EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5956 ISSN 1307-5543 – ejpam.com Published by New York Business Global Picture Fuzzy Modal Ideal Multifunctions Dali Shi1, M.N. Abu_Shugair2,∗, S.E. Abbas3, Ismail Ibedou4 1 Gugangzhou College of Technology and Business, China 2 Mathematics Department, College of Science, Jazan University, Jazan 45142, Saudi Arabia 3 Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt 4 Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt Abstract. This paper introduces the notion of a picture fuzzy modal topological structures (PFMTSs) via ideal. These structures are grounded on novel picture fuzzy topological opera- tors for closure and interior types, utilizing the two standard picture fuzzy modal operators □ and 3. The paper discusses several fundamental properties of picture fuzzy multifunctions PFMs via ideals. The results indicate that some properties considered satisfactory in the intuitionistic fuzzy modal topological structures, as defined by Atanassov in 2022, are not fulfilled. Also, we introduce many types of continuous multifunctions between picture fuzzy ideal topological spaces. 2020 Mathematics Subject Classifications: 94D05, 03E72, 03E75, 03B52, 03B20 Key Words and Phrases: Picture fuzzy multifunction, picture fuzzy modal topology, picture fuzzy operator 1. Introduction Fuzzification is a crucial tool for addressing humanistic systems in real-life problems. The seminal paper on fuzzy set theory was authored by Zadeh in 1965 ([1]). This theory of fuzzy sets (FSs) has been widely applied by many scholars. FSs theory described the positivism of an element ξ of a universal set ξ to a subset K ⊆ Ξ by the membership value ωK(ξ), and posited that the negativism of that element ξ ∈ Ξ to the set K is 1 − ωK(ξ). Atanassov in [2] based his theory of intuitionistic fuzzy sets (IFSs) on the notion that the negativism ϖK(ξ) of an element ξ ∈ Ξ to a subset K ⊆ Ξ may range from [0, 1] and need not be the complement of the positivism of that element ξ ∈ Ξ to K. The values ωK(ξ) and ϖK(ξ) represent the positivism and negativism of each ξ ∈ Ξ to K, respectively, with the condition that 0 ≤ ωK(ξ) + ϖK(ξ) ≤ 1. In this way, Atanassov encompassed all the FSs as a special case of his theory whenever ωK(ξ)+ϖK(ξ) = 1. IFSs are more meaningful ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5956 Email addresses: shidali@gzgs.edu.cn (Dali Shi), mabushqair@jazanu.edu.sa (M.N. Abu_Shugair), salaheldin_ahmed@science.sohag.edu.eg (S.E. Abbas), ismail.abdelaziz@fsc.bu.edu.eg (Ismail Ibedou) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 2 of 30 and applicable to real-life problems. Cuong in [3] introduced the theory of picture fuzzy sets (PFSs) by adding the neutralism of an element ξ ∈ Ξ to the subset K, represented by σK(ξ). This definition is conditioned with 0 ≤ ωK(ξ) + ϖK(ξ) + σK(ξ) ≤ 1. In case where σK(ξ) = 0 for all ξ ∈ Ξ, Then, we revert to intuitionistic sets K in IFS. Moreover, if ϖK(ξ) = 1 − ωK(ξ), then we revert to fuzzy set K in FS. There are several simple modifications for IFSs [4, 5], which we shall not discuss here. These modifications include pythagorean FSs [6], spherical FSs [7, 8], q-rung orthopair FSs [9] and q-rung orthopair PFSs [10], (ς, κ)-fuzzy local function, continuous multifunctions and double fuzzy ideal topological spaces [11, 12]. All these definitions, starting from FSs, have applications in image processing, decision theory, uncertainty modeling, and beyond, as in [5, 6, 9, 13–16]. In this paper, we merge the classical definitions of multifunctions in general topology and the standard modal logic [17–20] with the notion of PFSs, further expanding into the realm of PFMTSs. This exploration includes the creation of PFMTSs facilitated by the standard picture fuzzy operations of "union" (∪) and "intersection" (∩). Continuous functions between picture fuzzy topological spaces were discussed in [21]. Continuous multifunctions between picture fuzzy topological spaces were discussed in [22]. The motivations of this paper are as follow: Firstly, to present PFTSs related to the PFSs, and studying some important results including several modal operators. These results are given in Section 2. Secondly, to introduce PFTMs via ideals and their common results. Also, to define some types of continuity of picture fuzzy multifunctions. These results are given in Section 3. Finally, the conclusion and the future work are given in Section 4. The research on PFMTSs has several important applications in various domains: Deci- sion Making, Pattern Recognition, Artificial Intelligence, Information Retrieval and Data Mining. PFMTSs address critical gaps in handling uncertainty, imprecision, and neutrality, which are inherent in real-life problems across diverse domains. To bridge these gaps, PFSs were introduced, adding a neutrality component to the membership and non-membership values, thereby enabling a more nuanced representation of uncertainty. PFMTSs expand upon these concepts by the integration in modal logic and general topology using the PFSs. This integration introduces global operators, such as closure, interior, and modal operators (□ and 3), which modify classical topological and modal relationships. These global oper- ators facilitate a robust analysis of FSs under modal and topological constraints, providing a suitable tools for theoretical exploration and practical application. The study of PFMTS not only extends the theory of FSs but also establishes a wide platform for addressing mod- ern computational challenges. Its ability to integrate neutrality, positivity, and negativity within a unified framework lays the foundation for further exploration and application of PFMTS in dynamic systems, hybrid models, and emerging technologies, positioning it as a cornerstone of modern mathematical and computational innovation. PFMTSs have special important applications in decision making environments. Chel- lamani et. al [23], used picture fuzzy soft graphs to design a decision making scheme. Yang et. al [24], developed an adjustable soft discernibility matrix with the help of picture fuzzy soft sets and presented its applications in decision making. Joshi in [25–27] presented an innovative decision making process for a picture fuzzy environment with the help of the Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 3 of 30 concept R-norm and the VIKOR technique. More development of PFSs can be seen in [28–30]. In daily life, PFS theory provides more than one choice for any decision. As examples: (1) Suppose a person is suffering from some disease. Then, the positive, negative and neutral membership functions can be associated with curability bitterness and treatment of disease respectively. Refusal can be related to the insufficient economic conditions of the patient meaning that he cann’t afford the hospital expenses and refuses to be hospitalized. (2) Suppose a person has an allegation of a crime. Then, the positive, negative and neutral membership functions can be associated with maximum punishment, release and moderate punishment of the accused person respectively. Refusal can be related to the dismissal of the case due to reconciliation. keeping in mind the above literature and the importance of PFSs, as well as topological spaces, we reveal the study of PFMTSs. The major contri- butions of this paper are as follow: (a) The definition of some new notions of cl-PFMTS, int-int-PFMTS, cl-int-PFMTS, int- cl-PFMTS regarding the types of the topological operators "closure" and "interior" and any of the given modal operators. (b) The design of the various PFMs based on the stated notions. (c) The definition of the notion of continuous multifunctions in picture fuzzy topological spaces via ideals and an introduction to necessary and sufficient conditions of upper and lower PFM between two picture fuzzy ideal topological spaces. 2. Picture fuzzy operations Continuing from previous discussions and the notions given by Atanassov in [4, 31], let’s define a PFS K on the universal set Ξ. The set K consists of elements ξ ∈ Ξ, each described by degrees of positivism (ωK(ξ)), negativism (ϖK(ξ)), and neutralism (σK(ξ)) that lie within the interval [0, 1]. Specifically, K is represented as {⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}, where each component satisfies the condition 0 ≤ ωK(ξ)+ ϖK(ξ) + σK(ξ) ≤ 1 for every element ξ. The term πK(ξ) = 1 − (ωK(ξ) + ϖK(ξ) + σK(ξ)) indicates the degree of refusal membership value for each ξ in K, quantifying the extent to which ξ does not belong to K. This framework is pivotal for assessing and handling the nuances of membership within PFSs, enabling a more comprehensive analysis of elements based on their multiple affinities. Definition 2.1. [4, 31] Let Ξ be a nonempty set, K = {⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} and Q = {⟨ξ, ω Q(ξ), ϖ Q(ξ), σ Q(ξ)⟩ |ξ ∈ Ξ}. Then, (1) K ⊆ Q iff for all ξ ∈ Ξ, ωK(ξ) ≤ ω Q(ξ), ϖK(ξ) ≥ ϖ Q(ξ) and σK(ξ) ≤ σ Q(ξ) or σK(ξ) ≥ σ Q(ξ) (2) K ∪ Q = {⟨ξ, (ωK(ξ) ∨ ω Q(ξ)) , (ϖK(ξ) ∧ϖ Q(ξ)) , (σK(ξ) ∧ σ Q(ξ))⟩ |ξ ∈ Ξ} (3) K ∩ Q = {⟨ξ, (ωK(ξ) ∧ ω Q(ξ)) , (ϖK(ξ) ∨ϖ Q(ξ)) , (σK(ξ) ∧ σ Q(ξ))⟩ |ξ ∈ Ξ} (4)K ⊔ Q = {〈 ξ, ωK(ξ) + ω Q(ξ)− [(ωK(ξ).ω Q(ξ)) ∧ (σK(ξ).σ Q(ξ))] , ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ) 〉 |ξ ∈ Ξ } Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 4 of 30 (5)K ⊓ Q ={〈 ξ, ωK(ξ).ω Q(ξ), ϖK(ξ) +ϖ Q(ξ)− [(ϖK(ξ).ϖ Q(ξ)) ∧ (σK(ξ).σ Q (ξ))] , σK(ξ).σ Q(ξ) 〉 |ξ ∈ Ξ } (6)K@ Q = {〈 ξ, ωK (ξ)+ω Q (ξ) 2 , ϖK (ξ)+ϖ Q (ξ) 2 , σK (ξ)+σ Q (ξ) 2 〉 |ξ ∈ Ξ } (7) K# Q = {⟨ξ, ωK(ξ).ω Q(ξ), (ϖK(ξ) ∨ϖ Q(ξ)) , σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} (8) K ∗ Q = {⟨ξ, (ωK(ξ) ∨ ω Q(ξ)) , ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} (9) Ⅎ K = {⟨ξ,ϖK(ξ), ωK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} (10) K ⊼ Q= 0 if K ⊆ Q , and K ⊼ Q= K ∩ (Ⅎ Q) otherwise. Now, the definitions of standard two modal operators over PFSs are presented. □K = {⟨ξ, ωK(ξ), 1− ωK(ξ)− σK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} , 3K = {⟨ξ, 1−ϖK(ξ)− σK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} . We can see that □K ⊆ K ⊆ 3K in general, and □K ̸= K ̸= 3K for any proper set K in (PFS), that is, σK(ξ) ̸= 0. Otherwise, K is an IFS and still □K ⊆ K ⊆ 3K as usual in (IFS). Moreover, if K is non proper PFS and ϖK(ξ) = 1 − ωK(ξ), then K is a FS and □K = K = 3K. Thus, (FS) ⊆ (IFS) ⊆ (PFS). A PFS K = {⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} is called a picture fuzzy tautological set (PFTaut) iff for each ξ ∈ Ξ, ωK(ξ) ≥ ϖK(ξ). ♯ = {⟨ξ, 1, 0, 0⟩ |ξ ∈ Ξ}, ♭ = {⟨ξ, 0, 1, 0⟩ |ξ ∈ Ξ}, ♮ = {⟨ξ, 0, 0, 1⟩ |ξ ∈ Ξ}, 0 = {⟨ξ, 0, 0, 0⟩ |ξ ∈ Ξ} where ♭ ⊆ K ⊆ ♯ for all K ∈ (PFS). Normally, P (♭) = ♭ and P (♯) = {K|K ⊆ ♯} where K = {⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}. Therefore, (PFS) coincides with P (♯). (IFS) coin- cides with the set { K|K ⊆ Ξ} in which K = {⟨ξ, ωK(ξ), ϖK(ξ), 0⟩ |ξ ∈ Ξ}. Moreover, (FS) coincides with the set {K|K ⊆ Ξ} in which K = {⟨ξ, ωK(ξ), 1− ωK(ξ), 0⟩ |ξ ∈ Ξ} or K = {⟨ξ, 1−ϖK(ξ), ϖK(ξ), 0⟩ |ξ ∈ Ξ}. Any operation from the above is well defined if the sum of its three values (positivism, negativism and neutralism) is a number in [0, 1]. We will check for the definitions of operations K# Q and K ∗ Q. 0 ≤ ωK(ξ).ω Q(ξ) + [ϖK(ξ) ∨ϖ Q(ξ)] + σK(ξ).σ Q(ξ) ≤ [ωK(ξ) ∧ ω Q(ξ)] + [ϖK(ξ) ∨ϖ Q ] + [σK(ξ) ∧ σ Q(ξ)] ≤ [ωK(ξ) ∧ ω Q(ξ)] + [(1− ωK(ξ)− σK(ξ)) ∨ (1− ω Q(ξ)− σ Q(ξ))] + [σK(ξ) ∧ σ Q(ξ)] ≤ [ωK(ξ) ∧ ω Q(ξ)] + 1− [ωK(ξ) ∧ ω Q(ξ)]− [σK(ξ) ∧ σ Q(ξ)] + [σK(ξ) ∧ σ Q(ξ)] = 1. Also, it is clear that 0 ≤ (ωK(ξ) ∨ ω Q(ξ)) +ϖK(ξ).ϖ Q + σK(ξ).σ Q(ξ) ≤ 1. Now, we check the duality of the operations # and ∗: For K, Q ∈ P (♯), Ⅎ (ℲK#Ⅎ Q) = Ⅎ ({⟨ξ,ϖK(ξ), ωK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} # {⟨ξ,ϖ Q(ξ), ω Q(ξ), σ Q(ξ)⟩ |ξ ∈ Ξ}) = Ⅎ ({⟨ξ,ϖK(ξ).ϖ Q(ξ), [ωK(ξ) ∨ ω Q(ξ)] , σK(ξ).σ Q⟩ |ξ ∈ Ξ}) = {⟨ξ, [ωK(ξ) ∨ ω Q(ξ)] , ϖK(ξ).ϖ Q , σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} = K ∗ Q, and in the same manner, Ⅎ (ℲK ∗ Ⅎ Q) = K# Q. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 5 of 30 The operators “closure” and “interior” over PFSs are defined by: cl∩ (K) = {⟨ξ, ϵK ,ℵK , κK⟩ |ξ ∈ Ξ} and int∪ (K) = {⟨ξ, εK , ϑK , κK⟩ |ξ ∈ Ξ} where ϵK = ∨ ξ∈Ξ ωK(ξ), ℵK = ∧ ξ∈Ξ ϖK(ξ), κK = ∧ ξ∈Ξ σK(ξ) εK = ∧ ξ∈Ξ ωK(ξ), ϑK = ∨ ξ∈Ξ ϖK(ξ), κK = ∨ ξ∈Ξ σK(ξ). Theorem 2.1. For every K, Q ∈ P(♯), K ⊓ Q ⊆ K# Q ⊆ K ∩ Q ⊆ K@ Q ⊆ K ∪ Q ⊆ K ∗ Q ⊆ K ⊔ Q. Proof. K ⊓ Q ={〈 ξ, ωK(ξ).ω Q(ξ), ϖK(ξ) +ϖ Q(ξ)− (ϖK(ξ).ϖ Q(ξ) ∧ σK(ξ).σ Q(ξ)) , σK(ξ).σ Q(ξ) 〉 |ξ ∈ Ξ } ⊆ {⟨ξ, ωK(ξ).ω Q(ξ), (ϖK(ξ) ∨ϖ Q(ξ)) , σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ}= K# Q, K# Q = {⟨ξ, ωK(ξ).ω Q(ξ), (ϖK(ξ) ∨ϖ Q(ξ)) , σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} ⊆ {⟨ξ, (ωK(ξ) ∧ ω Q(ξ)) , (ϖK(ξ) ∨ϖ Q(ξ)) , (σK(ξ) ∧ σ Q(ξ)) |ξ ∈ Ξ}⟩ = K ∩ Q, K ∩ Q = {⟨ξ, (ωK(ξ) ∧ ω Q(ξ)) , (ϖK(ξ) ∨ϖ Q(ξ)) , (σK(ξ) ∧ σ Q (ξ)) |ξ ∈ Ξ}⟩ ⊆ {〈 ξ, ωK (ξ)+ω Q (ξ) 2 , ϖK (ξ)+ϖ Q (ξ) 2 , σK (ξ)+σ Q (ξ) 2 〉 |ξ ∈ Ξ } = K@ Q, K@ Q = {〈 ξ, ωK (ξ)+ω Q (ξ) 2 , ϖK (ξ)+ϖ Q (ξ) 2 , σK (ξ)+σ Q (ξ) 2 〉 |ξ ∈ Ξ } ⊆ {⟨ξ, (ωK(ξ) ∨ ω Q(ξ)) , (ϖK(ξ) ∧ϖ Q(ξ)) , (σK(ξ) ∧ σ Q(ξ))⟩ |ξ ∈ Ξ}= K ∪ Q, K ∪ Q = {⟨ξ, (ωK(ξ) ∨ ω Q(ξ)) , (ϖK(ξ) ∧ϖ Q(ξ)) , (σK(ξ) ∧ σ Q (ξ))⟩ |ξ ∈ Ξ} ⊆ {⟨ξ, (ωK(ξ) ∨ ω Q(ξ)) , ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ}= K ∗ Q, K ∗ Q = {⟨ξ, (ωK(ξ) ∨ ω Q(ξ)) , ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} ⊆ {〈 ξ, ωK(ξ) + ω Q(ξ)− (ωK(ξ).ω Q(ξ) ∧ σK(ξ).σ Q(ξ)) , ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ) 〉 |ξ ∈ Ξ } = K ⊔ Q. Now, we will construct the picture fuzzy implication operation on P(♯) as follows: K ↠ Q = {⟨ξ, (ϖK(ξ) ∨ ω Q(ξ)) , ωK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} . We checked these properties for the implication operation: ♯ ↠ ♯ = ♯, ♭ ↠ ♯ = ♯, ♮ ↠ ♯ = ♯, 0 ↠ ♯ = ♯, ♯ ↠ ♭ = ♭, ♭ ↠ ♭ = ♯, ♮ ↠ ♭ = 0, 0 ↠ ♭ = 0, ♯ ↠ ♮ = 0, ♭ ↠ ♮ = ♯, ♮ ↠ ♮ = ♮, 0 ↠ ♮ = 0, ♯ ↠ 0 = 0, ♭ ↠ 0 = ♯, ♮ ↠ 0 = 0, 0 ↠ 0 = 0. Following Atanassov in [31], we will give these nine axioms related with our new defined implication operation. Let Ξ be a nonempty set and K, Q, D ∈ P (♯). Then, Axiom 1: If K ⊆ Q, then Q ↠ D ⊆ K ↠ D. Axiom 2: If K ⊆ Q, then D ↠ K ⊆ D ↠ Q. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 6 of 30 Axiom 3: ♭ ↠ Q = ♯. Axiom 4: ♯ ↠ Q = Q. Axiom 5: K ↠ K = ♯. Axiom 6: K ↠ ( Q ↠ D) = Q ↠ (K ↠ D). Axiom 7: K ↠ Q = ♯ iff K ⊆ Q. Axiom 8: K ↠ Q = Ⅎ Q ↠ ℲK. Axiom 9: ↠ is a continuous function. We followed [31] in defining a number of axioms marked with an asterisk (∗) refering to the tautological operations, and (Axiom7∗) is given to show that "iff" in Axiom7 maybe not correct. Axiom 3*: ♭ ↠ Q is a PFTaut set. Axiom 4*: ♯ ↠ Q is a PFTaut set. Axiom 5*: K ↠ K is a PFTaut set. Axiom 7*: K ↠ Q = ♯ implies that K ⊆ Q, and K ⊆ Q implies that K ↠ Q is a PFTaut set. Theorem 2.2. For K, Q, D ∈ P(♯) the new implication (↠) satisfies Axioms 1, 2, 3, 3*, 5*, 6, 7*, 8, 9. Proof. (For Axiom1), let K ⊆ Q. Then, K ↠ D = {⟨ξ, (ϖK(ξ) ∨ ω D(ξ)) , ωK(ξ).ϖ D(ξ), σK(ξ).σ D(ξ)⟩ |ξ ∈ Ξ} , Q ↠ D = {⟨ξ, (ϖ Q(ξ) ∨ ω D(ξ)) , ω Q(ξ).ϖ D(ξ), σ Q(ξ).σ D(ξ)⟩ |ξ ∈ Ξ} , now (ϖK(ξ) ∨ ω D(ξ)) ≥ (ϖ Q(ξ) ∨ ω D(ξ)), ωK(ξ).ϖ D(ξ) ≤ ω Q(ξ).ϖ D(ξ), and σK(ξ).σ D(ξ) ≥ σ Q(ξ).σ D(ξ) or σK(ξ).σ D(ξ) ≥ σ Q(ξ).σ D(ξ). Therefore, Q ↠ D ⊆ K ↠ D. (For Axiom2), let K ⊆ Q. Then, D ↠ K = {⟨ξ, (ϖ D(ξ) ∨ ωK(ξ)) , ω D(ξ).ϖK(ξ), σ D(ξ).σK(ξ)⟩ |ξ ∈ Ξ} , D ↠ Q = {⟨ξ, (ϖ D(ξ) ∨ ω Q(ξ)) , ω D(ξ).ϖ Q(ξ), σ D(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} . That is, it follows (ϖ D(ξ) ∨ ωK(ξ)) ≤ (ϖ D(ξ) ∨ ω Q(ξ)), ω D(ξ).ϖK(ξ) ≥ ω D(ξ).ϖ Q(ξ), and σ D(ξ).σK(ξ) ≥ σ D(ξ).σ Q(ξ) or σ D(ξ).σK(ξ) ≤ σ D(ξ).σ Q(ξ). Thus, D ↠ Q ⊇ D ↠ K. (For Axiom3), ♭ ↠ Q = {⟨ξ, (1 ∨ ω Q(ξ)) , 0.ϖ Q(ξ), 0.σ Q(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, 1, 0, 0⟩ |ξ ∈ Ξ} = ♯, this also meaning ♭ ↠ Q is a PFTaut set, (Axiom3∗) is satisfied. (For Axiom4∗), ♯ ↠ Q = {⟨ξ, (0 ∨ ω Q(ξ)) , 1.ϖ Q(ξ), 0.σ Q(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, ω Q(ξ), ϖ Q , 0⟩ |ξ ∈ Ξ} ≠ Q, Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 7 of 30 that is, Axiom 4 is not satisfied, and also ♯ ↠ Q is not a PFTaut set, (Axiom4∗) is not satisfied in general. (For Axiom5∗), K ↠ K = {⟨ξ, (ϖK(ξ) ∨ ωK(ξ)) , ωK(ξ).ϖK(ξ), σK(ξ).σK(ξ)⟩ |ξ ∈ Ξ} , since (ϖK(ξ) ∨ ωK(ξ)) ≥ ωK(ξ).ϖK(ξ). Then, K ↠ K is a PFTaut set, (Axiom5∗) is satisfied while (Axiom5) is not valid. Because we can find elements ξ ∈ Ξ for which (ϖK(ξ) ∨ ωK(ξ)) < 1, ωK(ξ).ϖK(ξ) > 0. (For Axiom6), K ↠ ( Q ↠ D) = K ↠ {⟨ξ, (ϖ Q(ξ) ∨ ω D(ξ)) , ω Q(ξ).ϖ D(ξ), σ Q(ξ).σ D(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, (ϖK(ξ) ∨ (ϖ Q(ξ) ∨ ω D(ξ))) , ωK(ξ).ω Q(ξ).ϖ D(ξ), σK(ξ).σ Q(ξ).σ D(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, (ϖK(ξ) ∨ϖ Q(ξ) ∨ ω D(ξ)) , ωK(ξ).ω Q(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ).σ D(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, (ϖ Q(ξ) ∨ (ϖK(ξ) ∨ ω D(ξ))) , ω Q(ξ).ωK(ξ).ϖ D(ξ), σ Q(ξ).σK(ξ).σ D(ξ)⟩ |ξ ∈ Ξ} = Q ↠ {⟨ξ, (ϖK(ξ) ∨ ω D(ξ)) , ωK(ξ).ϖ D(ξ), σK(ξ).σ D(ξ)⟩ |ξ ∈ Ξ} = Q ↠ (K ↠ D) . (For Axiom7∗), if K ↠ Q = {⟨ξ, (ϖK(ξ) ∨ ω Q(ξ)) , ωK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} = ♯, then (ϖK(ξ) ∨ ω Q(ξ)) = 1, ωK(ξ).ϖ Q(ξ) = 0, σK(ξ).σ Q(ξ) = 0. Therefore, either K = ♭ and hence K ⊆ Q or Q = ♯ and it means again K ⊆ Q. Conversely, if we suppose K ⊆ Q, then ωK(ξ) ≤ ω Q(ξ), ϖK(ξ) ≥ ϖ Q(ξ), σK(ξ) ≥ σ Q(ξ) or σK(ξ) ≤ σ Q(ξ). Thus, (ϖK(ξ) ∨ ω Q(ξ)) + ωK(ξ).ϖ Q(ξ) + σK(ξ).σ Q(ξ) ≥ ω Q(ξ) + ωK(ξ).ϖ Q(ξ) + σK(ξ).σ Q(ξ) ≥ ω Q(ξ) +ϖ Q(ξ) + σ Q(ξ) ≥ 0, or (ϖK(ξ) ∨ ω Q(ξ)) + ωK(ξ).ϖ Q(ξ) + σK(ξ).σ Q(ξ) ≥ ϖK(ξ) + ωK(ξ).ϖ Q(ξ) + σK(ξ).σ Q(ξ) ≥ ϖK(ξ) + ωK(ξ) + σK(ξ) ≥ 0, and then K ↠ Q is a PFTaut set. But if K ⊆ Q, then (ϖK(ξ) ∨ ω Q(ξ)) ≥ ωK(ξ).ϖ Q(ξ), and so may not imply (ϖK(ξ) ∨ ω Q(ξ)) = 1, ωK(ξ).ϖ Q(ξ) = 0 and σK(ξ).σ Q(ξ) = 0. Hence, (Axiom7) is not satisfied in general. (For Axiom8), K ↠ Q = {⟨ξ, (ϖK(ξ) ∨ ω Q(ξ)) , ωK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, (ω Q(ξ) ∨ϖK(ξ)) , ϖ Q(ξ).ωK(ξ), σ Q(ξ).σK(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ,ϖ Q(ξ), ω Q(ξ), σ Q(ξ)⟩ |ξ ∈ Ξ} ↠ {⟨ξ,ϖK(ξ), ωK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} = Ⅎ Q ↠ ℲK. (For Axiom9), since the operations “max ” and “multiplication” as functions preserve the continuity, then ↠ is a continuous function. According to the above theorem, for all K ∈ P(♯), the implication operation ↠ satisfies the Axioms (1, 2, 3, 3*, 5*, 6, 7*, 8, 9). Moreover, in the intuitionistic fuzzy case, that is, if we take σK(ξ) = 0 for all K ∈ P(♯), then this implication operation ↠ satisfies also (Axiom4) as the case given in [31] for (IFSs) but still not satisfying (Axiom4∗). As another extension for the defined operations (∗) and (#), respectively, we introduce here these forms: W (K) = {⟨ξ, ϵK , ϱK ,𭟋K⟩ |ξ ∈ Ξ} ,Z (K) = {⟨ξ, φK , ϑK ,𭟋K⟩ |ξ ∈ Ξ} Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 8 of 30 where φK = ∏ ξ∈Ξ ωK(ξ), ϱK = ∏ ξ∈Ξ ϖK(ξ), 𭟋K = ∏ ξ∈Ξ σK(ξ). These new operations are well defined because 0 ≤ ϵK ≤ 1, 0 ≤ ϱK ≤ 1, 0 ≤ 𭟋K ≤ 1, 0 ≤ φK ≤ 1, 0 ≤ ϑK ≤ 1, 0 ≤ 𭟋K ≤ 1, 0 ≤ ϵK + ϱK +𭟋K ≤ ϵK + ℵK + κK ≤ 1, 0 ≤ φK + ϑK +𭟋K ≤ εK + ϑK + κK ≤ 1. These operations are dual to each other. For K ∈ P (♯), ℲZ (ℲK) = ℲZ {⟨ξ,ϖK(ξ), ωK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} = Ⅎ {⟨ξ, ϱK , ϵK ,𭟋K⟩ |ξ ∈ Ξ} = {⟨ξ, ϵK , ϱK ,𭟋K⟩ |ξ ∈ Ξ} = W (K) , and analogously, ℲW (ℲK) = Z (K). For any K ∈ P (♯), we have: Z (K) ⊆ int∪ (K) ⊆ K ⊆ cl∩ (K) ⊆ W (K). Let O and Q be topological operators such that for each PFS K ∈ P (♯): O (K) = ℲQ (ℲK) , Q (K) = ℲO (ℲK) . Let △,▽ : P (♯)×P (♯) → P (♯) be operations over Ξ such that for any two K, Q ∈ P (♯), K▽ Q = Ⅎ (ℲK△Ⅎ Q) , K△ Q = Ⅎ (ℲK▽Ⅎ Q) . Let ◦ and • : P (♯) → P (♯) be two modal operators over Ξ such that for any K ∈ P (♯): ◦K = Ⅎ • (ℲK), •K = Ⅎ ◦ (ℲK). In [31], Atanassov investigated (for IFSs) the notions of PFMTS and feeble PFMTS (PFFMTS, for short) and, moreover in [13], established extensions of those definitions named by cl-cl-PFMTS, int-int-PFMTS, cl-int-PFMTS and int-cl-PFMTS regarding the types of the topological operators “closure” and “interior”and any of the given modal oper- ators. In similar strategy, we will define (for PFSs) four certain cases. Theorem 2.3. ⟨P (♯) ,W, ∗,3⟩ is a cl-cl-PFFMTS for which in conditions CC4, CC5 and CC9, the relation “ = ” is changed to the relation “ ⊇ ”. Proof. Let K, Q ∈ P (♯). Then, we check in a sequential manner the validity of the nine conditions CC1 – CC9. The checks of conditions CC1 - CC4 are analogous, but different from those in [22], while of CC6 - CC8 are the same. We give them only here for completeness of the proof. (CC1) W (K ∗ Q) = W ({⟨ξ, ωK(ξ) ∨ ω Q(ξ), ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ}) = {〈 ξ, ∨ ξ∈Ξ (ωK(ξ) ∨ ω Q(ξ)) , ∏ ξ∈Ξ (ϖK(ξ).ϖ Q(ξ)) , ∏ ξ∈Ξ (σK(ξ).σ Q(ξ)) 〉 |ξ ∈ Ξ } Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 9 of 30 = {⟨ξ, ϵK ∨ ϵ Q , ϱK .ϱ Q ,𭟋K .𭟋 Q⟩ |ξ ∈ Ξ} = {⟨ξ, ϵK , ϱK ,𭟋K⟩ |ξ ∈ Ξ} ∗ {⟨ξ, ϵ Q , ϱ Q ,𭟋 Q⟩ |ξ ∈ Ξ} = W (K) ∗W ( Q), (CC2) K = {⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} ⊆ {⟨ξ, ϵK , ϱK ,𭟋K⟩ |ξ ∈ Ξ} = W (K), (CC3) W (♭) = W ({⟨ξ, 0, 1, 0⟩ |ξ ∈ Ξ}) = {〈 ξ, ∨ ξ∈Ξ 0, ∏ ξ∈Ξ 1, ∏ ξ∈Ξ 0 〉 |ξ ∈ Ξ } = {⟨ξ, 0, 1, 0⟩ |ξ ∈ Ξ} = ♭, (CC4) W (W (K)) = W ({⟨ξ, ϵK , ϱK ,𭟋K⟩ |ξ ∈ Ξ}) = {〈 ξ, ∨ ξ∈Ξ ϵK , ∏ ξ∈Ξ ϱK , ∏ ξ∈Ξ 𭟋K 〉 |ξ ∈ Ξ } = {〈 ξ, ϵK , (ϱK) E , (𭟋K) E 〉 |ξ ∈ Ξ } ⊇ {⟨ξ, ϵK , ϱK ,𭟋K⟩ |ξ ∈ Ξ} = W (K), (CC5) 3 (K ∗ Q) = 3 ({⟨ξ, ωK(ξ) ∨ ω Q(ξ), ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ}) = {⟨ξ, 1−ϖK(ξ).ϖ Q(ξ)− σK(ξ).σ Q (ξ), ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q⟩ |ξ ∈ Ξ} ⊇ {⟨ξ, 1− (ϖK(ξ) ∧ϖ Q(ξ))− (σK(ξ) ∧ σ Q(ξ)) , ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} ⊇ {⟨ξ, (1−ϖK(ξ)− σK(ξ)) ∨ (1−ϖ Q (ξ)− σ Q(ξ)) , ϖK(ξ).ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, 1−ϖK(ξ)− σK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} ∗ {⟨ξ, 1−ϖ Q(ξ)− σ Q(ξ), ϖ Q , σ Q(ξ)⟩ | ξ ∈ Ξ} = 3 (K) ∗3 ( Q), (CC6) K = {⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} ⊆ {⟨ξ, 1−ϖK(ξ)− σK(ξ), ϖK(ξ), σK(ξ)⟩ | ξ ∈ Ξ} = 3 (K), (CC7) 3 (♯) = 3({⟨ξ, 1, 0, 0⟩ |ξ ∈ Ξ}) = {⟨ξ, 1, 0, 0⟩ |ξ ∈ Ξ} = ♯, (CC8) 3 (3 (K)) = 3 ({⟨ξ, 1−ϖK(ξ)− σK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}) = {⟨ξ, 1−ϖK(ξ)− σK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} = 3 (K), (CC9) 3 (W (K)) = 3 ({⟨ξ, ϵK , ϱK ,𭟋K⟩ |ξ ∈ Ξ}) = {⟨ξ, 1− ϱK −𭟋K , ϱK ,𭟋K⟩ |ξ ∈ Ξ} ⊇ {⟨ξ, 1− ℵK − κK , ϱK ,𭟋K⟩ |ξ ∈ Ξ} ⊇ {〈 ξ, ∨ ξ∈Ξ (1−ϖK(ξ)− σK(ξ)) , ϱK ,𭟋K 〉 |ξ ∈ Ξ } = W ({⟨ξ, 1−ϖK(ξ)− σK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}) = W (3 (K)). The reason that operators W and 3 are from one type (“closure”) is in the validity of conditions CC2 and CC6. Theorem 2.4. ⟨P (♯) ,Z,#,□⟩ is an int-int-PFFMTS for which in conditions II4, II5 and II9, the relation “ = ” is changed to the relation “ ⊆ ”. Proof. Let K, Q ∈ P (♯). Then, we check in a sequential manner the validity of the conditions II1–II5, and II9 because the checks of the validity of conditions II6–II8 are given in [22] and are similar to these in the proof of Theorem 2.3. (II1) Z (K# Q) = Z ({⟨ξ, ωK(ξ).ω Q(ξ), ϖK(ξ) ∨ϖ Q(ξ), σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ}) = {〈 ξ, ∏ ξ∈Ξ (ωK(ξ).ω Q(ξ)) , ∨ ξ∈Ξ (ϖK(ξ) ∨ϖ Q(ξ)) , ∏ ξ∈Ξ (σK(ξ).σ Q(ξ)) 〉 |ξ ∈ Ξ } = {⟨ξ, φK .φ Q , ϑK ∨ ϑ Q ,𭟋K .𭟋 Q⟩ |ξ ∈ Ξ} = {⟨ξ, φK , ϑK ,𭟋K⟩ |ξ ∈ Ξ}# {⟨ξ, φ Q , ϑ Q ,𭟋 Q⟩ |ξ ∈ Ξ} = Z (K)#Z ( Q), (II2) K = {⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} ⊇ {⟨ξ, φK , ϑK ,𭟋K⟩ |ξ ∈ Ξ} = Z (K), Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 10 of 30 (II3) Z (♯) = Z ({〈 ξ, ∨ ξ∈Ξ 1, ∏ ξ∈Ξ 0, ∏ ξ∈Ξ 0 〉 |ξ ∈ Ξ }) = ♯, (II4) Z (Z (K)) = Z ({⟨ξ, φK , ϑK ,𭟋K⟩ |ξ ∈ Ξ}) = {〈 ξ, ∏ ξ∈Ξ φK , ∨ ξ∈Ξ ϑK , ∏ ξ∈Ξ 𭟋K 〉 |ξ ∈ Ξ } = {〈 ξ, (φK) E , ϑK , (𭟋K) E 〉 |ξ ∈ Ξ } ⊆ {⟨ξ, φK , ϑK ,𭟋K⟩ |ξ ∈ Ξ} = Z (K), (II5) □ (K# Q) = □ ({⟨ξ, ωK(ξ).ω Q(ξ), (ϖK(ξ) ∨ϖ Q(ξ)) , σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ}) = {⟨ξ, ωK(ξ).ω Q(ξ), 1− ωK(ξ).ω Q(ξ)− σK(ξ).σ Q(ξ), σK(ξ).σ Q⟩ |ξ ∈ Ξ} ⊆ {⟨ξ, ωK(ξ).ω Q(ξ), 1− (ωK(ξ) ∧ ω Q(ξ))− (σK(ξ) ∧ σ Q(ξ)) , σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} ⊆ {⟨ξ, ωK(ξ).ω Q(ξ), (1− ωK(ξ)− σK(ξ)) ∨ (1− ω Q(ξ)− σ Q(ξ)) , σK(ξ).σ Q(ξ)⟩ |ξ ∈ Ξ} = {⟨ξ, ωK(ξ), 1− ωK(ξ)− σK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}# {⟨ξ, ω Q(ξ), 1− ω Q(ξ)− σ Q , σ Q(ξ)⟩ |ξ ∈ Ξ} = □(K)#□ ( Q), (II6) □(K) = □ ({⟨ξ, ωK(ξ), ϖK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}) = {⟨ξ, ωK(ξ), 1− ωK(ξ)− σK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} ⊆ K. (II7) □(♭) = □ ({⟨ξ, 0, 1, 0⟩ |ξ ∈ Ξ}) = {⟨ξ, 0, 1, 0⟩ |ξ ∈ Ξ} = ♭. (II8) □(□(K)) = □({⟨ξ, ωK(ξ), 1− ωK(ξ)− σK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}) = {⟨ξ, ωK(ξ), 1− ωK(ξ)− σK(ξ), σK(ξ)⟩ |ξ ∈ Ξ} = □(K). (II9) □ (Z (K)) = □ ({⟨ξ, φK , ϑK ,𭟋K⟩ |ξ ∈ Ξ}) = {⟨ξ, φK , 1− φK −𭟋K ,𭟋K⟩ |ξ ∈ Ξ} ⊆ {⟨ξ, φK , 1− εK − κK ,𭟋K⟩ |ξ ∈ Ξ} ⊆ {〈 ξ, φK , ∨ ξ∈Ξ (1− ωK(ξ)− σK(ξ)) ,𭟋K 〉 |ξ ∈ Ξ } = Z ({⟨ξ, ωK(ξ), 1− ωK(ξ)− σK(ξ), σK(ξ)⟩ |ξ ∈ Ξ}) = Z (□ (K)). Theorem 2.5. ⟨P (♯) ,W, ∗,□⟩ is a cl-int-PFFMTS for which in condition CI4, the rela- tion “ = ” is changed to the relation “ ⊇ ”, and in conditions CI5 and CI9, the relation “ = ” is changed to the relation “ ⊆ ”. Theorem 2.6. ⟨P (♯) ,Z,#,3⟩ is an int-cl-PFFMTS for which in condition IC4, the relation “ = ” is changed to the relation “ ⊆ ”, and in condition IC5 and CI9, the relation “ = ” is changed to the relation “ ⊇ ”. 3. Picture fuzzy ideal multifunctions The map F : Ξ ↬ Υ is called a PFM for any (ξ, ζ) ∈ Ξ × Υ iff F(ξ) ∈ ( I3 )Υ for each ξ ∈ Ξ. The degree of membership of ζ ∈ F(ξ) is denoted by: F(ξ)(ζ) = ΨF(ξ, ζ). The domain of F, denoted by D (F) and the range of F, denoted by R (F), are defined by: for any ξ ∈ Ξ and ζ ∈ Υ, D (F) (ξ) = ⋃ ζ∈Υ ΨF(ξ, ζ) and R (F) (ζ) = ⋃ ξ∈Ξ ΨF(ξ, ζ). F is called crisp iff ΨF(ξ, ζ) = ⟨1, 0, 0⟩ ∀ξ ∈ Ξ and ζ ∈ Υ. F is called normalized PFM iff ∀ξ ∈ Ξ, there exists ζ0 ∈ Υ such that ΨF(ξ, ζ0) = ⟨1, 0, 0⟩. F is called surjective iff R (F) (ζ) = ⟨1, 0, 0⟩ ∀ζ ∈ Υ. The inverse of F denoted by F− : Υ → Ξ is a PFM defined by: F−(ζ)(ξ) = F(ξ)(ζ) = ΨF(ξ, ζ). One easily verifies that D (F−) = R (F) and D (F) = R(F−). The image F(K) of K ∈ ( I3 )Ξ, the lower inverse Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 11 of 30 Fl( Q) of Q ∈ ( I3 )Υ and the upper inverse Fu( Q) of Q ∈ ( I3 )Υ are defined respectively as follow: F(K)(ζ) = ⋃ ξ∈Ξ [ΨF(ξ, ζ) ∩K(ξ)] , Fl( Q)(ξ) = ⋃ ζ∈Υ [ΨF(ξ, ζ) ∩ Q(ζ)] , Fu( Q)(ξ) = ⋂ ζ∈Υ [ℲΨF(ξ, ζ) ∪ Q(ζ)] . Definition 3.1. A picture fuzzy topology on Ξ is a map τ : ( I3 )Ξ → I3 defined by τ(K) = ⟨ωτ (K), ϖτ (K), στ (K)⟩ on Ξ which satisfies the following properties: (1) τ(♭) = τ(♯) = ⟨1, 0, 0⟩ . (2) τ(K1 ∩K2) ≥ τ(K1) ∧ τ(K2), for each K1,K2 ∈ ( I3 )Ξ . (3) τ( ⋃ i∈Γ Ki) ≥ ∧ i∈Γ τ(Ki), for each Ki ∈ ( I3 )Ξ, i ∈ Γ. The pair (Ξ, τ) is called a picture fuzzy topological space in Šostak’s sense. For any K ∈ ( I3 )Ξ the number ωτ (K) is called the openness degree, ϖτ (K) is called the non openness degree, while στ (K) is called the neutral degree. For K ∈ ( I3 )Ξ, clτ (K, ⟨ς,κ, ϑ⟩) = ⋂ { Q ∈ ( I3 )Ξ : K ⊆ Q, τ(Ⅎ Q) ≥ ⟨ς,κ, ϑ⟩}, intτ (K, ⟨ς,κ, ϑ⟩) = ⋃ { Q ∈ ( I3 )Ξ : K ⊇ Q, τ( Q) ≥ ⟨ς,κ, ϑ⟩}. Definition 3.2. The map ℓP : ( I3 )Ξ → I3 is called picture fuzzy ideal on Ξ if it satisfies the following conditions for K, Q∈ ( I3 )Ξ : (1) ℓP (♭) = ⟨1, 0, 0⟩, ℓP (♯) = ⟨0, 1, 0⟩. (2) K ⊆ Q ⇒ ℓP (K) ≥ ℓP ( Q). (3) ℓP (K ∪ Q ) ≥ ℓP (K) ∧ ℓP ( Q). If ℓP1 and ℓP2 are picture fuzzy ideals on Ξ, we say that ℓP1 is finer than ℓP2 (ℓP2 is coarser than ℓP1 ), denoted by ℓP2 ⊆ ℓP1 , iff ℓP2 (K) ≤ ℓP1 (K) ∀K ∈ ( I3 )Ξ. Let us define the special picture fuzzy ideals ℓP0, ℓP1 by ℓP0 (K) = { ⟨1, 0, 0⟩ if K = ♭, ⟨0, 1, 0⟩ otherwise, and ℓP1 (K) = { ⟨0, 1, 0⟩ if K = ♯, ⟨1, 0, 0⟩ otherwise. Definition 3.3. Let (Ξ, τ, ℓP ) be a picture fuzzy ideal topological space, K ∈ ( I3 )Ξ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Then, the ⟨ς,κ, ϑ⟩-fuzzy local function Φ(K, ⟨ς,κ, ϑ⟩) of K defined as follows: Φ(K, ⟨ς,κ, ϑ⟩) = ⋂ { Q ∈ ( I3 )Ξ : ℓP (K⊼ Q ) ≥ ⟨ς,κ, ϑ⟩, τ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩}. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 12 of 30 Remark 3.1. (1) If we take ℓP = ℓP0 for each K ∈ ( I3 )Ξ we have Φ(K, ⟨ς,κ, ϑ⟩) = ⋂ { Q ∈ ( I3 )Ξ : K ⊆ Q , τ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩} = cl (K, ⟨ς,κ, ϑ⟩). (2) If we take ℓP = ℓP1 (resp. ℓP (K) ≥ ⟨ς,κ, ϑ⟩) for each K ∈ ( I3 )Ξ we have Φ(K, ⟨ς,κ, ϑ⟩) = ⟨0, 1, 0⟩. We will occasionally write Φ(K, ⟨ς,κ, ϑ⟩) or Φ(K, ℓP , ⟨ς,κ, ϑ⟩) for Φ(K, ℓP , τ, ⟨ς,κ, ϑ⟩). Theorem 3.1. Let (Ξ, τ, ℓP ) be a picture fuzzy ideal topological space and ℓP1 ,ℓP2 be two picture fuzzy ideals on Ξ. Then, for any set K, Q ∈ ( I3 )Ξ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. (1) Φ(♭, ⟨ς,κ, ϑ⟩) = ⟨0, 1, 0⟩ . (2) If K ⊆ CH, then Φ(K, ⟨ς,κ, ϑ⟩) ⊆ Φ( Q , ⟨ς,κ, ϑ⟩). (3) If ℓP2 ⊆ ℓP1 , then Φ(K, ℓP1 , ⟨ς,κ, ϑ⟩) ⊆ Φ(K, ℓP2 , ⟨ς,κ, ϑ⟩). (4) Φ(K, ⟨ς,κ, ϑ⟩) = clτ (Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ clτ (K, ⟨ς,κ, ϑ⟩). (5) Φ(Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ Φ(K, ⟨ς,κ, ϑ⟩) and Ⅎ (Φ(K, ⟨ς,κ, ϑ⟩)) ̸= Φ(Ⅎ K, ⟨ς,κ, ϑ⟩). (6) Φ(K ∪ Q , ⟨ς,κ, ϑ⟩) ⊇ Φ(K, ⟨ς,κ, ϑ⟩) ∪ Φ( Q , ⟨ς,κ, ϑ⟩) and Φ(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ Φ(K, ⟨ς,κ, ϑ⟩) ∩ Φ(CH, ⟨ς,κ, ϑ⟩). (7) If ℓP ( Q ) ≥ ⟨ς,κ, ϑ⟩, then Φ(K ∪ Q , ⟨ς,κ, ϑ⟩) ⊇ Φ(K, ⟨ς,κ, ϑ⟩). Proof. (1) From Definition 3.3, we have Φ(♭, ⟨ς,κ, ϑ⟩) = ⟨0, 1, 0⟩. (2) Suppose that K ⊆ Q and Φ(K, ⟨ς,κ, ϑ⟩) ⊈ Φ( Q , ⟨ς,κ, ϑ⟩). By the definition of Φ( Q , ⟨ς,κ, ϑ⟩), there exists D∈ ( I3 )Ξ with Φ( Q , ⟨ς,κ, ϑ⟩) ⊆ D, ℓP ( Q ⊼ D) ≥ ⟨ς,κ, ϑ⟩, τ(Ⅎ D) ≥ ⟨ς,κ, ϑ⟩ and Φ(K, ⟨ς,κ, ϑ⟩) ⊈ D. Also, K⊼ D⊆ Q ⊼ D, ℓP (K⊼ D) ≥ ℓP ( Q ⊼ D) ≥ ⟨ς,κ, ϑ⟩, hence Φ(K, ⟨ς,κ, ϑ⟩) ⊆ D, it is a contradiction. Thus, Φ(K, ⟨ς,κ, ϑ⟩) ⊆ Φ( Q , ⟨ς,κ, ϑ⟩). (3) Suppose that Φ(K, ℓP1 , ⟨ς,κ, ϑ⟩) ⊈ Φ(K, ℓP2 , ⟨ς,κ, ϑ⟩) if ℓP2 ⊆ ℓP1 . By the definition of Φ(K, ℓP2 , ⟨ς,κ, ϑ⟩) there exists D∈ ( I3 )Ξ with Φ(K, ℓP2 , ⟨ς,κ, ϑ⟩) ⊆ D, ℓP2 (K⊼ D) ≥ ⟨ς,κ, ϑ⟩, τ(Ⅎ D) ≥ ⟨ς,κ, ϑ⟩ such that Φ(K, ℓP1 , ⟨ς,κ, ϑ⟩) ⊈ D. Since ℓP2 ⊆ ℓP1 implies ℓP1 (K⊼ D) ≥ ℓP2 (K⊼ D) ≥ ⟨ς,κ, ϑ⟩. Hence, Φ(K, ℓP1 , ⟨ς,κ, ϑ⟩) ⊆D, it is a contradiction. Then, Φ(K, ℓP1 , ⟨ς,κ, ϑ⟩) ⊆ Φ(K, ℓP2 , ⟨ς,κ, ϑ⟩). (4) From Definition 3.3, we have Φ(K, ⟨ς,κ, ϑ⟩) = cl (Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩). Since ℓP0 ⊆ ℓP for any picture fuzzy ideal ℓP , Φ(K, ℓP , ⟨ς,κ, ϑ⟩) ⊆ Φ(K, ℓP0, ⟨ς,κ, ϑ⟩) = cl (K, ⟨ς,κ, ϑ⟩). Thus, Φ(K, ⟨ς,κ, ϑ⟩) = cl (Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ cl (K, ⟨ς,κ, ϑ⟩). (5) By (4), we have Φ(Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) = cl (Φ(Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ cl (Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) = Φ(K, ⟨ς,κ, ϑ⟩). In general the converse is not true as will be shown in Example 3.1. (6) Since K ⊆ K ∪ Q and Q ⊆ K ∪ Q, Φ(K, ⟨ς,κ, ϑ⟩) ⊆ Φ(K ∪ Q , ⟨ς,κ, ϑ⟩) and Φ( Q , ⟨ς,κ, ϑ⟩) ⊆ Φ(K ∪ Q , ⟨ς,κ, ϑ⟩). Thus, Φ(K, ⟨ς,κ, ϑ⟩) ∪ Φ( Q , ⟨ς,κ, ϑ⟩) ⊆ Φ(K ∪ Q , ⟨ς,κ, ϑ⟩). Also, K∩ Q ⊆ K and K∩ Q ⊆ Q, Φ(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ Φ(K, ⟨ς,κ, ϑ⟩) and Φ(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ Φ( Q , ⟨ς,κ, ϑ⟩). Thus, Φ(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ Φ(K, ⟨ς,κ, ϑ⟩)∩Φ( Q , ⟨ς,κ, ϑ⟩). (7) Since ℓP ( Q ) ⊇ ⟨ς,κ, ϑ⟩, Φ( Q , ⟨ς,κ, ϑ⟩) = ⟨0, 1, 0⟩ . Thus, Φ(K ∪ Q , ⟨ς,κ, ϑ⟩) ⊇ Φ(K, ⟨ς,κ, ϑ⟩) ∪ Φ( Q , ⟨ς,κ, ϑ⟩) ⊇ Φ(K, ⟨ς,κ, ϑ⟩). Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 13 of 30 The following example shows that generally Φ(Φ(K, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ̸= Φ(K, ⟨ς,κ, ϑ⟩), and Ⅎ (Φ(K, ⟨ς,κ, ϑ⟩)) ̸= Φ(ℲK, ⟨ς,κ, ϑ⟩) for any K ∈ ( I3 )Ξ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Example 3.1. Let Ξ = {ξ1, ξ2}, for K1 = {⟨ξ, 0.33, 0.33, 0.2⟩ | ξ ∈ Ξ}, K2 = {⟨ξ, 0.5, 0.3, 0.2⟩ | ξ ∈ Ξ}, K3 = {⟨ξ, 0.33, 0.33, 0.1⟩ |ξ ∈ Ξ} and Q = {⟨ξ, 0.4, 0.4, 0.2⟩ |ξ ∈ Ξ}. Define τ, ℓP :( I3 )Ξ → I3 as follows: τ(K) =  ⟨1, 0, 0⟩ if K ∈ {♭, ♯} , ⟨0.33, 0.33, 0.33⟩ if K = K1, ⟨0.5, 0.2, 0.1⟩ if K = K2, ⟨0, 1, 0⟩ otherwise, ℓP (K) =  ⟨1, 0, 0⟩ if K = ♭, ⟨0.75, 0.15, 0.1⟩ if ♭ ⊆ K ⊆ {⟨ξ, 0.33, 0.33, 0.33⟩ |ξ ∈ Ξ}, ⟨0.4, 0.3, 0.3⟩ if {⟨ξ, 0.33, 0.33, 0.33⟩ |ξ ∈ Ξ} ⊆ K < ♯, ⟨0, 1, 0⟩ otherwise. Then, ♭ = Φ(Φ( Q, ⟨0.33, 0.33, 0.33⟩), ⟨0.33, 0.33, 0.33⟩) ̸= Φ( Q, ⟨0.33, 0.33, 0.33⟩) = {⟨ξ, 0.33, 0.33, 0⟩ |ξ ∈ Ξ}, ♯ = Ⅎ (Φ(K3, ⟨0.33, 0.33, 0.33⟩)) ̸= Φ(ℲK3, ⟨0.33, 0.33, 0.33⟩) = ♭. Definition 3.4. Let (Ξ, τ, ℓP ) be a picture fuzzy ideal topological space. Then, for each K ∈ ( I3 )Ξ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, we define an operator cl∗ : ( I3 )Ξ × I3 → ( I3 )Ξ as follows: cl∗(K, ⟨ς,κ, ϑ⟩) = K ∪ Φ(K, ⟨ς,κ, ϑ⟩). Now, if ℓP = ℓP0 then, cl∗(K, ⟨ς,κ, ϑ⟩) = K∪Φ(K, ⟨ς,κ, ϑ⟩) = K∪ clτ (K, ⟨ς,κ, ϑ⟩) = clτ (K, ⟨ς,κ, ϑ⟩) . Theorem 3.2. Let (Ξ, τ, ℓP ) be a picture fuzzy ideal topological space. Then, for any fuzzy set K,K ∈ ( I3 )Ξ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the operator cl∗ : ( I3 )Ξ× I3 → ( I3 )Ξ satisfies the following properties: (1) cl∗(♭, ⟨ς,κ, ϑ⟩) = ♭. (2)K ⊆ cl∗(K, ⟨ς,κ, ϑ⟩) ⊆ clτ (K, ⟨ς,κ, ϑ⟩) . (3) If K ⊆ Q , then cl∗(K, ⟨ς,κ, ϑ⟩) ⊆ cl∗( Q , ⟨ς,κ, ϑ⟩). (4) cl∗(K ∪ Q , ⟨ς,κ, ϑ⟩) ⊇ cl∗(K, ⟨ς,κ, ϑ⟩) ∪ cl∗( Q , ⟨ς,κ, ϑ⟩). (5) cl∗(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ cl∗(K, ⟨ς,κ, ϑ⟩) ∩ cl∗( Q , ⟨ς,κ, ϑ⟩). Proof. (1) Since cl∗(♭, ⟨ς,κ, ϑ⟩) = ♭ ∪ Φ(♭, ⟨ς,κ, ϑ⟩) and Φ(♭, ⟨ς,κ, ϑ⟩) = ♭ implies cl∗(♭, ⟨ς,κ, ϑ⟩) = ♭. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 14 of 30 (2) cl∗(K, ⟨ς,κ, ϑ⟩) = K ∪Φ(K, ⟨ς,κ, ϑ⟩) implies K ⊆ cl∗(K, ⟨ς,κ, ϑ⟩). Since K ⊆ clτ (K, ⟨ς,κ, ϑ⟩) and from Theorem 3.1(4), we have Φ(K, ⟨ς,κ, ϑ⟩) ⊆ clτ (K, ⟨ς,κ, ϑ⟩) implies cl∗(K, ⟨ς,κ, ϑ⟩) ⊆ clτ (K, ⟨ς,κ, ϑ⟩). Thus, K ⊆ cl∗(K, ⟨ς,κ, ϑ⟩) ⊆ clτ (K, ⟨ς,κ, ϑ⟩). (3) From K ⊆Q and Theorem 3.1(2), we have K ∪Φ(K, ⟨ς,κ, ϑ⟩) ⊆Q ∪Φ(Q , ⟨ς,κ, ϑ⟩) and then, cl∗(K, ⟨ς,κ, ϑ⟩) ⊆ cl∗( Q , ⟨ς,κ, ϑ⟩). (4) Since K ⊆ K ∪ Q and Q ⊆ K ∪ Q, cl∗(K, ⟨ς,κ, ϑ⟩) ⊆ cl∗(K ∪ Q , ⟨ς,κ, ϑ⟩) and cl∗(Q , ⟨ς,κ, ϑ⟩) ⊆ cl∗(K ∪Q , ⟨ς,κ, ϑ⟩). Thus, cl∗(K, ⟨ς,κ, ϑ⟩)∪cl∗(Q , ⟨ς,κ, ϑ⟩) ⊆ cl∗(K ∪ Q , ⟨ς,κ, ϑ⟩). (5) K∩ Q ⊆ K and K∩ Q ⊆ Q, cl∗(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ cl∗(K, ⟨ς,κ, ϑ⟩) and cl∗(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ cl∗( Q , ⟨ς,κ, ϑ⟩). Thus, cl∗(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ cl∗(K, ⟨ς,κ, ϑ⟩) ∩ cl∗( Q , ⟨ς,κ, ϑ⟩). Theorem 3.3. Let (Ξ, τ, ℓP ) be a picture fuzzy ideal topological space. Then, for each K ∈ ( I3 )Ξ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, we define an operator int∗ : ( I3 )Ξ × I3 → ( I3 )Ξ as follows: int∗(K, ⟨ς,κ, ϑ⟩) = K ∩ Ⅎ (Φ(ℲK, ⟨ς,κ, ϑ⟩)) . For K, Q ∈ ( I3 )Ξ, the operator int∗ satisfies the following properties: (1) int∗(♯, ⟨ς,κ, ϑ⟩) = ♯. (2) intτ (K, ⟨ς,κ, ϑ⟩) ⊆ int∗(K, ⟨ς,κ, ϑ⟩) ⊆ K. (3) If K ⊆ Q , then int∗(K, ⟨ς,κ, ϑ⟩) ⊆ int∗( Q , ⟨ς,κ, ϑ⟩). (4) int∗(K∩ Q , ⟨ς,κ, ϑ⟩) ⊆ int∗(K, ⟨ς,κ, ϑ⟩) ∩ int∗( Q , ⟨ς,κ, ϑ⟩). (5) int∗(♯, ⟨ς,κ, ϑ⟩) = int (K, ⟨ς,κ, ϑ⟩) if ℓP = ℓP0. (6) int∗(Ⅎ K, ⟨ς,κ, ϑ⟩) = Ⅎ (cl∗(K, ⟨ς,κ, ϑ⟩)) . Proof. It is similarly proved as the proof of Theorem 3.2. Definition 3.5. Let F : (Ξ, τ) ↬ (Υ, σ) be a PFM, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Then, F is called: (1) PF uS-continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q) for each Q ∈( I3 )Υ , σ( Q) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ∩D (F) ⊆ Fu( Q). (2) PF lS-continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q) for each Q ∈( I3 )Υ, σ( Q) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fl( Q). (3) PF uA-continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q) for each Q ∈ ( I3 )Υ , σ( Q) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ∩D (F) ⊆ Fu(intσ(clσ( Q, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)). (4) PF lA-continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q) for each Q ∈( I3 )Υ, σ( Q) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 15 of 30 that K ⊆ Fl(intσ(clσ( Q, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)). (5) PF uS (resp. PF lS)-continuous iff it is PF uS (resp. PF lS)-continuous at every fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F). (6) PF uA (resp. PF lA)-continuous iff it is PF uA (resp. PF lA)-continuous at every fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F). Definition 3.6. Let F : (Ξ, τ, ℓP ) ↬ (Υ, σ) be a PFM, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Then, F is called: (1) PF u ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ∩D (F) ⊆ Φ(Fu( Q ), ⟨ς,κ, ϑ⟩). (2) PF l ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Φ(Fl( Q ), ⟨ς,κ, ϑ⟩). (3) PF uA ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q∈ ( I3 )Υ, σ(Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ∩D (F) ⊆ Fu(intσ(cl ∗( Q, ⟨ς,κ, ϑ⟩ , ⟨ς,κ, ϑ⟩)). (4) PF lA ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ) for each Q ∈ ( I3 )Υ, σ(Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)). (5) PF u ℓP -continuous (resp. PF l ℓP -continuous) iff it is PF u ℓP -continuous (resp. PF l ℓP -continuous) at every fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F). (6) PF uA ℓP -continuous (resp. PF lA ℓP -continuous) iff it is PF uA ℓP -continuous (resp. PF lA ℓP -continuous) at every fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F). Remark 3.2. (1) If F is normalized PFM, then F isPF u ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ, τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Φ(Fu( Q ), ⟨ς,κ, ϑ⟩). (2) If F is normalized PFM, then F is PF uA ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ, τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fu(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)). (3) PF u (resp. PF l) ℓP -continuity and PF u (resp. PF lS)-continuity are independent notions as it will be shown in Example 3.2. (4) PF uS (resp. PF lS)-continuity ⇒ PF uA (resp. PF lA) ℓP -continuity ⇒ PF uA (resp. PF lA) -continuity. (5) PF uA (resp. PF lA) ℓP0-continuity ⇔ PF uA (resp. PF lA) -continuity. Theorem 3.4. Let F : (Ξ, τ, ℓP ) ↬ (Υ, σ) be a PFM (resp. normalized PFM), then F is PF l (resp. PF u) Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 16 of 30 ℓP -continuous iff Fl ( Q) ⊆ intτ ( Φ(Fl( Q), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ ) (resp. Fu( Q) ⊆ intτ (Φ(Fu( Q), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)) for each Q ∈ ( I3 )Υ , σ( Q) ≥ ⟨ς,κ, ϑ⟩ , ς ∈ I0,κ ∈ I1 and ϑ ∈ I1 . Proof. (⇒) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ, σ( Q) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ F l( Q). Then, there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Φ(F l( Q), ⟨ς,κ, ϑ⟩). Thus, ξ⟨ς,κ,ϑ⟩ ∈ K ⊆ intτ (Φ(F l( Q), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) and hence, F l ( Q) ⊆ intτ (Φ(F l( Q), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩). (⇐) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ, σ( Q) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ F l( Q). Then, F l ( Q) ⊆ intτ (Φ(F l( Q), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) and hence, ξ⟨ς,κ,ϑ⟩ ∈ intτ (Φ(F l( Q), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ Φ(F l( Q), ⟨ς,κ, ϑ⟩). Thus, F is PF l ℓP -continuous. Other case is similarly proved. Example 3.2. Let Ξ = {ξ1, ξ2}, Υ = {ζ1, ζ2, ζ3} and F : Ξ ↬ Υ be a PFM defined by ΨF(ξ1, ζ1) = ⟨0.1, 0.3, 0.2⟩, ΨF(ξ1, ζ2) = ⟨0.2, 0.3, 0.4⟩, ΨF(ξ1, ζ3) = ⟨1, 0, 0⟩, ΨF(ξ2, ζ1) = ⟨0.4, 0.4, 0.1⟩, ΨF(ξ2, ζ2) = ⟨0.3, 0.33, 0.33⟩, ΨF(ξ2, ζ3) = ⟨1, 0, 0⟩. For K1 = {⟨ξ, 0.33, 0.33, 0⟩ | ξ ∈ Ξ}, K2 = {⟨ξ, 0.4, 0.4, 0.2⟩ | ξ ∈ Ξ} and Q1 = {⟨ζ, 0.33, 0.33, 0.33⟩ | ζ ∈ Υ} define picture fuzzy topologies τ1, τ2: ( I3 )Ξ → I3, σ: ( I3 )Υ → I3, and define picture fuzzy ideals ℓP1 ,ℓP2 :( I3 )Ξ → I3 as follow. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 17 of 30 τ1(K) =  ⟨1, 0, 0⟩ if K ∈ {♭, ♯} , ⟨0.5, 0.33, 0.17⟩ if K = K1, ⟨0, 1, 0⟩ otherwise, , τ2(K) =  ⟨1, 0, 0⟩ if K ∈ {♭, ♯} , ⟨0.5, 0.33, 0.1⟩ if K = K2, ⟨0, 1, 0⟩ otherwise, ℓP1 (K) =  ⟨1, 0, 0⟩ if K = ♭, ⟨0.4, 0.2, 0.4⟩ if ♭ ⊆ K ⊆ {⟨ξ, 0.4, 0.1, 0.5⟩ |ξ ∈ Ξ}, ⟨0, 1, 0⟩ otherwise, ℓP2 (K) =  ⟨1, 0, 0⟩ if K = ♭, ⟨0.4, 0.2, 0.15⟩ if ♭ ⊆ K ⊆ {⟨ξ, 0.2, 0.2, 0.6⟩ |ξ ∈ Ξ}, ⟨0, 1, 0⟩ otherwise, σ( Q) =  ⟨1, 0, 0⟩ if Q ∈ {♭, ♯} , ⟨0.33, 0.33, 0.33⟩ if Q = Q1, ⟨0, 1, 0⟩ otherwise. Then, (1) F : (Ξ, τ1, ℓ P 1 ) ↬ (Υ, σ) is PF uS (resp. PF lS)-continuous but it is not PF u (resp. PF l) ℓP -continuous because Fu( Q1) = K1 ⊆ intτ (Fu( Q1), ⟨0.33, 0.33, 0.33⟩) = K1. Fl( Q1) = K1 ⊆ intτ (Fl( Q1), ⟨0.33, 0.33, 0.33⟩) = K1. but Fu( Q1) = K1 ⊈ intτ (Φ(Fu( Q1), ⟨0.33, 0.33, 0.33⟩), ⟨0.33, 0.33, 0.33⟩) = ♭. Fl( Q1) = K1 ⊈ intτ (Φ(Fl( Q1), ⟨0.33, 0.33, 0.33⟩), ⟨0.33, 0.33, 0.33⟩) = ♭. (2) F : (Ξ, τ2, ℓ P 2 ) ↬ (Υ, σ) is PF u (resp. PF l) ℓP -continuous but it is not PF uS (resp. PF lS)-continuous because Fu( Q1) = K1 ⊆ intτ (Φ(Fu( Q1), ⟨0.33, 0.33, 0.33⟩), ⟨0.33, 0.33, 0.33⟩) = ♯. Fl( Q1) = K1 ⊆ intτ (Φ(Fl( Q1), ⟨0.33, 0.33, 0.33⟩), ⟨0.33, 0.33, 0.33⟩) = ♯. but Fu( Q1) = K1 ⊈ intτ (Fu( Q1), ⟨0.33, 0.33, 0.33⟩) = ♭. Fl( Q1) = K1 ⊈ intτ (Fl( Q1), ⟨0.33, 0.33, 0.33⟩) = ♭. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 18 of 30 Theorem 3.5. For a PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q ∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the following statements are equivalent: (1) F is PF lA ℓP -continuous. (2) Fl( Q) ⊆ intτ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) ) , ⟨ς,κ, ϑ⟩), if σ( Q) ≥ ⟨ς,κ, ϑ⟩ . (3) clτ (Fu(clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu ( Q), if σ(Ⅎ Q) ≥ ⟨ς,κ, ϑ⟩ . Proof. (1) =⇒ (2) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ). Then, there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)). Thus, ξ⟨ς,κ,ϑ⟩ ∈ K ⊆ intτFl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), and hence Fl ( Q) ⊆ intτ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩ ) . (2) =⇒ (3) Let Q ∈ ( I3 )Υ with σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩. Then, by (2) ℲFu ( Q) = Fl(Ⅎ Q) ⊆ intτ ( Fl(intσ(cl ∗ (Ⅎ Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩ ) = Ⅎclτ (Fu(clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) . Thus, clτ (Fu(clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu ( Q). (3) =⇒ (1) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ, σ( Q) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ). Then by (3), we have Ⅎ [ intτ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) ) , ⟨ς,κ, ϑ⟩) ] = clτ (Fu(clσ(int ∗ (Ⅎ Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu (Ⅎ Q ) = Ⅎ Fl( Q), and Fl( Q ) ⊆ intτ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) ) , ⟨ς,κ, ϑ⟩). Therefore, ξ⟨ς,κ,ϑ⟩ ∈ intτ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) ) , ⟨ς,κ, ϑ⟩) ⊆ Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)). Thus, F is PF lA ℓP -continuous. The following theorem is similarly proved as the proof of Theorem 3.5. Theorem 3.6. For a normalized PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q ∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the following statements are equivalent: (1) F is PF uA ℓP -continuous. (2) Fu( Q) ⊆ intτ (Fu(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)) , ⟨ς,κ, ϑ⟩), if σ( Q) ≥ ⟨ς,κ, ϑ⟩ . (3) clτ ( Fl(clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩ ) ⊆ Fl ( Q), if σ(Ⅎ Q) ≥ ⟨ς,κ, ϑ⟩ . Example 3.3. Let Ξ = {ξ1, ξ2}, Υ = {ζ1, ζ2, ζ3} and F : Ξ ↬ Υ be a PFM de- fined by ΨF(ξ1, ζ1) = ⟨1, 0, 0⟩, ΨF(ξ1, ζ2) = ⟨0.1, 0.2, 0.7⟩, ΨF(ξ1, ζ3) = ⟨0.3, 0.2, 0.4⟩, ΨF(ξ2, ζ1) = ⟨0.33, 0.3, 0.33⟩, ΨF(ξ2, ζ2) = ⟨1, 0, 0⟩, ΨF(ξ2, ζ3) = ⟨0.4, 0.2, 0.4⟩. For K1 = {⟨ξ, 0.4, 0.4, 0.2⟩ | ξ ∈ Ξ}, K2 = {⟨ξ, 0.32, 0.3, 0⟩ | ξ ∈ Ξ} and Q1 = {⟨ζ, 0.32, 0.3, 0.33⟩ | ζ ∈ Υ} define picture fuzzy topologies τ : ( I3 )Ξ → I3, σ:( I3 )Υ → I3, and picture fuzzy ideal ℓP : ( I3 )Υ → I3 as follows: Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 19 of 30 τ(K) =  ⟨1, 0, 0⟩ if K ∈ {♭, ♯} , ⟨0.55, 0.1, 0.11⟩ if K = K1, ⟨0, 1, 0⟩ otherwise, , σ( Q ) =  ⟨1, 0, 0⟩ if Q ∈ {♭, ♯} , ⟨0.32, 0.3, 0.33⟩ if Q = Q1, ⟨0, 1, 0⟩ otherwise, ℓP ( Q ) =  ⟨1, 0, 0⟩ if Q = ♭, ⟨0.44, 0.2, 0.3⟩ if ♭ ⊆ Q ⊆ {⟨ζ, 0.2, 0.2, 0.4⟩ |ζ ∈ Υ}, ⟨0, 1, 0⟩ otherwise. Then, F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uS (resp. PF lS) ℓP -continuous but is not PF uS (resp. PF lS)-continuous because K2 = Fu( Q1) ⊆ intτ (Fu(intσ(cl ∗ ( Q1, ⟨0.32, 0.3, 0.33⟩) , ⟨0.32, 0.3, 0.33⟩)) , ⟨0.32, 0.3, 0.33⟩) = ♯, K2 = Fl( Q1) ⊆ intτ ( Fl(intσ(cl ∗ ( Q1, ⟨0.32, 0.3, 0.33⟩) , ⟨0.32, 0.3, 0.33⟩) ) , ⟨0.32, 0.3, 0.33⟩) = ♯. but K2 = Fu( Q1) ⊈ intτ (Fu( Q1), ⟨0.32, 0.3, 0.33⟩) = ♭, K2 = Fl( Q1) ⊈ intτ ( Fl( Q1), ⟨0.32, 0.3, 0.33⟩ ) = ♭. Example 3.4. Let Ξ = {ξ1, ξ2}, Υ = {ζ1, ζ2, ζ3} and F : Ξ ↬ Υ be a PFM de- fined by ΨF(ξ1, ζ1) = ⟨1, 0, 0⟩, ΨF(ξ1, ζ2) = ⟨0.2, 0.6, 0.2⟩, ΨF(ξ1, ζ3) = ⟨0.25, 0.3, 0.4⟩, ΨF(ξ2, ζ1) = ⟨0.32, 0.31, 0.15⟩, ΨF(ξ2, ζ2) = ⟨0.2, 0.2, 0.3⟩ , ΨF(ξ2, ζ3) = ⟨1, 0, 0⟩. For K1 = {⟨ξ, 0.1, 0.35, 0.31⟩ | ξ ∈ Ξ}, K2 = {⟨ξ, 0.1, 0.45, 0.31⟩ | ξ ∈ Ξ}, Q1 = {⟨ζ, 0.05, 0.36, 0.51⟩ | ζ ∈ Υ}, Q2 = {⟨ζ, 0.1, 0.35, 0.51⟩ | ζ ∈ Υ}, Q3 = {⟨ζ, 0.05, 0.36, 0⟩ | ζ ∈ Υ} and Q4 = {⟨ζ, 0.1, 0.35, 0⟩ | ζ ∈ Υ}, and define picture fuzzy topologies τ : ( I3 )Ξ → I3, σ: ( I3 )Υ → I3, and picture fuzzy ideal ℓP : ( I3 )Υ → I3 as follows. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 20 of 30 τ(K) =  ⟨1, 0, 0⟩ if K ∈ {♭, ♯} , ⟨0.6, 0.1, 0.3⟩ if K ∈ {K1, K2}, ⟨0, 1, 0⟩ otherwise, , σ( Q) =  ⟨1, 0, 0⟩ if Q ∈ {♭, ♯} , ⟨0.35, 0.5, 0.15⟩ if Q = Q1, ⟨0.55, 0.2, 0.25⟩ if Q = Q2, ⟨0, 1, 0⟩ otherwise, ℓP ( Q ) =  ⟨1, 0, 0⟩ if Q = ♭, ⟨0.55, 0.15, 0.3⟩ if Q ∈ { Q3, Q4}, ⟨0, 1, 0⟩ otherwise. Then, F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uA (resp. PF lA)-continuous but is not PF uA (resp. PF lA) ℓP -continuous because {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊆ intτ (Fu(intσ(clσ( Q1, ⟨0.35, 0.5, 0.15⟩), ⟨0.35, 0.5, 0.15⟩)) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}, {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ} = Fu( Q2) ⊆ intτ (Fu(intσ(clσ( Q2, ⟨0.35, 0.5, 0.15⟩), ⟨0.35, 0.5, 0.15⟩)) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}, {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊆ intτ ( Fl(intσ(clσ( Q1, ⟨0.35, 0.5, 0.15⟩), ⟨0.35, 0.5, 0.15⟩) ) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}, {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ} = Fl( Q2) ⊆ intτ ( Fl(intσ(clσ( Q2, ⟨0.35, 0.5, 0.15⟩), ⟨0.35, 0.5, 0.15⟩) ) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}, but {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊈ intτ (Fu(intσ(cl ∗( Q1, ⟨0.35, 0.5, 0.15⟩), ⟨0.35, 0.5, 0.15⟩)) , ⟨0.35, 0.5, 0.15⟩) = ♭, {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊈ intτ ( Fl(intσ(cl ∗( Q1, ⟨0.35, 0.5, 0.15⟩), ⟨0.35, 0.5, 0.15⟩) ) , ⟨0.35, 0.5, 0.15⟩) = ♭. Theorem 3.7. For a PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q ∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the following statements are equivalent: (1) F is PF lA ℓP -continuous. (2) τ ( Fl ( Q) ) ≥ ⟨ς,κ, ϑ⟩, if Q = intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). (3) τ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)) ) ≥ ⟨ς,κ, ϑ⟩ if σ( Q ) ≥ ⟨ς,κ, ϑ⟩. Proof. (1) =⇒ (2) If Q = intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), then σ( Q ) ≥ ⟨ς,κ, ϑ⟩. By Theorem 3.5(2), Fl( Q) ⊆ intτ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) ) , ⟨ς,κ, ϑ⟩) = intτ ( Fl( Q), ⟨ς,κ, ϑ⟩ ) . Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 21 of 30 Thus, τ ( Fl ( Q) ) ≥ ⟨ς,κ, ϑ⟩. (2) ⇔ (3) Obvious. (3) =⇒ (1) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ, σ( Q) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ). Then, by (3) and Q ⊆ intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), τ ( Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)) ) ≥ ⟨ς,κ, ϑ⟩, and ξ⟨ς,κ,ϑ⟩ ∈ Fl ( Q) ⊆ Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)). Thus, F is PF lA ℓP -continuous. The following theorems are similarly proved as the proof of Theorem 3.7. Theorem 3.8. For a PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q ∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the following statements are equivalent: (1) F is PF lA ℓP -continuous. (2) τ (ℲFu ( Q)) ≥ ⟨ς,κ, ϑ⟩, if Q = clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). (3) τ (ℲFu(clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩))) ≥ ⟨ς,κ, ϑ⟩ if σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩. Theorem 3.9. For a normalized PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q ∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the following statements are equivalent: (1) F is PF uA ℓP -continuous. (2) τ (Fu ( Q)) ≥ ⟨ς,κ, ϑ⟩, if Q = intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). (3) τ (Fu(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩))) ≥ ⟨ς,κ, ϑ⟩ if σ( Q ) ≥ ⟨ς,κ, ϑ⟩. Theorem 3.10. For a normalized PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q ∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the following statements are equivalent: (1) F is PF uA ℓP -continuous. (2) τ ( ℲFl ( Q) ) ≥ ⟨ς,κ, ϑ⟩ if Q = clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). (3) τ ( ℲFl(clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)) ) ≥ ⟨ς,κ, ϑ⟩ if σ(Ⅎ Q) ≥ ⟨ς,κ, ϑ⟩. Theorem 3.11. Let F : (Ξ, τ) ↬ (Υ, σ, ℓP ) be a PFM. Then, F is PF lA ℓP -continuous iff clτ (Fu ( Q) , ⟨ς,κ, ϑ⟩) ⊆ Fu(clσ( Q , ⟨ς,κ, ϑ⟩)) for any Q ∈ ( I3 )Υ with Q ⊆ clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Proof. (⇒) Let F be a PF lA ℓP -continuous. Then, for any Q ∈ ( I3 )Υ with Q⊆ clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩) =D(say), where D= clσ(int ∗ ( D, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). By Theorem 3.7, τ (Ⅎ Fu ( D)) ≥ ⟨ς,κ, ϑ⟩, and thus clτ (Fu ( Q) , ⟨ς,κ, ϑ⟩) ⊆ clτ (Fu ( D) , ⟨ς,κ, ϑ⟩) = Fu(clσ(int ∗ ( D, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)) ⊆ Fu(clσ( Q, ⟨ς,κ, ϑ⟩)). (⇐) Let Q ∈ ( I3 )Υ with Q = clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). Then, Q⊆ clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), and clτ (Fu ( Q) , ⟨ς,κ, ϑ⟩) ⊆ Fu(clσ( Q, ⟨ς,κ, ϑ⟩)) = Fu ( Q) . Therefore, we obtain τ (Ⅎ Fu ( Q)) ≥ ⟨ς,κ, ϑ⟩. Thus, by Theorem 3.7, F is PF lA ℓP -continuous. The following theorem is similarly proved as the proof of Theorem 3.11. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 22 of 30 Theorem 3.12. Let F : (Ξ, τ) ↬ (Υ, σ, ℓP ) be a normalized PFM. Then, F is PF uA ℓP -continuous iff clτ ( Fl ( Q) , ⟨ς,κ, ϑ⟩ ) ⊆ Fl(clσ( Q , ⟨ς,κ, ϑ⟩)) for any Q ∈ ( I3 )Υ with Q ⊆ clσ(int ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Definition 3.7. Let F : (Ξ, τ) ↬ (Υ, σ, ℓP ) be a PFM, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Then, F is called: (1) PF uW ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ, τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ∩D (F) ⊆ Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)). (2) PF lW ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ, τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)). (3) PF uAW ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ∩D (F) ⊆ clτ (Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). (4) PF lAW ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ) for each Q ∈ ( I3 )Υ, σ(Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). (5) PF uW ℓP -continuous (resp. PF lW ℓP -continuous) iff it is PF uW ℓP -continuous (resp. PF lW ℓP -continuous) at every fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F). (6) PF uAW ℓP -continuous (resp. PF lAW ℓP -continuous) iff it is PF uAW ℓP - continuous (resp. PF lAW ℓP -continuous) at every fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F). Remark 3.3. (1) If F is normalized PFM, then F is PF uW ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)). (2) If F is normalized PFM, then F is PF uAW ℓP -continuous at a fuzzy point ξ⟨ς,κ,ϑ⟩ ∈ D (F) iff ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ) for each Q ∈ ( I3 )Υ, σ( Q ) ≥ ⟨ς,κ, ϑ⟩ there exists K ∈( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ clτ (Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). (3) PF uA (resp. PF lA) ℓP -continuity ⇒ PF uW (resp. PF lW ) ℓP -continuity ⇒ PF uW (resp. PF lW )-continuity. (4) PF uW (resp. PF lW ) ℓP0-continuity ⇔ PF uW (resp. PF lW )-continuity. (5) PF uW (resp. PF lW ) ℓP -continuity ⇒ PF uAW (resp. PF lAW ) ℓP -continuity ⇒ PF uAW (resp. PF lAW )-continuity. (6) PF uAW (resp. PF lAW ) ℓP0-continuity ⇔ PF uAW (resp. PF lAW )-continuity. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 23 of 30 Theorem 3.13. A PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF lW ℓP -continuous iff Fl( Q ) ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for each Q ∈ ( I3 )Υwith σ( Q) ≥ ⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Proof. (⇒) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υwith σ( Q ) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ). Then, there exists K ∈ ( I3 )Ξ, τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)). Thus, ξ⟨ς,κ,ϑ⟩ ∈ K ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), and hence Fl ( Q) ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). (⇐) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υwith σ( Q ) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ). Then, ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ) ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, ξ⟨ς,κ,ϑ⟩ ∈ K ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)). Hence, F is PF uW ℓP -continuous. The following theorem is similarly proved as the proof of Theorem 3.13. Theorem 3.14. A normalized PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uW ℓP -continuous iff Fu( Q) ⊆ intτ (Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for each Q ∈ ( I3 )Υwith σ( Q) ≥ ⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. The following examples shows that generally PF uW ℓP -continuous and PF lW ℓP - continuous (resp. PF uW continuous and PF lW continuous) multifunction need not be either PF uA ℓP -continuous (resp. PF uW ℓP -continuous) multifunction or PF lA ℓP -continuous (resp. PF lW ℓP -continuous) multifunction. Example 3.5. From Example 3.4, F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uW (resp. PF lW )- continuous but is not PF uW (resp. PF lW ) ℓP -continuous because {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊆ intτ (Fu(clσ( Q1, ⟨0.35, 0.5, 0.15⟩)) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}, {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ} = Fu( Q2) ⊆ intτ (Fu(clσ( Q2, ⟨0.35, 0.5, 0.15⟩)) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}, {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊆ intτ ( Fl(clσ( Q1, ⟨0.35, 0.5, 0.15⟩) ) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}, ⟨0.1, 0.35, 0⟩ = Fl( Q2) ⊆ intτ ( Fl(clσ( Q2, ⟨0.35, 0.5, 0.15⟩) ) , ⟨0.35, 0.5, 0.15⟩) = {⟨ξ, 0.1, 0.35, 0⟩ |ξ ∈ Ξ}. but {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fu(Q1)⊈ intτ (Fu(cl∗( Q1, ⟨0.35, 0.5, 0.15⟩)) , ⟨0.35, 0.5, 0.15⟩) = ♭, {⟨ξ, 0.05, 0.36, 0⟩ |ξ ∈ Ξ} = Fl(Q1)⊈ intτ ( Fl(cl∗( Q1, ⟨0.35, 0.5, 0.15⟩) ) , ⟨0.35, 0.5, 0.15⟩) = ♭. Example 3.6. From the Example 3.4, for K1 = {⟨ξ, 0.6, 0.3, 0.1⟩ | ξ ∈ Ξ}, Q 1 = {⟨ζ, 0.3, 0.6, 0.1⟩ | ζ ∈ Υ}, define picture fuzzy topologies τ : ( I3 )Ξ → I3, σ: ( I3 )Υ → I3, and picture fuzzy ideal ℓP : ( I3 )Υ → I3 as follows: Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 24 of 30 τ(K) =  ⟨1, 0, 0⟩ if K ∈ {♭, ♯} , ⟨0.6, 0.2, 0.2⟩ if K = K1, ⟨0, 1, 0⟩ otherwise, , σ( Q ) =  ⟨1, 0, 0⟩ if Q ∈ {♭, ♯} , ⟨0.5, 0.3, 0.1⟩ if Q = Q 1, ⟨0, 1, 0⟩ otherwise, ℓP ( Q ) =  ⟨1, 0, 0⟩ if Q = ♭, ⟨0.7, 0.15, 0.15⟩ if {⟨ζ, 0.4, 0.5, 0.1⟩ |ζ ∈ Υ} ⊆ Q ⊆ ♯, ⟨0, 1, 0⟩ otherwise. F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uW (resp. PF lW ) ℓP -continuous but is not PF uA (resp. PF lA) ℓP -continuous because {⟨ξ, 0.3, 0.6, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊆ intτ (Fu(cl∗( Q1, ⟨0.5, 0.3, 0.1⟩)) , ⟨0.5, 0.3, 0.1⟩) = {⟨ξ, 0.3, 0.6, 0⟩ |ξ ∈ Ξ}, {⟨ξ, 0.3, 0.6, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊆ intτ ( Fl(cl∗( Q1, ⟨0.5, 0.3, 0.1⟩) ) , ⟨0.5, 0.3, 0.1⟩) = {⟨ξ, 0.3, 0.6, 0⟩ |ξ ∈ Ξ}, but {⟨ξ, 0.3, 0.6, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊈ intτ (Fu(intσ(cl ∗( Q1, ⟨0.5, 0.3, 0.1⟩), ⟨0.5, 0.3, 0.1⟩)) , ⟨0.5, 0.3, 0.1⟩) = ♭, {⟨ξ, 0.3, 0.6, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊈ intτ ( Fl(intσ(cl ∗( Q1, ⟨0.5, 0.3, 0.1⟩), ⟨0.5, 0.3, 0.1⟩) ) , ⟨0.5, 0.3, 0.1⟩) = ♭. Theorem 3.15. A PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF lW ℓP -continuous iff clτ (Fu(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu( Q ) for each Q ∈ ( I3 )Υ with σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Proof. (⇒) Let Q ∈ ( I3 )Υwith σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩ . Then, by Theorem 3.13, Ⅎ Fu( Q) = Fl(Ⅎ Q) ⊆ intτ (Fl(cl∗ (Ⅎ Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) = Ⅎ clτ (Fu(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, clτ (Fu(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu( Q). (⇐) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υwith σ( Q ) ≥ ⟨ς,κ, ϑ⟩ and ξt ∈ Fl( Q ). Then, Ⅎintτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) = clτ (Fu(int∗ (Ⅎ Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fu(Ⅎ Q) = Ⅎ Fl( Q), and hence, Fl( Q) ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, it is PF lW ℓP -continuous. The following theorem is similarly proved as the proof of Theorem 3.15. Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 25 of 30 Theorem 3.16. A normalized PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uW ℓP -continuous iff clτ (Fl(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) ⊆ Fl( Q ) for each Q ∈ ( I3 )Υwith σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Theorem 3.17. If F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is normalized PF uW ℓP -continuous and F (K) ⊆ intσ(cl ∗(F (K) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) for each K ∈ ( I3 )Ξ. Then, F is PF uA ℓP - continuous. Proof. Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ,σ( Q ) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fu( Q ). Then, there exists K ∈ ( I3 )Ξ with τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ Fu(cl∗( Q , ⟨ς,κ, ϑ⟩)), then F (K) ⊆ F (Fu(cl∗( Q, ⟨ς,κ, ϑ⟩))) ⊆ cl∗(Q , ⟨ς,κ, ϑ⟩). Since F (K) ⊆ intσ(cl ∗(F (K) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ intσ(cl ∗( Q , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), hence K ⊆ Fu (F (K)) ⊆ Fu (intσ(cl ∗( Q, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)). Then, F is PF uA ℓP -continuous. Theorem 3.18. Let F : (Ξ, τ) ↬ (Υ, σ, ℓP ) be a PF lW ℓP -continuous. Then, Fl( Q) ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for any Q ∈ ( I3 )Υ with Q ⊆ intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Proof. Let F be a PF lW ℓP -continuous and Q ∈ ( I3 )Υ with Q ⊆ intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩). Then, if ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ) ⊆ Fl(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩)), there exists K ∈ ( I3 )Ξ, τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ (Fl(cl∗(intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩)) ⊆ Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)). Thus, K ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), and Fl( Q) ⊆ intτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). The following theorem is similarly proved as the proof of Theorem 3.18. Theorem 3.19. Let F : (Ξ, τ) ↬ (Υ, σ, ℓP ) be a normalized PF uW ℓP -continuous. Then, Fu( Q) ⊆ intτ (Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩) for any Q ∈ ( I3 )Υ with Q ⊆ intσ(cl ∗ ( Q, ⟨ς,κ, ϑ⟩) , ⟨ς,κ, ϑ⟩), ς ∈ I0,κ ∈ I1 and ϑ ∈ I1. Theorem 3.20. For a PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q ∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1,the following statements are equivalent: (1) F is PF lAW ℓP -continuous. (2) Fl( Q) ⊆ intτ (clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩),if σ( Q) ≥ ⟨ς,κ, ϑ⟩ . (3) clτ (intτ (Fu(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ Fu ( Q), if σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩ . Proof. (1) =⇒ (2) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ,σ( Q ) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ). Then, there exists K ∈ ( I3 )Ξ , τ(K) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ K such that K ⊆ clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 26 of 30 ξt ∈ K ⊆ intτ (clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), and hence Fl ( Q) ⊆ intτ (clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩). (2) =⇒ (3) Let Q ∈ ( I3 )Υ with σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩. Then, by (2) Ⅎ Fu ( Q) = Fl(Ⅎ Q) ⊆ intτ (clτ (Fl(cl∗ (Ⅎ Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) = Ⅎclτ (intτ (Fu(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ , thus, clτ (intτ (Fu(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ ⊆ Fu ( Q). (3) =⇒ (1) Let ξ⟨ς,κ,ϑ⟩ ∈ D (F), Q ∈ ( I3 )Υ, σ( Q) ≥ ⟨ς,κ, ϑ⟩ and ξ⟨ς,κ,ϑ⟩ ∈ Fl( Q ). Then, by (3), we have Ⅎintτ (clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) = clτ (intτ (Fu(int∗ (Ⅎ Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ ⊆ Fu (Ⅎ Q) = Ⅎ Fl( Q), and hence Fl( Q) ⊆ intτ (clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩). Therefore, ξ⟨ς,κ,ϑ⟩ ∈ intτ (clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩) ⊆ clτ (Fl(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, F is PF lAWℓP -continuous. The following theorem is similarly proved as the proof of Theorem 3.20. Theorem 3.21. For a normalized PFM F : (Ξ, τ) ↬ (Υ, σ, ℓP ), Q∈ ( I3 )Υ, ς ∈ I0,κ ∈ I1 and ϑ ∈ I1, the following statements are equivalent: (1) F is PF uAW ℓP -continuous. (2) Fu( Q) ⊆ intτ (clτ (Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩),if σ( Q) ≥ ⟨ς,κ, ϑ⟩ . (3) clτ ( intτ (Fl(int∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩ ) ⊆ Fl ( Q), if σ(Ⅎ Q ) ≥ ⟨ς,κ, ϑ⟩. The following example shows that generally PF uAW continuous and PF lAW con- tinuous (resp. PF uAW ℓP -continuous and PF lAW ℓP -continuous) need not be either PF uAW ℓP -continuous (resp. PF uW ℓP -continuous) or PF lAW ℓP -continuous (resp. PF lW ℓP -continuous). Example 3.7. Let Ξ = {ξ1, ξ2}, Υ = {ζ1, ζ2, ζ3} and F : Ξ ↬ Υ be a PFM defined by ΨF(ξ1, ζ1) = ⟨0.1, 0.3, 0.6⟩, ΨF(ξ1, ζ2) = ⟨1, 0, 0⟩, ΨF(ξ1, ζ3) = ⟨0.23, 0.12, 0.4⟩, ΨF(ξ2, ζ1) = ⟨0.31, 0.23, 0.43⟩, ΨF(ξ2, ζ2) = ⟨0.45, 0.1, 0.4⟩, ΨF(ξ2, ζ3) = ⟨1, 0, 0⟩. For K1 = {⟨ξ, 0.3, 0.2, 0.1⟩ | ξ ∈ Ξ} and Q 1 = {⟨ζ, 0.2, 0.5, 0.3⟩ | ζ ∈ Υ} define picture fuzzy topologies τ : ( I3 )Ξ → I3, σ: ( I3 )Υ → I3, and picture fuzzy ideal ℓP : ( I3 )Υ → I3 as follows: Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 27 of 30 τ(K) =  ⟨1, 0, 0⟩ if K ∈ {♭, ♯} , ⟨0.35, 0.3, 0.3⟩ if K = K1, ⟨0, 1, 0⟩ otherwise, , σ( Q ) =  ⟨1, 0, 0⟩ if Q ∈ {♭, ♯} , ⟨0.31, 0.31, 0.18⟩ if Q = Q1, ⟨0, 1, 0⟩ otherwise, ℓP ( Q ) =  ⟨1, 0, 0⟩ if Q = ♭, ⟨0.36, 0.31, 0.2⟩ if ♭ ⊆ Q ⊆ {⟨ζ, 0.2, 0.5, 0.21⟩ |ζ ∈ Υ}, ⟨0, 1, 0⟩ otherwise. Then, (1) F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uAW (resp. PF lAW )-continuous but is not PF uAW (resp. PF lAW ) ℓP -continuous because {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊆ intτ (clτ (Fu(clσ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩), ⟨0.31, 0.31, 0.18⟩) = ♯, {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊆ intτ (clτ (Fl(clσ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩), ⟨0.31, 0.31, 0.18⟩) = ♯, but {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊈ intτ (clτ (Fu(cl∗ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩), ⟨0.31, 0.31, 0.18⟩) = ♭, {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊈ intτ (clτ (Fl(cl∗ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩), ⟨0.31, 0.31, 0.18⟩) = ♭, (2) For K1 = {⟨ξ, 0.3, 0.2, 0.1⟩ | ξ ∈ Ξ}, then F : (Ξ, τ) ↬ (Υ, σ, ℓP ) is PF uAW (resp. PF lAW ) ℓP -continuous but is not PF uW (resp. PF lW ) ℓP -continuous because {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fu( Q1) ⊆ intτ (clτ (Fu(cl∗ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩), ⟨0.31, 0.31, 0.18⟩) = ♯, {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fl( Q1) ⊆ intτ (clτ (Fl(cl∗ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩), ⟨0.31, 0.31, 0.18⟩) = ♯, but {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fu(Q1)⊈ intτ (Fu(cl∗ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩) = ♭, {⟨ξ, 0.2, 0.5, 0⟩ |ξ ∈ Ξ} = Fl(Q1)⊈ intτ (Fl(cl∗ ( Q1, ⟨0.31, 0.31, 0.18⟩)), ⟨0.31, 0.31, 0.18⟩) = ♭, Theorem 3.22. Let F : (Ξ, τ) ↬ (Υ, σ, ℓP ) be a normalized PFM, F be PF uAW ℓP - continuous and PF lA ℓP -continuous. Then, F is PF uW ℓP -continuous. Proof. Let Q ∈ ( I3 )Υwith σ( Q ) ≥ ⟨ς,κ, ϑ⟩ and F be PF uAW ℓP -continuous. Then, by Theorem 3.21(2), Fu( Q) ⊆ intτ (clτ (Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩). Dali Shi et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5956 28 of 30 Since clσ(Q, ⟨ς,κ, ϑ⟩) = clσ(int ∗(clσ(Q, ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), ⟨ς,κ, ϑ⟩), it follows from The- orem 3.8(2) that τ (ℲFu (clσ( Q, ⟨ς,κ, ϑ⟩))) ≥ ⟨ς,κ, ϑ⟩, then τ (ℲFu (cl∗( Q, ⟨ς,κ, ϑ⟩))) ≥ ⟨ς,κ, ϑ⟩, and Fu( Q) ⊆ intτ (Fu(cl∗ ( Q, ⟨ς,κ, ϑ⟩)), ⟨ς,κ, ϑ⟩). Thus, by Theorem 3.4, F is PF uW ℓP - continuous. The following theorem is similarly proved as the proof of Theorem 3.22. Theorem 3.23. Let F : (Ξ, τ) ↬ (Υ, σ, ℓP ) be a normalized PFM, F be PF lAW ℓP - continuous and PF uA ℓP -continuous. Then, F is PF lW ℓP -continuous. 4. Conclusion In this paper, we introduced a definition of picture fuzzy implication operation in PFSs. Two operations based on the product form of the essential degrees of positivism, negativism and neutralism are investigated. Two other operations in (PFSs) are established based on the sum and the product of these essential degrees. Depending on the previous four picture fuzzy operations, we defined four PFMTSs over PFSs. Although the variety of objects defined in this paper, all structures defined here did not satisfy some of the Kuratowski closure conditions or the Kuratowski interior conditions. These constructed structures are called "feeble" (PFFMTSs) standing for not all required conditions for a topological (closure or interior) operator are satisfied. Still, based on the simple operations □ and 3, we got accurate definitions of "closure" and "interior" operators for PFMTSs. So, □ and 3 are the standard two modal operators over PFSs introducing two PFMTSs. In future research work, we will extend the work to some other fuzzy environment and derive some more properties for different topological spaces. Further, we will expand our work to consider the picture topological space to be temporal as well as in the corresponding results. Conflicts of interest: The authors declare that they have not any conflicts of interest. 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