Some new de morgan picture operator triples in picture fuzzy logic

Journal of Computer Science and Cybernetics, V.33, N.2 (2017), 143–164 DOI 10.15625/1813-9663/33/2/10706 SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC BUI CONG CUONG1, ROAN THI NGAN2, LE BA LONG3 1Institute of Mathematics; 1bccuong@gmail.com 2Basic Science Faculty, Ha Noi University of Natural Resources and Environment 3Post and Telecommunication Institute of Technology
Abstract. A new concept of picture fuzzy sets (PFS) were introduced in 2013, which are direct extensions of the fuzzy sets and the intuitionistic fuzzy sets. Then some operations on PFS with some properties are considered in [7, 5]. Some basic operators of fuzzy logic as negation, t-norms, t-conorms for picture fuzzy sets firstly are defined and studied in [6, 9]. This paper is devoted to some classes of representable picture fuzzy t-norms and representable picture fuzzy t-conorms on PFS and a basic algebra structure of Picture Fuzzy Logic De Morgan triples of picture operators. Keywords. Picture fuzzy sets, Picture fuzzy t-norms, Picture fuzzy t-conorm, De Morgan picture operator triple. 1. INTRODUCTION Recently, Bui Cong Cuong and Kreinovich (2013) first defined “picture fuzzy sets” [7, 5], which are a generalization of the Zadeh’s fuzzy sets [27, 28] and the Antanassov’s intuition- istic fuzzy sets [2, 1]. This concept is particularly effective in approaching the practical problems in relation to the synthesis of ideas, such as decisions making problems, voting analysis, fuzzy clustering, financial forecasting. The basic definitions and basic operators in the picture fuzzy sets theory were given in [3, 4, 7, 5]. The new basic logic connectives on the PFS firstly were presented in [6, 9]. These new concepts are supporting to new computing procedures in computational intelligence problems and in other applications (see [14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26]). In this paper we study some algebraic properties of the picture fuzzy t-norms and the picture fuzzy t-conorms on PFS, which are basic operators of the Picture Fuzzy Logics. Some classes of the representable picture fuzzy t-norms and the representable picture fuzzy t-conorms were first given in [1, 8] will be presented. Then a basic algebra structure on PFS – De Morgan picture operator triples will be considered and some new De Morgan picture operator triples will be presented. We first recall some basic notions of the picture fuzzy sets. Definition 1.1. [7] A picture fuzzy set A on a universe X is an object of the form A = {(x, µA(x), ηA(x), νA(x)) |x ∈ X} , c© 2017 Vietnam Academy of Science & Technology 144 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG where µA(x), ηA(x), νA(x) 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 ≤ µA(x), ηA(x), νA(x) ≤ 1 and µA(x) + ηA(x) + νA(x) ≤ 1,∀x ∈ X. Then, ∀x ∈ X : 1− (µA(x) + ηA(x) + νA(x)) is called the degree of refusal membership of x in A. Consider the set D∗ = { x = (x1, x2, x3)|x ∈ [0,1]3, x1 + x2 + x3 ≤ 1 } . From now on, we will assume that if x ∈ D∗, then x1, x2 and x3 denote, respectively, the first, the second and the third component of x, i.e. , x = (x1, x2, x3). We have a lattice (D∗,≤1), where ≤1 defined for ∀x, y ∈ D∗ (x ≤1 y)⇔ (x1 y3) ∨ ({x1 = y1, x3 = y3, x2 ≤ y2}) , (x = y)⇔ (x1 = y1, x2 = y2, x3 = y3), ∀x, y ∈ D∗ We define the first, second and third projection mapping pr1, pr2 and pr3 on D ∗ as pr1(x) = x1 and pr2(x) = x2 and pr3(x) = x3, on all x ∈ D∗. Note that, if for x, y ∈ D∗ that neither x ≤1 y nor y ≤1 x, then x and y are incomparable w.r.t ≤1, and denoted as x‖≤1 y. From now on, we denote u ∧ v = min(u, v), u ∨ v = max(u, v) for all u, v ∈ R1. For each x, y ∈ D∗, we define inf(x, y) = { min(x, y), if x ≤1 y or y ≤1 x (x1 ∧ y1, 1− x1 ∧ y1 − x3 ∨ y3, x3 ∨ y3), else sup(x, y) = { max(x, y), if x ≤1 y or y ≤1 x (x1 ∨ y1, 0, x3 ∧ y3), else Proposition 1.2. With these operators (D∗,≤1) is a complete lattice. Proof. See [6, 9]. Using this lattice, we easily see that every picture fuzzy set A = {(x, µA(x), ηA(x), νA(x)) |x ∈ X} corresponds an D∗− fuzzy set [12] mapping, i.e., we have a mapping A : X → D∗ : x→ {(x, µA(x), ηA(x), νA(x)) |x ∈ X}. Interpreting picture fuzzy sets as D∗−fuzzy sets gives way to greater flexibility in calcu- lating with membership degrees, since the triple of numbers formed by the three degrees is an element of D∗, and often allows to obtain more compact formulas.  SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 145 2. PICTURE FUZZY NEGATION OPERATOR Now we consider some basic fuzzy operators of the Picture Fuzzy Logics. Picture fuzzy negations are an extension of the fuzzy negations [22] and the intuitionistic fuzzy negations [2]. They are defined as follows. Definition 2.1.A mapping N : D∗ → D∗ satisfying conditions N(0D∗) = 1D∗ and N(1D∗) = 0D∗ and N is nonincreasing is called a picture fuzzy negation operator. If N(N(x)) = x for all x ∈ D∗, then N is called an involutive negation operator. Definition 2.2. Let f1, f2 : D ∗ → D∗ be mappings on D∗. We say that the mapping f2 is greater than f1 if f1(x) ≤1 f2(x), ∀x ∈ D∗, and we denote that as f1 ≤ f2. We write f1 < f2, if f1 ≤ f2, and f1 6= f2. Let x = (x1, x2, x3) ∈ D∗. We first give 2 drastic picture negation operators nd (x) = { 0D∗ if x 6= 0D∗ 1D∗ if x = 0D∗ and nd2 (x) = { 1D∗ if x 6= 1D∗ 0D∗ if x = 1D∗ . Proposition 2.3. nd and nd2 are picture negation operators and for each picture negation operator n(x), nd(x) ≤1 n(x) ≤1 nd2(x), ∀x ∈ D∗. Definition 2.4. The mapping n0 : D ∗ → D∗ defined by n0(x) = (x3, 0, x1), for each x ∈ D∗. Proposition 2.5. n0 is a picture fuzzy negation operator. It is called the simple picture negation. Proof. Indeed, 1D∗ = (1, 0, 0) ∈ D∗ then n0(1D∗) = n0(1, 0, 0) = (0, 0, 1) = 0D∗ . Analo- gously, n0(0D∗) = n0(0, 0, 1) = (1, 0, 0) = 1D∗ ∈ D∗. Let x, y ∈ D∗ and x ≤1 y. Consider 3 subsets B1 = {(x1 < y1) ∧ (x3 ≥ y3)}, B2 = {(x1 = y1) ∧ (x3 > y3)}, B3 = {(x1 = y1) ∧ (x3 = y3) ∧ (x2 ≤ y2)}. We have to consider 4 following cases Case 1a. x1 < y1 and x3 = y3 then (n0 (y) , n0 (x)) ∈ B2 ⇒ n0 (y)≤1n0 (x), Case 1b. x1 y3 then (n0 (y) , n0 (x)) ∈ B1 ⇒ n0 (y)≤1n0 (x), Case 2. x1 = y1 and x3 > y3 then (n0 (y) , n0 (x)) ∈ B1 ⇒ n0 (y)≤1n0 (x), Case 3. x1 = y1 and x3 = y3 then (n0 (y) , n0 (x)) ∈ B3 ⇒ n0 (y)≤1n0 (x). It shows that the mapping n0(x) = (x3, 0, x1) is non-increasing and the operator n0(x) is a picture negation operator.  Definition 2.6. Let x = (x1, x2, x3) ∈ D∗. Denote x4 = 1 − (x1 + x2 + x3). The mapping NS is by NS(x) = (x3, x4, x1), for each x ∈ D∗. Proposition 2.7. NS is an involutive picture negation operator and is called the picture standard negation operator. Proof. It is analogous to the proof of the Proposition 2.5 Some other picture fuzzy negations were given in [9]. 146 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG 3. PICTURE FUZZY T-NORMS AND PICTURE FUZZY T-CONORMS Fuzzy t-norms on [0, 1] and fuzzy t-conorms on [0, 1] were defined and considered in [22, 13]. In 2004 , G.Deschrijver et al. [10] introduced the notion of intuitionistic fuzzy t-norms and t-conorms and investigated under which conditions a similar representation theorem could be obtained. For further usage, we define L∗ = {x ∈ D∗ |x2 = 0} . We can consider the set L∗ defined by L∗ = {u = (u1, u3) ∣∣u ∈ [0, 1]2, u1 + u3 ≤ 1} . Consider the order relation u ≤ v on L∗, defined by u ≤ v ⇔ ((u1 ≤ v1)∧ (u3 ≥ v3)), for all u, v ∈ L∗. We define the first, and the second projection mapping pr1 and pr3 on L ∗, as pr1(u) = u1 and pr3(u) = u3, on all u ∈ L∗. The units of L∗ are 1L∗ = (1, 0) and 0L∗ = (0, 1). Definition 3.1. [10] An intuitionistic fuzzy t-norm is a commutative, associative, increasing (L∗)2 → L∗ mapping T satisfying T (1L∗ , u) = u, for all u ∈ L∗. Definition 3.2. [10] An intuitionistic fuzzy t-conorm is a commutative, associative, increas- ing (L∗)2 → L∗ mapping S satisfying S(v, 0L∗) = v, for all v ∈ L∗. Definition 3.3. [10] An intuitionistic fuzzy t-norm T is called t-representable iff there exist a fuzzy t-norm t1 on [0, 1] and a fuzzy t-conorm s3 on [0, 1] satisfying for all u, v ∈ L∗, T (u, v) = (t1(u1, v1), s3(u3, v3)). Definition 3.4. [10] An intuitionistic fuzzy t-conorm S is called t-representable iff there exist a fuzzy t-norm t1 on [0,1] and a fuzzy t-conorm s3 on [0,1] satisfying for all u, v ∈ L∗, S(u, v) = (s3(u1, v1), t1(u3, v3)). Now we define picture fuzzy t-norms and picture fuzzy t-conorms, which are classes of conjunction operators and classes of disjunction operators - main basic operators of the picture fuzzy logics. Picture fuzzy t-norms are direct extensions of the fuzzy t-norms in [28, 22, 13] and of the intuitionistic fuzzy t-norms in [2], and they are important operators in [11]. Let x = (x1, x2, x3) ∈ D∗. Denote I(x) = {y ∈ D∗ : y = (x1, y2, x3), 0 ≤1 y2 ≤1 x2}. Definition 3.5. A mapping T : D∗ ×D∗ → D∗ is a picture fuzzy t-norm if the mapping T satisfies the following conditions T (x, y) = T (y, x), ∀x, y ∈ D∗ (commutative), T (x, T (y, z)) = T (T (x, y), z), ∀x, y, z ∈ D∗ (associativity), T (x, y) ≤1 T (x, z), ∀x, y, z ∈ D∗, y ≤1 z (monotonicity), T (1D∗ , x) ∈ I(x), ∀x ∈ D∗ (boundary condition). Fisrt we present some picture fuzzy t-norms on picture fuzzy sets. Definition 3.6. A picture fuzzy t-norm T is called representable iff there exist two fuzzy t-norms t1, t2 on [0,1] and a fuzzy t-conorm s3 on [0,1] satisfying T (x, y) = (t1 (x1, y1) , t2 (x2, y2) , s3 (x3, y3)) , ∀x, y ∈ D∗ SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 147 . We give some representable picture fuzzy t-norms, for all x, y ∈ D∗ 1. Tmin (x, y) = (min (x1, y1) ,min (x2, y2) ,max (x3, y3)). 2. T02 (x, y) = (min (x1, y1) , x2y2,max (x3, y3)). 3. T03 (x, y) = (x1y1, x2y2,max (x3, y3)). 4. T04 (x, y) = (x1y1, x2y2, x3 + y3 − x3y3) . 5. T05 (x, y) =({ x1 ∧ y1 if x1 ∨ y1 = 1 0ifx1 ∨ y1 < 1 , { x2 ∧ y2 ifx2 ∨ y2 = 1 0ifx2 ∨ y2 < 1 , { x3 ∨ y3 ifx3 ∧ y3 = 0 1ifx3 ∧ y3 6= 0 ) . 6. T06 (x, y) = (max (0, x1 + y1 − 1) ,max (0, x2 + y2 − 1) ,min (1, x3 + y3)). 7. T07 (x, y) = (max (0, x1 + y1 − 1) ,max (0, x2 + y2 − 1) , x3 + y3 − x3y3). 8. T08 (x, y) =( max { 1 2 (x1 + y1 − 1 + x1y1) , 0 } ,max { 1 2 (x2 + y2 − 1 + x2y2) , 0 } , x3 + y3 − x3y3 ) . 9. T09 (x, y) = (x1y1,max (0, x2 + y2 − 1) , x3 + y3 − x3y3). 10. T010 (x, y) = (max (0, x1 + y1 − 1) , x2y2, x3 + y3 − x3y3). In this part we give some detailed proofs of picture t - norms. Proposition 3.7. Let x, y ∈ D∗, x = (x1, x2, x3), y = (y1, y2, y3). The mapping Tmin is a picture fuzzy t-norm. Proof. Let x, y ∈ D∗, then x1 + x2 ≤ 1− x3, and y1 + y2 ≤ 1− y3, (x1 ∧ y) + (x2 ∧ y2) ≤ min(1− x3, 1− y3) = 1−max(x3, y3), (x1 ∧ y) + (x2 ∧ y2) + max(x3, y3) ≤ 1, Tmin(x, y) = ((x1 ∧ y), (x2 ∧ y2),max(x3, y3)) ∈ D∗. The mapping Tmin is a picture fuzzy t-norm, since other conditions easily are verified.  Let x, y ∈ D∗, x = (x1, x2, x3), y = (y1, y2, y3). Proposition 3.8. The mapping T02 (x, y) = (min (x1, y1) , x2y2, max (x3, y3)) is a picture fuzzy t-norm. Proof. We remark that x2.y2 ≤ x2 ∧ y2 ⇒ ((x1 ∧ y1) + x2.y2 + max(x3, y3)) ≤ T (x, y) ∈ D∗. It implies that the mapping T02 (x, y) = (min (x1, y1) , x2y2, max (x3, y3)) is a picture fuzzy t-norm.  Proposition 3.9. The mapping T04 (x, y) = (x1y1, x2y2, x3 + y3 − x3y3) is a picture fuzzy t-norm. 148 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG Proof. We have x1y1 + x2y2 ≤ (1− x2 − x3) (1− y2 − y3) + x2y2 = (1− y2 − y3 − x2 + x2y2 + x2y3 − x3 + x3y2 + x3y3) + x2y2 = (1− x3 − y3 + x3y3) + (x2y2 + x2y3 + x2y2 + x3y2 − x2 − y2) = (1− x3 − y3 + x3y3) + (x2 (y2 + y3 − 1) + y2 (x2 + x3 − 1)) ≤ 1− x3 − y3 + x3y3 ⇒ x1y1 + x2y2 + x3 + y3 − x3y3 ≤ 1.  Proposition 3.10. Let mapping t2 is a fuzzy t-norm on [0, 1], then the mapping T0t2 (x, y) = (min (x1, y1) , t2(x2, y2),max (x3, y3)) , is a picture fuzzy t - norm. Proof. See the proof of the Proposition 3.8. Definition 3.11. A mapping S : D∗ ×D∗ → D∗ is a picture fuzzy t-conorm if S satisfies all of the following conditions 1. S (x, y) = S (y, x) , ∀x, y ∈ D∗ (commutative), 2. S (x, S (y, z)) = S (S (x, y) , z) ,∀x, y, z ∈ D∗ (associativity), 3. S (x, y)≤1S (x, z) , ∀x, y, z ∈ D∗, y≤1z (monotonicity), 4. S (0D∗ , x) ∈ I (x) , ∀x ∈ D∗ (boundary condition). Definition 3.12. A picture fuzzy t-conorm S is called representable iff there exist two fuzzy t-norms t1, t2 on [0,1] and a fuzzy t-conorm s3 on [0,1] satisfying. Some examples of representable picture fuzzy t-conorms, for all x, y ∈ D∗. 1. Smax (x, y)= (max (x1, y1),min (x2, y2),min (x3, y3)). 2. S02 (x, y) = (max (x1, y1), x2y2,min (x3, y3)). 3. S03 (x, y) = (max (x1, y1), x2y2, x3y3). 4. S04 (x, y) = (x1 + y1 − x1y1, x2y2, x3y3). 5. S05 (x, y) = ( x1 ∨ y1, { x2 ∧ y2 if x2 ∨ y2 = 1 0 if x2 ∨ y2 < 1 , x3 ∧ y3 ) . 6. S06 (x, y) = ({ x1 ∨ y1 if x1 ∧ y1 = 0 1 if x1 ∧ y1 6= 0 , x2 ∧ y2, { x3 ∧ y3 if x3 ∨ y3 = 1 0 if x3 ∨ y3 < 1 ) . SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 149 Proposition 3.13. For any representable picture fuzzy t-norm T we have T05 (x, y) ≤1 T (x, y) ≤1 Tmin (x, y) ,∀x, y ∈ D∗. Proposition 3.14. For any representable picture fuzzy t-conorm S we have S05 (x, y) ≤1 S (x, y) ≤1 S06 (x, y), ∀x, y ∈ D∗. Proposition 3.15. Assume T (u, v) is a t- representable intuitionistic fuzzy t-norm T (u, v) = (t1 (u1, v1), s3 (u3, v3)),∀u = (u1, u3), v = (v1, v3) ∈ L∗ where, t1 is a fuzzy t-norm on [0, 1], s3 is a fuzzy t-conorm on [0, 1]. Assume t2 is a t-norm on [0, 1] satisfying. 0 ≤ t1 (x1, y1) + t2 (x2, y2) + s3 (x3, y3) ≤ 1,∀x, y ∈ D∗ then T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗ is a representable picture fuzzy t-norm. Proposition 3.16. Assume S (u, v) is a t-representable intuitionistic fuzzy t-conorm S (u, v) = (s3 (u1, v1), t1 (u3, v3)), ∀ u = (u1, u3), v = (v1, v3) ∈ L∗ where, t1 is a fuzzy t-norm on [0, 1], s3 is a fuzzy t-conorm on [0, 1]. Assume t2 is a t-norm on [0, 1] satisfies 0 ≤ t1 (x1, y1) + t2 (x2, y2) + s3 (x3, y3) ≤ 1,∀x, y ∈ D∗ then S (x, y) = (s3 (x1, y1), t2 (x2, y2), t1 (x3, y3)),∀x, y ∈ D∗ is a representable picture fuzzy t-conorm. Now we define some new concepts for the picture fuzzy logic. Definition 3.17. A picture fuzzy t-norm T is called Achimerdean iff ∀x ∈ D∗\ {0D∗ , 1D∗} , T (x, x) <1 x. Definition 3.18. A picture fuzzy t-norm T is called 1. Nilpotent iff: ∃x, y ∈ D∗\ {0D∗} , T (x, y) = 0D∗ . 2. Strict iff: ∀x, y ∈ D∗\ {0D∗} , T (x, y) 6= 0D∗ . With these defitions we have the following propositions. Proposition 3.19. Let T ∗ = {Nilpotent picture t− norms}, T ∗∗ = {strict picture t− norms} then T ∗ ∩ T ∗∗ = ∅. Definition 3.20. A picture fuzzy t-conorm S is called Achimerdean iff ∀x ∈ D∗\ {0D∗ , 1D∗} , S (x, x) >1 x. Definition 3.21. A picture fuzzy t-conorm S is called 150 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG 1. Nilpotent iff: ∃x, y ∈ D∗\ {1D∗} , S (x, y) = 1D∗ . 2. Strict iff: ∀x, y ∈ D∗\ {1D∗} , S (x, y) 6= 1D∗ . Proposition 3.22. Let S∗ = {nilpotent picture fuzzy t− conorms}, S∗∗ = {strict picture t− conorms} then S∗ ∩ S∗∗ = ∅. Proposition 3.23. Assume T is a representable picture fuzzy t-norm T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗, and t1, t2, s3 are Archimedean on [0, 1], then T is Archimedean. Proof. For all x ∈ D∗\ {0D∗ , 1D∗}, we have T (x, x) = (t1 (x1, x1), t2 (x2, x2), s3 (x3, x3)). Since t1, t2, s3 are Archimedean on [0,1]. It follows that t1 (x1, x1) x3, so T (x, x) <1 x. Thus T is Archimedean.  Proposition 3.24. Assume S is a representable picture fuzzy t-conorm S (x, y) = (s3 (x1, y1), t2 (x2, y2), t1 (x3, y3)),∀x, y ∈ D∗. and t1, t2, s3 are Archimedean on [0, 1], then S is Archimedean. 4. SOME SUBCLASSES OF REPRESENTABLE PICTURE FUZZY T-NORMS We can give some subclasses of representable picture fuzzy t-norms. 4.1. Strict-strict-strict t-norms subclass, denoted by ∆sss Definition 4.1. A picture fuzzy t-norm T is called strict-strict-strict iff T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)),∀x, y ∈ D∗, where t1, t2 are strict fuzzy t-norms on [0,1] and s3 is a strict fuzzy t-conorm on [0,1]. Example 4.1. T1(x, y) = (x1y1, x2y2, x3 + y3 − x3y3), T2(x, y) = ( x1y1 λ1 + (1− λ1)(x1 + y1 − x1y1) , x2y2 λ2 + (1− λ2)(x2 + y2 − x2y2) , (x a 3 + y a 3 − xa3ya3) 1 a ) , λ1, λ2, a ∈ [1,+∞), SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 151 4.2. Nipoltent-nipoltent-nipoltent t-norms subclass, denoted by ∆nnn Definition 4.2. A picture fuzzy t-norm T is called nipoltent-nipoltent-nipoltent iff T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗, where t1, t2 are nipoltent fuzzy t-norms on [0,1] and s3 is a nipoltent fuzzy t-conorm on [0,1]. Examples 4.2. T3(x, y) = (0 ∨ (x1 + y1 − 1), 0 ∨ (x2 + y2 − 1), 1 ∧ (x3 + y3)), T4(x, y) = (((x1 + y1 − 1)(1 + λ1)− λ1x1y1) ∨ 0, ((x2 + y2 − 1)(1 + λ2)− λ2x2y2) ∨ 0, 1 ∧ (xa3 + ya3) 1 a ), λ1, λ2 ∈ [0,+∞), a ≥ 1, T5(x, y) = ((0 ∨ (xa1 + ya1 − 1)) 1 a , (0 ∨ (xb2 + yb2 − 1)) 1 b , 1 ∧ (xc3 + yc3) 1 c ), a, b, c ≥ 1, T6(x, y) = (( 1 a(x1 + y1 − 1 + (a− 1)x1y1) ∨ 0), ( 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0), 1 ∧ (xc3 + yc3) 1 c ), a, b ∈ (0, 1]; c ≥ 1, T7(x, y) = (( 1 a (x1 + y1 − 1 + (a− 1)x1y1) ∨ 0), ((x2 + y2 − 1)(1 + λ)− λx2y2) ∨ 0, 1 ∧ (xb3 + yb3) 1 b ), a ∈ (0, 1], λ ≥ 0, b ≥ 1, T8(x, y) = (((x1 + y1 − 1)(1 + λ)− λx1y1) ∨ 0, (1 a (x2 + y2 − 1 + (a− 1)x2y2) ∨ 0), 1 ∧ (xb3 + yb3) 1 b ), a ∈ (0, 1], b ≥ 1, λ ≥ 0, T9(x, y) = (( 1 a (x1 + y1 − 1 + (a− 1)x1y1) ∨ 0), 0 ∨ (xb2 + yb2 − 1) 1 b , 1 ∧ (xc3 + yc3) 1 c ), a ∈ (0, 1], b, c ≥ 1, T10(x, y) = (0 ∨ (xa1 + ya1 − 1) 1 a , ( 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0), 1 ∧ (xc3 + yc3) 1 c ), b ∈ (0, 1], a, c ≥ 1, T11(x, y) = (((x1 + y1 − 1)(1 + λ)− λx1y1) ∨ 0, 0 ∨ (xa2 + ya2 − 1) 1 a , 1 ∧ (xb3 + yb3) 1 b ), λ ≥ 0, a, b ≥ 1, T12(x, y) = (0 ∨ (xa1 + ya1 − 1) 1 a , ((x2 + y2 − 1)(1 + λ)− λx2y2) ∨ 0, 1 ∧ (xb3 + yb3) 1 b ), λ ≥ 0, a, b ≥ 1. 152 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG 4.3. Nipoltent-nipoltent-strict t-norms subclass, denoted by ∆nns Definition 4.3. A picture fuzzy t-norm T is called nipoltent-nipoltent-strict iff T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗, where t1, t2 are nipoltent fuzzy t-norms on [0,1] and s3 is a strict fuzzy t-conorm on [0,1]. Examples 4.3. T13(x, y) = (0 ∨ (x1 + y1 − 1), 0 ∨ (x2 + y2 − 1), x3 + y3 − x3y3), T14(x, y) = ( 1 2 (x1 + y1 − 1 + x1y1) ∨ 0, 1 2 (x2 + y2 − 1 + x2y2) ∨ 0, (xa3 + ya3 − xa3ya3) 1 a ), a ≥ 1. T15(x, y) = (((x1 + y1 − 1)(1 + λ1)− λ1x1y1) ∨ 0, ((x2 + y2 − 1)(1 + λ2)− λ2x2y2) ∨ 0, (xa3 + y a 3 − xa3ya3) 1 a ), λ1, λ2 ∈ [0,+∞), a ≥ 1, T16(x, y) = (0 ∨ (xa1 + ya1 − 1) 1 a , (xb2 + y b 2 − 1) 1 b ∨ 0, (xc3 + yc3 − xc3yc3) 1 c ), a, b, c ≥ 1, T17(x, y) = ( 1 a(x1 + y1 − 1 + (a− 1)x1y1) ∨ 0, 0 ∨ (xb2 + yb2 − 1) 1 b , (xc3 + y c 3 − xc3yc3) 1 c ), a ∈ (0, 1]; b, c ≥ 1, T18(x, y) = ( 1 a(x1 + y1 − 1 + (a− 1)x1y1) ∨ 0, 1b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0, (xc3 + y c 3 − xc3yc3) 1 c ), a, b ∈ (0, 1]; c ≥ 1, T19(x, y) = ( 1 a (x1 + y1 − 1 + (a− 1)x1y1) ∨ 0, ((x2 + y2 − 1)(1 + b)− bx2y2) ∨ 0, (xc3 + y c 3 − xc3yc3) 1 c ), a ∈ (0, 1]; b ≥ 0; c ≥ 1, T20(x, y) = (0 ∨ (xa1 + ya1 − 1) 1 a , 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0, (xc3 + y c 3 − xc3yc3) 1 c ), b ∈ (0, 1]; a, c ≥ 1, T21(x, y) = (((x1 + y1 − 1)(1 + a)− ax1y1) ∨ 0, 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0, (xc3 + y c 3 − xc3yc3) 1 c ), a ≥ 0; b ∈ (0, 1]; c ≥ 1, T22(x, y) = (((x1 + y1 − 1)(1 + λ)− λx1y1) ∨ 0, 0 ∨ (xa2 + ya2 − 1) 1 a , (xb3 + y b 3 − xb3yb3) 1 b ), λ ≥ 0, a, b ≥ 1, T23(x, y) = (0 ∨ (xa1 + ya1 − 1) 1 a , ((x2 + y2 − 1)(1 + λ)− λx2y2) ∨ 0, (xb3 + y b 3 − xb3yb3) 1 b ), λ ≥ 0, a, b ≥ 1. SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 153 4.4. Strict-nipoltent-strict t-norms subclass, denoted by ∆sns Definition 4.4. A picture fuzzy t-norm T is called strict -nipoltent-strict iff T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗, where t1 is a strict fuzzy t-norm on [0,1], t2 is a nipoltent fuzzy t-norm on [0,1] and s3 is a strict fuzzy t-conorm on [0,1]. Example 4.4. T24(x, y) = (x1y1, 0 ∨ (x2 + y2 − 1), x3 + y3 − x3y3) , T25(x, y) = ( x1y1 λ1 + (1− λ1)(x1 + y1 − x1y1) , ((x2 + y2 − 1)(1 + λ2)− λ2x2y2) ∨ 0, (xa3 + y a 3 − xa3ya3) 1 a ), λ1 ≥ 1, λ2 ≥ 0, a ≥ 1, T26(x, y) = ( x1y1 λ1 + (1− λ1)(x1 + y1 − x1y1) , 0 ∨ (x a 2 + y a 2 − 1) 1 a , (xb3 + y b 3 − xb3yb3) 1 b ), λ1 ≥ 1, a, b ≥ 1, T27(x, y) = ( x1y1 λ1 + (1− λ1)(x1 + y1 − x1y1) , 1 a (x2 + y2 − 1 + (a− 1)x2y2) ∨ 0, (xb3 + y b 3 − xb3yb3) 1 b ), λ1, b ≥ 1, a ∈ (0, 1]. 4.5. Nipoltent-strict-strict t-norms subclass, denoted by ∆nss Definition 4.5. A picture fuzzy t-norm T is called nipoltent-strict-strict iff T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗, where t1 is a nipoltent fuzzy t-norm on [0,1], t2 is a strict fuzzy t-norm on [0,1] and s3 is a strict fuzzy t-conorm on [0,1]. Example 4.5. T28(x, y) = (0 ∨ (x1 + y1 − 1), x2y2, x3 + y3 − x3y3), T29(x, y) = ( 1 a (x1 + y1 − 1 + (a− 1)x1y1) ∨ 0, x2y2 λ+ (1− λ)(x2 + y2 − x2y2) , (xb3 + y b 3 − xb3yb3) 1 b ), a ∈ (0, 1]; b, λ ≥ 1, T30(x, y) = (0 ∨ (xa1 + ya1 − 1) 1 a , x2y2 λ+ (1− λ)(x2 + y2 − x2y2) , (xb3 + y b 3 − xb3yb3) 1 b ), a, b, λ ≥ 1, T31(x, y) = (((x1 + y1 − 1)(1 + λ1)− λ1x1y1, x2y2 λ2 + (1− λ2)(x2 + y2 − x2y2) , (xa3 + y a 3 − xa3ya3) 1 a ), a, λ2 ≥ 1, λ1 ∈ (0, 1]. Proposition 4.6. There doesn’t exist representable picture fuzzy t-norm T 154 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗, where t1 or t2 is a strict fuzzy t-norm on [0, 1], and s3 is a nipoltent fuzzy t-conorm on [0, 1]. Proof. Assume T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)), ∀x, y ∈ D∗, with t1 is a strict t-norm and there exist x3, y3 ∈ (0, 1) such that S3 (x3, y3) = 1. Let x1, x2 6= 0|x1 +x2 +x3 ≤ 1; y1, y2 6= 0|y1 + y2 + y3 ≤ 1, and since t1 is strict t-norm then t1 (x1, y1) > 0. Let x = (x1, x2, x3), y = (y1, y2, y3), we have a contradiction t1 (x1, y1) + t2 (x2, y2) + s3 (x3, y3) > 1. Similarly, if t2 is strict t-norm and s3 is nipoltent t-conorm then we have a contradiction.  Proposition 4.7. If T belongs to one of four classes ∆sss, ∆nns, ∆sns, ∆nss then T is strict. Proof. Assume for all x, y ∈ D∗, s3 is a strict fuzzy t-conorm on [0,1], T is a representable picture fuzzy t-norm T (x, y) = (t1 (x1, y1), t2 (x2, y2), s3 (x3, y3)) and T is nipoltent. Then ∃x, y ∈ D∗\ {0D∗} , T (x, y) = 0D∗ , and it implies t1 (x1, y1) = 0, t2 (x2, y2) = 0, s3 (x3, y3) = 1. Since s3 is a strict fuzzy t-conorm on [0, 1], then x3 = 1 or y3 = 1, which is a contradiction.  Proposition 4.8. If T belongs to the class ∆nnn then T is a nipoltent picture fuzzy t-norm. Proof. Assume T ∈ ∆nnn,∀x, y ∈ D∗ : T (x, y) = (t1 (x1, y1) , t2 (x2, y2) , s3 (x3, y3)) . Since t1, t2 are nipoltent fuzzy t-norms on [0, 1], we have ∃x1, y1, x2, y2|t1 (x1, y1) = 0, t2 (x2, y2) = 0. Since t1, t2 are not decreasing, so ∀x′1 ≤ x1, y ′ 1 ≤ y1; x′2 ≤ x2, y′2 ≤ y2|t1 (x′1, y′1) = 0, t2 (x′2, y′2) = 0. Since s is a nipoltent fuzzy t-conorm on [0,1] so ∃x3, y3 6= 1|s3 (x3, y3) = 1. Let x = (x′1, x′2, x3) , y = (y′1, y′2, y3) ∈ D∗. Then T (x, y) = (t1 (x ′ 1, y ′ 1) , t2 (x ′ 2, y ′ 2) , s3 (x3, y3)) = 0D∗ . T is a nipoltent picture fuzzy t-norm.  5. SOME SUBCLASSES OF REPRESENTABLE PICTURE FUZZY T-CONORMS Similarly to the Section 4, we can give some subclasses of representable picture fuzzy t-conorms. 5.1. Strict-strict-strict t-conorms subclass, denoted by ∇sss Definition 5.1. A picture fuzzy t-conorm S is called strict-strict-strict iff S (x, y) = (s3 (x1, y1) , t2 (x2, y2) , t1 (x3, y3)) ,∀x, y ∈ D∗. where t1, t2 are strict fuzzy t-norms on [0, 1] and s3 is a strict fuzzy t-conorm on [0, 1]. Examples 5.1. S1 (x, y) = (x1 + y1 − x1y1, x2y2, x3y3) , SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 155 S2(x, y) = ( (xa1 + y a 1 − xa1ya1) 1 a , x2y2 λ1 + (1− λ1)(x2 + y2 − x2y2) , x3y3 λ2 + (1− λ2)(x3 + y3 − x3y3) ) , with λ1, λ2, a ∈ [1,+∞). 5.2. Nipoltent-nipoltent-nipoltent t-conorms subclass, denoted by ∇nnn Definition 5.2. A picture fuzzy t-conorm S is called nipoltent-nipoltent-nipoltent iff S (x, y) = (s3 (x1, y1) , t2 (x2, y2) , t1 (x3, y3)) , ∀x, y ∈ D∗, where t1, t2 are nipoltent fuzzy t-norms on [0, 1] and s3 is a nipoltent fuzzy t-conorm on [0, 1]. Examples 5.2. S3(x, y) = (1 ∧ (x1 + y1), 0 ∨ (x2 + y2 − 1), 0 ∨ (x3 + y3 − 1)), S4(x, y) = (1 ∧ (xa1 + ya1) 1 a , ((x2 + y2 − 1)(1 + λ1)− λ1x2y2) ∨ 0, ((x3 + y3 − 1)(1 + λ2)− λ2x3y3) ∨ 0), λ1, λ2 ∈ [0,+∞), a ≥ 1, S5(x, y) = (1 ∧ (xa1 + ya1) 1 a , (0 ∨ (xb2 + yb2 − 1)) 1 b , 0 ∨ (xc3 + yc3 − 1) 1 c ), a, b, c ≥ 1, S6(x, y) = (1 ∧ (xa1 + ya1) 1 a , ( 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0), ( 1 c (x3 + y3 − 1 + (c− 1)x3y3) ∨ 0)), a ≥ 1; b, c ∈ (0, 1], S7(x, y) = (1 ∧ (xa1 + ya1) 1 a , ( 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0), ((x3 + y3 − 1)(1 + λ)− λx3y3) ∨ 0), a ≥ 1, b ∈ (0, 1], λ ≥ 0, S8(x, y) = (1 ∧ (xa1 + ya1) 1 a , ((x2 + y2 − 1)(1 + λ)− λx2y2) ∨ 0, ( 1 b (x3 + y3 − 1 + (b− 1)x3y3) ∨ 0)), a ≥ 1, b ∈ (0, 1], λ ≥ 0, S9(x, y) = (1 ∧ (xa1 + ya1) 1 a , ( 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0), 0 ∨ (xc3 + yc3 − 1) 1 c ), b ∈ (0, 1], a, c ≥ 1, S10(x, y) = (1 ∧ (xa1 + ya1) 1 a , 0 ∨ (xb2 + yb2 − 1) 1 b , ( 1 c (x3 + y3 − 1 + (c− 1)x3y3) ∨ 0)), c ∈ (0, 1], a, b ≥ 1, S11(x, y) = (1 ∧ (xa1 + ya1) 1 a , ((x2 + y2 − 1)(1 + λ)− λx2y2) ∨ 0, 0 ∨ (xb3 + yb3 − 1) 1 b ), λ ≥ 0, a, b ≥ 1, S12(x, y) = (1 ∧ (xa1 + ya1) 1 a , 0 ∨ (xb2 + yb2 − 1) 1 b , ((x3 + y3 − 1)(1 + λ)− λx3y3) ∨ 0), λ ≥ 0, a, b ≥ 1. 156 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG 5.3. Strict-nipoltent-nipoltent t-conorms subclass, denoted by ∇snn Definition 5.3. A picture fuzzy t-conorm S is called strict-nipoltent-nipoltent iff S (x, y) = (s3 (x1, y1) , t2 (x2, y2) , t1 (x3, y3)) , ∀x, y ∈ D∗, where t1, t2 are nipoltent fuzzy t-norms on [0,1] and s3 is a strict fuzzy t-conorm on [0,1]. Examples 5.3. S13(x, y) = (x1 + y1 − x1y1, 0 ∨ (x2 + y2 − 1), 0 ∨ (x3 + y3 − 1)), S14(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 1 2 (x2 + y2 − 1 + x2y2) ∨ 0, 1 2 (x3 + y3 − 1 + x3y3) ∨ 0), a ≥ 1, S15(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , ((x2 + y2 − 1)(1 + λ1)− λ1x2y2) ∨ 0, ((x3 + y3 − 1)(1 + λ2)− λ2x3y3) ∨ 0), λ1, λ2 ∈ [0,+∞), a ≥ 1, S16(x, y) = ((x c 1 + y c 1 − xc1yc1) 1 c , 0 ∨ (xa2 + ya2 − 1) 1 a , 0 ∨ (xb3 + yb3 − 1) 1 b ), a, b, c ≥ 1, S17(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0, 0 ∨ (xc3 + yc3 − 1) 1 c ), b ∈ (0, 1]; a, c ≥ 1, S18(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0, 1 c (x3 + y3 − 1 + (c− 1)x3y3) ∨ 0), b, c ∈ (0, 1]; a ≥ 1, S19(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 1 b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0, ((x3 + y3 − 1)(1 + c)− cx3y3) ∨ 0), a ≥ 1, b ∈ (0, 1]; c ≥ 0, S20(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , (xb2 + y b 2 − 1) 1 b ∨ 0, 1 c (x3 + y3 − 1 + (c− 1)x3y3) ∨ 0), c ∈ (0, 1]; a, b ≥ 1, S21(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , ((x2 + y2 − 1)(1 + b)− bx2y2) ∨ 0, 1 c (x3 + y3 − 1 + (c− 1)x3y3) ∨ 0), a ≥ 1; b ≥ 0; c ∈ (0, 1], S22(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , ((x2 + y2 − 1)(1 + λ)− λx2y2) ∨ 0, 0 ∨ (xb3 + yb3 − 1) 1 b ), λ ≥ 0, a, b ≥ 1, S23(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 0 ∨ (xb2 + yb2 − 1) 1 b , ((x3 + y3 − 1)(1 + λ)− λx3y3) ∨ 0), λ ≥ 0, a, b ≥ 1. SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 157 5.4. Strict-nipoltent-strict t-conorms subclass, denoted by ∇sns Definition 5.4. A picture fuzzy t-conorm S is called strict-nipoltent-strict iff S (x, y) = (s3 (x1, y1) , t2 (x2, y2) , t1 (x3, y3)) , ∀x, y ∈ D∗, where t1 is a strict fuzzy t-norm on [0, 1], t2 is a nipoltent fuzzy t-norm on [0, 1] and s3 is a strict fuzzy t-conorm on [0, 1]. Examples 5.4. S24(x, y) = (x1 + y1 − x1y1, 0 ∨ (x2 + y2 − 1), x3y3), S25(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , ((x2 + y2 − 1)(1 + λ1)− λ1x2y2) ∨ 0, x3y3 λ2 + (1− λ2)(x3 + y3 − x3y3)), λ1 ≥ 0;λ2, a ≥ 1, S26(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 0 ∨ (xb2 + yb2 − 1) 1 b , x3y3 λ1 + (1− λ1)(x3 + y3 − x3y3)), λ1 ≥ 1; a, b ≥ 1, S27(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 1b (x2 + y2 − 1 + (b− 1)x2y2) ∨ 0,x3y3 λ1 + (1− λ1)(x3 + y3 − x3y3)), a, λ1 ≥ 1; b ∈ (0, 1]. 5.5. Strict-strict-nipoltent t-conorms subclass, denoted by ∇ssn Definition 5.5. A picture fuzzy t-conorm S is called strict-strict-nipoltent iff S (x, y) = (s3 (x1, y1) , t2 (x2, y2) , t1 (x3, y3)) , ∀x, y ∈ D∗, where t1 is a nipoltent fuzzy t-norm on [0, 1], t2 is a strict fuzzy t-norm on [0, 1] and s3 is a strict fuzzy t-conorm on [0, 1]. Examples 5.5. S28(x, y) = (x1 + y1 − x1y1, x2y2, 0 ∨ (x3 + y3 − 1)), S29(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , x2y2 λ+ (1− λ)(x2 + y2 − x2y2) , 1 b (x3 + y3 − 1 + (b− 1)x3y3) ∨ 0), a, λ ≥ 1; b ∈ (0, 1], S30(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , x2y2 λ+ (1− λ)(x2 + y2 − x2y2) , 0 ∨ (xb3 + yb3 − 1) 1 b ), a, b, λ ≥ 1, S31(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , x2y2 λ+ (1− λ)(x2 + y2 − x2y2) , ((x2 + y2 − 1)(1 + b)− bx2y2) ∨ 0), a, λ ≥ 1; b ≥ 0. Proposition 5.6. There doesn’t exist representable picture fuzzy t-conorm S S (x, y) = (s3 (x1, y1) , t2 (x2, y2) , t1 (x3, y3)) , ∀x, y ∈ D∗, 158 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG here t1 or t2 is strict fuzzy t-norm on [0, 1] and s3 is a nipoltent fuzzy t-conorm on [0, 1]. Proposition 5.7. If S belongs to one of four classes ∇sss, ∇snn, ∇sns, ∇ssn then S is strict. Proposition 5.8. If S belongs to the class ∇nnn then S is nipoltent. 6. SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC De Morgan picture operator triples is a basic algebra of the Picture Fuzzy Logic. The notion of t-norm plays the role of intersection , or in logical terms, “and”. The duality of that notion is that of union , or “or” . In the case of sets, union and intersection are related via complements. The well-known De Morgan formulas do that. They are (A ∪B)C = (AC ∩BC), (A ∩B)C = (AC ∪BC). Let T (x, y) be a picture fuzzy t-norm and let S(x, y) be a picture fuzzy t-conorm and n(x) be a picture negation operator, The De Morgan formulas now become the new equations n(S(x, y)) = T (n(x), n(y)), ∀x, y ∈ D∗, (a, ∗) n(T (x, y)) = S(n(x), n(y)), ∀x, y ∈ D∗. (b, ∗) Definition 6.1. The triple of operators (T, S, n) is called a De Morgan picture operator triple if they satisfy both the equation (a, ∗) and the equation (b, ∗). Then we say that T and S are dual corresponding to the negation operator n(x). With an involutive picture negation operator, De Morgan triples of picture fuzzy opera- tors satisfy the following equations S(x, y) = n(T (n(x), n(y))), ∀x, y ∈ D∗, (a, ∗∗) and T (x, y) = n(S(n(x), n(y))), ∀x, y ∈ D∗. (b, ∗∗) Some De Morgan picture operator triples were given in [9]. Now we give some new De Morgan picture operator triples (T, S, n0) corresponding the picture negation operator n0(x). Proposition 6.2. The triple (Tmin, Smax, n0) corresponding the picture negation operator n0(x) is a De Morgan picture operator triple. Proof. We have Tmin (x, y) = (min (x1, y1) ,min (x2, y2) ,max (x3, y3)) . And Smax (x, y) = (max (x1, y1) ,min (x2, y2) ,min (x3, y3)) . n0(Smax (x, y)) = n0 (max (x1, y1) ,min (x2, y2) ,min (x3, y3)) = (min(x3, y3), 0,max(x1, y1)), and n0(x) = (x3, 0, x1), n0(y) = (y3, 0, y1)⇒ Tmin (n0(x), n0(y)) = (min (x3, y3) ,min (0, 0) ,max (x1, y1)) = (min(x3, y3), 0,max(x1, y1)) = n0(Smax(x, y)). SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 159 It means that we have the equation (a, ∗). Analogously n0(Tmin (x, y)) = (max (x3, y3) , 0,min (x1, y1)) , Smax (n0(x), n0(y)) = (max (x3, y3) ,min (0, 0) ,mn (x1, y1)) = (max(x3, y3), 0,min(x1, y1)) = n0(Tmin(x, y)). We have the equation (b, ∗).  Proposition 6.3. Consider the picture t-norm T02 (x, y) = (min (x1, y1) , x2y2,max (x3, y3)) and the picture t-conorm S02 (x, y) = (max (x1, y1) , x2y2,min (x3, y3)) . The triple (T02, S02, n0) is a De Morgan picture operator triple. Proof. The proof is analogous to the proof of the Proposition 6.2. Proposition 6.4. Let t2(x, y), t3(x, y) be fuzzy t-norms on [0, 1]. Consider the picture fuzzy t-norm Tmin,t2 (x, y) = (min (x1, y1) , t2(x2y2),max (x3, y3)) and the picture fuzzy t-conrm Smax,t3 (x, y) = (max (x1, y1) , t2(x2y2),min (x3, y3)) . The triple of operators (Tmin,t2 , Smax,t3 , n0) corresponding the picture negation operator n0(x) is a De Morgan picture operator triple. Proof. Smax,t3 (x, y) = (max (x1, y1) , t3 (x2, y2) ,min (x3, y3)) . n0(Smax,t3 (x, y)) = n0 (max (x1, y1) , t3 (x2, y2) ,min (x3, y3)) = (min(x3, y3), 0,max(x1, y1)), and n0(x) = (x3, 0, x1), n0(y) = (y3, 0, y1)⇒ Tmin,t2 (n0(x), n0(y)) = (min (x3, y3) , t2 (0, 0) ,max (x1, y1)) = (min(x3, y3), 0,max(x1, y1)) = n0(Smax,t3(x, y)). It means that we have the equation (a, ∗). Analogously n0(Tmin,t2 (x, y)) = (max (x3, y3) , 0,min (x1, y1)) , Smax,t3 (n0(x), n0(y)) = (max (x3, y3) , t3 (0, 0) ,mn (x1, y1)) = (max(x3, y3), 0,min(x1, y1)) = n0(Tmin,t(x, y)). We have the equation (b, ∗).  We easily receive the following proposition. Proposition 6.5. Consider the picture t-norm T04 and the picture t-conorm S04 T04(x, y) = (x1y1, x2y2, x3 + y3 − x3y3), and S04 (x, y) = (x1 + y1 − x1y1, x2y2, x3y3) . The triple (T04, S04, n0) is a De Morgan picture operator triple. 160 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG Proof. n0(S04 (x, y)) = n0 (x1 + y1 − x1y1, x2y2, x3y3) = (x3y3, 0, x1 + y1 − x1y1), and n0(x) = (x3, 0, x1), n0(y) = (y3, 0, y1)⇒ T04 (n0(x), n0(y)) = (x3y3, 0.0, x1 + y1 − x1y1) = (x3y3, 0, x1 + y1 − x1y1) = n0(S04(x, y)). It means that we have the equation (a, ∗). Analogously n0(T04 (x, y)) = (x3 + y3 − x3y3, 0, x1y1) , S04 (n0(x), n0(y)) = (x3 + y3 − x3y3, 0.0, x1y1) = (x3 + y3 − x3y3, 0, x1y1) = n0(T04(x, y)). We have the equation (b, ∗).  Now we consider the case where picture t-norm T belongs to the nilpotent, nilpotent, nilpotent subclass ∆nnn and S belongs to the subclass ∇nnn. Proposition 6.6. Consider the picture t-norm T3 and the picture t-conorm S3 T3(x, y) = (0 ∨ (x1 + y1 − 1), 0 ∨ (x2 + y2 − 1), 1 ∧ (x3 + y3)), S3(x, y) = (1 ∧ (x1 + y1), 0 ∨ (x2 + y2 − 1), 0 ∨ (x3 + y3 − 1)). The triple (T3, S3, n0) is a De Morgan picture operator triple. Proof. n0(S3(x, y)) = n0((1 ∧ (x1 + y1), 0 ∨ (x2 + y2 − 1), 0 ∨ (x3 + y3 − 1))) = (0 ∨ (x3 + y3 − 1), 0, (1 ∧ (x1 + y1)), n0(x) = (x3, 0, x1), n0(y) = (y3, 0, y1)⇒ T3 (n0(x), n0(y)) = (((x3 + y3 − 1) ∨ 0), (0 + 0− 1) ∨ 0), 1 ∧ (x1 + y1)) = ((0 ∨ (x3 + y3 − 1), 0, 1 ∧ (x1 + y1)) = n0(S3(x, y)). It means that we have the equation (a, ∗). Analogously n0(T3 (x, y)) = (1 ∧ (x3 + y3)), 0, (0 ∨ (x1+y1 − 1)), S3 (n0(x), n0(y)) = (1 ∧ (x3 + y3), (0 ∨ (0 + 0− 1), (0 ∨ (x1 + y1 − 1)) = (1 ∧ (x3 + y3), 0, (0 ∨ (x1 + y1 − 1)) = n0(T3(x, y)). We have the equation (b, ∗).  Proposition 6.7. Consider the picture t-norm T4 of subclass ∆nnn and S4 belongs to the subclass ∇nnn. T4(x, y) = (((x1 + y1 − 1)(1 + λ1)− λ1x1y1) ∨ 0, ((x2 + y2 − 1)(1 + λ2)− λ2x2y2) ∨ 0, 1 ∧ (xa3 + y a 3) 1 a ), S4(x, y) = (1∧ (xa1 + ya1) 1 a , ((x2 + y2− 1)(1 +λ2)−λ2x2y2)∨ 0, ((x3 + y3− 1)(1 + λ1) − λ1x3y3) ∨ 0), where λ1, λ2 ∈ [0,+∞), a ≥ 1. The triple (T4, S4, n0) is a De Morgan picture operator triple. SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 161 Proof. n0(S4(x, y)) = n0((1 ∧ (xa1 + ya1) 1 a , ((x2 + y2 − 1)(1 + λ2)− λ2x2y2) ∨ 0, ((x3 + y3 − 1)(1 + λ1)− λ1x3y3) ∨ 0)) = ((x3 + y3 − 1)(1 + λ1)− λ1x3y3) ∨ 0), 0, (1 ∧ (xa1 + ya1) 1 a ), T4(n0(x), n0(y)) = (((x3 + y3 − 1)(1 + λ1)− λ1x3y3) ∨ 0, ((0 + 0− 1)(1 + λ2)− λ20.0) ∨ 0, 1 ∧ (xa1 + ya1) 1 a ) = (((x3 + y3 − 1)(1 + λ1)− λ1x3y3) ∨ 0, 0, 1 ∧ (xa1 + ya1) 1 a ) = n0(S4(x, y)). It means that we have the equation (a, ∗). Analogously n0(T4(x, y)) = ((1 ∧ (xa3 + ya3) 1 a ), 0, (x1 + y1 − 1)(1 + λ1)− λ1x1y1) ∨ 0)), S4(n0(x), n0(y)) = ((1 ∧ (xa3 + ya3) 1 a , ((0 + 0− 1)(1 + λ2)− λ20.0)) ∨ 0, ((x1 + y1 − 1)(1 + λ1)− λ1x1y13) ∨ 0)) = ((1 ∧ (xa3 + ya3) 1 a , 0, ((x1 + y1 − 1)(1 + λ1)− λ1x1y13) ∨ 0)) = n0(T4(x, y)). We have the equation (b, ∗).  Proposition 6.8. We consider the case picture t-norm T11 belongs to ∆nns - the nilpotent, nilpotent, strict subclass and S11 belongs to the subclass ∇snn. T11(x, y) = (0 ∨ (x1 + y1 − 1), 0 ∨ (x2 + y2 − 1), x3 + y3 − x3y3), S11(x, y) = (x1 + y1 − x1y1, 0 ∨ (x2 + y2 − 1), 0 ∨ (x3 + y3 − 1)). The triple (T11, S11, n0) is a De Morgan picture operator triple. Proof. n0(S11(x, y)) = n((x1 + y1 − x1y1, 0 ∨ (x2 + y2 − 1), 0 ∨ (x3 + y3 − 1))) = ((0 ∨ (x3 + y3 − 1)), 0, (x1 + y1 − x1y1)), T11(n0(x), n0(y)) = (0 ∨ (x3 + y3 − 1), 0 ∨ (0 + 0− 1), x1 + y1 − x1y1) = (0 ∨ (x3 + y3 − 1), 0, x1 + y1 − x1y1) = n0(S11(x, y)). It means that we have the equation (a, ∗). Analogously n0(T11(x, y)) = n0((0 ∨ (x1 + y1 − 1), 0 ∨ (x2 + y2 − 1), x3 + y3 − x3y3)) = ((x3 + y3 − x3y3), 0, (0 ∨ (x1 + y1 − 1)), S11(n0(x), n0(y)) = ((x3 + y3 − x3y3), 0 ∨ (0 + 0− 1), 0 ∨ (x1 + y1 − 1)) = ((x3 + y3 − x3y3), 0, (0 ∨ (x1 + y1 − 1)) = n0(T11(x, y)), we have the equation (b, ∗).  Now we consider the case where picture t-norm T12 belongs the nilpotent, nilpotent, strict subclass ∆nns and the picture t-conorm S12 belongs to the subclass ∇snn. 162 BUI CONG CUONG, ROAN THI NGAN, LE BA LONG Proposition 6.9. Consider T12(x, y) = ( 1 2 (x1 + y1 − 1 + x1y1) ∨ 0, 1 2 (x2 + y2 − 1 + x2y2) ∨ 0, (xa3 + ya3 − xa3ya3) 1 a ), S12(x, y) = ((x a 1 + y a 1 − xa1ya1) 1 a , 1 2 (x2 + y2 − 1 + x2y2) ∨ 0, 1 2 (x3 + y3 − 1 + x3y3) ∨ 0), where a ≥ 1. The triple (T12, S12, n0) is a De Morgan picture operator triple. Proof. n0(S12(x, y)) = n0(((x a 1 + y a 1 − xa1ya1) 1 a , 1 2 (x2 + y2 − 1 + x2y2) ∨ 0, 1 2 (x3 + y3 − 1 + x3y3) ∨ 0)) = (( 1 2 (x3 + y3 − 1 + x3y3) ∨ 0), 0, (xa1 + ya1 − xa1ya1) 1 a ), T12(n0(x), n0(y)) = (( 1 2 (x3 + y31 − 1 + x3y3) ∨ 0), 1 2 (0 + 0− 1 + 0.0) ∨ 0, (xa1 + ya1 − xa1ya1) 1 a )) = (( 1 2 (x3 + y31 − 1 + x3y3) ∨ 0), 0, (xa1 + ya1 − xa1ya1) 1 a ) = n0(S12(x, y)). It means that we have the equation (a, ∗). Analogously n0(T12(x, y)) = n0(( 1 2 (x1 + y1 − 1 + x1y1) ∨ 0), 1 2 (x2 + y2 − 1 + x2y2) ∨ 0, (xa3 + ya3 − xa3ya3) 1 a )) = ((xa3 + y a 3 − xa3ya3) 1 a ), 0, ( 1 2 (x1 + y1 − 1 + x1y1) ∨ 0)) = S12(n0(x), n0(y)), S12(n0(x), n0(y)) = ((xa3 + y a 3 − xa3ya3) 1 a , ( 1 2 (0 + 0− 1 + 0.0) ∨ 0), (1 2 (x1 + y1 − 1 + x1y3) ∨ 0)) = ((xa3 + y a 3 − xa3ya3) 1 a , 0, ( 1 2 (x1 + y1 − 1 + x1y3) ∨ 0)) = n0(T12(x, y)). It means that we have the equation (b, ∗).  Some other De Morgan picture operator triples can be seen in [9, 8]. 7. CONCLUSION Conjunction operations (fuzzy t-norms) and disjunction operations (fuzzy t-conorms) are basic operators of the fuzzy logics [22, 13]. Picture fuzzy t-norms and picture fuzzy t-conorms firstly were defined and studied in 2015 [6, 9]. In this paper we give some algebraic properties of the picture fuzzy t-norms and the picture fuzzy t-conrms on picture fuzzy sets, including some classes of representable picture fuzzy t-norms and and some classes of representable picture fuzzy t-conorms. Then we study the De Morgan picture operator triples of the Picture Fuzzy Logics. Some new classes of De Morgan picture operator triples were presented. In the following papers new other issues of the Picture Fuzzy Logic should be considered. SOME NEW DE MORGAN PICTURE OPERATOR TRIPLES IN PICTURE FUZZY LOGIC 163 Acknowledgment This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.01-2017.02. REFERENCES [1] K. 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Received on September 19, 2017 Revised on December 01, 2017