Article Search
닫기

Original Article

Split Viewer

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 100-105

Published online March 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.1.100

© The Korean Institute of Intelligent Systems

Double Fuzzy -Irresolute Multifunctions

Veeraiyan Suganya1, Pitchayan Gomathi Sundari2, and Neelamegarajan Rajesh2

1Department of Mathematics, Vivekanandha College of Arts and Science for Women, Tamilnadu, India
2Department of Mathematics, Rajah Serfoji Government College (affliated to Bharathdasan University), Tamilnadu, India

Correspondence to :
Neelamegarajan Rajesh (nrajesh_topology@yahoo.co.in)

Received: May 27, 2021; Revised: May 27, 2021; Accepted: July 15, 2021

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

In this study, we introduce and investigate the concepts of double fuzzy upper and fuzzy lower α-irresolute multifunctions. Several characterizations and properties of these multifunctions are established in double fuzzy topological spaces.

Keywords: Double fuzzy topological space, (r, s)-fuzzy α-open set, Double fuzzy α-irresolute multifunctions

The fuzzy concept has penetrated nearly all branches of mathematics since the concept has been defined by Zadeh [1]. Fuzzy sets have applications in several fields, such as information [2] and control [3]. The theory of fuzzy topological spaces was first defined and developed by Chang [2], and since then various notions of general topology have been generalized to the fuzzy topological spaces defined by Chang. Šostak [4] and Kubiak [5] developed fuzzy topology as an extension of the fuzzy topology introduced by Chang [2]. It has since been developed in several ways. Šostak [6] published a survey article on the areas of development of fuzzy topological spaces. Topologists often refer to the concept of fuzzy topology developed by Chang as “”and the concept of fuzzy topology proposed by Kubiak-Šostak as “”, where L is an appropriate lattice. In a previous study [3], The author introduced the idea of intuitionistic fuzzy sets; subsequently, Coker [7,8] introduced the concept of intuitionistic fuzzy topological spaces. Furthermore, as a generalization of fuzzy topological spaces, Montal and Samanta [9] introduced the concept of the intuitionistic gradation of openness. In 2005, Garcia and Rodabaugh [10] proposed the termination of the term intuitionistic. They proved that the term intuitionistic is unsuitable in mathematics and related applications and replaced it with double. Several topologists have studied various concepts in a double fuzzy topological space [1122]. In this study, we introduce and examine the concepts of double fuzzy upper and fuzzy lower α-irresolute multifunctions. The characterizations and properties of these multifunctions are established in double fuzzy topological spaces.

Throughout this paper, let X be a non-empty set and I be the unit interval [0, 1]; I0 = (0, 1] and I1 = [0, 1). The family of all fuzzy sets in X is denoted by IX. We denote the smallest and greatest fuzzy sets in X by 0̄ and 1̄, respectively. For a fuzzy set λIX, 1̄ – λ denotes the complement. Considering a function f : IX –→ IY and its inverse f−1 : IY –→ IX, they can be defined by f(λ)(y) = ∨f(x)=yλ(x) and f−1(μ)(x) = μ(f(x)) for each λIX, μIY and xX. All other notations are standard notations of the fuzzy set theory.

Definition 1 [23]

Let F : (X, τ) → (Y, σ), then F is called a fuzzy multifunction if and only if F(x) ∈ IY for each xX. The degree of membership ofy in F(x) is denoted by F(x)(y) = GF (x, y) for any (x, y) ∈ X × Y. The domain of F, denoted by dom(F), and the range of F, denoted by rng(F), for any xX and yY are defined by dom(F)(x)=yYGF(x,y) and rng(F)(x)=xXGF(x,y), respectively.

Definition 2 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then F is termed:

  • (1) normalized if and only if for each xX, there exists a y0Y such that GF (x, y0) = 1̄.

  • (2) crisp if and only if GF (x, y0) = 1̄ for each xX and yY.

Definition 3 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then:

  • (1) the lower inverse of μIY is a fuzzy set Fl(μ) ∈ IX defined by Fl(μ)(x)=yY(GF(x,y)μ(y)).

  • (2) the upper inverse of μIY is a fuzzy set Fu(μ) ∈ IX defined by Fu(μ)(x)=yY(GFc(x,y)μ(y)).

Definition 4 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then

  • (1) F(λ1) ≤ F(λ2) if λ1λ2.

  • (2) Fl(μ1) ≤ Fl(μ2) and Fu(μ1) ≤ Fu(μ2) if μ1μ2.

  • (3) Fl(μc) ≤ (Fu(μ))c.

  • (4) Fu(μc) ≤ (Fl(μ))c.

  • (5) F(Fu(μ)) ≤ μ if F is a crisp.

  • (6) Fu(F(λ)) ≥ λ if F is a crisp.

Definition 5 [23]

Let F : (X, τ) → (Y, σ) and H : (Y, σ) → (Z, η) be two fuzzy multifunctions. Then, the composition HF is defined as ((HF)(x))(z)=yY(GF(x,y)GH(y,z)).

Theorem 1 [23]

Let F : (X, τ) → (Y, σ) and H : (Y, σ) → (Z, η) be two fuzzy multifunctions. Thus, the following can be obtained:

  • (1) HF = F(H).

  • (2) (HF)u = Fu(Hu).

  • (3) (HF)l = Fl(Hl).

Theorem 2 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then, the following can be obtained:

  • (1) (iΓFi)(λ)=iΓFi(λ).

  • (2) (iΓFi)l(μ)=iΓFil(μ).

  • (3) (iΓFi)u(μ)=iΓFiu(μ).

Definition 6

A fuzzy point xt in X is a fuzzy set that takes the value tI0 at x and zero elsewhere such that xtλ if and only if tλ(x).

Definition 7 [8,9]

A double fuzzy topology on X is a pair of maps τ, τ : IXI, which satisfies the following properties:

  • (1) τ (λ) ≤ 1̄ – τ (λ) for each λIX.

  • (2) τ (λ1λ2) ≥ τ (λ1) ∧ τ (λ2) and τ (λ1λ2) ≤ τ (λ1) ∨ τ (λ2) for each λ1, λ2IX.

  • (3) τ(iΓλi)iΓτ(λi) and τ(iΓλi)iΓτ(λi) for each λiIXandi ∈ Γ.

The triplet (X, τ, τ) is called a double fuzzy topological space.

Definition 8 [8,9]

A fuzzy set λ is termed (r, s)-fuzzy open if τ (λ) ≥ r and τ (λ) ≤ s; λ is termed (r, s)-fuzzy closed if and only if 1̄ – λ is an (r, s)-fuzzy open set.

Theorem 3 [24,25]

Let (X, τ, τ) be a double fuzzy topological space. Then, the double fuzzy closure and double fuzzy interior operators of λIX are defined by: Cτ,τ (λ, r, s) = ∧{μIX | λμ, τ (1̄ – μ) ≥ randτ(1̄ – μ) ≤ s}, Iτ,τ (λ, r, s) = ∨{μIX | μλ, τ (μ) ≥ r, τ(μ) ≤ s},, respectively, where rI0 and sI1 such that r + s ≤ 1.

Definition 9

Let (X, τ, τ) be a double fuzzy topological space. For each λIX, rI0, and sI1,

  • (1) λ is called (r, s)-fuzzy α-open if λIτ,τ (Cτ,τ (Iτ,τ (λ, r, s), r, s), r, s).

  • (2) λ is called (r, s)-fuzzy α-closed if 1̄ – λ is (r, s)-fuzzy α-open.

  • (3) αCτ,τ (λ, r, s) = ∧{μIX : μλ and μ is (r, s)-fuzzy α-closed} is called (r, s)-fuzzy α-closure of λ.

  • (4) αIτ,τ (λ, r, s) = ∨{μIX : μλ and μ is (r, s)-fuzzy α-open} is called (r, s)-fuzzy α-interior of λ.

Definition 10 [?]

Let F : (X, τ, τ) → (Y, σ, σ) be a fuzzy multifunction. Then, F is called:

  • (1) double fuzzy upper α-continuous at a fuzzy point xtdom(F) if xtFu(μ) for each μIY, σ(μ) ≥ r, and σ(μ) ≤ s, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λdom(F) ≤ Fu(μ).

  • (2) double fuzzy lower α-continuous at a fuzzy point xtdom(F) if xtFu(μ) for each μIY, σ(μ) ≥ r and σ(μ) ≤ s, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λFl(μ).

  • (3) double fuzzy upper (lower) α-continuous if it is fuzzy upper (lower) α-continuous at every xtdom(F).

Definition 11

Let F : (X, τ, τ) → (Y, σ, σ) be a fuzzy multifunction. Then, F is called:

  • (1) double fuzzy upper α-irresolute at a fuzzy point xtdom(F) if xtFu(μ) for each (r, s)-fuzzy α-open set μIY, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λdom(F) ≤ Fu(μ).

  • (2) double fuzzy lower α-irresolute at a fuzzy point xtdom(F) if xtFu(μ) for each (r, s)-fuzzy α-open set μIY, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λFl(μ).

  • (3) double fuzzy upper (lower) α-irresolute if it is fuzzy upper (lower) α-irresolute at every xtdom(F).

Proposition 1

If F is normalized, then F is a double fuzzy upper α-irresolute at a fuzzy point xtdom(F) if and only if xtFu(μ) for each (r, s)-fuzzy α-open set μIY, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λFu(μ).

Theorem 4

For a fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ), the following statements are equivalent:

  • (1) F is double fuzzy lower α-irresolute,

  • (2) Fl(μ) is (r, s)-fuzzy α-open for any (r, s)-fuzzy α-open set μIY,

  • (3) Fu(μ) is (r, s)-fuzzy α-closed for any (r, s)-fuzzy α-closed set μIY,

  • (4) Cτ,τ (Iτ,τ (Cτ,τ (Fu(μ), r, s), r, s), r, s) ≤ Fu(αCσ,σ (μ, r, s)) for any μIY,

  • (5) αCτ,τ (Fu(μ), r, s) ≤ Fu(αCσ,σ (μ, r, s)) for any μIY.

Proof

(1) ⇒ (2): Let xtdom(F), (r, s)-fuzzy α-open set μIY, and xtFl(μ). Then, there exists an (r, s)-fuzzy α-open set λIX and xtλ with λFl(μ), and xtIτ,τ (Cτ,τ (Iτ,τ (Fl(μ), r, s), r, s), r, s). Therefore, Fl(μ) ≤ Iτ,τ (Cτ,τ (Iτ,τ (Fl(μ), r, s), r, s), r, s). Thus, Fl(μ) is (r, s)-fuzzy α-open.

(2) ⇒ (3): Let μIY be an (r, s)-fuzzy α-closed set; hence, from (2), Fl(1̄ – μ) = 1̄ – Fu(μ) is (r, s)-fuzzy α-open. Accordingly, Fu(μ) is (r, s)-fuzzy α-closed.

(3) ⇒ (4): Let μIY ; hence, from (3), Fu(αCσ,σ (μ, r, s)) is (r, s)-fuzzy α-closed. Thus, we obtain:

Cτ,τ(Iτ,τ(Cτ,τ(Fu(μ),r,s),r,s),r,s)Cτ,τ(Iτ,τ(Cτ,τ(Fu(αCσ,σ(μ,r,s)),r,s),r,s),r,s)Fu(αCσ,σ(μ,r,s)).

(4) ⇒ (5): Let μIY. Hence, we have

αCτ,τ(Fu(μ),r,s)=Fu(μ)Cτ,τ(Iτ,τ(Cτ,τ(Fu(Cσ,σ(μ,r,s)),r,s),r,s),r,s)Fu(Cσ,σ(μ,r,s)).

(5) ⇒ (3): Let μIY be an (r, s)-fuzzy α-closed set; hence, from (5), αCτ,τ (Fu(μ), r, s) ≤ Fu(αCσ,σ (μ, r, s)) = Fu(μ). This demonstrates that Fu(μ) is (r, s)-fuzzy α-closed.

(3) ⇒ (2): Let μIY be an (r, s)-fuzzy α-open set; hence, from (3), Fu(1̄ – μ) = 1̄ – Fl(μ) is (r, s)-fuzzy α-closed. Thus, Fl(μ) is (r, s)-fuzzy α-open.

(2) ⇒ (1): Let xtdom(F) and μIY be an(r, s)-fuzzy α-open set with xtFl(μ); from (2), it can be inferred that Fl(μ) is (r, s)-fuzzy α-open. Thus, F is double fuzzy lower α-irresolute.

We state the following result without proof in view of the above theorem.

Theorem 5

For a fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ), the following statements are equivalent:

  • (1) F is double fuzzy upper α-irresolute.

  • (2) Fu(μ) is r-fuzzy α-open for any (r, s)-fuzzy α-open set μIY.

  • (3) Fl(μ) is r-fuzzy α-closed for any (r, s)-fuzzy α-closed set μIY.

  • (4) Cτ,τ (Iτ,τ (Cτ,τ (Fl(μ), r, s), r, s), r, s) ≤ Fl(αCσ,σ (μ, r, s)) for any μIY.

  • (5) αCτ,τ (Fl(μ), r, s) ≤ Fl(αCσ,σ (μ, r, s)) for any μIY.

Corollary 1

Let F : (X, τ, τ) → (Y, σ, σ) be a fuzzy multifunction. Then, the following can be postulated:

  • (1) F as normalized implies that it is double fuzzy upper α-irresolute at a fuzzy point xt if and only if xtIτ,τ (Cτ,τ (Iτ,τ (Fu(μ), r, s), r, s), r, s) for (r, s)-fuzzy α-open sets μIY and xtFu(μ).

  • (2) F is double fuzzy lower α-irresolute at a fuzzy point xt if and only if xtIτ,τ (Cτ,τ (Iτ,τ (Fl(μ), r, s), r, s), r, s) for (r, s)-fuzzy α-open sets μIY and xtFl(μ).

Proposition 2

For a fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ), we obtain the following:

  • (1) Every double fuzzy upper α-continuous multifunction is double fuzzy upper α-irresolute.

  • (2) Every double fuzzy lower α-continuous multifunction is double fuzzy lower α-irresolute.

The following example demonstrates that the converses of the above propositions are not true.

Example 1

Let X = {x1, x2}, Y = {y1, y2, y3}, and F : (X, τ, τ) → (Y, σ, σ) be fuzzy multifunctions defined by GF (x1, y1) = 0.2, GF (x1, y2) = 1̄, GF (x1, y3) = 0.3, GF (x2, y1) = 0.5, GF (x2, y2) = 0.3, and GF (x2, y3) = 1̄. We define fuzzy topologies τ, τ : IXI, and σ, σ : IYI as follows:

τ(λ)={1,if λ=0¯or 1¯,12if λ0.5¯,0,otherwise,         τ(λ)={0,if λ=0¯or 1¯,12if λ0.5¯,1,otherwise,σ(λ)={1,if λ=0¯or 1¯12,if λ=0.5¯13,if λ=0.4¯0,otherwise,         σ(λ)={0,if λ=0¯or 1¯,12,if λ=0.5¯,23,if λ=0.4¯,   1,otherwise.

Then, F is (13,23)-fuzzy upper (lower) α-irresolute, but not (13,23)-fuzzy upper (lower) α-continuous.

Theorem 6

Let F: (X, τ, τ)→(Y, σ, σ) and H: (Y, σ, σ) → (Z, η, η) be fuzzy multifunctions. Thus, the following can be concluded:

  • (1) If F and H are double fuzzy lower α-irresolute, then, HF is also double fuzzy lower α-irresolute.

  • (2) If F is double fuzzy lower α-irresolute and H is double fuzzy lower α-continuous, then, HF is also double fuzzy lower α-irresolute.

Proof

Follows from the respective definitions.

Theorem 7

Let {Fi : i ∈ Γ} be a family of double fuzzy lower α-irresolute functions. Then, (iΓFi)l(μ)=iΓFil(μ) is double fuzzy lower α-irresolute.

Proof

Let μIY ; then, (iΓFi)l(μ)=iΓFil(μ). Because {Fi : i ∈ Γ} is a family of double fuzzy lower α-irresolute multifunctions, Fil(μ) is (r, s)-fuzzy α-open for any (r, s)-fuzzy α-open set μIY. Furthermore, we can conclude that (iΓFi)l(μ)=iΓFil(μ) is (r, s)-fuzzy α-open for any (r, s)-fuzzy α-open set μ. Hence, iΓFi is double fuzzy lower α-irresolute.

Definition 12

A fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ) is called (r, s)-fuzzy α-compact-valued if and only if F(xt) is (r, s)-fuzzy α-compact for each xt ∈ dom (F).

Theorem 8

Let F : (X, τ, τ) → (Y, σ, σ) be a crisp double fuzzy upper α-irresolute and α-compact-valued multifunction. If λ is (r, s)-fuzzy α-compact, then, F(λ) is (r, s)-fuzzy α-compact.

Proof

Let λIX be an (r, s)-fuzzy α-compact set and {γi : γi is an (r, s)-fuzzy α-open set, i ∈ Γ} be a family covering F(λ), that is, F(λ)iΓγi. As λ=xtλxt, we have F(λ)=F(xtλxt)=xtλF(xt)iΓγi. It follows that for each xtλ,F(xt)iΓγi. Because F is (r, s)-α-compact-valued, there exists a finite subset Γxt of Γ such that F(xt)nΓxtγn=γxt. Subsequently, we have: xtFu(F(xt)) ≤ Fu(γxt) and λ=xtλxtxtλFu(F(xt))Fu(γxt). Because γxt is (r, s)-fuzzy α-open, we can conclude that Fu(γxt) is (r, s)-fuzzy α-open. Hence, {Fu(γxt) : Fu(γxt) is (r, s)-fuzzy α-open, xtλ} is a family covering the set λ. Because λ is (r, s)-fuzzy α-compact, there exists a finite index set N such that λnNFu(γxtn). Subsequently, F(λ)F(nNFu(γxtn))=nNF(Fu(γxtn)nNγxtn. Thus, F(λ) is (r, s)-fuzzy α-compact.

There is no potential conflict of interest relevant to this study.

  1. Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
    CrossRef
  2. Chang, CL (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 24, 182-190.
    CrossRef
  3. Atanassov, KT (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems. 61, 137-142. https://doi.org/10.1016/0165-0114(94)90229-1
    CrossRef
  4. AP Sostak (1985). On a fuzzy topological structure. Proceedings of the 13th Winter School on Abstract Analysis. Palermo, Italy: Circolo Matematico di Palermo, pp. 89-103
  5. Kubiak, T 1985. On fuzzy topologies. PhD dissertation. Adam Mickiewicz University. Poznan, Poland.
  6. Sostak, AP (1996). Basic structures of fuzzy topology. Journal of Mathematical Sciences. 78, 662-701.
    CrossRef
  7. Coker, D (1996). An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces. Journal of Fuzzy Mathematics. 4, 749-764.
  8. Coker, D (1997). An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems. 88, 81-89. https://doi.org/10.1016/S0165-0114(96)00076-0
    CrossRef
  9. Mondal, TK, and Samanta, SK (2002). On intuitionistic gradation of openness. Fuzzy Sets and Systems. 131, 323-336. https://doi.org/10.1016/S0165-0114(01)00235-4
    CrossRef
  10. Garcia, JG, and Rodabaugh, SE (2005). Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies. Fuzzy Sets and Systems. 156, 445-484. https://doi.org/10.1016/j.fss.2005.05.023
    CrossRef
  11. Abd-Allah, MA, El-Saady, K, and Ghareeb, A (2009). (r, s)-Fuzzy F-open sets and (r, s)-fuzzy F-closed spaces. Chaos, Solitons & Fractals. 42, 649-656. https://doi.org/10.1016/j.chaos.2009.01.031
    CrossRef
  12. Mohammed, FM, Noorani, MSM, and Ghareeb, A (2016). Generalized fuzzy b-closed and generalized ★-fuzzy b-closed sets in double fuzzy topological spaces. Egyptian Journal of Basic and Applied Sciences. 3, 61-67.
    CrossRef
  13. Mohammed, FM, Noorani, MSM, and Ghareeb, A (2014). Somewhat slightly generalized double fuzzy semicontinuous functions. International Journal of Mathematics and Mathematical Sciences. 2014. article no. 756376
    CrossRef
  14. Mohammed, FM, Noorani, MSM, and Ghareeb, A (2017). New notions from (alpha,beta)-generalised fuzzy preopen sets. Gazi University Journal of Science. 30, 311-331.
  15. Ghareeb, A (2011). Normality of double fuzzy topological spaces. Applied Mathematics Letters. 24, 533-540. https://doi.org/10.1016/j.aml.2010.11.008
    CrossRef
  16. Ghareeb, A (2012). Weak forms of continuity in I-double gradation fuzzy topological spaces. SpringerPlus. 1. article no. 19
    CrossRef
  17. El-Saady, K, and Ghareeb, A (2012). Several types of (r, s)-fuzzy compactness defined by an (r, s)-fuzzy regular semiopen sets. Annals of fuzzy Mathematics and Informatics. 3, 159-169.
  18. Lee, EP (2004). Semiopen sets on intuitionistic fuzzy topological spaces in Sostak’s sense. Journal of the Korean Institute of Intelligent Systems. 14, 234-238. https://doi.org/10.5391/JKIIS.2004.14.2.234
    CrossRef
  19. Lee, SO, and Lee, EP (2005). Fuzzy (r, s)-preopen sets. International Journal of Fuzzy Logic and Intelligent Systems. 5, 136-139. https://doi.org/10.5391/IJFIS.2005.5.2.136
    CrossRef
  20. Lee, SO, and Lee, EP (2006). Fuzzy strongly (r, s)-semiopen sets. International Journal of Fuzzy Logic and Intelligent Systems. 6, 299-303. https://doi.org/10.5391/IJFIS.2006.6.4.299
    CrossRef
  21. Lee, SJ, and Kim, JT (2007). Fuzzy (r, s)-irresolute maps. International Journal of Fuzzy Logic and Intelligent Systems. 7, 49-57. https://doi.org/10.5391/IJFIS.2007.7.1.049
    CrossRef
  22. Zahran, AM, Abd-Allah, MA, and Ghareeb, A (2010). Several types of double fuzzy irresolute functions. International Journal of Computational Cognition. 8, 19-23.
  23. Abbas, SE, and Taha, IM (2020). On upper and lower contra-continuous fuzzy multifunctions. Punjab University Journal of Mathematics. 47, 104-117.
  24. Coker, D, and Demirci, M (1996). An introduction to intuitionistic fuzzy topological spaces in Sostak’s sense. Busefal. 67, 67-76.
  25. Lee, EP, and Im, YB (2001). Mated fuzzy topological spaces. Journal of the Korean Institute of Intelligent Systems. 11, 161-165.
  26. Suganya, V, Gomathisundari, P, and Rajesh, N (2022). Double fuzzy α-continuous multifunctions. International Journal of Fuzzy Logic and Intelligent Systems. Manuscript submitted

Veeraiyan Suganya received her M.Phil. in Mathematics from Periyar University, Salem, Tamil Nadu, India. She is currently an Assistant Professor at the Department of Mathematics, Vivekanandha College of Arts and Science for Women, Tiruchengode, Tamil Nadu, India. Her research interests include fuzzy topology and soft set theory.

E-mail: nrajesh topology@yahoo.co.in

Pitchayan Gomathi Sundari received her Ph.D. in Mathematics from Bharathidasan University, Trichy, Tamil Nadu, India, in 2014. She is currently an Assistant Professor at the Department of Mathematics, Rajah Serfoji Government College, Thanjavur, Tamil Nadu, India. Her research interests include general topology, fuzzy topology, and soft set theory.

E-mail: suganjo115sai@gmail.com

Neelamegarajan Rajesh received his Ph.D. degree in Mathematics from Madurai Kamaraj University, Tamil Nadu, India, in 2007. He is currently an Assistant Professor at the Department of Mathematics, Rajah Serfoji Government College, Thanjavur, Tamil Nadu, India. His research interests include general topology, fuzzy topology, and soft set theory.

E-mail: gomathi.rsg@gmail.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(1): 100-105

Published online March 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.1.100

Copyright © The Korean Institute of Intelligent Systems.

Double Fuzzy -Irresolute Multifunctions

Veeraiyan Suganya1, Pitchayan Gomathi Sundari2, and Neelamegarajan Rajesh2

1Department of Mathematics, Vivekanandha College of Arts and Science for Women, Tamilnadu, India
2Department of Mathematics, Rajah Serfoji Government College (affliated to Bharathdasan University), Tamilnadu, India

Correspondence to:Neelamegarajan Rajesh (nrajesh_topology@yahoo.co.in)

Received: May 27, 2021; Revised: May 27, 2021; Accepted: July 15, 2021

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

Abstract

In this study, we introduce and investigate the concepts of double fuzzy upper and fuzzy lower α-irresolute multifunctions. Several characterizations and properties of these multifunctions are established in double fuzzy topological spaces.

Keywords: Double fuzzy topological space, (r, s)-fuzzy &alpha,-open set, Double fuzzy &alpha,-irresolute multifunctions

1. Introduction

The fuzzy concept has penetrated nearly all branches of mathematics since the concept has been defined by Zadeh [1]. Fuzzy sets have applications in several fields, such as information [2] and control [3]. The theory of fuzzy topological spaces was first defined and developed by Chang [2], and since then various notions of general topology have been generalized to the fuzzy topological spaces defined by Chang. Šostak [4] and Kubiak [5] developed fuzzy topology as an extension of the fuzzy topology introduced by Chang [2]. It has since been developed in several ways. Šostak [6] published a survey article on the areas of development of fuzzy topological spaces. Topologists often refer to the concept of fuzzy topology developed by Chang as “”and the concept of fuzzy topology proposed by Kubiak-Šostak as “”, where L is an appropriate lattice. In a previous study [3], The author introduced the idea of intuitionistic fuzzy sets; subsequently, Coker [7,8] introduced the concept of intuitionistic fuzzy topological spaces. Furthermore, as a generalization of fuzzy topological spaces, Montal and Samanta [9] introduced the concept of the intuitionistic gradation of openness. In 2005, Garcia and Rodabaugh [10] proposed the termination of the term intuitionistic. They proved that the term intuitionistic is unsuitable in mathematics and related applications and replaced it with double. Several topologists have studied various concepts in a double fuzzy topological space [1122]. In this study, we introduce and examine the concepts of double fuzzy upper and fuzzy lower α-irresolute multifunctions. The characterizations and properties of these multifunctions are established in double fuzzy topological spaces.

2. Preliminaries

Throughout this paper, let X be a non-empty set and I be the unit interval [0, 1]; I0 = (0, 1] and I1 = [0, 1). The family of all fuzzy sets in X is denoted by IX. We denote the smallest and greatest fuzzy sets in X by 0̄ and 1̄, respectively. For a fuzzy set λIX, 1̄ – λ denotes the complement. Considering a function f : IX –→ IY and its inverse f−1 : IY –→ IX, they can be defined by f(λ)(y) = ∨f(x)=yλ(x) and f−1(μ)(x) = μ(f(x)) for each λIX, μIY and xX. All other notations are standard notations of the fuzzy set theory.

Definition 1 [23]

Let F : (X, τ) → (Y, σ), then F is called a fuzzy multifunction if and only if F(x) ∈ IY for each xX. The degree of membership ofy in F(x) is denoted by F(x)(y) = GF (x, y) for any (x, y) ∈ X × Y. The domain of F, denoted by dom(F), and the range of F, denoted by rng(F), for any xX and yY are defined by dom(F)(x)=yYGF(x,y) and rng(F)(x)=xXGF(x,y), respectively.

Definition 2 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then F is termed:

  • (1) normalized if and only if for each xX, there exists a y0Y such that GF (x, y0) = 1̄.

  • (2) crisp if and only if GF (x, y0) = 1̄ for each xX and yY.

Definition 3 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then:

  • (1) the lower inverse of μIY is a fuzzy set Fl(μ) ∈ IX defined by Fl(μ)(x)=yY(GF(x,y)μ(y)).

  • (2) the upper inverse of μIY is a fuzzy set Fu(μ) ∈ IX defined by Fu(μ)(x)=yY(GFc(x,y)μ(y)).

Definition 4 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then

  • (1) F(λ1) ≤ F(λ2) if λ1λ2.

  • (2) Fl(μ1) ≤ Fl(μ2) and Fu(μ1) ≤ Fu(μ2) if μ1μ2.

  • (3) Fl(μc) ≤ (Fu(μ))c.

  • (4) Fu(μc) ≤ (Fl(μ))c.

  • (5) F(Fu(μ)) ≤ μ if F is a crisp.

  • (6) Fu(F(λ)) ≥ λ if F is a crisp.

Definition 5 [23]

Let F : (X, τ) → (Y, σ) and H : (Y, σ) → (Z, η) be two fuzzy multifunctions. Then, the composition HF is defined as ((HF)(x))(z)=yY(GF(x,y)GH(y,z)).

Theorem 1 [23]

Let F : (X, τ) → (Y, σ) and H : (Y, σ) → (Z, η) be two fuzzy multifunctions. Thus, the following can be obtained:

  • (1) HF = F(H).

  • (2) (HF)u = Fu(Hu).

  • (3) (HF)l = Fl(Hl).

Theorem 2 [23]

Let F : (X, τ) → (Y, σ) be a fuzzy multifunction. Then, the following can be obtained:

  • (1) (iΓFi)(λ)=iΓFi(λ).

  • (2) (iΓFi)l(μ)=iΓFil(μ).

  • (3) (iΓFi)u(μ)=iΓFiu(μ).

Definition 6

A fuzzy point xt in X is a fuzzy set that takes the value tI0 at x and zero elsewhere such that xtλ if and only if tλ(x).

Definition 7 [8,9]

A double fuzzy topology on X is a pair of maps τ, τ : IXI, which satisfies the following properties:

  • (1) τ (λ) ≤ 1̄ – τ (λ) for each λIX.

  • (2) τ (λ1λ2) ≥ τ (λ1) ∧ τ (λ2) and τ (λ1λ2) ≤ τ (λ1) ∨ τ (λ2) for each λ1, λ2IX.

  • (3) τ(iΓλi)iΓτ(λi) and τ(iΓλi)iΓτ(λi) for each λiIXandi ∈ Γ.

The triplet (X, τ, τ) is called a double fuzzy topological space.

Definition 8 [8,9]

A fuzzy set λ is termed (r, s)-fuzzy open if τ (λ) ≥ r and τ (λ) ≤ s; λ is termed (r, s)-fuzzy closed if and only if 1̄ – λ is an (r, s)-fuzzy open set.

Theorem 3 [24,25]

Let (X, τ, τ) be a double fuzzy topological space. Then, the double fuzzy closure and double fuzzy interior operators of λIX are defined by: Cτ,τ (λ, r, s) = ∧{μIX | λμ, τ (1̄ – μ) ≥ randτ(1̄ – μ) ≤ s}, Iτ,τ (λ, r, s) = ∨{μIX | μλ, τ (μ) ≥ r, τ(μ) ≤ s},, respectively, where rI0 and sI1 such that r + s ≤ 1.

Definition 9

Let (X, τ, τ) be a double fuzzy topological space. For each λIX, rI0, and sI1,

  • (1) λ is called (r, s)-fuzzy α-open if λIτ,τ (Cτ,τ (Iτ,τ (λ, r, s), r, s), r, s).

  • (2) λ is called (r, s)-fuzzy α-closed if 1̄ – λ is (r, s)-fuzzy α-open.

  • (3) αCτ,τ (λ, r, s) = ∧{μIX : μλ and μ is (r, s)-fuzzy α-closed} is called (r, s)-fuzzy α-closure of λ.

  • (4) αIτ,τ (λ, r, s) = ∨{μIX : μλ and μ is (r, s)-fuzzy α-open} is called (r, s)-fuzzy α-interior of λ.

Definition 10 [?]

Let F : (X, τ, τ) → (Y, σ, σ) be a fuzzy multifunction. Then, F is called:

  • (1) double fuzzy upper α-continuous at a fuzzy point xtdom(F) if xtFu(μ) for each μIY, σ(μ) ≥ r, and σ(μ) ≤ s, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λdom(F) ≤ Fu(μ).

  • (2) double fuzzy lower α-continuous at a fuzzy point xtdom(F) if xtFu(μ) for each μIY, σ(μ) ≥ r and σ(μ) ≤ s, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λFl(μ).

  • (3) double fuzzy upper (lower) α-continuous if it is fuzzy upper (lower) α-continuous at every xtdom(F).

3. Double Fuzzy α-Irresolute Multifunctions

Definition 11

Let F : (X, τ, τ) → (Y, σ, σ) be a fuzzy multifunction. Then, F is called:

  • (1) double fuzzy upper α-irresolute at a fuzzy point xtdom(F) if xtFu(μ) for each (r, s)-fuzzy α-open set μIY, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λdom(F) ≤ Fu(μ).

  • (2) double fuzzy lower α-irresolute at a fuzzy point xtdom(F) if xtFu(μ) for each (r, s)-fuzzy α-open set μIY, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λFl(μ).

  • (3) double fuzzy upper (lower) α-irresolute if it is fuzzy upper (lower) α-irresolute at every xtdom(F).

Proposition 1

If F is normalized, then F is a double fuzzy upper α-irresolute at a fuzzy point xtdom(F) if and only if xtFu(μ) for each (r, s)-fuzzy α-open set μIY, there exists an (r, s)-fuzzy α-open set λIX and xtλ such that λFu(μ).

Theorem 4

For a fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ), the following statements are equivalent:

  • (1) F is double fuzzy lower α-irresolute,

  • (2) Fl(μ) is (r, s)-fuzzy α-open for any (r, s)-fuzzy α-open set μIY,

  • (3) Fu(μ) is (r, s)-fuzzy α-closed for any (r, s)-fuzzy α-closed set μIY,

  • (4) Cτ,τ (Iτ,τ (Cτ,τ (Fu(μ), r, s), r, s), r, s) ≤ Fu(αCσ,σ (μ, r, s)) for any μIY,

  • (5) αCτ,τ (Fu(μ), r, s) ≤ Fu(αCσ,σ (μ, r, s)) for any μIY.

Proof

(1) ⇒ (2): Let xtdom(F), (r, s)-fuzzy α-open set μIY, and xtFl(μ). Then, there exists an (r, s)-fuzzy α-open set λIX and xtλ with λFl(μ), and xtIτ,τ (Cτ,τ (Iτ,τ (Fl(μ), r, s), r, s), r, s). Therefore, Fl(μ) ≤ Iτ,τ (Cτ,τ (Iτ,τ (Fl(μ), r, s), r, s), r, s). Thus, Fl(μ) is (r, s)-fuzzy α-open.

(2) ⇒ (3): Let μIY be an (r, s)-fuzzy α-closed set; hence, from (2), Fl(1̄ – μ) = 1̄ – Fu(μ) is (r, s)-fuzzy α-open. Accordingly, Fu(μ) is (r, s)-fuzzy α-closed.

(3) ⇒ (4): Let μIY ; hence, from (3), Fu(αCσ,σ (μ, r, s)) is (r, s)-fuzzy α-closed. Thus, we obtain:

Cτ,τ(Iτ,τ(Cτ,τ(Fu(μ),r,s),r,s),r,s)Cτ,τ(Iτ,τ(Cτ,τ(Fu(αCσ,σ(μ,r,s)),r,s),r,s),r,s)Fu(αCσ,σ(μ,r,s)).

(4) ⇒ (5): Let μIY. Hence, we have

αCτ,τ(Fu(μ),r,s)=Fu(μ)Cτ,τ(Iτ,τ(Cτ,τ(Fu(Cσ,σ(μ,r,s)),r,s),r,s),r,s)Fu(Cσ,σ(μ,r,s)).

(5) ⇒ (3): Let μIY be an (r, s)-fuzzy α-closed set; hence, from (5), αCτ,τ (Fu(μ), r, s) ≤ Fu(αCσ,σ (μ, r, s)) = Fu(μ). This demonstrates that Fu(μ) is (r, s)-fuzzy α-closed.

(3) ⇒ (2): Let μIY be an (r, s)-fuzzy α-open set; hence, from (3), Fu(1̄ – μ) = 1̄ – Fl(μ) is (r, s)-fuzzy α-closed. Thus, Fl(μ) is (r, s)-fuzzy α-open.

(2) ⇒ (1): Let xtdom(F) and μIY be an(r, s)-fuzzy α-open set with xtFl(μ); from (2), it can be inferred that Fl(μ) is (r, s)-fuzzy α-open. Thus, F is double fuzzy lower α-irresolute.

We state the following result without proof in view of the above theorem.

Theorem 5

For a fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ), the following statements are equivalent:

  • (1) F is double fuzzy upper α-irresolute.

  • (2) Fu(μ) is r-fuzzy α-open for any (r, s)-fuzzy α-open set μIY.

  • (3) Fl(μ) is r-fuzzy α-closed for any (r, s)-fuzzy α-closed set μIY.

  • (4) Cτ,τ (Iτ,τ (Cτ,τ (Fl(μ), r, s), r, s), r, s) ≤ Fl(αCσ,σ (μ, r, s)) for any μIY.

  • (5) αCτ,τ (Fl(μ), r, s) ≤ Fl(αCσ,σ (μ, r, s)) for any μIY.

Corollary 1

Let F : (X, τ, τ) → (Y, σ, σ) be a fuzzy multifunction. Then, the following can be postulated:

  • (1) F as normalized implies that it is double fuzzy upper α-irresolute at a fuzzy point xt if and only if xtIτ,τ (Cτ,τ (Iτ,τ (Fu(μ), r, s), r, s), r, s) for (r, s)-fuzzy α-open sets μIY and xtFu(μ).

  • (2) F is double fuzzy lower α-irresolute at a fuzzy point xt if and only if xtIτ,τ (Cτ,τ (Iτ,τ (Fl(μ), r, s), r, s), r, s) for (r, s)-fuzzy α-open sets μIY and xtFl(μ).

Proposition 2

For a fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ), we obtain the following:

  • (1) Every double fuzzy upper α-continuous multifunction is double fuzzy upper α-irresolute.

  • (2) Every double fuzzy lower α-continuous multifunction is double fuzzy lower α-irresolute.

The following example demonstrates that the converses of the above propositions are not true.

Example 1

Let X = {x1, x2}, Y = {y1, y2, y3}, and F : (X, τ, τ) → (Y, σ, σ) be fuzzy multifunctions defined by GF (x1, y1) = 0.2, GF (x1, y2) = 1̄, GF (x1, y3) = 0.3, GF (x2, y1) = 0.5, GF (x2, y2) = 0.3, and GF (x2, y3) = 1̄. We define fuzzy topologies τ, τ : IXI, and σ, σ : IYI as follows:

τ(λ)={1,if λ=0¯or 1¯,12if λ0.5¯,0,otherwise,         τ(λ)={0,if λ=0¯or 1¯,12if λ0.5¯,1,otherwise,σ(λ)={1,if λ=0¯or 1¯12,if λ=0.5¯13,if λ=0.4¯0,otherwise,         σ(λ)={0,if λ=0¯or 1¯,12,if λ=0.5¯,23,if λ=0.4¯,   1,otherwise.

Then, F is (13,23)-fuzzy upper (lower) α-irresolute, but not (13,23)-fuzzy upper (lower) α-continuous.

Theorem 6

Let F: (X, τ, τ)→(Y, σ, σ) and H: (Y, σ, σ) → (Z, η, η) be fuzzy multifunctions. Thus, the following can be concluded:

  • (1) If F and H are double fuzzy lower α-irresolute, then, HF is also double fuzzy lower α-irresolute.

  • (2) If F is double fuzzy lower α-irresolute and H is double fuzzy lower α-continuous, then, HF is also double fuzzy lower α-irresolute.

Proof

Follows from the respective definitions.

Theorem 7

Let {Fi : i ∈ Γ} be a family of double fuzzy lower α-irresolute functions. Then, (iΓFi)l(μ)=iΓFil(μ) is double fuzzy lower α-irresolute.

Proof

Let μIY ; then, (iΓFi)l(μ)=iΓFil(μ). Because {Fi : i ∈ Γ} is a family of double fuzzy lower α-irresolute multifunctions, Fil(μ) is (r, s)-fuzzy α-open for any (r, s)-fuzzy α-open set μIY. Furthermore, we can conclude that (iΓFi)l(μ)=iΓFil(μ) is (r, s)-fuzzy α-open for any (r, s)-fuzzy α-open set μ. Hence, iΓFi is double fuzzy lower α-irresolute.

Definition 12

A fuzzy multifunction F : (X, τ, τ) → (Y, σ, σ) is called (r, s)-fuzzy α-compact-valued if and only if F(xt) is (r, s)-fuzzy α-compact for each xt ∈ dom (F).

Theorem 8

Let F : (X, τ, τ) → (Y, σ, σ) be a crisp double fuzzy upper α-irresolute and α-compact-valued multifunction. If λ is (r, s)-fuzzy α-compact, then, F(λ) is (r, s)-fuzzy α-compact.

Proof

Let λIX be an (r, s)-fuzzy α-compact set and {γi : γi is an (r, s)-fuzzy α-open set, i ∈ Γ} be a family covering F(λ), that is, F(λ)iΓγi. As λ=xtλxt, we have F(λ)=F(xtλxt)=xtλF(xt)iΓγi. It follows that for each xtλ,F(xt)iΓγi. Because F is (r, s)-α-compact-valued, there exists a finite subset Γxt of Γ such that F(xt)nΓxtγn=γxt. Subsequently, we have: xtFu(F(xt)) ≤ Fu(γxt) and λ=xtλxtxtλFu(F(xt))Fu(γxt). Because γxt is (r, s)-fuzzy α-open, we can conclude that Fu(γxt) is (r, s)-fuzzy α-open. Hence, {Fu(γxt) : Fu(γxt) is (r, s)-fuzzy α-open, xtλ} is a family covering the set λ. Because λ is (r, s)-fuzzy α-compact, there exists a finite index set N such that λnNFu(γxtn). Subsequently, F(λ)F(nNFu(γxtn))=nNF(Fu(γxtn)nNγxtn. Thus, F(λ) is (r, s)-fuzzy α-compact.

References

  1. Zadeh, LA (1965). Fuzzy sets. Information and Control. 8, 338-353. https://doi.org/10.1016/S0019-9958(65)90241-X
    CrossRef
  2. Chang, CL (1968). Fuzzy topological spaces. Journal of Mathematical Analysis and Applications. 24, 182-190.
    CrossRef
  3. Atanassov, KT (1994). New operations defined over the intuitionistic fuzzy sets. Fuzzy Sets and Systems. 61, 137-142. https://doi.org/10.1016/0165-0114(94)90229-1
    CrossRef
  4. AP Sostak (1985). On a fuzzy topological structure. Proceedings of the 13th Winter School on Abstract Analysis. Palermo, Italy: Circolo Matematico di Palermo, pp. 89-103
  5. Kubiak, T 1985. On fuzzy topologies. PhD dissertation. Adam Mickiewicz University. Poznan, Poland.
  6. Sostak, AP (1996). Basic structures of fuzzy topology. Journal of Mathematical Sciences. 78, 662-701.
    CrossRef
  7. Coker, D (1996). An introduction to fuzzy subspaces in intuitionistic fuzzy topological spaces. Journal of Fuzzy Mathematics. 4, 749-764.
  8. Coker, D (1997). An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems. 88, 81-89. https://doi.org/10.1016/S0165-0114(96)00076-0
    CrossRef
  9. Mondal, TK, and Samanta, SK (2002). On intuitionistic gradation of openness. Fuzzy Sets and Systems. 131, 323-336. https://doi.org/10.1016/S0165-0114(01)00235-4
    CrossRef
  10. Garcia, JG, and Rodabaugh, SE (2005). Order-theoretic, topological, categorical redundancies of interval-valued sets, grey sets, vague sets, interval-valued “intuitionistic” sets, “intuitionistic” fuzzy sets and topologies. Fuzzy Sets and Systems. 156, 445-484. https://doi.org/10.1016/j.fss.2005.05.023
    CrossRef
  11. Abd-Allah, MA, El-Saady, K, and Ghareeb, A (2009). (r, s)-Fuzzy F-open sets and (r, s)-fuzzy F-closed spaces. Chaos, Solitons & Fractals. 42, 649-656. https://doi.org/10.1016/j.chaos.2009.01.031
    CrossRef
  12. Mohammed, FM, Noorani, MSM, and Ghareeb, A (2016). Generalized fuzzy b-closed and generalized ★-fuzzy b-closed sets in double fuzzy topological spaces. Egyptian Journal of Basic and Applied Sciences. 3, 61-67.
    CrossRef
  13. Mohammed, FM, Noorani, MSM, and Ghareeb, A (2014). Somewhat slightly generalized double fuzzy semicontinuous functions. International Journal of Mathematics and Mathematical Sciences. 2014. article no. 756376
    CrossRef
  14. Mohammed, FM, Noorani, MSM, and Ghareeb, A (2017). New notions from (alpha,beta)-generalised fuzzy preopen sets. Gazi University Journal of Science. 30, 311-331.
  15. Ghareeb, A (2011). Normality of double fuzzy topological spaces. Applied Mathematics Letters. 24, 533-540. https://doi.org/10.1016/j.aml.2010.11.008
    CrossRef
  16. Ghareeb, A (2012). Weak forms of continuity in I-double gradation fuzzy topological spaces. SpringerPlus. 1. article no. 19
    CrossRef
  17. El-Saady, K, and Ghareeb, A (2012). Several types of (r, s)-fuzzy compactness defined by an (r, s)-fuzzy regular semiopen sets. Annals of fuzzy Mathematics and Informatics. 3, 159-169.
  18. Lee, EP (2004). Semiopen sets on intuitionistic fuzzy topological spaces in Sostak’s sense. Journal of the Korean Institute of Intelligent Systems. 14, 234-238. https://doi.org/10.5391/JKIIS.2004.14.2.234
    CrossRef
  19. Lee, SO, and Lee, EP (2005). Fuzzy (r, s)-preopen sets. International Journal of Fuzzy Logic and Intelligent Systems. 5, 136-139. https://doi.org/10.5391/IJFIS.2005.5.2.136
    CrossRef
  20. Lee, SO, and Lee, EP (2006). Fuzzy strongly (r, s)-semiopen sets. International Journal of Fuzzy Logic and Intelligent Systems. 6, 299-303. https://doi.org/10.5391/IJFIS.2006.6.4.299
    CrossRef
  21. Lee, SJ, and Kim, JT (2007). Fuzzy (r, s)-irresolute maps. International Journal of Fuzzy Logic and Intelligent Systems. 7, 49-57. https://doi.org/10.5391/IJFIS.2007.7.1.049
    CrossRef
  22. Zahran, AM, Abd-Allah, MA, and Ghareeb, A (2010). Several types of double fuzzy irresolute functions. International Journal of Computational Cognition. 8, 19-23.
  23. Abbas, SE, and Taha, IM (2020). On upper and lower contra-continuous fuzzy multifunctions. Punjab University Journal of Mathematics. 47, 104-117.
  24. Coker, D, and Demirci, M (1996). An introduction to intuitionistic fuzzy topological spaces in Sostak’s sense. Busefal. 67, 67-76.
  25. Lee, EP, and Im, YB (2001). Mated fuzzy topological spaces. Journal of the Korean Institute of Intelligent Systems. 11, 161-165.
  26. Suganya, V, Gomathisundari, P, and Rajesh, N (2022). Double fuzzy α-continuous multifunctions. International Journal of Fuzzy Logic and Intelligent Systems. Manuscript submitted