THE -COMPOSITION OF INTUITIONISTIC FUZZY IMPLICATIONS: AN ALGEBRAIC STUDY OF POWERS AND FAMILIES

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1 Volume 9, No. 1, Jan (Special ssue) nternational Journal of Mathematical Archive-9(1), 2018, Available online through SSN THE -COMPOSTON OF NTUTONSTC FUZZY MPLCATONS: AN ALGEBRAC STUDY OF POWERS AND FAMLES VSHNU SNGH 1 & SHV PRASAD YADAV 2 Department of Mathematics, ndian nstitute of Technology Roorkee, Roorkee , ndia. vishnumsc.singh@gmail.com 1, spyorfma@gmail.com 2 ABSTRACT Viewing a binary operation between pair of fuzzy implications (Fs), the pair of nth powers of self Fs and the pair of a F and its n th power proposed by Vemuri and Jayaram [15, 16, 17] is extended to the binary operation between pair of intuitionistic fuzzy implications (Fs), the pair of n th powers of self Fs and the pair of a F and its n th power. Finally, the basic properties like neutrality, ordering, exchange principles, the powers w.r.t. and their convergence, and the closures of some families of Fs w.r.to the operation are studied. Keywords: ntuitionistic fuzzy logic connectives, ntuitionistic fuzzy implications, Algebras, Semi-group, Monoid, Lattice. AMS Classification: 03E NTRODUCTON Fs are the generalizations of the Fs to the multi-valued setting. They play an important role in decision theory, control theory, approximate reasoning, expert systems, etc. The different ways of obtaining Fs and the better understanding of the behavior of different models for Fs, and their algebraic structures are studied in the book [3]. Recently, some authors studied Fs from different perspectives [3, 16]. Yager [19] studied some new classes of implication operators and their role in approximate reasoning. Naturally, investigating the common properties of some important Fs used in fuzzy logic is meaningful. The implicative represen- tation of the rule base is less frequent in applications, but more challenging. n this case, each single rule is understood as an individual condition/restriction imposed on a modeled input-output relation. Different rules correspond to the different restrictions and are required to hold simultaneously. The view of the stored knowledge is logically driven and the related fuzzy relation can be seen as a kind of theory in the logical sense. The issue of coherence (logical consistency) of the rule base is critical in this case [15]. Deschrijver and Kerre [6] established the relationships between fuzzy sets, L-fuzzy sets [10], interval-valued fuzzy sets [7], intuitionistic fuzzy sets (FSs), and interval-valued FSs. The major difference among the three classes lies in the choice of F operators. n general, fuzzy systems choose t-norm as implication operators, such as the min or product operator, while Boolean fuzzy systems utilize genuine multi-valued implications that mainly contain R-implication, S-implication and QL-implication [1]. Deschrijver and Kerre [8] introduced some aggregation operators on the lattice L and defined the special classes of binary aggregation operators based on t-norms on the unit interval. Liu and Wang [11] gave corrections of some limitations of the article [8]. Deschrijver et al. [5] introduced the concepts of F t-norm and t-conorm and investigated under which conditions a similar representation theorem can be obtained. Cornelis et al. [4] constructed a representation theorem for Lukasiewicz implicators on the lattice, which serves as the underlying algebraic structure for both FSs and interval-valued fuzzy sets. Shi et al. [12] investigated constructive methods for F operators. Atanassov [2] introduced five new F operations on FSs containing multiplication and studied their properties. n [9], the Atanassov s operator was considered together with representable Atanassov s ntuitionistic De Morgan triples in order to generate new De Morgan triples, providing an extension of the notions of De Morgan triples and automorphisms on unitary interval for FSs. Many approximate reasoning and interesting results in F environment are reported [18]. n this work, our aim is two-fold. On the one hand, we are interested to develop a binary operation between two Fs and studied their properties and algebraic structures. On the other hand, we are interested to the same binary operation among n-th power self Fs and studied the same properties as in two different Fs. nternational Journal of Mathematical Archive- 9(1), Jan

2 The rest of the paper is organized as follows. Section 2 is devoted to introduce the basic definitions and useful theorems relevant to the proposed work. n Section 3, the monoid of Fs are discussed. The basic properties of Fs w.r.to - composition are introduced in Section 4 and 5. n Section 6, 7, 8 and 9, self composition w.r.to - [n], convergence of [n], closure of [n], w.r.to the basic properties and closure of [n] w.r.t. functional equations are discussed. Finally, we conclude this paper in Section 10. n Table 1, the functions sg and sg are defined by, 1, a > 0, 0, a > 0, sg(a) = and (a) sg = 0, a 0, 1, a 0. Table-1: List of some Fs. Name Formula Zadeh 1 ( ) ZD1 (x, y) = (max(x 2, min(x 1, y 1 )), min (x 1, y 2 )) Zadeh 2 ( ) ZD2 (x, y) = (max(x 2, min(x 1, y 1 )), min (x 1, max (x 2, y 2 ))) Gaines-Rescher ( ) GR (x, y) = (1 sg(x 1 y 1 ), y 2 sg(x 1 y 1 )) Gödel ( ) GD (x, y) = (1 (x 1 y 1 )sg(x 1 y 1 ), y 2 sg(x 1 y 1 )) Fodor s 1 ( ) FD1 (x, y) = (sg (x 1 y 1 ) + sg(x 1 y 1 )max (x 2, y 1 ), sg(x 1 y 1 )min (x 1, y 2 )) Kleene-Dienes ( ) KD (x, y) = (max(x 2, y 1 ), min (x 1, y 2 )) Lukasiewicz ( ) LK (x, y) = (min(1, x 2 + y 1 ), max (0, x 1 + y 2 1)) Reichenbach ( ) RB (x, y) = (x 2 + x 1 y 1, x 1 y 2 ) Klir and Yuan 1 ( ) KY1 (x, y) = (x 2 + x 1 y 1, x 1 x 2 + x 2 1 y 2 ) Atanassov 1 ( ) A1 (x, y) = (1 (1 y 1 )sg(x 1 y 1 ), y 2 sg(x 1 y 1 )sg(y 2 x 2 )) Atanassov 2 ( ) A2 (x, y) = (max(x 2, y 1 ), 1 max (x 2, y 1 )) Atanassov and ( ) AK (x, y) = (x 2 + y 1 x 2 y 1, x 1 y 2 ) Kolev 2. PRELMNARES n this section, some basic definitions, arithmetic operations and notion of fuzzy and F inequalities are presented, which are taken from the articles Definition 1: Let X be a universe of discourse. Then an FS A in X is defined by the set A = < x, μ A (x), ν A (x) > x X, μ A, ν A : X [0, 1] are functions such that 0 μ A (x) + ν A (x) 1, x X. The value μ A (x) represents the degree of membership and ν A (x) represents the degree of non-membership of x X being in A. The degree of hesitation for the element x X being ina is denoted by π A (x) and is defined by π A (x) = 1 μ A (x) ν A (x) [0, 1], x X. Deschrijver and Kerre [6] have shown that FSs can also be seen as L-fuzzy sets in the sense of Goguen [10]. Consider the set L and operation L defined by L = {(x 1, x 2 ): (x 1, x 2 ) [0, 1] 2 & x 1 + x 2 1} (x 1, x 2 ) L (y 1, y 2 ) x 1 y 1 & x 2 y 2, (x 1, x 2 ), (y 1, y 2 ) L. Then, (L, L ) is a complete lattice [8]. For each nonempty A L L, we have supa = (sup x 1 : x 1 [0, 1] & ( x 2 [0, 1 x 1 ]) (x 1, x 2 ) A, inf {x 2 : x 2 [0, 1] & ( x 1 [0, 1 x 2 ])((x 1, x 2 ) A)} infa = (inf x 1 : x 1 [0, 1] & ( x 2 [0, 1 x 1 ]) (x 1, x 2 ) A, sup {x 2 : x 2 [0, 1] & ( x 1 [0, 1 x 2 ])((x 1, x 2 ) A)} Definition 2 ([5], Definition 2.2): A function : L 2 L is called an F if for x, x, x, y, y, y in L, it satisfies the following conditions: if x L x, then (x, y) L (x, y), i. e., (., y) is decreasing (11) if y L y, then (x, y ) L (x, y ), i. e., (x,. ) is increasing (12) (0 L, 0 L ) = 1 L, (1 L, 1 L ) = 1 L, (1 L, 0 L ) = 0 L (13) We also define the following set for further usage: D = {(x 1, x 2 ): (x 1, x 2 ) L & x 1 + x 2 = 1}, and the first and second projection mapping pr 1 and pr 2 on L, defined as pr 1 (x 1, x 2 ) = x 1 and pr 2 (x 1, x 2 ) = x 2, for all (x 1, x 2 ) L. Definition 3: An F : L 2 L is said to satisfy (i). the left neutrality property (NP), if (1 L, y) = y, y L; 2018, JMA. All Rights Reserved 72

3 (ii). the ordering property (OP), if x L y (x, y) = 1 L, x, y L; (iii). the identity principle (P), if (x, x) = 1 L, x L; (iv). the exchange principle (EP), if x, (y, z) = y, (x, z), x, y, z L. Definition 4 ([5], Definition 3.1): A function N: L L is called an F negation if N(0 L ) = 1 L, N(1 L ) = 0 L, N is decreasing. (2.1) Definition 5: A function T: L 2 L is called a triangular norm (shortly t-norm) if it satisfies the commutative, associative, increasing in both components and T(x, 1 L ) = x, x L. A function S: L 2 L is called a triangular conorm (shortly t-conorm) if it satisfies the commutative, associative, increasing in both components and S (x, 0 L ) = x, x L. Definition 6 ([4], Definition 5): (t-representability) A t-norm T on L (respectively t-conorm S) is called t-representable if there exists a t-norm T and a t-conorm S on [0, 1] (respectively a t-conorm S and a t-norm T on [0, 1]) such that, for x = (x1, x2), y = (y1, y2) L, T (x, y) = (T (x1, y1), S(x2, y2)), S (x, y) = (S (x1, y1), T (x2, y2)) T and S (respectively S and T ) are called the representants of T (respectively S). Theorem 1 ([4]: Theorem 2). Given a t-norm T and t-conorm S on [0, 1] satisfying T (a, b) 1 S(1 a, 1 b) for all a, b [0, 1], the mappings T and S defined by T (x, y) = (T (x1, y1), S(x2, y2)), S (x, y) = (S (x1, y1), T (x2, y2)) for x = (x1, x2), y = (y1, y2) L are a t-norm and a t-conorm on L, respectively. 3. MONOD OF Fs n this section, we introduced the monoid structure of Fs. Definition 7: For any two Fs, J, we define J as J (x, y) = (x, J (x, y)), x, y L. (3.1) Example 1: Let us consider two F operators, viz., Gaines-Reschere: ( ) GR (x, y) = (1 sg(x 1 y 1 ), y 2 sg(x 1 y 1 )) and Reichenbach: ( ) RB (x, y) = (x 2 + x 1 y 1, x 1 y 2 ) (see Table 1). Then (( ) GR ( ) RB )(x, y) = 1 sg(x 1 (1 y 1 ) x 2 ), x 1 y 2 sg(x 1 (1 y 1 ) x 2 ). Theorem 2: The function J is an F. Next aim is to investigate the algebraic structure of on the operation. Theorem 3: (,) forms a monoid whose identity element is given by (x, y) = 1 L, x = 0 L, y, otherwise. Remark 1: Let us take Reichenbach F ( ) RB (x, y) = (x 2 + x 1 y 1, x 1 y 2 ). Then (( ) RB ( ) RB )(x, y) = (x 2 + x 1 (x 2 + x 1 y 1 ), x 1 2 y 2 ), which is not same as ( ) RB (x, y). Thus is not idempotent in and consequently, (, ) is a non-idempotent monoid. Theorem 4: An is invertible w.r.t. if and only if 1 (x, y) = L, x = 0 L, φ(y), x > L 0 L. where the function φ: L L is an increasing bijection. (3.2) Definition 8: The pair of Fs (, J ) are said to be mutually exchangeable (ME) if (x, J (y, z)) = J (y, (x, z)), x, y, z L. 2018, JMA. All Rights Reserved 73

4 Example 2: Let us suppose for fixed ρ = (ρ 1, ρ 2 ), σ = (σ 1, σ 2 ) L. Let us consider two F operators (x, y) = 1 L, x L ρ, and J y, x > L ρ, (x, y) = 1 L, x L σ, y 2, x > L σ, where x, y L. t easy to check that (, J) satis ies ME. Remark 2: f, J are ME, then, J are commutative w.r.t.. To see this, let us suppose x = y in (ME), which then becomes (x, J (y, z)) = J (y, (x, z)), i.e., ( J )(x, z) = (J )(x, z), y, z L. Theorem 5. Let, J satisfy EP and ME. Then J satisfies EP. Definition 9: An F is said to satisfy the law of importation (L) w.r.t. a t-norm T if (x, (y, z)) = (T(x, y), z), x, y, z L. Remark 3: Note that even if, J satisfy L w.r.to the same t-norm T, J may not satisfy L w.r.t. any t-norm or may satisfy L w.r.t. the same t-norm or even a different t-norm T. Theorem 6: Let, J satisfy L w.r.to the same t-norm T. f, J satisfy ME, then J satisfies L w.r.t. the same t- norm T. 4. THE -COMPOSTON w.r.to THE BASC PROPERTES n this section, as mentioned before, we do the following: Given, J satisfying a certain properties P, we now investigate whether J also satisfies the same property or not. f not, then we attempt to characterize, J such that J also satisfies the same property. Towards this end we have the following result. Theorem 7: Let, J and φ Φ. Then( J ) φ = ( ) φ (J ) φ. Theorem 8: f, J satisfy (NP) ((P), self conjugacy, continuity), then J satisfies the same property. 5. THE - COMPOSTON w.r.t. PROPERTY OP While the composition preserves the properties NP, P, self-congugacy and continuity, this is not true with either the OP or EP. Theorem 9: Let, J satisfy OP. Then the following statements are equivalent: (i) J satisfies OP. (ii) J satisfies the following for all x, y L: x > L J (x, y), whenever x > L y. (iii) J (x, y) L y, whenever x > L y. Proof: Let, J satisfy OP. (i) (ii): J satisfies OP. Then ( J ) (x, y) = 1 L x L y, i.e., (x, J (x, y)) = 1 L x L y, i.e., x L J (x, y) x L y, which implies that x > L J (x, y), for all x > L y (ii) (iii): Let J satisfy (5.1). f x > L y, then there exists ε = (ε1, ε2) > L 0 L, arbitrarily small, such that x > L y + ε > L y. Now, from the antitonicity of J in the first variable and (5.1), we have J (x, y) L J (y + ε, y) < L y + ε. Since ε = (ε1, ε2) > L 0L is arbitrary, we see that x > L J (x, y) L y for all x > L y. (iii) (i): Let J satisfy x > L J (x, y) L y for all x > L y. Let x L y. Then, since J satisfies OP, J (x, y) = 1 L and consequently, ( J )(x, y) = (x, J (x, y)) =1 L. Let x > L y. Then we have J (x, y) L y < L x. From OP of, it follows that (x, J (x, y)) < L 1 L. n other words, we have x > L y ( J )(x, y) < L 1 L and hence J satisfies OP. 2018, JMA. All Rights Reserved 74

5 [n] 6. SELF COMPOSTON w.r.t. Since we have analyzed the associativity of binary operation on F, one can define the self composition of Fs w.r.t. the same binary operation. [n] Definition 10: Let. For any n N, the n-th power of w.r.to is denoted by [n] (x, y), n = 1, (x, y) = x, [n 1] (x, y) = [n 1] x, (x, y), n 2. where x, y L. Noting that if, then [n] for every n N. and is defined as follows: (6.1) Example 3: Let us consider Reichenbach F operator ( ) RB (x, y) = (x 2 + x 1 y 1, x 1 y 2 ). (see Table 1). Then the ( [n] ) RB (x, y) is given by ( [n] (x 2 + x 1 y 1, x 1 y 2 ), n = 1, ) RB (x, y) = (x 2 + x 1 x 2 + x 2 1 x x n 1 y 1, x n 1 y 2 ), n 2, where x, y L. [n] 7. CONVERGENCE OF n this section, we investigate the convergence of [n] w.r.t. operation. Theorem 10: Let satisfy w.r.t. a t-norm T. Then (i) [n] (x, y) = x [n] T, y, where x [n] T = T(x, x [n 1] T ) and x [1] T = x for any x L and n N. (ii) Further, let T be Archimedean, i.e., for any x, y L {0 L, 1 L } there exists an n N such that x [n] T < T y. Then lim [n] n (x, y) = 1 L, x < L 1 L, (x, y), x = 1 L. (iii) f, in addition, satisfies NP then lim [n] n (x, y) = 1 L, x < L 1 L, y, x = 1 L. Proof: (i) Since satisfies the L, [2] (x, y) = x, (x, y) = (T(x, x), y). Thus [2] (x, y) = (x [2] T, y). By mathematical induction, we get [n] (x, y) = (x [n] T, y). (ii) Take ε = (ε 1, 1 ε 1 ) > L 0 L and x < L 1 L. Since T is Archimedean, for any ε = (ε 1, 1 ε 1 ) > L 0 L m N s.t. x [n] T < L ε. Thus, for any y L, [n] (x, y) = (x [n] T, y) (0 L, y) as n and lim n [n] (x, y) = 1 L. f x = 1 L, then [2] (1 L, y) = 1 L, (1 L, y) = (T(1 L, 1 L ), y) = (1 L, y) and in general [n] (1 L, y) = (1 L, y). (iii) Follows from (ii) and the fact that (1 L, y) = y. 8. CLOSURE OF [n] w.r.to THE BASC PROPERTES n this section, given an satisfying a particular property P, we investigate whether all the powers [n] of satisfy the same property or not. From Theorem 9, we see that if satisfies any one of the properties NP, P, selfconjugacy and continuity then [n] satisfies the same for all n N. Hence it is enough to investigate the preservation of OP and EP. Below we prove that if satisfies EP, then the (, [n] ) satisfies ME for any n N. Theorem 11: f satisfies (EP) then the pair (, [n] ) satisfies (ME) for all n N, i.e., x, [n] (y, z) = [n] y, (x, z), x, y, z L. (8.1) Proof: The theorem is proved with the help of mathematical induction on n. For n = 1, satisfies (8.1), from the EP of. Let us suppose that satisfies (8.1) for n = k-1. Now, for n = k, x, [k] (y, z) = x, y, [k 1] (y, z) = x, [k 1] y, (y, z) = [k 1] y, x, (y, z) = [k 1] y, y, (x, z) = [k] y, (x, z), x, y, z L. 2018, JMA. All Rights Reserved 75

6 Thus the pair (, [n] ) satisfies (ME) for all n N. Theorem 12: f satisfies EP then [n] satisfies EP for all n N. Proof: A direct verification provides the proof. Theorem 13: Let satisfy OP. Then the following statements are equivalent: (i) [2] satisfies OP. (ii) satisfies the following for all x, y L: x > L (x, y), whenever x > L y. (iii) [n] satisfies OP for all n N. Proof: The same as Theorem10 9. CLOSURE OF [n] w.r.t. FUNCTONAL EQUATONS Theorem 14: f satisfies L w.r.t. t-norm T, then [n] also satisfies L w.r.t. t-norm T. Proof: This theorem is proved with the help of mathematical induction on n. For n = 1, [n] satisfies L. Let us suppose ~ that [n] satisfies (L) w.r.t. the t-norm T for n = k-1. Then [k 1] (T(x, y), z) = [k 1] x, [k 1] (y, z), x, y, z L. (9.1) From Theorem 10(i), recall that if satisfies (L) w.r.to a t-norm T, then [k] (x, y) = (x [k] ). Now, for n = k, [k] [k] [k 1] [k 1] (T(x, y), z) = T(x, y) T, z = T(T(x, y), T(x, y) T ), z = T(x, y), ( T(x, y) T, z) = T(x, y), [k 1] (T(x, y), z) = x, y, [k 1] (T(x, y), z) = x, y, [k 1] x, [k 1] (y, z) = x, [k 1] x, y, [k 1] (y, z) = [k] x, [k] (y, z). Thus [n] satisfies L for all n N. 10. CONCLUSONS AND FUTURE SCOPE n this paper, we have investigated -composition between between the pair of Fs, the pair of n th powers of self Fs and the pair of a F and its n th power, which is a generalization of -composition in Fs [15, 16, 17]. Specifically, we have studied the algebraic structures and shown that some other properties like NP, P, continuity are preserved but fail to preserve OP and EP. Finally, we have also proposed a new concept of ME, a generalization of EP to a pair of Fs and n th power of self Fs. n future, based on the composition, we plan to work on interval valued Fs w.r.to two different as well as n th order self. REFERENCES 1. Alcalde, C., Burusco, A., Fuentes-Gonza lez, R., A constructive method for the definition of interval-valued fuzzy implication operators, Fuzzy Sets Systems, 153 (2005), Atanassov, K., Szmidt, E., Kacprzyk, J., ntuitionistic fuzzy implication 187, Notes on ntuitionistic Fuzzy Sets, 23 (2017), Baczynski, M., Jayaram, B., Fuzzy mplications, Studies in Fuzziness and Soft Computing, vol. 231, Springer, Berlin (2008). 4. Cornelis, C., Deschrijver, G., Kerre, E.E., mplication in intuitionistic fuzzy and interval- valued fuzzy set theory: construction, classification, application. nternational Journal of Approximate Reasoning, 35 (2004), Deschrijver, G., Cornelis, C., Kerre, E. E., On the representation of intuitionistic fuzzy t- norms and t- conorms. EEE transactions on fuzzy systems, 12 (2004), , JMA. All Rights Reserved 76

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