Next Article in Journal
Decomposition of Dynamical Signals into Jumps, Oscillatory Patterns, and Possible Outliers
Next Article in Special Issue
The Effect of Prudence on the Optimal Allocation in Possibilistic and Mixed Models
Previous Article in Journal
A Generalized Fejér–Hadamard Inequality for Harmonically Convex Functions via Generalized Fractional Integral Operator and Related Results
Previous Article in Special Issue
On Generalized Roughness in LA-Semigroups
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets

1
Department of Bioinformatics and Mathematical Modelling, Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, 105 Acad. G. Bonchev Str., 1113 Sofia, Bulgaria
2
Intelligent Systems Laboratory, Prof. Asen Zlatarov University, 8010 Bourgas, Bulgaria
Mathematics 2018, 6(7), 123; https://doi.org/10.3390/math6070123
Submission received: 30 May 2018 / Revised: 4 July 2018 / Accepted: 4 July 2018 / Published: 13 July 2018
(This article belongs to the Special Issue Fuzzy Mathematics)

Abstract

:
The definition of the most extended modal operator of first type over interval-valued intuitionistic fuzzy sets is given, and some of its basic properties are studied.

1. Introduction

Intuitionistic fuzzy sets (IFSs; see [1,2,3,4,5]) were introduced in 1983 as an extension of the fuzzy sets defined by Lotfi Zadeh (4.2.1921–6.9.2017) in [6]. In recent years, the IFSs have also been extended: intuitionistic L -fuzzy sets [7], IFSs of second [8] and nth [9,10,11,12] types, temporal IFSs [4,5,13], multidimensional IFSs [5,14], and others. Interval-valued intuitionistic fuzzy sets (IVIFSs) are the most detailed described extension of IFSs. They appeared in 1988, when Georgi Gargov (7.4.1947–9.11.1996) and the author read Gorzalczany’s paper [15] on the interval-valued fuzzy set (IVFS). The idea of IVIFS was announced in [16,17] and extended in [4,18], where the proof that IFSs and IVIFSs are equipollent generalizations of the notion of the fuzzy set is given.
Over IVIFS, many (more than the ones over IFSs) relations, operations, and operators are defined. Here, similar to the IFS case, the standard modal operators ☐ and ◇ have analogues, but their extensions—the intuitionistic fuzzy extended modal operators of the first type—already have two different forms. In the IFS case, there is an operator that includes as a partial case all other extended modal operators. In the present paper, we construct a similar operator for the case of IVIFSs and study its properties.

2. Preliminaries

Let us have a fixed universe E and its subset A. The set
A = { x , M A ( x ) , N A ( x ) x E } ,
where M A ( x ) [ 0 , 1 ] and N A ( x ) [ 0 , 1 ] are closed intervals and for all x E :
sup M A ( x ) + sup N A ( x ) 1
is called IVIFS, and functions M A : E P ( [ 0 , 1 ] ) and N A : E P ( [ 0 , 1 ] ) represent the set of degrees of membership (validity, etc.) and the set of degrees of non-membership (non-validity, etc.) of element x E to a fixed set A E , where P ( Z ) = { Y | Y Z } for an arbitrary set Z .
Obviously, both intervals have the representation:
M A ( x ) = [ inf M A ( x ) , sup M A ( x ) ] ,
N A ( x ) = [ inf N A ( x ) , sup N A ( x ) ] .
Therefore, when
inf M A ( x ) = sup M A ( x ) = μ A ( x ) and inf N A ( x ) = sup N A ( x ) = ν A ( x ) ,
the IVIFS A is transformed to an IFS.
We must mention that in [19,20] the second geometrical interpretation of the IFSs is given (see Figure 1).
IVIFSs have geometrical interpretations similar to, but more complex than, those of the IFSs. For example, the analogue of the geometrical interpretation from Figure 1 is shown in Figure 2.
Obviously, each IVFS A can be represented by an IVIFS as
A = { x , M A ( x ) , N A ( x ) x E }
= { x , M A ( x ) , [ 1 sup M A ( x ) , 1 inf M A ( x ) ] x E } .
The geometrical interpretation of the IVFS A is shown in Figure 3. It has the form of a section lying on the triangle’s hypotenuse.
Modal-type operators are defined similarly to those defined for IFSs, but here they have two forms: shorter and longer. The shorter form is:
A = { x , M A ( x ) , [ inf N A ( x ) , 1 sup M A ( x ) ] x E } , A = { x , [ inf M A ( x ) , 1 sup N A ( x ) ] , N A ( x ) x E } , D α ( A ) = { x , [ inf M A ( x ) , sup M A ( x ) + α ( 1 sup M A ( x ) sup N A ( x ) ) ] , [ inf N A ( x ) , sup N A ( x ) + ( 1 α ) ( 1 sup M A ( x ) sup N A ( x ) ) ] x E } , F α , β ( A ) = { x , [ inf M A ( x ) , sup M A ( x ) + α ( 1 sup M A ( x ) sup N A ( x ) ) ] , [ inf N A ( x ) , sup N A ( x ) + β ( 1 sup M A ( x ) sup N A ( x ) ) ] x E } , f o r α + β 1 , G α , β ( A ) = { x , [ α inf M A ( x ) , α sup M A ( x ) ] , [ β inf N A ( x ) , β sup N A ( x ) ] x E } , H α , β ( A ) = { x , [ α inf M ( x ) , α sup M A ( x ) ] , [ inf N A ( x ) , sup N A ( x ) + β ( 1 sup M A ( x ) sup N A ( x ) ) ] x E } , H α , β ( A ) = { x , [ α inf M A ( x ) , α sup M A ( x ) ] , [ inf N A ( x ) , sup N A ( x ) + β ( 1 α sup M A ( x ) sup N A ( x ) ) ] x E } , J α , β ( A ) = { x , [ inf M A ( x ) , sup M A ( x ) + α ( 1 sup M A ( x ) sup N A ( x ) ) ] , [ β inf N A ( x ) , β sup N A ( x ) ] x E } , J α , β ( A ) = { x , [ inf M A ( x ) , sup M A ( x ) + α ( 1 sup M A ( x ) β sup N A ( x ) ) ] , [ β inf N A ( x ) , β . sup N A ( x ) ] x E } ,
where α , β [ 0 , 1 ] .
Obviously, as in the case of IFSs, the operator D α is an extension of the intuitionistic fuzzy forms of (standard) modal logic operators ☐ and ◇, and it is a partial case of F α , β .
The longer form of these operators (operators ☐, ◇, and D do not have two forms—only the one above) is (see [4]):
F ¯ α γ β δ ( A ) = { x , [ inf M A ( x ) + α ( 1 sup M A ( x ) sup N A ( x ) ) , sup M A ( x ) + β ( 1 sup M A ( x ) sup N A ( x ) ) ] , [ inf N A ( x ) + γ ( 1 sup M A ( x ) sup N A ( x ) ) , sup N A ( x ) + δ ( 1 sup M A ( x ) sup N A ( x ) ) ] x E } where β + δ 1 , G ¯ α γ β δ ( A ) = { x , [ α inf M A ( x ) , β sup M A ( x ) ] , [ γ inf N A ( x ) , δ sup N A ( x ) ] x E } , H ¯ α γ β δ ( A ) = { x , [ α inf M A ( x ) , β sup M A ( x ) ] , [ inf N A ( x ) + γ ( 1 sup M A ( x ) sup N A ( x ) ) , sup N A ( x ) + δ ( 1 sup M A ( x ) sup N A ( x ) ) ] x E } , H ¯ α γ β δ ( A ) = { x , [ α inf M A ( x ) , β sup M A ( x ) ] , [ inf N A ( x ) + γ ( 1 β sup M A ( x ) sup N A ( x ) ) , sup N A ( x ) + δ ( 1 β sup M A ( x ) sup N A ( x ) ) ] x E } , J ¯ α γ β δ = { x , [ inf M A ( x ) + α ( 1 sup M A ( x ) sup N A ( x ) ) , sup M A ( x ) + β ( 1 sup M A ( x ) sup N A ( x ) ) ] , [ γ inf N A ( x ) , δ sup N A ( x ) ] x E } , J ¯ α γ β δ ( A ) = { x , [ inf M A ( x ) + α ( 1 δ sup M A ( x ) sup N A ( x ) ) , sup M A ( x ) + β ( 1 sup M A ( x ) δ sup N A ( x ) ) ] , [ γ . inf N A ( x ) , δ . sup N A ( x ) ] x E } ,
where α , β , γ , δ [ 0 , 1 ] such that α β and γ δ .
Figure 4 shows to which region of the triangle the element x E (represented by the small rectangular region in the triangle) will be transformed by the operators F , G , . . . , irrespective of whether they have two or four indices.

3. Operator X

Now, we introduce the new operator
X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A )
= { x , [ a 1 inf M A ( x ) + b 1 ( 1 inf M A ( x ) c 1 inf N A ( x ) ) ,
a 2 sup M A ( x ) + b 2 ( 1 sup M A ( x ) c 2 sup N A ( x ) ) ] ,
[ d 1 inf N A ( x ) + e 1 ( 1 f 1 inf M A ( x ) inf N A ( x ) ) ,
d 2 sup N A ( x ) + e 2 ( 1 f 2 sup M A ( x ) sup N A ( x ) ) ] | x E } ,
where a 1 , b 1 , c 1 , d 1 , e 1 , f 1 , a 2 , b 2 , c 2 , d 2 , e 2 , f 2 [ 0 , 1 ] , the following three conditions are valid for i = 1 , 2 :
a i + e i e i f i 1 ,
b i + d i b i c i 1 ,
b i + e i 1 ,
and
a 1 a 2 , b 1 b 2 , c 1 c 2 , d 1 d 2 , e 1 e 2 , f 1 f 2 .
Theorem 1.
For every IVIFS A and for every a 1 , b 1 , c 1 , d 1 , e 1 , f 1 , a 2 , b 2 , c 2 , d 2 , e 2 , f 2 [ 0 , 1 ] that satisfy (2)–(5), X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A ) is an IVIFS.
Proof. 
Let a 1 , b 1 , c 1 , d 1 , e 1 , f 1 , a 2 , b 2 , c 2 , d 2 , e 2 , f 2 [ 0 , 1 ] satisfy (2)–(5) and let A be a fixed IVIFS. Then, from (5) it follows that
a 1 inf M A ( x ) + b 1 ( 1 inf M A ( x ) c 1 inf N A ( x ) )
a 2 sup M A ( x ) + b 2 ( 1 sup M A ( x ) c 2 sup N A ( x ) )
and
d 1 inf N A ( x ) + e 1 ( 1 f 1 inf M A ( x ) inf N A ( x ) )
d 2 sup N A ( x ) + e 2 ( 1 f 2 sup M A ( x ) sup N A ( x ) ) .
Now, from (5) it is clear that it will be enough to check that
X = a 2 sup M A ( x ) + b 2 ( 1 sup M A ( x ) c 2 sup N A ( x ) )
+ d 2 sup N A ( x ) + e 2 ( 1 f 2 sup M A ( x ) sup N A ( x ) )
= ( a 2 b 2 e 2 f 2 ) sup M A ( x ) + ( d 2 e 2 b 2 c 2 ) sup N A ( x ) + b 2 + e 2 1 .
In fact, from (2),
a 2 b 2 e 2 f 2 1 b 2 e 2
and from (3):
d 2 e 2 b 2 c 2 1 b 2 e 2 .
Then, from (1),
X ( 1 b 2 e 2 ) ( sup M A ( x ) + sup N A ( x ) ) + b 2 + e 2
1 b 2 e 2 + b 2 + e 2 = 1 .
Finally, when sup M A ( x ) = inf N A ( x ) = 0 and from (4),
X = b 2 ( 1 0 0 ) + e 2 ( 1 0 0 ) = b 2 + e 2 1 .
Therefore, the definition of the IVIFS is correct. ☐
All of the operators described above can be represented by the operator X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 at suitably chosen values of its parameters. These representations are the following:
A = X 1 0 r 1 1 0 s 1 1 0 r 2 1 1 1 ( A ) , A = X 1 0 r 1 1 0 s 1 1 1 1 1 0 s 2 ( A ) , D α ( A ) = X 1 0 r 1 1 α 1 1 0 r 2 1 1 α 1 ( A ) , F α , β ( A ) = X 1 0 r 1 1 0 s 1 1 α 1 1 β 1 ( A ) , G α , β ( A ) = X α 0 r 1 β 0 s 1 α 0 r 2 β 0 s 2 ( A ) , H α , β ( A ) = X α 0 r 1 1 0 s 1 α 0 r 2 1 β 1 ( A ) , H α , β ( A ) = X α 0 r 1 α 0 s 1 1 0 r 2 1 β α ( A ) , J α , β ( A ) = X 1 0 r 1 β 0 s 1 1 α 1 β 0 s 2 ( A ) , J α , β ( A ) = X 1 0 r 1 β 0 s 1 1 α β β 0 s 2 ( A ) , F ¯ α γ β δ ( A ) = X 1 α 1 1 γ 1 1 β 1 1 δ s 1 ( A ) ,
G ¯ α γ β δ ( A ) = X α 0 r 1 β 0 s 1 γ 0 r 2 δ 0 s 2 ( A ) , H ¯ α γ β δ ( A ) = X α 0 r 1 1 γ 1 β 0 r 2 1 δ 1 ( A ) , H ¯ α γ β δ ( A ) = X α 0 r 1 1 γ 1 β 0 r 2 1 δ β ( A ) , J ¯ α γ β δ ( A ) = X 1 α 1 γ 0 s 1 1 β 1 δ 0 s 2 ( A ) , J ¯ α γ β δ ( A ) = X 1 α δ γ 0 s 1 1 β δ δ 0 s 2 ( A ) ,
where r 1 , r 2 , s 1 , s 2 are arbitrary real numbers in the interval [ 0 , 1 ] .
Three of the operations, defined over two IVIFSs A and B, are the following:
¬ A = { x , N A ( x ) , M A ( x ) x E } , A B = { x , [ min ( inf M A ( x ) , inf M B ( x ) ) , min ( sup M A ( x ) , sup M B ( x ) ) ] , [ max ( inf N A ( x ) , inf N B ( x ) ) , max ( sup N A ( x ) , sup N B ( x ) ) ] x E } , A B = { x , [ max ( inf M A ( x ) , inf M B ( x ) ) , max ( sup M A ( x ) , sup M B ( x ) ) ] , [ min ( inf N A ( x ) , inf N B ( x ) ) , min ( sup N A ( x ) , sup N B ( x ) ) ] x E } .
For any two IVIFSs A and B, the following relations hold:
A B iff x E , inf M A ( x ) inf M B ( x ) , inf N A ( x ) inf N B ( x ) , sup M A ( x ) sup M B ( x ) and sup N A ( x ) sup N B ( x ) ) , A B iff B A , A = B iff A B and B A .
Theorem 2.
For every two IVIFSs A and B and for every a 1 , b 1 , c 1 , d 1 , e 1 , f 1 , a 2 , b 2 , c 2 , d 2 , e 2 , f 2 [ 0 , 1 ] that satisfy (2)–(5),
(a) 
¬ X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( ¬ A ) = X d , e , f , a , b , c ( A ) ,
(b) 
X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A B )
X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A ) X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( B ) ,
(c) 
X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A B )
X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A ) X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( B ) .
Proof. 
(c) Let a 1 , b 1 , c 1 , d 1 , e 1 , f 1 , a 2 , b 2 , c 2 , d 2 , e 2 , f 2 [ 0 , 1 ] satisfy (2)–(5) , and let A and B be fixed IVIFSs. First, we obtain:
Y = X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A B )
= X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( { x , [ max ( inf M A ( x ) , inf M B ( x ) ) ,
max ( sup M A ( x ) , sup M B ( x ) ) ] ,
[ min ( inf N A ( x ) , inf N B ( x ) ) , min ( sup N A ( x ) , sup N B ( x ) ) ] x E } )
= { x , [ a 1 max ( inf M A ( x ) , inf M B ( x ) ) + b 1 ( 1 max ( inf M A ( x ) , inf M B ( x ) )
c 1 min ( inf N A ( x ) , inf N B ( x ) ) ) , a 2 max ( sup M A ( x ) sup M B ( x ) )
+ b 2 ( 1 max ( sup M A ( x ) sup M B ( x ) ) c 2 min ( sup N A ( x ) , sup N B ( x ) ) ) ] ,
[ d 1 min ( inf N A ( x ) , inf N B ( x ) ) + e 1 ( 1 f 1 max ( inf M A ( x ) , inf M B ( x ) )
min ( inf N A ( x ) , inf N B ( x ) ) ) , d 2 min ( sup N A ( x ) , sup N B ( x ) )
+ e 2 ( 1 f 2 max ( sup M A ( x ) sup M B ( x ) ) min ( sup N A ( x ) , sup N B ( x ) ) ) ] | x E } .
Second, we calculate:
Z = X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( A ) X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 ( B )
= { x , [ a 1 inf M A ( x ) + b 1 ( 1 inf M A ( x ) c 1 inf N A ( x ) ) ,
a 2 sup M A ( x ) + b 2 ( 1 sup M A ( x ) c 2 sup N A ( x ) ) ] ,
[ d 1 inf N A ( x ) + e 1 ( 1 f 1 inf M A ( x ) inf N A ( x ) ) ,
d 2 sup N A ( x ) + e 2 ( 1 f 2 sup M A ( x ) sup N A ( x ) ) ] | x E }
{ x , [ a 1 inf M B ( x ) + b 1 ( 1 inf M B ( x ) c 1 inf N B ( x ) ) ,
a 2 sup M B ( x ) + b 2 ( 1 sup M B ( x ) c 2 sup N B ( x ) ) ] ,
[ d 1 inf N B ( x ) + e 1 ( 1 f 1 inf M B ( x ) inf N B ( x ) ) ,
d 2 sup N B ( x ) + e 2 ( 1 f 2 sup M B ( x ) sup N B ( x ) ) ] | x E }
= { x , [ max ( a 1 inf M A ( x ) + b 1 ( 1 inf M A ( x ) c 1 inf N A ( x ) ) ,
a 1 inf M B ( x ) + b 1 ( 1 inf M B ( x ) c 1 inf N B ( x ) ) ) ,
max ( a 2 sup M A ( x ) + b 2 ( 1 sup M A ( x ) c 2 sup N A ( x ) ) ,
a 2 sup M B ( x ) + b 2 ( 1 sup M B ( x ) c 2 sup N B ( x ) ) ) ] ,
[ min ( d 1 inf N A ( x ) + e 1 ( 1 f 1 inf M A ( x ) inf N A ( x ) ) ,
d 1 inf N B ( x ) + e 1 ( 1 f 1 inf M B ( x ) inf N B ( x ) ) ) ,
min ( d 2 sup N A ( x ) + e 2 ( 1 f 2 sup M A ( x ) sup N A ( x ) ) ,
d 2 sup N B ( x ) + e 2 ( 1 f 2 sup M B ( x ) sup N B ( x ) ) ) ] x E } .
Let
P = a 1 max ( inf M A ( x ) , inf M B ( x ) ) + b 1 ( 1 max ( inf M A ( x ) , inf M B ( x ) )
c 1 min ( inf N A ( x ) , inf N B ( x ) ) ) max ( a 1 inf M A ( x ) + b 1 ( 1 inf M A ( x ) c 1 inf N A ( x ) ) ,
a 1 inf M B ( x ) + b 1 ( 1 inf M B ( x ) c 1 inf N B ( x ) ) )
= a 1 max ( inf M A ( x ) , inf M B ( x ) ) + b 1 b 1 max ( inf M A ( x ) , inf M B ( x ) )
b 1 c 1 min ( inf N A ( x ) , inf N B ( x ) ) ) max ( ( a 1 b 1 ) inf M A ( x ) + b 1 b 1 c 1 inf N A ( x ) ,
( a 1 b 1 ) inf M B ( x ) + b 1 b 1 c 1 inf N B ( x ) )
= a 1 max ( inf M A ( x ) , inf M B ( x ) ) b 1 max ( inf M A ( x ) , inf M B ( x ) )
b 1 c 1 min ( inf N A ( x ) , inf N B ( x ) ) max ( ( a 1 b 1 ) inf M A ( x ) b 1 c 1 inf N A ( x ) ,
( a 1 b 1 ) inf M B ( x ) b 1 c 1 inf N B ( x ) ) .
Let inf M A ( x ) inf M B ( x ) . Then
P = ( a 1 b 1 ) inf M A ( x ) b 1 c 1 min ( inf N A ( x ) , inf N B ( x ) ) max ( ( a 1 b 1 ) inf M A ( x )
b 1 c 1 inf N A ( x ) , ( a 1 b 1 ) inf M B ( x ) b 1 c 1 inf N B ( x ) ) .
Let ( a 1 b 1 ) inf M A ( x ) b 1 c 1 inf N A ( x ) ( a 1 b 1 ) inf M B ( x ) b 1 c 1 inf N B ( x ) . Then
P = ( a 1 b 1 ) inf M A ( x ) b 1 c 1 min ( inf N A ( x ) , inf N B ( x ) ) ( a 1 b 1 ) inf M A ( x )
+ b 1 c 1 inf N A ( x )
= b 1 c 1 inf N A ( x ) b 1 c 1 min ( inf N A ( x ) , inf N B ( x ) ) 0 .
If ( a 1 b 1 ) inf M A ( x ) b 1 c 1 inf N A ( x ) < ( a 1 b 1 ) inf M B ( x ) b 1 c 1 inf N B ( x ) . Then
P = ( a 1 b 1 ) inf M A ( x ) b 1 c 1 min ( inf N A ( x ) , inf N B ( x ) ) ( a 1 b 1 ) inf M B ( x )
+ b 1 c 1 inf N B ( x ) ) .
= b 1 c 1 inf N B ( x ) b 1 c 1 min ( inf N A ( x ) , inf N B ( x ) ) 0 .
Therefore, the inf M A -component of IVIFS Y is higher than or equal to the inf M A -component of IVIFS Z. In the same manner, it can be checked that the same inequality is valid for the sup M A -components of these IVIFSs. On the other hand, we can check that that the inf N A - and sup N A -components of IVIFS Y are, respectively, lower than or equal to the inf N A and sup N A -components of IVIFS Z. Therefore, the inequality (c) is valid. ☐

4. Conclusions

In the near future, the author plans to study some other properties of the new operator X a 1 b 1 c 1 d 1 e 1 f 1 a 2 b 2 c 2 d 2 e 2 f 2 .
In [21], it is shown that the IFSs are a suitable tool for the evaluation of data mining processes and objects. In the near future, we plan to discuss the possibilities of using IVIFSs as a similar tool.

Funding

This research was funded by the Bulgarian National Science Fund under Grant Ref. No. DN-02-10/2016.

Conflicts of Interest

The author declares no conflict of interest.

References

  1. Atanassov, K. Intuitionistic Fuzzy Sets. 1983, VII ITKR Session. Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84. Available online: http://www.biomed.bas.bg/bioautomation/2016/vol_20.s1/files/20.s1_03.pdf (accessed on 13 July 2018).
  2. Atanassov, K. Intuitionistic fuzzy sets. Int. J. Bioautom. 2016, 20, S1–S6. [Google Scholar]
  3. Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. [Google Scholar] [CrossRef]
  4. Atanassov, K. Intuitionistic Fuzzy Sets; Springer: Heidelberg, Germany, 1999. [Google Scholar]
  5. Atanassov, K. On Intuitionistic Fuzzy Sets Theory; Springer: Berlin, Germany, 2012. [Google Scholar]
  6. Zadeh, L. Fuzzy sets. Inf. Control 1965, 8, 338–353. [Google Scholar] [CrossRef] [Green Version]
  7. Atanassov, K.; Stoeva, S. Intuitionistic L-fuzzy sets. In Cybernetics and Systems Research 2; Trappl, R., Ed.; Elsevier Sci. Publ.: Amsterdam, The Nertherlands, 1984; pp. 539–540. [Google Scholar]
  8. Atanassov, K. A second type of intuitionistic fuzzy sets. BUSEFAL 1993, 56, 66–70. [Google Scholar]
  9. Atanassov, K.; Szmidt, E.; Kacprzyk, J.; Vassilev, P. On intuitionistic fuzzy pairs of n-th type. Issues Intuit. Fuzzy Sets Gen. Nets 2017, 13, 136–142. [Google Scholar]
  10. Atanassov, K.T.; Vassilev, P. On the Intuitionistic Fuzzy Sets of n-th Type. In Advances in Data Analysis with Computational Intelligence Methods; Studies in Computational Intelligence; Gawęda, A., Kacprzyk, J., Rutkowski, L., Yen, G., Eds.; Springer: Cham, Switzerland, 2018; Volume 738, pp. 265–274. [Google Scholar]
  11. Parvathi, R.; Vassilev, P.; Atanassov, K. A note on the bijective correspondence between intuitionistic fuzzy sets and intuitionistic fuzzy sets of p-th type. In New Developments in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics; SRI PAS IBS PAN: Warsaw, Poland, 2012; Volume 1, pp. 143–147. [Google Scholar]
  12. Vassilev, P.; Parvathi, R.; Atanassov, K. Note on intuitionistic fuzzy sets of p-th type. Issues Intuit. Fuzzy Sets Gen. Nets 2008, 6, 43–50. [Google Scholar]
  13. Atanassov, K. Temporal Intuitionistic Fuzzy Sets; Comptes Rendus de l’Academie bulgare des Sciences; Bulgarian Academy of Sciences: Sofia, Bulgaria, 1991; pp. 5–7. [Google Scholar]
  14. Atanassov, K.; Szmidt, E.; Kacprzyk, J. On intuitionistic fuzzy multi-dimensional sets. Issues Intuit. Fuzzy Sets Gen. Nets 2008, 7, 1–6. [Google Scholar]
  15. Gorzalczany, M. Interval-valued fuzzy fuzzy inference method—Some basic properties. Fuzzy Sets Syst. 1989, 31, 243–251. [Google Scholar] [CrossRef]
  16. Atanassov, K. Review and New Results on Intuitionistic Fuzzy Sets. 1988, IM-MFAIS-1-88, Mathematical Foundations of Artificial Intelligence Seminar. Available online: http://www.biomed.bas.bg/bioautomation/2016/vol_20.s1/files/20.s1_03.pdf (accessed on 13 July 2018 ).
  17. Atanassov, K. Review and new results on intuitionistic fuzzy sets. Int. J. Bioautom. 2016, 20, S7–S16. [Google Scholar]
  18. Atanassov, K.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst. 1989, 31, 343–349. [Google Scholar] [CrossRef]
  19. Atanassov, K. Geometrical Interpretation of the Elements of the Intuitionistic Fuzzy Objects. 1989, IM-MFAIS-1-89, Mathematical Foundations of Artificial Intelligence Seminar. Available online: http://biomed.bas.bg/bioautomation/2016/vol_20.s1/files/20.s1_05.pdf (accessed on 13 July 2018).
  20. Atanassov, K. Geometrical interpretation of the elements of the intuitionistic fuzzy objects. Int. J. Bioautom. 2016, 20, S27–S42. [Google Scholar]
  21. Atanassov, K. Intuitionistic fuzzy logics as tools for evaluation of Data Mining processes. Knowl.-Based Syst. 2015, 80, 122–130. [Google Scholar] [CrossRef]
Figure 1. The second geometrical interpretation of an intuitionistic fuzzy set (IFS).
Figure 1. The second geometrical interpretation of an intuitionistic fuzzy set (IFS).
Mathematics 06 00123 g001
Figure 2. The second geometrical interpretation of an interval-valued intuitionistic fuzzy set (IVIFS).
Figure 2. The second geometrical interpretation of an interval-valued intuitionistic fuzzy set (IVIFS).
Mathematics 06 00123 g002
Figure 3. The second geometrical interpretation of an IVFS.
Figure 3. The second geometrical interpretation of an IVFS.
Mathematics 06 00123 g003
Figure 4. Region of transformation by the application of the operators.
Figure 4. Region of transformation by the application of the operators.
Mathematics 06 00123 g004

Share and Cite

MDPI and ACS Style

Atanassov, K. On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets. Mathematics 2018, 6, 123. https://doi.org/10.3390/math6070123

AMA Style

Atanassov K. On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets. Mathematics. 2018; 6(7):123. https://doi.org/10.3390/math6070123

Chicago/Turabian Style

Atanassov, Krassimir. 2018. "On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets" Mathematics 6, no. 7: 123. https://doi.org/10.3390/math6070123

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

Article Metrics

Back to TopTop