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Axioms
, Volume 12 (5) – Apr 22, 2023

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axioms Article 1,2 3 4, Krassimir Atanassov , Nora Angelova , Tania Pencheva * Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Bl. 105, 1113 Soﬁa, Bulgaria Intelligent Systems Laboratory, Prof. Dr. Assen Zlatarov University, 1 “Prof. Yakimov” Blvd., 8010 Burgas, Bulgaria Faculty of Mathematics and Informatics, Soﬁa University, 5 James Bouchier Blvd., 1164 Soﬁa, Bulgaria Department of QSAR and Molecular Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, Acad. Georgi Bonchev Str., Bl. 105, 1113 Soﬁa, Bulgaria * Correspondence: tania.pencheva@biomed.bas.bg Abstract: The concept of an Intuitionistic Fuzzy Modal Topological Structure (IFMTS) was introduced previously, and some of its properties were studied. So far, there are two different IFMTSs based on the classical intuitionistic fuzzy operations: “union” ([) and “intersection” (\). In the present paper, two new IFMTSs are developed. They are based on new intuitionistic fuzzy topological operators from closure and interior types, introduced here for the ﬁrst time, and on the two standard intuitionistic fuzzy modal operators and}. Some basic properties of the new IFMTSs are discussed. The newly presented IFMTSs could be considered as a basis for the next research on the IFMTSs. Some ideas for the future development of the IFMTS theory and open problems are formulated, related to the existence of other intuitionistic fuzzy operations that can generate new intuitionistic fuzzy topological operators and, respectively, new IFMTSs. Keywords: Intuitionistic Fuzzy Modal Topology; Intuitionistic Fuzzy Operator; Intuitionistic Fuzzy Set MSC: 03E72; 54A40 Citation: Atanassov, K.; Angelova, 1. Introduction N.; Pencheva, T. On Two Intuitionistic Fuzzy Modal The present paper is a continuation of the research in [1], in which the combination of Topological Structures. Axioms 2023, the ideas and deﬁnitions from the areas of (general) topology (see, e.g., [2–4]), of (standard) 12, 408. https://doi.org/10.3390/ modal logic (see, e.g., [5–8]) and of intuitionistic fuzziness (see, e.g., [9,10]) is introduced axioms12050408 for ﬁrst time. Presented there are two examples of Intuitionistic Fuzzy Modal Topological Structures (IFMTSs), based on the classical intuitionistic fuzzy operations “union” ([) Academic Editors: Eunsuk Yang and and “intersection” (\). The reason for the present research is the authors’ desire for new Xiaohong Zhang examples of such structures to be constructed that are different from those presented in [1]. Received: 28 February 2023 The structures constructed in this investigation are based on the deﬁnitions of two new Revised: 18 April 2023 intuitionistic fuzzy topological operators from closure and interior types (in the terms Accepted: 20 April 2023 of [3]). To date, and besides [1], the authors have recognized only the operations and Published: 22 April 2023 topological operators presented in this paper as being able to satisfy all the conditions for IFTMS construction. In this line of reasoning, looking for new operators is still an open problem, as formulated in the Conclusion as well. In Section 2, short remarks over Intuitionistic Fuzzy Sets (IFSs) are given, Section 3 Copyright: © 2023 by the authors. presents the deﬁnitions of the new operators, while in Section 4, the new structures are Licensee MDPI, Basel, Switzerland. introduced and some of their basic properties are discussed. In the Conclusion, some new This article is an open access article distributed under the terms and directions of the development of the present ideas are discussed. conditions of the Creative Commons As mentioned in [1], the Intuitionistic Fuzzy Topology (IFT) has been developed very Attribution (CC BY) license (https:// actively during the last years. The ﬁrst steps in this process were published in [11–37], but creativecommons.org/licenses/by/ until now there has been no systematic research on IFT development. 4.0/). Axioms 2023, 12, 408. https://doi.org/10.3390/axioms12050408 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 408 2 of 13 2. Short Remarks over IFSs In 1983, the fuzzy sets of Lotﬁ Zadeh (1921–2017) [38] were extended to Intuitionistic Fuzzy Sets (IFSs, see, e.g., [9,10,39]). All results that are valid for the fuzzy sets can be transformed here as well. Let a set E be ﬁxed. An IFS A in E is an object of the following form: A = fhx, m (x), n (x)ijx 2 Eg, A A where functions m : E ! [0, 1] and n : E ! [0, 1] deﬁne the degree of membership and A A the degree of non-membership of the element x 2 E, respectively, and for every x 2 E, 0 m (x) + n (x) 1. A A Let the universe E be given everywhere below. One of the geometrical interpretations of the IFSs is shown in Figure 1 (see [9,10]). h0, 1i n (x) m (x) h1, 0i Figure 1. A geometrical interpretation of an element x 2 E. Following [10,40], it is important to mention that the functions m, n (and also p, known as the degree of uncertainty and equal to 1 m n) can be continuous or discrete with respect to the concrete cases. If the universe E and the three functions are constructive objects, then the operations over the IFSs with the universe E preserve the constructiveness. For every two IFSs A and B, a number of relations and operations are defined (see, e.g., [9,10]). The most important six of them are as follows: A B iff (8x 2 E)(m (x) m (x) & n (x) n (x)); A B A B A B iff B A; A = B iff (8x 2 E)(m (x) = m (x) & n (x) = n (x)); A B A B :A = fhx, n (x), m (x)ijx 2 Eg; A A A\ B = A\ B = fhx, min(m (x), m (x)), max(n (x), n (x))ijx 2 Eg; 4 A B A B A[ B = A[ B = fhx, max(m (x), m (x)), min(n (x), n (x))ijx 2 Eg, 4 A B A B while the next two operations are introduced in [41]: A\ B = fhx, min(m (x), m (x)), 1 min(m (x), m (x))ijx 2 Eg; 33 A B A B A[ B = fhx, 1 min(n (x), n (x)), min(n (x), n (x))ijx 2 Eg. 33 A B A B The used subscripts 4 and 33 are introduced in [10,41] and express the subsequent number of the respective operation. The last two operations do not have a direct analogue with those from the fuzzy set theory. The geometrical interpretations of these new operations are given for the ﬁrst time in Figures 2 and 3. Axioms 2023, 12, 408 3 of 13 In Section 3, the aforementioned last two operations will become a basis for the deﬁnitions of two new intuitionistic fuzzy topological operators. The ﬁrst two (simplest) analogues of the topological operators closure and interior, deﬁned over IFSs, are presented as follows (see, e.g., [9,10]): C(A) = fhx, sup m (y), inf n (y)ijx 2 Eg, A A y2E y2E I(A) = fhx, inf m (y), sup n (y)ijx 2 Eg. A A y2E y2E h0, 1i 1 min(m (x), m (x)) A B n (x) n (x) m (x) h1, 0i m (x) = min(m (x), m (x)) A A B Figure 2. A geometrical interpretation of the operation \ . h0, 1i n (x) n (x) = min(m (x), m (x)) B A B m (x) m (x) h1, 0i A B 1 min(m (x), m (x)) A B Figure 3. A geometrical interpretation of the operation [ . The geometrical interpretations of both intuitionistic fuzzy topological operators C(A) and I(A) are given in Figures 4 and 5. Over the IFSs, different modal operators can be deﬁned. They do not have analogues in fuzzy set theory. Axioms 2023, 12, 408 4 of 13 The ﬁrst two (simplest) modal operators, deﬁned over the IFSs, are as follows (see, e.g., [5–8]): A = fhx, m (x), 1 m (x)ijx 2 Eg; A A }A = fhx, 1 n (x), n (x)ijx 2 Eg. A A The geometrical interpretation of both intuitionistic fuzzy modal operators and} is given in Figure 6. Let O = fhx, 0, 1ijx 2 Eg, E = fhx, 1, 0ijx 2 Eg. Let for each set X P(X) = fYjY Xg. Then, for sets O and E , we obtain that P(O ) = fO g, P(E ) = fAjA E g. h0, 1i x x 2 4 r x n (x) C(A) h0, 0i h1, 0i m (x) C(A) Figure 4. Geometrical interpretation of the topological operator C . h0, 1i r @ n (x) I(A) x r r x h0, 0i h1, 0i m (x) I(A) Figure 5. Geometrical interpretation of the topological operator I . Axioms 2023, 12, 408 5 of 13 h0, 1i r x }x r r n (x) m (x) h1, 0i Figure 6. Geometrical interpretation of the modal operators and }. 3. Deﬁnitions of Two New Intuitionistic Fuzzy Topological Operators Using the method for the construction of the ﬁrst two intuitionistic fuzzy topological operators C and I on the basis of the standard operations [ and \ (see [9,10]), below, we construct two new intuitionistic fuzzy topological operators. At the moment, the ﬁrst two topological operators have no indices, but for completeness, we can index them as C and I . The next operators are indexed using the number of intuitionistic fuzzy operations that generate them. Thus, the operators newly presented here are as follows for each IFS A: C (A) = fhx, 1 inf n (y), inf n (y)ijx 2 Eg; 33 A A y2E y2E I (A) = fhx, inf m (y), 1 inf m (y)ijx 2 Eg. 33 A A y2E y2E The geometrical interpretations of the two intuitionistic fuzzy topological operators are given in Figures 7 and 8. Theorem 1. For each IFS A: (a) :C (:A) = I (A), 33 33 (b) :I (:A) = C (A). 33 33 Proof. For case (a), we obtain :C (:A) = :C (fhx, n (y), m (y)ijx 2 Eg); 33 33 A A = :fhx, 1 inf m (y), inf m (y)ijx 2 Eg; A A y2E y2E = fhx, inf m (y), 1 inf m (y)ijx 2 Eg; A A y2E y2E = I (A). The check of (b) is similar. Axioms 2023, 12, 408 6 of 13 h0, 1i x x 2 4 r x n (x) C(A) h0, 0i h1, 0i m (x) C(A) Figure 7. Geometrical interpretation of the topological operator C . h0, 1i n (x) @ I(A) r @ x @ r x r @ h0, 0i h1, 0i m (x) I(A) Figure 8. Geometrical interpretation of the topological operator I . Theorem 2. For each IFS A: I (A) I(A) C(A) C (A). 33 33 Proof. From the deﬁnitions of operators C and I , for each x 2 E, we have the equalities (see, e.g., [9,10]) 0 sup m (y) + inf n (y) 1, A A y2E y2E 0 inf m (y) + sup n (y) 1. A A y2E y2E Therefore, inf m (y) sup m (y) 1 inf n (y) A A A y2E y2E y2E and 1 inf m (y) sup n (y) inf m (y) A A A y2E y2E y2E from which the validity of the assertion follows. Axioms 2023, 12, 408 7 of 13 4. Deﬁnitions of the Two New Intuitionistic Fuzzy Modal Topological Structures and Their Examples By analogy with the deﬁnition of Kazimierz Kuratowski (1896–1980) [3], in [1], we gave the deﬁnition of the IFMTS, which is an essential extension of the one in [3]. Here, we repeat the deﬁnition given in [1], but with a small correction as follows. The object hP(E ),O, D,i, is called a cl-IFMTS when the following are true: - E is a ﬁxed universe, - D : P(E )P(E ) ! P(E ) is an operation over E, - O : P(E ) ! P(E ) is an operator over E, generated by an operation D, - is a modal operator, and for every two IFSs A, B 2 P(E ), the following conditions are valid: C1 O(ADB) = O(A)DO(B), C2 A O(A), C3 O(O ) = O , C4 O(O(A)) = O(A), C5 (ArB) = Ar B, C6 A A, C7 E = E , C8 A = A, C9 O(A) = O(A). In the deﬁnition in [1], an operation r was added in order to be dual according to the De Morgan Law with operation D, and for this reason, it is omitted in the present deﬁnitions. Having in mind the results from Section 3, we formulate and prove the following theorems. Theorem 3. hP(E ),C ,[ ,}i is a cl-IFMTS. 33 33 Proof. Let the IFSs A, B 2 P(E ) be given. Then, we check sequentially the validity of the nine conditions C1–C9. C1. C (A[ B) = C (fhx, m (x), n (x)ijx 2 Eg[ fhx, m (x), n (x)ijx 2 Eg) 33 33 33 A A 33 B B = C (fhx, 1 min(n (x), n (x)), min(n (x), n (x))ijx 2 Eg) 33 A B A B = fhx, 1 inf(min(n (x), n (x))), inf min(n (x), n (x))ijx 2 Eg A B A B y2E y2E = fhx, 1 min( inf n (x), inf n (x)), min( inf n (x), inf n (x))ijx 2 Eg A B A B y2E y2E y2E y2E = fhx, 1 inf n (x), inf n (x)ijx 2 Eg A A y2E y2E [ fhx, 1 inf n (x), inf n (x)ijx 2 Eg 33 B B y2E y2E = C (A)[ C (B); 33 33 33 Axioms 2023, 12, 408 8 of 13 C2. A = fhx, m (x), n (x)ijx 2 Eg A A fhx, sup m (x), inf n (x)ijx 2 Eg A A y2E y2E fhx, sup(1 n (x)), inf n (x)ijx 2 Eg A A y2E y2E = fhx, 1 inf n (x), inf n (x)ijx 2 Eg A A y2E y2E = C (A); C3. C (O ) = C (fhx, 0, 1ijx 2 Eg) 33 33 = fhx, 1 inf 1, inf 1ijx 2 Eg y2E y2E = fhx, 0, 1ijx 2 Eg = O ; C4. Having in mind that inf n (y) is a constant, we obtain that y2E C (C (A)) = C (fhx, 1 inf n (x), inf n (x)ijx 2 Eg) 33 33 33 A A y2E y2E = fhx, 1 inf inf n (y), inf inf n (y)ijx 2 Eg A A z2E y2E z2E y2E = fhx, 1 inf n (y), inf n (y)ijx 2 Eg A A y2E y2E = C (A); C5. }(A[ B) = }(fhx, 1 min(n (x), n (x)), min(n (x), n (x))ijx 2 Eg); 33 A B A B = fhx, 1 min(n (x), n (x)), min(n (x), n (x))ijx 2 Eg; A B A B = fhx, m (x), n (x)ijx 2 Eg[ fhx, m (x), n (x)ijx 2 Eg 33 B B A A = }A[ }B; C6. A = fhx, m (x), n (x)ijx 2 Eg A A fhx, m (x), 1 n (x)ijx 2 Eg A A = }fhx, m (x), n (x)ijx 2 Eg A A = }A; C7. }E = }fhx, 1, 0ijx 2 Eg) = fhx, 1 0, 0ijx 2 Eg = (fhx, 1, 0ijx 2 Eg) = E ; Axioms 2023, 12, 408 9 of 13 C8. }}A = }fhx, 1 n (x), n (x)ijx 2 Eg A A = fhx, 1 n (x), n (x)ijx 2 Eg A A = }A; C9. }C (A) = }fhx, sup m (y), inf n (y)ijx 2 Eg) 33 A A y2E y2E = fhx, sup m (y), 1 sup n (y)ijx 2 Eg) A A y2E y2E = C (fhx, 1 n (x), n (x)ijx 2 Eg) 33 A A = C (}A). This completes the proof of Theorem 3. By analogy with the deﬁnition of cl-IFMTS and of Kuratowski’s deﬁnition for inclusion operator, here we state that the object hP(E ),Q,r,i, is called a in-IFMTS when the following are true: - E is a ﬁxed universe, - r : P(E )P(E ) ! P(E ) is an operation over E, - Q : P(E ) ! P(E ) is an operator over E, generated by operation r, - is a modal operator, and for every two IFSs A, B 2 P(E ), the following conditions are valid: I1 Q(ArB) = Q(A)rQ(B), I2 Q(A) A, I3 Q(E ) = E , I4 Q(Q(A)) = Q(A), I5 (ArB) = Ar B, I6 A A, I7 O = O , I8 A = A, I9 Q(A) = Q(A). Theorem 4. hP(E ),I ,\ ,i is a in-IFMTS. 33 33 Proof. Let the IFSs A, B 2 P(E ) be given. Then, we check sequentially the validity of the nine conditions I1–I9. Axioms 2023, 12, 408 10 of 13 I1. I (A\ B) = I (fhx, min(m (x), m (x)), 1 min(m (x), m (x)))ijx 2 Eg) 33 33 33 A B A B = fhx, inf(min(m (x), m (x)), 1 inf(min(m (x), m (x)))ijx 2 Eg B B A A y2E y2E = fhx, min( inf m (x), inf m (x)), 1 min( inf m (x), inf m (x))ijx 2 Eg A B A B y2E y2E y2E y2E = fhx, inf m (x), 1 inf m (x)ijx 2 Eg A A y2E y2E \ fhx, inf m (x), 1 inf m (x)ijx 2 Eg 33 B B y2E y2E = I (A)\ I (B); 33 33 33 I2. I (A) = fhx, inf m (x), 1 inf m (x)ijx 2 Eg 33 A A y2E y2E = fhx, inf m (x), sup(1 m (x))ijx 2 Eg A A y2E y2E fhx, inf m (x), sup n (x)ijx 2 Eg A A y2E y2E fhx, m (x), n (x)ijx 2 Eg A A = A; I3. I (E ) = I (fhx, 1, 0ijx 2 Eg) 33 33 = fhx, inf 1, 1 inf 1ijx 2 Eg y2E y2E = fhx, 1, 0ijx 2 Eg = E ; I4. Having in mind that sup n (y) is a constant, we obtain that y2E I (I (A)) = I (fhx, inf m (x), 1 inf m (x)ijx 2 Eg) 33 33 33 A A y2E y2E = fhx, inf inf m (x), 1 inf inf m (x)ijx 2 Eg A A z2E y2E z2E y2E = fhx, inf m (x), 1 inf m (x)ijx 2 Eg A A y2E y2E = I (A); I5. (A\ B) = fhx, min(m (x), m (x)), 1 min(m (x), m (x))ijx 2 Eg 33 A B A B = fhx, min(m (x), m (x)), 1 min(m (x), m (x))ijx 2 Eg A B A B = fhx, m (x), 1 m (x)ijx 2 Eg\ fhx, m (x), 1 m (x)ijx 2 Eg 33 B B A A = A\ B; I6. A = fhx, m (x), 1 m (x)ijx 2 Eg A A fhx, m (x), n (x)ijx 2 Eg A A = A; Axioms 2023, 12, 408 11 of 13 I7. O = fhx, 0, 1ijx 2 Eg = fhx, 0, 1 0ijx 2 Eg = (fhx, 0, 1ijx 2 Eg) = O ; I8. A = fhx, m (x), 1 m (x)ijx 2 Eg A A = fhx, m (x), 1 m (x)ijx 2 Eg A A = A; I9. I (A) = fhx, inf m (x), 1 inf m (x)ijx 2 Eg 33 A A y2E y2E = fhx, inf m (x), 1 inf m (x)ijx 2 Eg A A y2E y2E = I (fhx, m (x), n (x)ijx 2 Eg) 33 A A = I (A). This completes the proof of Theorem 4. 5. Conclusions or Ideas for the Future In the present paper, two new IFMTSs are developed, which are based on the new Intuitionistic Fuzzy Topological Operators from closure and interior types and are different from the standard ones from [1] and recognized so far as the only ones able to satisfy the conditions of IFTMS constrictions. The basic properties of the new IFMTSs have been presented. The ideas described in [1] and in the present investigation open some directions for the future development of the forms of the Intuitionistic Fuzzy Topological Structures (IFTSs). They can have not only the IFMTSs forms discussed in [1] and here but, e.g., those of Intuitionistic Fuzzy Temporal Topological Structures, introduced in [42]. An open problem is to search for new intuitionistic fuzzy conjunctions and disjunc- tions to generate new IFMTSs. On the other hand, in some papers, e.g., [43], some of the conditions of the IFMTSs were changed with weaker ones (e.g., such that in which the relation “=” was changed with the relation “” or “”), or in which some of the conditions were omitted. These IFMTSs were called “feeble” IFMTSs. We use the word “feeble” to eliminate the conﬂict with the well-known concept of a weak topology. At the moment, six different types of such structures are introduced. So, an Open Problem is to search for other feeble IFMTSs. Another open problem is what kinds of IFMTSs can be constructed on the basis of the extended modal operators (see [10]). In addition, in the IFSs theory, there are two other types of modal operators. Thus, another open problem is whether some of these modal operators can generate new IFMTSs. Another direction of our research, mentioned in [1], is related to the extension of the forms of the topological operators discussed above. If, e.g., A(E) P(E ), then we can construct the topological operators: C(A(E)) = fhx, sup m (x), inf n (x)ijx 2 Eg, A A A2A(E) A2A(E) I(A(E)) = fhx, inf m (x), sup n (x)ijx 2 Eg, A A A2A(E) A2A(E) C (A(E)) = fhx, 1 inf n (y), inf n (y)ijA 2 A(E)g, A A A2A(E) A2A(E) Axioms 2023, 12, 408 12 of 13 I (A(E)) = fhx, inf m (y), 1 inf m (y)ijA 2 A(E)g. 33 A A A2A(E) A2A(E) We can study the properties of these four new topological operators. The next possible direction is related to the use of the extended topological operators, discussed in [10]. In practice, all of the above research can be re-written (in a more detailed form) for these extended topological operators, and we hope that new results, speciﬁc for them, will arise. Finally, we must mention that in the IFSs theory, there are also level operators, which can be used as well for the construction of new IFTSs. In the near future, the possibility to apply the above mentioned IFMTSs in different areas of data mining, intercriteria analysis and others will be investigated. It is very worth noting that the so deﬁned conditions of IFMTS are conditions for each MTS, when the basic set is an arbitrary one, but not speciﬁcally an IFS. Author Contributions: Conceptualization, K.A. and N.A.; methodology, K.A. and N.A.; software, N.A.; validation, K.A., N.A. and T.P.; formal analysis, K.A., N.A. and T.P.; investigation, K.A., N.A. and T.P.; writing—original draft preparation, K.A., N.A. and T.P.; writing—review and editing, K.A., N.A. and T.P.; visualization, K.A. and N.A.; supervision, K.A.; project administration, K.A.; funding acquisition, K.A. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Bulgarian National Science Fund grant number KP-06- N22-1/2018 “Theoretical Research and Applications of InterCriteria Analysis”. Data Availability Statement: Not applicable. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Atanassov, K. Intuitionistic Fuzzy Modal Topological Structure. Mathematics 2022, 10, 3313. [CrossRef] 2. Bourbaki, N. Éléments De Mathématique, Livre III: Topologie Générale, Chapitre 1: Structures Topologiques, Chapitre 2: Structures Uniformes; Herman: Paris, France, 1960. (In French) 3. Kuratowski, K. Topology; Academic Press: New York, NY, USA, 1966; Volume 1. 4. Munkres, J. Topology; Prentice Hall Inc.: New Jersey, NJ, USA, 2000. 5. Blackburn, P.; van Benthem, J.; Wolter, F. Handbook of Modal Logic; Elsevier: New York, NY, USA, 2006. 6. Feys, R. Modal Logics; Gauthier: Paris, France, 1965. 7. Fitting, M.; Mendelsohn, R. First Order Modal Logic; Kluwer: Dordrecht, The Netherlands, 1998. 8. 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Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 22, 2023

**Keywords: **Intuitionistic Fuzzy Modal Topology; Intuitionistic Fuzzy Operator; Intuitionistic Fuzzy Set

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