, (1_{ρ}) = {1}, {(3)}_{ρ} = {3} .

However, ρ is not complete because

{0} = {3} {3} = {(3)}_{ρ} {(3)}_{ρ} \neq {(3 \circ 3)}_{ρ} = {(0)}_{ρ} = {0, 2} .

Example 4.7

Consider a UP-algebra where is defined in Table 30.

If we define a PFS P = (μ_P, ν_P) as presented in Table 31, then P = (μ_P, ν_P) is a PFIUPF of . Let

ρ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 3), (3, 0)} .

Then ρ is an equivalence relation on . Thus,

{(0)}_{ρ} = {(3)}_{ρ} = {0, 3}, (1_{ρ}) = {1}, {(2)}_{ρ} = {2} .

Example 4.8

From Example 4.7, if we define a PFS P = (μ_P, ν_P) as presented in Table 32, then P = (μ_P, ν_P) is a PFCUPF of . Let

ρ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 3), (3, 0)} .

Then ρ is an equivalence relation on . Thus,

{(0)}_{ρ} = {(3)}_{ρ} = {0, 3}, (1_{ρ}) = {1}, {(2)}_{ρ} = {2} .

Given that μ̄_P(2) = 0.1 ≱ 0.2 = min{0.8, 0.2} = min{max {0.8, 0.1}, 0.2} = min{μ̄_P(3), μ̄_P(1)} = min{μ̄_P(1 ∘ ((2 ∘ 3) ∘ 2)), μ̄_P(1)}, we find that ρ⁺(P) is not a PFCUPF of .

Example 4.9

From Example 4.7, if we define a PFS P = (μ_P, ν_P) as presented in Table 33, then P = (μ_P, ν_P) is a PFSUPF of . Let

ρ = {(0, 0), (1, 1), (2, 2), (3, 3), (0, 3), (3, 0)} .

Then ρ is an equivalence relation on . Thus,

{(0)}_{ρ} = {(3)}_{ρ} = {0, 3}, (1_{ρ}) = {1}, {(2)}_{ρ} = {2} .

Open Problem

Is the upper approximation ρ⁺(P) a PFIUPF (resp., PFCUPF, PFSUPF) of if P is a PFIUPF (resp., PFCUPF, PFSUPF) of and ρ is congruent?

5. Conclusion and Future Works

Some important topics for consideration in our future study of UP-algebras include:

(1) to study the roughness of PFSs, as defined by Pawlak [12],
(2) to study the soft set theory of PFSs based on the concept of fuzzy soft sets defined by Maji et al. [?],
(3) to extend the study results of PFSs in UP-algebras to Fermatean fuzzy sets, which are defined by Senapati and Yager [?], and
(4) to apply the results from this study to our research on PFS operators, based on Fermatean fuzzy sets guidelines in the studies [?, ?, ?].

Abstract
1. Introduction and Preliminaries
2. PFSs in UP-Algebras
3. Sufficient Conditions for PFSs in UP-Algebras
4. Approximations
5. Conclusion and Future Works

Fig 1.

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Figure 1.

New PFSs in UP-algebras.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 56-78https://doi.org/10.5391/IJFIS.2023.23.1.56

Fig 2.

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Figure 2.

Sufficient conditions for new PFSs in UP-algebras.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 56-78https://doi.org/10.5391/IJFIS.2023.23.1.56

Fig 3.

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Figure 3.

Eight types of PFSs in UP-algebras.

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 56-78https://doi.org/10.5391/IJFIS.2023.23.1.56

Table 1 . Cayley table for Example 2.6.

∘	1	2	3	4
0	1	2	3	4
1	0	2	3	4
2	0	0	3	3
3	1	2	0	3
4	1	2	0	0

Table 2 . A PFS for Example 2.6.

	0	1	2	3	4
μ_P	0.8	0.7	0.5	0.3	0.3
ν_P	0	0.2	0.3	0.5	0.5

Table 3 . Cayley table for Example 2.8.

∘	1	2	3	4
0	1	2	3	4
1	0	2	3	4
2	0	0	3	4
3	0	0	0	4
4	1	2	3	0

Table 4 . A PFS for Example 2.8.

	0	1	2	3	4
μ_P	1	0.6	0.6	0.6	0.4
ν_P	0	0	0.1	0.1	0.4

Table 5 . Cayley table for Example 2.10.

∘	1	2	3	4
0	1	2	3	4
1	0	2	2	4
2	0	0	2	4
3	0	0	0	4
4	1	2	3	0

Table 6 . A PFS for Example 2.10.

	0	1	2	3	4
μ_P	0.9	0.5	0.2	0.2	0.2
ν_P	0.3	0.3	0.4	0.4	0.4

Table 7 . Cayley table for Example 2.12.

∘	1	2	3	4
0	1	2	3	4
1	0	2	3	4
2	0	0	3	4
3	0	1	0	4
4	0	0	0	0

Table 8 . A PFS for Example 2.12.

	0	1	2	3	4
μ_P	0.6	0.5	0.2	0.1	0.1
ν_P	0.3	0.4	0.5	0.6	0.8

Table 9 . Cayley table for Example 2.14.

∘	1	2	3	4
0	1	2	3	4
1	0	0	0	4
2	1	0	0	4
3	1	2	0	4
4	1	2	3	0

Table 10 . A PFS for Example 2.14.

	0	1	2	3	4
μ_P	0.5	0.4	0.4	0.4	0.3
ν_P	0.4	0.5	0.5	0.5	0.8

Table 11 . Cayley table for Example 2.15.

∘	1	2	3	4
0	1	2	3	4
1	0	1	2	4
2	0	0	2	4
3	0	0	0	4
4	0	0	2	0

Table 12 . A PFS for Example 2.15.

	0	1	2	3	4
μ_P	0.9	0.9	0.9	0.9	0.3
ν_P	0.3	0.3	0.3	0.3	0.6

Table 13 . Cayley table for Example 2.16.

∘	1	2	3	4
0	1	2	3	4
1	0	1	2	4
2	0	0	1	4
3	0	0	0	4
4	1	2	3	0

Table 14 . A PFS for Example 2.16.

	0	1	2	3	4
μ_P	0.5	0.2	0.2	0.2	0.1
ν_P	0.2	0.6	0.6	0.6	0.8

Table 15 . Cayley table for Example 2.17.

∘	1	2	3
0	1	2	3
1	0	2	2
2	0	0	2
3	0	0	0

Table 16 . A PFS for Example 2.17.

	0	1	2	3
μ_P	0.5	0.2	0.1	0.1
ν_P	0.4	0.7	0.9	0.9

Table 17 . A PFS for Example 2.18.

	0	1	2	3	4
μ_P	0.8	0.1	0.2	0.6	0.1
ν_P	0.1	0.7	0.6	0.2	0.7

Table 18 . A PFS for Example 2.19.

	0	1	2	3	4
μ_P	0.5	0.2	0.3	0.4	0.2
ν_P	0.5	0.9	0.8	0.6	0.9

Table 19 . Cayley table for Example 2.20.

∘	1	2	3	4
0	1	2	3	4
1	0	0	0	0
2	1	0	0	4
3	1	2	0	4
4	1	2	3	0

Table 20 . A PFS for Example 2.20.

	0	1	2	3	4
μ_P	0.7	0.1	0.4	0.6	0.1
ν_P	0.2	0.7	0.6	0.4	0.7

Table 21 . Cayley table for Example 2.21.

∘	1	2	3
0	1	2	3
1	0	2	3
2	0	0	3
3	0	0	0

Table 22 . A PFS for Example 2.21.

	0	1	2	3
μ_P	0.6	0.4	0.2	0.2
ν_P	0.3	0.5	0.9	0.9

Table 23 . A PFS for Example 2.22.

	0	1	2	3
μ_P	0.7	0.7	0.4	0.4
ν_P	0.5	0.5	0.6	0.6

Table 24 . A PFS for Example 2.23.

	0	1	2	3
μ_P	0.5	0.5	0.1	0.1
ν_P	0.6	0.6	0.8	0.8

Table 25 . Cayley table for Example 4.4.

∘	1	2	3
0	1	2	3
1	0	2	0
2	1	0	3
3	1	2	0

Table 26 . A PFS for Example 4.4.

	0	1	2	3
μ_P	1	0.1	0.3	0.3
ν_P	0	0.5	0.2	0.2

Table 27 . Cayley table for Example 4.5.

∘	1	2	3
0	1	2	3
1	0	2	3
2	1	0	3
3	1	2	0

Table 28 . A PFS for Example 4.5.

	0	1	2	3
μ_P	0.8	0.3	0.5	0.8
ν_P	0.2	0.9	0.7	0.2

Table 29 . A PFS for Example 4.6.

	0	1	2	3
μ_P	1	0.2	0.1	0.5
ν_P	0	0.6	0.9	0.4

Table 30 . Cayley table for Example 4.7.

∘	1	2	3
0	1	2	3
1	0	3	3
2	0	0	0
3	1	1	0

Table 31 . A PFS for Example 4.7.

	0	1	2	3
μ_P	0.6	0.5	0.3	0.3
ν_P	0.4	0.5	0.7	0.7

Table 32 . A PFS for Example 4.8.

	0	1	2	3
μ_P	0.8	0.2	0.1	0.1
ν_P	0.2	0.6	0.9	0.9

Table 33 . A PFS for Example 4.9.

	0	1	2	3
μ_P	0.9	0.8	0.2	0.2
ν_P	0.3	0.4	08	0.8

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International Journalof Fuzzy Logic andIntelligent Systems

Original Article

Pythagorean Fuzzy Implicative/Comparative/Shift UP-Filters of UP-Algebras with Approximations

Abstract

1. Introduction and Preliminaries

Definition 1.1 [5]

Definition 1.2 [17]

Definition 1.3 [17]

Definition 1.4 [17]

Proposition 1.5 [28]

Proposition 1.6 [28]

Definition 1.7

2. PFSs in UP-Algebras

Definition 2.1

Definition 2.2

Definition 2.3 [28]

Definition 2.4

Theorem 2.5

Example 2.6

Theorem 2.7

Example 2.8

Theorem 2.9

Example 2.10

Theorem 2.11

Example 2.12

Theorem 2.13

Example 2.16

Example 2.17

Example 2.18

Example 2.19

Example 2.20

Example 2.21

Example 2.22

Example 2.23

Theorem 2.24

3. Sufficient Conditions for PFSs in UP-Algebras

Theorem 3.1

Theorem 3.2

Theorem 3.3

Theorem 3.4

Theorem 3.5

Theorem 3.6

Theorem 3.7

Theorem 3.8

Theorem 3.9

4. Approximations

Definition 4.1

Proposition 4.2

Theorem 4.3

Example 4.4

Example 4.5

Example 4.6

Example 4.7

Example 4.8

Example 4.9

Open Problem

5. Conclusion and Future Works

Acknowledgments

Conflict of Interest

Figures

Tables

References

Biographies

Article

Original Article

Pythagorean Fuzzy Implicative/Comparative/Shift UP-Filters of UP-Algebras with Approximations

Abstract

1. Introduction and Preliminaries

Definition 1.1 [5]

Definition 1.2 [17]

Definition 1.3 [17]

Definition 1.4 [17]

Proposition 1.5 [28]

Proposition 1.6 [28]

Definition 1.7

2. PFSs in UP-Algebras

Definition 2.1

Definition 2.2

Definition 2.3 [28]

Definition 2.4

International Journal
of Fuzzy Logic and
Intelligent Systems

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