An investigation on Pythagorean fuzzy fraction dense space using Pythagorean fuzzy frames

Abstract

The concept of frame is a generalisation of the concept of category of topological space open subsets. As a result, each frame acts as an open set in this context and the Pythagorean fuzzy sets is defined as a frame. The primary goal of this research unit is to investigate the behaviour of Pythagorean fuzzy frames. Pythagorean fuzzy structure space is defined using Pythagorean fuzzy frames. Pythagorean fuzzy closed sets, Pythagorean fuzzy dense set, Pythagorean fuzzy nowhere dense set, Pythagorean fuzzy somewhere dense set is established in order to investigate the Pythagorean fuzzy frames defined in Pythagorean fuzzy structure space. Further, Pythagorean fuzzy continuous function is explored in this manuscript. Separation axioms of the Pythagorean fuzzy structure space is established in order to comprehend the Pythagorean fuzzy frame. Additionally Pythagorean fuzzy fraction dense space and Pythagorean fuzzy space is defined and explored to examine the behaviour of defined Pythagorean fuzzy frames.

Introduction

The idea of uncertainty has been one of the most important developments in science and mathematics in the twenty-first century. The traditional perspective, which holds that uncertainty is undesirable in research and should be avoided at all costs has gradually given way to an alternate approach which is tolerant of uncertainty and holds that science cannot escape it. In an effort to address these challenges Zadeh developed the idea of fuzzy sets in 1965 to mathematically describe ambiguity. He did this by giving each member of a given set a certain grade of membership. A fuzzy set can be mathematically defined by giving each feasible individual in the universal of discourse a value that represents their degree of participation in the fuzzy set. The non-membership function was then introduced by Atanassov¹. Pythagorean fuzzy sets were introduced by Yager² as an extension of intuitionistic fuzzy sets. These Pythagorean fuzzy sets are described in this article as being framed under a few criteria. A category of open subsets in a space that may be more general than a topological space is comparable to a frame. Anything that has a collection of open subsets that function essentially like topological space open subsets can be used to define this. Currently, studies have emphasized the concept of frame category. This attempt to visualize frames makes use of the Pythagorean fuzzy set.

In this study, the novelty lies in the introduction and comprehensive examination of Pythagorean fuzzy frames as a generalization of open sets in topological spaces. Several classical topological notions such as closed sets, dense sets, nowhere dense sets, and somewhere dense sets are extended to the Pythagorean fuzzy environment within the newly developed Pythagorean fuzzy structure space. Moreover, the study introduces the separation axioms and investigates the characteristics of fraction dense spaces, spaces and Pythagorean fuzzy continuous functions which have not been previously explored in this setting. This novel study provides a new and meaningful perspective for analyzing and generalizing topological properties within the Pythagorean fuzzy context.

Review of literature

The term frame was introduced by Duffin and Schaeffer³ in non-harmonic fourier series. Later Dowker and Papert⁴ first studied frames in topology. He defined the complete lattice as the open subsets of a topological space⁵. They also demonstrated that the non-tautological statement of point-set topology can be verified in frame theory, or topology without points. Structured frames has been studied by Frith⁶. He established the category of uniform frames and quasi uniform frames. He also investigated the links between various frames. Later paracompactness is studied using frames by Pultr and Ulehla⁷. This study defined frames as paracompact and properties of paracompact frames were examined. This study also proved the frames are normal. Closure and compactness of frames were also studied by Masuret⁸. Rajesh and Thrivikraman⁹ investigated frames in fuzzy and intuitionistics fuzzy contexts. Various properties of fuzzy frames and intuitionistic frames were discussed in this study. Later Lattice valued fuzzy frames(L-Frames) were discussed by El-Saady¹⁰. This study defined the concept of L-fuzzy sub-frames of a given ordinary frame related to traditional frames analogously to how L-fuzzy topological spaces related to L-topological spaces. Some properties of L-fuzzy sub-frames are explored. The notion of the existence of L-fuzzy sub-frames of a particular ordinary frame was put forward in this work in the same way as L-fuzzy topological spaces were defined in relation to L-topological spaces. Also¹¹ Studies in categorical topology have examined the relationship between topological spaces and frames in presheaf toposes of Msets, exploring internalizations, functorial connections, and conditions for adjunctions, with special focus on sobriety and spatiality when M is a group. Later fuzzy frames were studied via fuzzy posets Yao¹². Yao’s intention was to define an L-frame using an L-ordered set that included more restrictions. All of these works illustrate how frames have been investigated in a variety of circumstances. Frames are explored in fraction dense space in this article. Zhang¹³ investigated a general frame in intuitionistic rough set and defined intuitionistic fuzzy relation and its properties using lattice. Thumbakara¹⁴defined intuitionistic fuzzy frame and coframe and also explored intuitionistic fuzzy filters in coframe context¹⁵. have explored ideals generated by frame homomorphisms where structures are used to form frame congruences and sublocales and the resulting locale is analyzed for compactness conditions linking algebraic and topological properties¹⁶, have explored semilattice-based structures such as S-bases, D-bases, and L-bases to generalize frame completeness properties and unify classical classes like zero-dimensional, completely regular, and coherent frames.. These classical study paved way to define a structure space using frames in Pythagorean fuzzy context in the present study. Furthermore, continuity is defined in the present study based on^17–21 and these are the basic study to define the fuzzy continuous function. So these references are reviewed to define continuity in Pythagorean fuzzy frame. Separation axioms are also included based on the study of^16,22–26. Since these separation axioms gives the detail study of open sets. The separation axioms defined in the study is based on above mentioned study. The concept of fraction dense space was given by Hager and Martinez²⁷ in algebra. It was insisted that Fraction-dense algebras arise naturally in the consideration of quotient rings, and they give rise to an interesting class of topological spaces.

Contribution of the study

The paramount goal of this scholarly article is to analyse the behaviour of Pythagorean fuzzy frames which is defined as Pythagorean fuzzy sets in various spaces.

• (i)The paper constructively deal with the an introduction of Pythagorean fuzzy frames, which is novel and represents an extension of existing intuitionistic fuzzy frames.
• (ii)Pythagorean fuzzy structure space is established using Pythagorean fuzzy frames. Then Pythagorean fuzzy closed sets is defined. Also continuity, separation axioms of the Pythagorean fuzzy structure space are meticulously investigated in order to study the Pythagorean fuzzy open sets of the Pythagorean fuzzy structure space.
• (iii)Pythagorean fuzzy structure space is carving the path toward the conceptualization of Pythagorean fuzzy fraction dense space.
• (iv)The conceptual framework Pythagorean Fuzzy fraction dense space enhances the investigation of how the frames behave as sets in the new space established.
• (v)The Pythagorean fuzzy space is defined to study the relationship between the PFRCS and Pythagorean fuzzy closed sets.

Structure of the paper

In this study, section “Preliminaries”, consists of the basic definitions used for the study. Section “Pythagorean fuzzy frame (PFF)”, gives the definition for Pythagorean fuzzy frames and Pythagorean fuzzy structure space is explained. Also Pythagorean fuzzy closed set is defined in the Pythagorean fuzzy structure space. Various properties of Pythagorean fuzzy structure space is discussed. In section 6, Pythagorean fuzzy fraction dense space is defined. In section 7, Pythagorean fuzzy space defined and the characterisations are explored. The flowchart provides the framework of the study Fig. 1.

Fig. 1.

Framework of the Pythagorean fuzzy fraction dense space.

Preliminaries

This section provides the basic definition for this study and the nomenclature used for this study is given in the Table 1.

Table 1.

Nomenclature of this study.

Expansion	Abbreviation
Pythagorean fuzzy frame	PFF
Pythagorean fuzzy structure space	PFSS
Pythagorean fuzzy open set	PFOS
Pythagorean fuzzy closed set	PFCS
Pythagorean fuzzy closed set	PFCS
Pythagorean fuzzy closed set	PFCS
Pythagorean fuzzy regular open set	PFROS
Pythagorean fuzzy regular closed set	PFRCS
Pythagorean fuzzy dense set	PFDS
Pythagorean fuzzy nowhere dense set	PFnWDS
Pythagorean somewhere dense set	PFsWDS
Pythagorean fuzzy cs-dense set	PFcsDS
Pythagorean fuzzy continuous function	PFCF
Pythagorean fuzzy fraction dense space	PFFDS
Pythagorean fuzzy space	PF S

Definition 1

²⁸A lattice is the partial ordered elements of the power set of a universal set (or any subset of ) can be ordered by the set inclusion S in which the join (least upper bound, supremum) and meet (greatest lower bound, infimum) of any pair of sets is given by and , respectively.

Definition 2

⁵A partially ordered set(poset) is a set L with a relation , such that

1. if and then and

2. if and then .

Definition 3

⁵A complete lattice is a partially ordered set such that every subset A of L has a least upper bound. The least upper bound is unique and usually called the join of A and written as or in terms of elements, or .

Definition 4

²⁹A frame is a complete lattice L satisfying the distributivity law for any subset and any .

Definition 5

³⁰A Pythagorean fuzzy set R of is a pair such that for any where the fuzzy set , are the membership value and non-membership value respectively and is the strength of commitment at a point.

Definition 6

³⁰Let be a family of Pythagorean fuzzy set of . If

• (i)
• (ii), we have where I is an arbitrary index set .
• (iii), then , where and , then is called a Pythagorean fuzzy topology on and the pair be a Pythagorean fuzzy topological space.

Definition 7

³⁰Let and be two Pythagorean fuzzy sets of a non-empty set . Then,

• (iv) or if and

Definition 8

³¹Let be a Pythagorean fuzzy topological space and be a Pythagorean fuzzy set in . Then the Pythagorean fuzzy interior and Pythagorean fuzzy closure are defined by,

• (i)int(R)= {G|G is a PFOS in and }
• (ii)cl(R)= {K|K is a PFCS in and}

Pythagorean fuzzy frame (PFF)

In this section PFF and Pythagorean fuzzy structure space(PFSS) is defined using PFFs. Further Pythagorean fuzzy closed set(PFCS) is defined in Pythagorean fuzzy structure space. Various sets Pythagorean fuzzy dense set, Pythagorean fuzzy nowhere dense sets, Pythagorean fuzzy somewhere dense set, Pythagorean fuzzy cs-dense set is also defined and the continuous function of the defined PFSS is discussed.

Definition 9

Let be the frame in , then the Pythagorean fuzzy set is said to be PFF in , if it satisfies the following conditions:

• (i) for every arbitrary .
• (ii) for every .
• (iii) for all where and are unit and zero element of the frame .

Example 1

Let on where be the frame and where

is a PFF of .

Definition 10

Let be the frame of any non-empty set and let be a collection of PFFs. If this collection satisfies the following axioms

• (i),
• (ii)for any , where have
• (iii)for any , then () is called Pythagorean fuzzy structure space(PFSS). Each member in () is Pythagorean fuzzy open set(PFOS) and its complement is called Pythagorean fuzzy closed set(PFCS).

Example 2

Consider the frame . The PFFs are defined as , , where,

.

Therefore the collection of PFFs is a PF structure. Then the structure is a PFSS.

Definition 11

Pythagorean fuzzy closure and Pythagorean fuzzy interior of a PFS is defined by,

is closed in is open in

Definition 12

A PFF P of a PFSS is called PFCS if whenever where is a PFOS and . The counterpart of PFCS is the PFOS.

Notation: will indicate the assortment of all PFCS in .

Definition 13

A PFF P of a PFSS is called PFCS if whenever where is a PFOS and . The counterpart of PFCS is the PFOS.

Definition 14

The collection is PFCS in . The counterpart of PFCS is PFOS.

Notation: will indicate the assortment of all PFCS in .

Definition 15

Let be PFSS. A PFF R is called a Pythagorean fuzzy regular open set (PFROS) if and only if ; A PFS S is called a Pythagorean fuzzy regular closed (PFRCS) if and only if

Proposition 1

Let be PFSS, Then

• (i)The closure of PFOS is a PFRCS.
• (ii)The interior of PFCS is a PFROS.

Proof

(i) Let K be a PFOS in . Clearly, implies that . Now K is open implies that and hence . Thus is a PFRCS.

(ii) The proof is similar to (i) .

Proposition 2

For a PFF E of a PFSS . Then

• (i).
• (ii).

Proof

(i)

where .

(ii)

where .

Definition 16

A PFF K is a PFSS is called

• (i)PFDS if there exists no PFCS G in such that that is in .
• (ii)PFnWDS if there exists no non-zero PFOS F in such that that is in .
• (iii)PFsWDS if there exists a non-zero PFOS G in such that that is in and is called a complement of PFsWDS in and is denoted as PFcs-DS in .

Proposition 3

If K is a PFsWDS in a PFFDS , then there exists a PFRCS N in such that .

Proof

Let E be a PFsWDS in . Then, there exists PFOS in such that . Now . Since F is a PFOS by Proposition 6.2.51, the closure of F is a PFRCS in . Let . Then for the PFsWDS in K in there exists a PFRCS N in such that .

Proposition 4

If K is a PFcs-DS in PFFDS then,

• (i) is not a PFDS in .
• (ii)There exists a PFROS in such that .

Proof

(i) Let K be a PFcs-DS in . Then is a PFsWDS in . Thus

in . This implies that . So . Hence is not a PFDS in .

(ii) By (i) is not a PFDS in . Then there exists a PFCS F in such that . Thus . That is in . Since F is a PFCS in . By Proposition 6.2.51 is a PFROS in . Let . Then there exists a PFROS N in such that .

Continuous function in PFSS

Definition 17

Let and be any two PFSS and let be a function. If for any of , is a in , then is said to be a Pythagorean fuzzy continuous function (PFCF).

Proposition 5

Let and be any two PFSS and let be PFCF. Then for every PFF in , .

Proof

Let K be a PFF in . Since is a PFCS and is a PFCF. is a PFCS and . Now . Therefore, .

Proposition 6

Let and be any two PFSS. If E is PFCS in and if be a PFCF then is a PFCS in .

Proof

Let K be a PFOS in . If then in . Since E is a PFCS and is a PFOS in . Then implies . By assumption, is PFCS in and . Hence is PFCS.

Proposition 7

Let and be two PFSS and . Then the following statements are equivalent:

• (i) is PFCF.
• (ii) for each E in .
• (iii) for each E in .

Proof

from (i) is PFCF. Let E be a Pythagorean fuzzy frame. By the definition of PFCF is a PFF in . is a PFF in then is a PFF in . Therefore, .

. Given . Let E be a PFF in . Let E be a PFOS in . Since , . By (i) . Therefore . Hence . Therefore is a PFF in . Hence is PFCF.

Given is PFCF. Let E is a PFF in and . By Proposition 6.2.55, . Thus .

. To prove is PFCF. It is enough to prove the inverse image of each PFF in is a PFF in . Let E be a PFF in . To show that is PFF in . Since . but . Hence . Therefore is PFF in . This proves is a PFCF.

Separation axioms on Pythagorean fuzzy structure space

In this section, separation axioms are discussed on Pythagorean fuzzy structure space in detail. Four different spaces are defined and the characterisations are investigated.

Definition 18

A is called

• (i) space a) if for all there exists a such that or .
• (ii) space b) if for all there exists a such that or .
• (iii) space c) if for all there exists a such that or .
• (iv) space d) if for all there exists a such that or .

Proposition 8

Let be a . Then the following implications hold and it is given in Fig. 2

Fig. 2.

This diagram depicts the implication of space.

Proof

To prove . Let be a by definition of for all there exists such that implies , which is . Hence .

Similarly,

,

,

.

Remark 1

The converse of the above implications is not true. It can be seen through the following Examples 6.3.31, 6.3.32, 6.3.33.

Example 3

Consider the frame . The PFFs are defined as , ,

, where,

.

.

Then is a but not .

Example 4

Consider the frame . The PFFs are defined as , where,

Then is a but not .

Example 5

Consider the frame . The PFFs are defined as , where,

Then is a but not .

Proposition 9

Let be a , and be the characteristic function of Q and then

• (i) is is
• (ii) is is
• (iii) is is
• (iv) is is

Proof

Let is a space a). Let . Let then as . Since is a)space then there exists such that and Then R are in Q such that . Then Q is also a) space.

The proof of (ii), (iii), (iv) is obvious.

Definition 19

A is called

• (i) space a) if for all there exists a such that and
• (ii) space b) if for all there exists a such that and
• (iii) space c) if for all there exists a such that and
• (iv) space d) if for all there exists a such that and

Proposition 10

Let be a . Then the following implications hold and it is given in Fig. 3

Fig. 3.

This diagram depicts the implication of space.

Proof

To prove . Let be a by definition of for all there exists such that and implies , which is . Hence . Similarly,

,

,

.

Remark 2

None of the above implications are true. It can be proved by the following similar examples.

Proposition 11

Let be a , and be the characteristic function of Q and then

• (i) is is
• (ii) is is
• (iii) is is
• (iv) is is

Proof

Let is a space a). Let . Let then as . Since is a)space then there exists such that and . Then R, S are in Q such that and . Then Q is also a) space.

The proof of (ii), (iii), (iv) is obvious.

Definition 20

A is called

• (i) space a) if for all there exists a such that and
• (ii) space b) if for all there exists a such that and where
• (iii) space c) if for all there exists a such that and where
• (iv) space d) if for all there exists a such that and where

Proposition 12

Let be a . Then the following implications hold and it is given in Fig. 4

Fig. 4.

This diagram depicts the implication of space.

Proof

To prove . Let be a by definition of for all there exists such that and implies and where which is . Hence . Similarly,

,

,

.

Pythagorean fuzzy fraction dense space (PFFDS)

In this section we define Pythagorean fuzzy fraction dense space using the Pythagorean fuzzy structure space.

Definition 21

A PFSS () is called Pythagorean fuzzy fraction dense space (PFFDS) if for each M in , where F is a in .

Example 6

Consider the frame . The PFFs are defined as , ,

where,

.

Therefore the collection of PFFs . Then the structure is a PFSS. Let are the PFCS in which is defined as,

Then the = is PFS. Therefore, PFSS is called PFFDS.

Proposition 13

A PFSS is a PFDS if and only if for each PFRCS F in , where N is a PFCS in .

Proof

Let F be a PFRCS in . Then in . Let . Then K is PFOS in . Since is a PFFDS, where N is a PFCS in . Thus and in . Conversely, let T be a PFOS in . Then is a PF RCS in . By Proposition 4.10 where K is a PFCS in and then is PFFDS.

Proposition 14

If is a PFFDS and L is a PFROS in then where R is a PF in .

Proof

Let L is a PFROS in . is a PFRCS in . Since is a PFFDS by Proposition 6.3 where K is a PFCS in . Then by Proposition 4.11. Let where K is a PFOS in . Hence where R is a PFOS in .

Proposition 15

If K is a PFOS in PFFDS , then there exists a PFCS G in such that .

Proof

Let K be a PFOS in . Since is a PFFDS. where S is a PFCS in . implies .

Proposition 16

If K is a PFOS in a PFDS in , then there exists a PFCS S in such that .

Proof

Let K be a PFOS in . Now in . Since is a PFFDS, . Then K is PFCS in . Thus there exists a PFCS K in such that .

Proposition 17

If K is a PFOS in PFFDS in , then there exists PFCS G and S in such that .

Proof

Let K be a PFOS in . Since is a PFFDS, by Proposition 6.5, there exists a PFCS G in such that . Also by Proposition 6.6, there exists a PFCS E in such that . Then . Since in , for a PFOS K in , .

Proposition 18

If Q is a PFCS in PFFDS , then there exists a PFOS R in such that .

Proof

Let Q is a PFCS. Then is a PFOS in . Since is a PFFDS by Proposition 6.5 there exists a PFCS G in , . Then, by Proposition 4.11. This implies that . Let . Then is a PFOS and in .

Proposition 19

If Q is a PFCS in PFFDS , then there exists a PFOS M in such that .

Proof

Let Q is a PFCS in . Then is PFOS in . Since is PFFDS by Proposition 6.6, there exists a PFCS in such that . then, and by Proposition 4.11 . Let and M is a PFOS in and in .

Proposition 20

If Q is a PFCS in PFFDS in , then there exists PFOS M and R in such that in .

Proof

Let Q be a PFCS in . Since is PFFDS, by Proposition 6.8 there exists a PFOS R in such that . Also by Proposition 6.9, there exists a PFOS M in such that , then in . This implies that in .

Proposition 21

If L is a PFROS in a PFFDS then there exists a PFOS R with in such that .

Proof

Let L be a PFROS in . Since is a PFFDS, by Proposition 6.4, there exists a PFOS R in such that . Now . This implies and thus . Thus there is a PFOS R with in such that .

Corollary 1

If L is a PFROS in a PFFDS , then there exists a PFsWDS R in such that .

Proof

Let L be a PF ROS in .Since is a PFFDS,by Proposition 6.11 there exists a PFOS R with in such that . Now implies that R is a PFsWDS in .

Proposition 22

If M is a PFRCS in PFFDS , then there exists a PFCS Q in such that .

Proof

Let M be a PFRCS in . Then is a PFROS in . Since is a PFFDS by Proposition 6.11 there exists a PFOS R in such that then . Let Let Q is a PFCS in . Hence there exists a PFCS Q in such that .

Corollary 2

If M is a PFRCS in PFFDS , then there exists a PFcs-DS Q in such that .

Proof

Let M be a PFRCS in . Then is a PFROS in . Since is a PFFDS by Corollary 6.12, there exists a PFsWDS R in such that in . Then . Let . Then Q is a PFcs-DS in and .

Proposition 23

If is a PFFDS, then there exists a PFCS K and PFOS R in such that .

Proof

Let L be a PFROS in . Since is PFFDS by Proposition 6.11 there exists a PFOS R in such that and then in . Since a PFROS is a PFOS in , by Proposition 6.5. there exists a PFCS K in such that . Then and thus in .

The following Propositions from Proposition 6.16 to Proposition 6.23 shows that PFCS are not PFnWDS and the PFOS are not PFDS in PFFDS.

Proposition 24

If Q is a PFCS in PFFDS , then Q is not a PFnWDS in .

Proof

Let Q be a PFCS in . Since is a PFFDS,by Proposition 6.9 there exists a PFOS M in such that . Then and implies in . Hence Q is not a PFnWDS in .

Proposition 25

If K is a PFOS in PFFDS then K is not a PFDS in .

Proof

Let K is a PFOS in . Suppose that in . Then . This implies that the PFCS is a PFnWDS in the PFFDS , a contradiction by Proposition 6.16. Hence K is a not a PFDS in .

Proposition 26

If Q is a PFCS in PFFDS there exists a PFRCS K in such that .

Proof

Let Q is a PFCS in , then . Since is a PFFDS, Proposition 6.16, Q is a not a PFnWDS in and then in . Now and then there exists a PFOS K in such that . Then and is a PFRCS in by Proposition 4.10. Let . Hence there exists a PFRCS K in such that .

Proposition 27

If K is a PFOS in a PFFDS then there exists a PFROS L in such that .

Proof

Let K is a PFOS in . Then is a PFCS in . Since is a PFFDS by Proposition 6.18 there exists a PFRCS K in such that . Then . Let and L is a PFROS in and .

Proposition 28

If K is a PFnWDS in PFFDS , then there exists a PFcs-DS Q in such that .

Proof

Let K be a PFnWDS in and then is a PFCS in . Since is a PFFDS, by Proposition 6.18 there exists a PFRCS N in such that . By Corollary 6.14 there exists a PFcs-DS Q in such that and then .

Proposition 29

If K is a PFnWDS in a PFFDS , then there exists no non-zero PFRCS N in such that .

Proof

Let K is a PFnWDS in and . Since is a PFFDS by Proposition 6.18. there exists a PFRCS N in such that , then and . This implies that and . Thus, there exists no non-zero PFRCS K in such that .

Proposition 30

Let K is a PFFDS , then there exists a PFOS R in such that .

Proof

Let K be a PFROS in . Then in . Now is a PFCS in . Since is a PFFDS, by Proposition 6.9, there exists a PFOS R in such that . Then and in .

Proposition 31

If K is a PFROS in a PFFDS , then there exists a PFOS R and T in such that .

Proof

Let K be a PFROS in . Since is a PFFDS, by Proposition 6.11, there exists a PFOS in such that . Also by Proposition 6.22, there exists a PFOS T in such that . Thus in .

Pythagorean fuzzy fraction dense space and Pythagorean fuzzy space (PFS)

In this section Pythagorean fuzzy space is defined and it is proved that PFRCS are PFCS, also PFROS are PFOS in PFFDS and PFS.

Definition 22

A PFFDS is called PFS if each PFOS in is PFOS in .

Proposition 32

If F is a PF RCS in a PFFDS and PFS , then F is a PFCS in .

Proof

Let F be a PFRCS in . Since is a PFFDS, by Proposition 6.3. where K is a PFCS in . Since is a PFS, PFCS K is a PFCS and then in . Hence PFRCS F is a PFCS in .

Corollary 3

If K is PFROS in PFFDS and PFS , then K is a PFOS in .

Proof

Let K be a PFROS in . Then is a PFCS in . Since is a PFFDS, by Proposition 7.2. is PFCS in and thus K is a PFOS in .

Proposition 33

If F is a PFCS in PFFDS and PFS , then F is a PFsWDS in .

Proof

Let F be a PFCS in . Since is a PFS, the PFCS, F is a PFCS and then by Proposition 6.16 F is not a PFnWDS in . Thus in . Hence F is a PFsWDS in .

Corollary 4

If K is a PFOS in PFFDS and PFS , then K is a PFcs-DS in .

Proof

Let K is a PFOS in PFFDS . Then is a PFCS in . Since is a PFFDS by Proposition 7.4. is PFsWDS in and thus K is a PFcs-DS in .

Proposition 34

If F is a PFOS in PFFDS and PFS , then there exists a PFRCS N in such that .

Proof

Let F is a PFOS in . Since is PFFDS and PFS by Proposition 7.4. F is PFsWDS in . By Proposition 4.13. there exists a PFRCS N in such that .

Corollary 5

If K is a PFOS in PFFDS and PFS , then there exists a PFROS T in such that .

Proof

Let K be a PFOS in . Then is a PFCS in . Since is PFFDS and PFS by Proposition 7.6., there exists a PFRCS N in such that . This implies and . Let . Then T is a PFROS in and .

Proposition 35

If K is a PFOS in a PFFDS and PFS, then

• (i) in .
• (ii)K is not a PFDS in .

Proof

(i) Let K be a PFOS in . Then is a PFCS in . Since is PFFDS and PFS, by Proposition 7.4 is a PFsWDS and in . Then and in .

(ii) Since is a PFS, the PFOS K is a PFOS in . Also since is a PFFDS by Proposition 6.6, there exists a PFCS N in such that and is PFCS in , implies that in . Hence K is not a PFDS in .

Proposition 36

If K is a PFOS in a PFFDS and PFS , then is a PFCS in .

Proof

Let K be a PFOS in . Since is a PFFDS , where F is PFCS in . Also is a PFS, the PFCS F is a PFCS in and then . Then , where F is a PFCS in . Hence is a PFCS in .

Corollary 6

If F is a PFCS in PFFDS and PFS , then is a PFOS in .

Proof

Let F be PFCS in . Then is a PFOS in PFFDS and PFS , by Proposition 7.9. is PFCS in . By Proposition 4.11. . Hence is a PFOS in .

Comparison with existing theory

In the existing literature, the concept of Pythagorean fuzzy frames has been primarily studied as an extension of classical fuzzy and intuitionistic fuzzy frameworks, focusing mainly on the representation of uncertainty and partial truth in algebraic or decision-theoretic contexts. Earlier studies have concentrated on defining Pythagorean fuzzy sets, relations and topological spaces establishing their basic properties and operations. However, the structural development of Pythagorean fuzzy frames particularly from a topological viewpoint has remained limited.

The current study advances this theoretical foundation by introducing a Pythagorean fuzzy structure space which generalizes the notion of open sets in topology using frame theory. Unlike previous works that treated fuzzy frames as abstract algebraic constructs this research systematically defines and investigates new topological components within the Pythagorean fuzzy frame setting such as Pythagorean fuzzy closed sets, dense, nowhere dense and somewhere dense sets, continuous functions and separation axioms.

Furthermore, the study introduces and explores new spaces Pythagorean fuzzy fraction dense spaces and Pythagorean fuzzy spaces which have not been previously formulated in existing theories. These additions enrich the structural and functional understanding of Pythagorean fuzzy frames bridging the gap between abstract lattice theoretic foundations and topological generalizations.

In summary, while existing theories laid the groundwork for representing uncertainty using Pythagorean fuzzy sets and relations, the current study extends this framework by developing a comprehensive topological model based on frames offering a more generalized and flexible approach for analyzing fuzzy structures.

Conclusion and future work

In this study, Pythagorean fuzzy frames, Pythagorean fuzzy structure space, Pythagorean fuzzy continuous function, separation axioms on Pythagorean fuzzy structure space is established. Various sets are investigated in Pythagorean fuzzy structure space. Pythagorean fuzzy fraction dense space is introduced to study the defined sets in structure space. Several characterizations of PFFDS are investigated. Also the relationship between PF CS, PFOS, PF, PFCS are are explored in PFFDS and PFS. In future, relationship between PFFDS and various other spaces like baire space, sober space, cellular spaces can be investigated. Pythagorean fuzzy coframes, Pythagorean fuzzy subframes can be discussed in various Pythagorean fuzzy topological spaces.

The system of open sets of a space where the relationships between these sets meet certain lattice qualities is represented by an algebraic structure called a frame in topology. Each open set can be described in terms of degrees of membership and non-membership expressing uncertainty and partial truth conditions that are prevalent in many real-world systems by extending this concept into a Pythagorean fuzzy frame. For instance, a medical diagnosis system that groups patients according to test results and symptoms. A symptom may be categorized as present or absence using traditional crisp sets. In reality though, a lot of symptoms are not absolute; a patient may display them to some degree. Each symptom (like an open set) can be represented with Pythagorean fuzzy membership values using Pythagorean fuzzy frames which capture the degree to which a symptom confirms or refutes a diagnosis. The Pythagorean fuzzy component captures the uncertainty present in medical assessment while the frame structure guarantees the algebraic preservation of links among various symptoms (intersections, unions, and inclusion relations). This makes decision-making more precise and adaptable particularly in cases where clinical data are inconsistent or lacking. Similarly, this approach can be extended to: Risk analysis, where uncertain parameters influence investment or safety decisions. Artificial intelligence, where fuzzy frames support learning and reasoning under uncertainty. Engineering control systems, where approximate measurements require adaptive modeling.

Author contributions

1. N.B. Idea, Writing, Methodology and discussions 2. G.K. Idea, Drafting and Editing

Funding

Open access funding provided by Vellore Institute of Technology. The authors received no funding for this work.

Data availability

No datasets were generated or analysed during the current study.

Competing interests

The authors declare no competing interests.