Some Algebraic Properties of Picture Fuzzy t-Norms and Picture Fuzzy t – Conorms

Some Algebraic Properties of Picture Fuzzy t-Norms and Picture Fuzzy t – Conorms on Standard Neutrosophic Sets

Bui Cong Cuong1*, Roan Thi Ngan2 and Le Chi Ngoc3

1* Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam E-mail: bccuong@gmail.com, 1* Corresponding Author

2 Basic Science Faculty, Hanoi University of Natural Resources and Environment, E-mail: rtngan@hunre.edu.vn

3 School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Email: lechingoc@yahoo.com

Abstract

In 2013 we introduced a new notion of picture fuzzy sets (PFS), which are direct extensions of the fuzzy sets and the intuitonistic fuzzy sets. Then some operations on PFS with some properties are considered in [ 9,10 ]. In [15] these sets are considered as standard neutrosophic sets. Some basic operators of fuzzy logic as negation, t-norms ,t-conorms for picture fuzzy sets firstly are defined and studied in [13,14]. This paper is devoted to some algebraic properties of picture fuzzy t-norms and picture fuzzy t-conorms on standard neutrosophic sets.

Key words: Picture fuzzy sets,, Picture fuzzy t-norms, Picture fuzzy t-conorm , Algebraic property

1. Introduction

Recently, Bui Cong Cuong and Kreinovich (2013) first defined "picture fuzzy sets" [9,10], which are a generalization of the Zadeh’ fuzzy sets [1, 2] and the Antanassov’s intuitionistic fuzzy sets [3,4]. This concept is particularly effective in approaching the practical problems in relation to the synthesis of ideas, make decisions, such as voting, financial forecasting, estimation of risks in business. The new concept are supporting to new algorithms in computational intelligence problems [18].

In this paper we study some algebraic properties of the picture fuzzy t-norms and the picture fuzzy t-conorms on the standard neutrosophic sets , which are basic operators of the neutrosophic logics. Some classifications of the representable picture fuzzy t-norms and the representable picture fuzzy t-conorms will be presented.

We first recall some basic notions of the picture fuzzy sets.

Definition 1.1. [9] A picture fuzzy set A on a universe X is an object of the form

where (),(),() AAAxxx are respectively called the degree of positive membership, the degree of neutral membership, the degree of negative membership of x in A, and the following conditions are satisfied: 0(),(),()1 AAAxxx  and ()()()1,. AAAxxxxX  Then, : xX 1(()()()) AAAxxx is called the degree of refusal membership of x in A. Consider the set   *3 123123 (,,)|[0,1],1.Dxxxxxxxx  From now on, we will assume that if * : xD  , then 12 , xx and 3x denote, respectively, the first, the second and the third component of x, i.e. , 123 (,,).xxxx  We have a lattice * 1 (,) D  , where 1 defined by * ,:xyD 1 (). xy          11331133113322 ,,,,, xyxyxyxyxyxyxy 

 113322,,.xyxyxyxy 

We define the first, second and third projection mapping 1pr , then 2pr and 3pr on *D , defined as 11 () prxx  and 22 () prxx  and 33 () prxx  , on all *xD  .

Note that, if for * , xyD  that neither 1 xy  nor 1 yx  , then x and y are incomparable w.r.t 1 , denoted as 1 xy  From now on , we denote min(,),max(,) uvuvuvuv  for all 1 , uvR  . For each * , xyD  , we define 11 11113333

min(,), inf(,) (,1,), xyifxyoryx xy xyxyxyxyelse       11 1133

max(,), sup(,) (,0,), xyifxyoryx xy xyxyelse      

Proposition 1.2 With these operators * 1 (,) D  is a complete lattice. Proof. The units of these lattice are * 1(1,0,0) D  and * 0(0,0,1) D  . For each nonempty *AD  , we have

123 inf(inf,inf,inf) AprAprAprA  , where   11123 infinf[0,1](,,), prAxxxxxA  ,   33123 infsup[0,1](,,), prAxxxxxA  Denote   22123:(inf,,inf), BxprAxprAA  22 2 13

inf, inf 1infinf, BifB prA prAprAelse      and 123 sup(sup,sup,sup), AprAprAprA  , where   11123 supsup[0,1](,,), prAxxxxxA  ,   33123 sup[0,1](,,), prAinfxxxxxA  Denote   12123[0,1]:(sup,,sup), BxprAxprAA  11 2 sup, sup 0, BifB prA else      . Using this lattice, we easily see that every picture fuzzy set {(,(),(),())} AAA AxxxxxX   corresponds an *D fuzzy set [12] mapping, i.e., we have a mapping * ::{(,(),(),())}. AAA AXDxxxxxxX  

2. Picture fuzzy t-norms and picture fuzzy t-conorms

Now we consider some basic fuzzy operators of the Picture Fuzzy Logics. Picture fuzzy negations form an extension of the fuzzy negations [5] and the intuitionistic fuzzy negations [4]. They are defined as follows.

Definition 2.1. A mapping ** : NDD  satisfying conditions **(0)1DD N  and **(1)0DD N  and N is nonincreasing is called a picture fuzzy negation. If (()) NNxx  for all *xD  , then N is called an involutive negation

Let * 123 (,,) xxxxD  . The mapping 0N defined by 031 ()(,0,)Nxxx  , for each *xD  , is a picture fuzzy negation. Denote 41231().xxxx 

The mapping SN defined by 341 ()(,,) S Nxxxx  , for each *xD  , is an involutive negation and is called the picture fuzzy standard negation. Some picture fuzzy negations were given and studied in [13, 14]. Fuzzy t-norms on [0,1] and fuzzy t-conorms on [0,1] were defined and considered in [5,6]. In 2004 , G.Deschrijver et al.[11 ] introduced the notion of intuitionistic fuzzy t-norms and tconorms and investigated under which conditions a similar representation theorem could be obtained.

For further usage, we define ** 2 {0}LxDx We can considerthe set *L defined by *2 1313 {(,)[0,1],1}Luuuuuu 

Consider the order relation uv  on *L , defined by 1133 (()())uvuvuv  , for all * , uvL  . We define the first, and second projection mapping 1pr and 3pr on *L , defined as 11 () pruu  and 33 () pruu  ,on all *uL  The units of *L are * 1(1,0) L  and * 0(0,1). L 

Definition 2.2 [12]. An intuitionistic fuzzy t-norm is a commutative, associative, increasing *2*()LL  mapping T satisfying * (1,) L Tuu  , for all *uL  .

Definition 2.3. [12]. An intuitionistic fuzzy t-conorm is a commutative, associative, increasing *2*()LL  mapping S satisfying * (,0) L Svv  , for all *vL  .

Definition 2.4.[12]. An intuitionistic fuzzy t-norm T is called t-representable iff there exist a fuzzy t-norm 1t on [0,1] and a fuzzy t-conorm 3s on [0,1] satisfying for all * , uvL  , 111333 (,)((,),(,)) Tuvtuvsuv 

Definition 2.5. [12]. An intuitionistic fuzzy t-conorm S is called t-representable iff there exist a fuzzy t-norm 1t on [0,1] and a fuzzy t-conorm 3s on [0,1] satisfying for all * , uvL  ,

311133 (,)((,),(,)) Suvsuvtuv 

Now we define picture fuzzy t-norms and picture fuzzy t-conorms It means that we will give some classes of conjunction operators and and some classes of disjunction operators ,which are basic operators of the neutrosophic logics

Picture fuzzy t-norms are direct extensions of the fuzzy t-norms in [2, 5, 6] and of the intuitionistic fuzzy t-norms in [4].

Let * 123 (,,) xxxxD  . Denote   * 1231212():(,,),0 IxyDyxyxyx 

Definition 2.6 A mapping *** :DDTD  is a picture fuzzy t-norm if the mapping T satisfies the following conditions: 1. *(,)(,),,D TxyTyxxy (commutative), 2. *(,(,))((,),),,,D TxTyzTTxyzxyz  (associativity) 3. * 11(,)(,),,,D, TxyTxzxyzyz  (monotonicity) 4. * *(1,)(),D D TxIxx ( boundary condition). Fisrt we give a special picture fuzzy t-norm on picture fuzzy sets For all * ,:xyD 

Definition 2.7. We say that * 12112ker(,)(,),, TisweathanTifTxyTxyxyD  then we write 12, TT  We write 12, TT  if 12, TT  and 12TT 

Proposition 2.8. For any picture fuzzy t-norm * 1inf (,)(,),, TxyTxyxyD  It means that for any picture fuzzy t-norm we have infTT 

Proof. Let T be a picture fuzzy t-norm, we have * * 11 (,)(,1),, D TxyTxxxyD 

and * * 11 (,)(,1),, D TxyTyyxyD 

It implies for any picture fuzzy t-norm T we have * * 1 (,)(,1)inf(,),, D TxyTxxyxyD 

It means infTT 

Definition 2.9. A picture fuzzy t-norm T is called representable iff there exist two fuzzy t-norms 1t , 2t on [0,1] and a fuzzy t-conorm 3s on [0,1] satisfy:

* 111222333 ,,,,,,,,. TxytxytxysxyxyD  We give some representable picture fuzzy t-norms, for all * ,:xyD 

min112233 ,min,,min,,max,. Txyxyxyxy

02112233 ,min,,,max,. Txyxyxyxy

Now we give a new special picture fuzzy t-conorm. For all * , xyD          1 1133 sup ,0,,|| ,sup,. max,, xyxyifxy Sxyxy xyelse        

Proposition 2.11 For any picture fuzzy t-conorm * 1sup (,)(,),, SxySxyxyD  It means that for any picture fuzzy t-conorm S we have sup SS  Proof. Let S be a picture fuzzy t-conorm, we have * * 11 (,)(,0),, D SxySxxxyD  and * * 11 (,)(,0),, D SxySyyxyD  It implies for any S be a picture fuzzy t-norm, * * 11 (,)(,1)sup(,),, D SxySxxyxyD  It means sup SS 

Definition 2.12. A picture fuzzy t-conorm S is called representable iff there exist two fuzzy tnorms 1t , 2t on [0,1] and a fuzzy t-conorm 3s on [0,1] satisfy:

* 311222133 ,,,,,,,,. SxysxytxytxyxyD  Some examples of representable picture fuzzy t-conorms, for all * ,:xyD 

      * 051106,,,. SxySx,ySx,yxyD 

Proposition 2.15. Assume   , Tuv is a t- representable intuitionistic fuzzy t-norm:             * 1113331313 ,,,,,,,, TuvtuvsuvuuuvvvL 

where, 1t is a fuzzy t-norm on [0,1], 3s is a fuzzy t-conorm on [0,1]. Assume 2t is a t-norm on [0,1] satisfies:       * 111222333 0,,,1,, txy+txy+sxyxyD  then:           * 111222333 ,,,,,,,, TxytxytxysxyxyD  is a representable picture fuzzy t-norm.

Proposition 2.16. Assume   , Suv is a t-representable intuitionistic fuzzy t-conorm: 

    * 3111331313 ,,,,,,,, SuvsuvtuvuuuvvvL  where, 1t is a fuzzy t-norm on [0,1], 3s is a fuzzy t-conorm on [0,1]. Assume 2t is a t-norm on [0,1] satisfies:       * 111222333 0,,,1,, txy+txy+sxyxyD  then:           * 311222133 ,,,,,,,, SxysxytxytxyxyD  is a representable picture fuzzy t-conorm. Now we define some new concepts for the neutrosophic logics. Definition 2.17. A picture fuzzy t-norm T is called Achimerdean iff:     ** * 1 \0,1,,. DD xDTxxx  Definition 2.18 A picture fuzzy t-norm T is called:

nilpotent iff:     ** * ,\0,,0. DDxyDTxy 

strict iff:     ** * ,\0,,0. DDxyDTxy  With these defitions we have the following proposition: Proposition 2.19. Let   * , Tnilpotentpicturefuzzytnorms 

  ** Tstrictpicturetnorms  . Then ***TT

Definition 2.20. A picture fuzzy t-conorm S is called Achimerdean iff:     ** * 1 \0,1,,. DD xDSxxx 

Definition 2.21. A picture fuzzy t-conorm S is called:  nilpotent iff:     ** * ,\1,,1. DDxyDSxy 

strict iff:     ** * ,\1,,1. DDxyDSxy  Proposition 2.22. Let   * , Snilpotentpicturefuzzytconorms    ** Sstrictpicturetconorms  . Then ***SS

Proposition 2.23. Assume T is a representable picture fuzzy t-norm: 



 

 * 111222333 ,,,,,,,,, TxytxytxysxyxyD  and 12,,tt 3s are Archimedean on [0,1] [5], then T is Archimedean. Proof. For all   ** * \0,1DD xD  , we have:           111222333 ,,,,,,. Txxtxxtxxsxx  Since 12,,tt 3s are Archimedean on [0,1] . It follows that     11113333 ,,,, txxxsxxx  so   1 ,. Txxx  Thus T is Archimedean

Proposition 2.24 Assume S is a representable picture fuzzy t-conorm:           * 311222133 ,,,,,,,,. SxysxytxytxyxyD  and 12,,tt 3s are Archimedean on [0,1], then S is Archimedean.

3 A classification of representable picture fuzzy t-norms

We can give a classification of representable picture fuzzy t-norms to subclasses as follows: 3.1. Strict-strict-strict t-norms subclass, denoted by sss  Definition 3.1. A picture fuzzy t-norm T is called strict-strict-strict iff

          * 111222333 ,,,,,,,,, TxytxytxysxyxyD  where 1t , 2t are strict fuzzy t-norms on [0,1] and 3s is a strict fuzzy t-conorm on [0,1].

xyxy Txy xyxyxyxy xyxya

(,)(,, (1)()(1)() ()),,,[1,), aaaa a

 

   3.2.Nipoltent-nipoltent-nipoltent t-norms subclass, denoted by nnn  : Definition 3.2. A picture fuzzy t-norm T is called nipoltent-nipoltent-nipoltent iff           * 111222333 ,,,,,,,,, TxytxytxysxyxyD  where 1t , 2t are nipoltent fuzzy t-norms on [0,1] and 3s is a nipoltent fuzzy t-conorm on [0,1]. Example 3.2. 3112233 (,)(0(1),0(1),1()), Txyxyxyxy  4111111222222 1 3312

(,)(((1)(1))0,((1)(1))0, 1()),,[0,),1, aa a

Txyxyxyxyxy xya

 

11 (,)(((1(1))0),((1(1))0), 1()),,(0,1];1, cc c

  111 5112233 (,)((0(1)),(0(1)),1()),,,1, aabbcc abc Txyxyxyxyabc  611112222 1 33

Txyxyaxyxybxy ab xyabc  711112222 1 33

 1 (,)(((1(1))0),((1)(1))0, 1()),(0,1],0,1, bb b

Txyxyaxyxyxy a xyab

 

1 (,)(((1)(1))0,((1(1))0), 1()),(0,1],1,0, bb b

Txyxyxyxyaxy a xyab

  811112222 1 33

 

 

Txyxyaxyxy a xyabc

1 9111122 1 33

bb b cc c

1 (,)(((1(1))0),0(1), 1()),(0,1],,1,

Txyxyxybxy b xybac

  1 10112222 1 33

1 (,)(0(1),((1(1))0), 1()),(0,1],,1.

aa a cc c

Txyxyxyxy xyab

  1 11111122 1 33

 

(,)(((1)(1))0,0(1), 1()),0,,1,

aa a bb b

  1 12112222 1 33

  3.3. Nipoltent-nipoltent-strict t-norms subclass, denoted by nns 

aa a bb b

 

Txyxyxyxy xyab

(,)(0(1),((1)(1))0, 1()),0,,1.

Definition 3.3. A picture fuzzy t-norm T is called nipoltent-nipoltent-strict iff 

* 111222333 ,,,,,,,,, TxytxytxysxyxyD  where 1t , 2t are nipoltent fuzzy t-norms on [0,1] and 3s is a strict fuzzy t-conorm on [0,1]. Examples 3.3 1311223333 (,)(0(1),0(1),). Txyxyxyxyxy  1 14111122223333 11 (,)((1)0,(1)0,()),1,

Txyxyaxyxybxy ab xyxyabc

11 (,)((1(1))0,(1(1))0, ()),,(0,1];1, cccc c

1811112222 1 3333

Txyxyaxyxybbxy a xyxyabc

  1911112222 1 3333

1 (,)((1(1))0,((1)(1))0, ()),(0,1];0;1, cccc c

1 (,)(0(1),(1(1))0, ()),(0,1];,1,

Txyxyxybxy b xyxybac

  1 20112222 1 3333

aa a cccc c

Txyxyaaxyxybxy b xyxyabc

  2111112222 1 3333

1 (,)(((1)(1))0,(1(1))0, ()),0;(0,1];1, cccc c

Txyxyxyxy xyxyab

  1 22111122 1 3333

 

(,)(((1)(1))0,0(1), ()),0,,1,

aa a bbbb b

  1 23112222 1 3333

aa a bbbb b

 

(,)(0(1),((1)(1))0, ()),0,,1.

Txyxyxyxy xyxyab

  3.4. Strict-nipoltent-strict t-norms subclass, denoted by sns  Definition 3.4. A picture fuzzy t-norm T is called strict -nipoltent-strict iff

* 111222333 ,,,,,,,,, TxytxytxysxyxyD  where 1t is a strict fuzzy t-norm on [0,1], 2t is a nipoltent fuzzy t-norm on [0,1] and 3s is a strict fuzzy t-conorm on [0,1]. Example 3.4. 2411223333 (,)(,0(1),). Txyxyxyxyxy 



xy Txyxy xyxy xyxyab

(,)(,0(1), (1)() ()),1,,1,

   1 11 2622 111111 1 33331

aa a bbbb b

 

   11 272222 111111 1 33331          

   3.5. Nipoltent-strict-strict t-norms subclass, denoted by nss  Definition 3.5. A picture fuzzy t-norm T is called nipoltent-strict-strict iff  

1 (,)(,(1(1))0, (1)() ()),,1,(0,1]. bbbb b 22 291111 2222 1 3333

xy Txyxyaxy xyxya xyxyba b

  xy Txyxyaxy

* 111222333 ,,,,,,,,, TxytxytxysxyxyD  where 1t is a nipoltent fuzzy t-norm on [0,1], 2t is a strict fuzzy t-norm on [0,1] and 3s is a strict fuzzy t-conorm on [0,1]. Example 3.5. 2811223333 (,)(0(1),,), Txyxyxyxyxy     1 22 3011 2222 1 3333

1 (,)((1(1))0,, (1)() ()),(0,1];,1, bbbb (,)(0(1),, (1)() ()),,,1,

axyxy xyxyab xy Txyxy xyxy xyxyab

aa a bbbb b

 

xy Txyxyxy xyxy xyxya

   22 31111111 222222 1 333321

  

(,)(((1)(1),, (1)() ()),,1,(0,1]. aaaa a

  

Proposition 3.6 There doesn’t exist t-representable picture fuzzy t-norm T :           * 111222333 ,,,,,,,,, TxytxytxysxyxyD  where 1t or 2t is a strict fuzzy t-norm on [0,1], and 3s is a nipoltent fuzzy t-conorm on [0,1].

Proof. Assume           * 111222333 ,,,,,,,,, TxytxytxysxyxyD  with 1t is a strict t-norm and exist   33,0,1xy  such that   333,1.Sxy  Let 1212312123 ,0|1;,0|1, xxxxxyyyyy  and since 1t is strict t-norm then:  111,0.txy  Let     123123 ,,,,,,xxxxyyyy  we have a contradition:       111222333,,,1.txytxysxy

Similarly, if 2t is strict t-norm and 3s is nipoltent t-conorm then we have a contraddition. Proposition 3.7. If T belongs to one of four classes sss  , nns  , sns  , nss  then T is strict.

Proof. Assume for all * ,,xyD  3s is a strict fuzzy t-conorm on [0,1], T is representable picture fuzzy t-norm:           111222333 ,,,,,, Txytxytxysxy  and T is nipoltent. Then     ** * ,\0,,0, DDxyDTxy  and it implies       111222333,0,,0,,1.txytxysxy

Since 3s is a strict fuzzy t-conorm on [0,1],then 3 1 x  or 3 1 y  , which is a contradition.

Proposition 3.8 If T belongs to the class nnn  then T is a nipoltent picture fuzzy t-norm.

Proof. Assume * ,,: nnn TxyD            111222333 ,,,,,, Txytxytxysxy  Since 12 , tt are nipoltent fuzzy t-norms on [0,1], we have

1122111222 ,,,|,0,,0. xyxytxytxy  Since 12 , tt are not decreasing, so:

4.1. Strict-strict-strict t-conorms subclass, denoted by sss 

aaaa a xy Sxyxyxy xyxy xy a xyxy t-conorm on [0,1]. Example 4.2 3112233 (,)(1(),0(1),0(1)), Sxyxyxyxy  1 411221122 33223312

   (,)(1(),((1)(1))0, ((1)(1))0),,[0,),1,

nnn  : Definition 4.2. A picture fuzzy t-conorm S is called nipoltent-nipoltent-nipoltent iff aabbab cc c

    4.2.Nipoltent-nipoltent-nipoltent aa a Sxyxyxyxy xyxya     11 51122 1 33

t-conorms subclass, denoted by (,)(1(),(0(1)), 0(1)),,,1,

311222133 ,,,,,,,,. SxysxytxytxyxyD  where 1t , 2t are nipoltent fuzzy t-norms on [0,1] and 3s is a nipoltent fuzzy Sxyxyxy xyabc

 

1 (,)(1(),((1(1))0), 1 ((1(1))0)),1;,(0,1],

aa a Sxyxyxybxy b xycxyabc c

1 6112222 3333

aa a Sxyxyxybxy b xyxyab

  1 7112222 3333

1 (,)(1(),((1(1))0), ((1)(1))0),1,(0,1],0,

  1 8112222 3333

aa a Sxyxyxyxy xybxyab b

(,)(1(),((1)(1))0, 1 ((1(1))0)),1,(0,1],0,

 

Sxyxyxybxy b xybac

  1 9112222 1 33

1 (,)(1(),((1(1))0), 0(1)),(0,1],,1,

aa a cc c

(,)(1(),0(1), 1 ((1(1))0)),(0,1],,1,

  11 101122 3333

aabbabSxyxyxy xycxycab c

Sxyxyxyxy xyab

  1 11112222 1 33

aa a bb b

 

(,)(1(),((1)(1))0, 0(1)),0,,1,

  11 121122 3333

(,)(1(),0(1), ((1)(1))0),0,,1. aabbabSxyxyxy xyxyab    4.3.Strict-nipoltent-nipoltent t-conorms subclass, denoted by snn  Definition 4.3. A picture fuzzy t-conorm S is called strict-nipoltent-nipoltent iff

Sxyxyxyxybxy b xybac

aaaa a cc c

1 (,)((),(1(1))0, 1 (1(1))0),,(0,1];1,

aaaa a Sxyxyxyxybxy b xycxybca c

  1 1811112222 3333

  1 1911112222 3333

1 (,)((),(1(1))0, ((1)(1))0),1,(0,1];0,

  11 20111122 3333 aaaa a bb b

aaaa a Sxyxyxyxybxy b xyccxyabc  1 2211112222 1 33

aaaabbabSxyxyxyxy xycxycab c   11 23111122 3333

(,)((),(1)0, 1 (1(1))0),(0,1];,1,  

  1 2111112222 3333 xyxyab    4.4.Strict-nipoltent-strict

Sxyxyxyxyxy xyab

(,)((),0(1), ((1)(1))0),0,,1. aaaabbabSxyxyxyxy

A

(,)((),((1)(1))0, 0(1)),0,,1,

(,)((),((1)(1))0, 1 (1(1))0),1;0;(0,1], subclass,

t-conorms

denoted

aaaa a Sxyxyxyxybbxy xycxyabc c by sns  Definition 4.4.

 picture

fuzzy t-conorm S is called strict-nipoltent-strict iff

on

on

Proposition 4.6. There doesn’t exist representable picture fuzzy t-conorm S :

311222133 ,,,,,,,,. SxysxytxytxyxyD  where 1t or 2t is strict fuzzy t-norm on [0,1] and 3s is a nipoltent fuzzy t-conorm on [0,1].

Proposition 4.7. If S belongs to one of four classes sss  , snn  , sns  , ssn  then S is strict.

Proposition 4.8. If S belongs to the class nnn  then S is nipoltent.

5. Conclusion

t-norms and t-conorms are basic operators of the fuzzy logics [5,6]. Picture fuzzy t-norms and picture fuzzy t-conorms firstly defined and studied in 2015 [13]. In this paper we give some algebraic properties of the picture fuzzy t-norms and the picture fuzzy t-conrms on standard neutrsophic sets , including some classifications of the class of representable picture fuzzy t-norms and and of the class of representable picture fuzzy t-conorms. Another important operators of picture fuzzy logics should be considered in the future papers.

Acknowledgment

This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.01-2017. .

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