IOSR Journal of Mathematcs (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. PP 8-34 www.osrournals.org Matrx-Norm Aggregaton Operators Shna Vad, Sunl Jacob John Department of Mathematcs, Natonal Insttute of Technology Calcut, Kerala Department of Mathematcs, Natonal Insttute of Technology Calcut, Kerala Abstract: Multple set theory s a new mathematcal approach to handle vagueness together wth multplcty. Multple set s expert n formalsng numerous propertes of obects multfarously. Ths paper proposes a generalzaton for t-norms and t-conorms on multple sets. Recently, a study on the concept of unnorm aggregaton operators on bounded lattce has been done. Ths paper dscusses unnorm aggregaton operators on the bounded lattce M, called matrx-norm aggregaton operators, where M M ([0,]) denotes the collecton of all n k matrces wth entres from[0,]. The paper also ntroduces matrx-norm aggregaton operator nduced by unnorm aggregaton operators on[0,]. Fnally, the paper nvestgates some propertes of matrx-norm aggregaton operators. Keywords: aggregaton operator, multple set, t-norm, t-conorm, unnorm. nk I. Introducton Uncertanty, vagueness, nexactness, etc. are nevtable part of our daly lfe. Many theores have been ntroduced to represent these concepts mathematcally, whch ncludes fuzzy set [], vague set [], ntutonstc fuzzy set [3], mult fuzzy set [4], fuzzy mult set [5] and many more. Recently, Shna, John and Thomas ntroduced multple sets [6-7] to handle vagueness and multplcty together. Multple set s expert n formalsng numerous propertes of obects multfarously. Multple set assgns a membershp matrx for each obect n the unversal set and each raw n membershp matrx characterzes ts varous propertes where each entres n the raw corresponds obect s multplcty. Trangular norms and conorms (t-norms and t-conorms) [8-0] are mportant notons n fuzzy set theory and are wdely used n many contexts. Also varous other set-based t-norms and t-conorms are developed, such as, on ntutonstc fuzzy set [], mult fuzzy set [] and has found many applcatons n varous felds. As a contnuaton, Shna and John ntroduced t-norms and t-conorms on multple sets, called multple t-norms and t-conorms [3-4] and t s proven that they generalzes the standard operatons of unon and ntersecton on multple sets. In 996, Yager and Rybalov [5] ntroduced the concept of unnorm aggregaton operators (smply, unnorms) as a generalzaton of fuzzy t-norms and t-conorms. Later, an exhaustve study of unnorm operators s establshed through many papers [6-]. Unnorms have proven to be useful n many felds lke fuzzy logc, expert systems, neural networks, aggregaton and fuzzy systems [-3]. In 05, Karacal and Mesar [4], generalzed the concepts of unnorms by defnng them on bounded lattces nstead of [0,] and n 07, Karacal, rtugrul and Mesar [5] proposed a characterzaton of unnorms on a general bounded lattce by means of sextuple of a t-norms, t-conorms and four aggregaton functon on the bounded lattce. Ths paper provdes a study on matrx-norm aggregaton operators-unnorms on the bounded lattce M M ([0,]), the set of all n k matrces wth entres from [0,]. The paper nvestgates that matrxnorms generalzes multple t-norms and t-conorms. The paper also dscusses varous propertes of matrx-norms. II. Basc Concepts Ths secton descrbes some basc defntons and propertes related to multple sets and unnorms.. Multple sets In 05, Shna, John and Thomas, ntroduced the concept of multple set to handle vagueness and multplcty of obects smulaneously. The membershp matrx assgned to each obects helps the multple set to formalze numerous propertes of obects at a tme. ach raw n membershp matrx characterzes ts varous propertes where each entres n the raw corresponds obect s multplcty. nk Natonal Conference On Dscrete Mathematcs & Computng (NCDMC-07) 8 Page
Matrx-Norm Aggregaton Operators Defnton.. [7] Let X be a non-empty crsp set called the unversal set. A multple set A of order ( nk, ) over X s an obect of the form {( x, A( x )); x X} matrx A( x ) = k k A ( x) A ( x) A ( x) A ( x) A ( x) A ( x) k n n n A ( x) A ( x) A ( x) Î, where for each x X ts membershp value s an n k where A, A,..., A n are fuzzy membershp functons and for each,,..., n, k A ( x), A ( x),..., A ( x ) are membershp values of the fuzzy membershp functon A for the element x X, wrtten n decreasng order. The matrx Ax ( ) s called the membershp matrx. The unversal set X can be vewed as a multple set of order ( nk, ) over X for whch the membershp matrx for each x X s an n k matrx wth all entres one. Also, the empty set can be vewed as a multple set of order ( nk, ) over X for whch the membershp matrx for each x X s an n k matrx wth all entres zero. The set of all multple sets of order ( nk, ) over X s denoted by MS, ( ) nk X. Let M M nk ([0,]) be the set of all n k matrces wth entres from [0,]. It s notced that a multple set A of order ( nk, ) over X can be vewed as a functon A : X M whch maps each x X to ts ( nk, ) membershp matrx Ax ( ). From ts structure, t s clear that multple set generalzes the concepts of fuzzy sets [], mult fuzzy sets [4], fuzzy multsets [5] and multsets [6]. The notons of subset and equalty between multple sets s defned as follows; Defnton.. [7] Gven two multple sets A and B n MS, ( ) nk X, then A s a subset of B, denoted as A B, f and only f A ( x) B ( x) for every x X, =,,...,n and =,,...,k. Also, A s equal to B, denoted as A B, f and only f A B and B A, that s, f and only f A ( x) B ( x) for every x X, =,,...,n and =,,...,k. Gven two multple sets A and B n Defnton..3 [7] The unon of A and B, denoted as A B, for each x X s gven by A B ( x) A B ( x), every =,,...,n and =,,...,k. MS, ( X ), standard multple set operatons are defned as follows; nk Defnton..4 [7] The ntersecton of A and B, denoted as A B, matrx for each x X s gven by A B ( x) A B ( x), for every =,,...,n and =,,...,k. s a multple set whose membershp matrx where ( ) A B x max A ( ), ( ) x B x for s a multple set whose membershp where ( ) A B x mn A ( ), ( ) x B x Defnton..5 [7] The complement of A, denoted as A s a multple set whose membershp matrx for each x X A ( x) A ( x) k A ( x) A ( x) for every =,,...,n and s gven by where =,,...,k. There exst a broad class of functons whose members qualfy as generalzatons of standard operatons of ntersecton and unon on multple sets. ach of these classes s characterzed by properly defned axoms and they are called multple t-norm and t-conorm, respectvely. Natonal Conference On Dscrete Mathematcs & Computng (NCDMC-07) 9 Page
Matrx-Norm Aggregaton Operators Defnton..6 [4] A multple t-norm T s a bnary operaton on M that satsfes the followng axoms, for all A, B, D M ; (T) Monotoncty: T( A, B) T( A, D), whenever B D. (T) Commutatvty: T( A, B) T( B, A) (T3) Assocatvty: T( A, T( B, D)) T( T( A, B), D) (T4) Boundary condton: T( A, ) A where denotes the n k matrx wth all entres. A specal type of multple t-norms are ntroduced wth the ad of fuzzy t-norms as follows; Defnton..7 [4] Let t be fuzzy t-norm. Defne the bnary operaton T on M as follows: A [ a ] and B [ b ] n M are mapped to C [ c ] n M by c t( a, b ) for all,,..., n and,,..., k. Then T s a multple t-norm on M, called the multple t-norm nduced by fuzzy t-norm t. Some examples for multple t-norms nduced by fuzzy t-norms are gven as follows; () Standard multple ntersecton: Multple t-norm MIN nduced by fuzzy t-norm t( a, b) mn( a, b). () Algebrac product: Multple t-norm AP nduced by fuzzy t-norm t( a, b) ab. a when b (3) Drastc ntersecton: Multple t-norm DI nduced by fuzzy t-norm t( a, b) b when a 0 otherwse Defnton..8 [4] A multple t-conorm S s a bnary operaton on M that satsfes the followng axoms, for all A, B, D M ; (S) Monotoncty: S( A, B) S( A, D), whenever B D. (S) Commutatvty: S( A, B) S( B, A) (S3) Assocatvty: S( A, S( B, D)) S( S( A, B), D) (S4) Boundary condton: S( A, 0 ) A where 0 denotes the n k matrx wth all entres 0. A specal type of multple t-conorms are ntroduced wth the ad of fuzzy t-conorms as follows; Defnton..7 [4] Let s be fuzzy t-conorm. Defne the bnary operaton S on M as follows: A [ a ] and B [ b ] n M are mapped to C [ c ] n M by c s( a, b ) for all,,..., n and,,..., k. Then S s a multple t-conorm on M, called the multple t-conorm nduced by fuzzy t-conorm s. Some examples for multple t-conorms nduced by fuzzy t-conorms are gven as follows; () Standard multple unon: Multple t-conorm MAX nduced by fuzzy t-conorm s( a, b) max( a, b). () Algebrac sum: Multple t-conorm AS nduced by fuzzy t-conorm s( a, b) a b ab. (3) Drastc unon: Multple t-conorm DU nduced by fuzzy t-conorm a when b 0 s( a, b) b when a 0 otherwse. Unnorm Aggregaton Operator In 996, Yager and Rybalov ntroduced unnorm aggregaton operators as a generalzaton of the fuzzy t-norm and t-conorm. Unnorms allow for an dentty element lyng anywhere n the unt nterval rather than at one or zero as n the case of t-norms and t-conorms. Defnton.. [5] A unnorm s a mappng u : [0,] [0,] [0,] havng the followng propertes; (u) Commutatvty: u( a, b) u( b, a) (u) Monotoncty: u( a, b) u( c, d) f a c and b d (u3) Assocatvty: u( a, u( b, c)) u( u( a, b), c) Natonal Conference On Dscrete Mathematcs & Computng (NCDMC-07) 30 Page
Matrx-Norm Aggregaton Operators (u4) There exst some elements e [0,] called the dentty element or neutral element such that for all a [0,], u( a, e) a. Followng are some examples for unnorms [9]; () u ( a, b) mn( a, b) wth the neutral element. () u ( a, b) max( a, b) wth the neutral element 0. (3) 3 (, ) ab u a b wth the neutral element e [0,]. e (4) 4 (, ) a b ab e u a b wth the neutral element e [0,]. e (5) u 5 ( a, b) max 0, a b e (6) u a b mn a b e wth the neutral element e [0,]. 6 (, ), wth the neutral element e [0,]. (7) 7 (, ) ab u a b, where a a, wth the neutral element e 0.5. ab ab In 05, Kamacal and Mesar generalzed the concepts of unnorms by defnng them on bounded lattce nstead of [0,]. Defnton.. [4] Let L,, 0, be a bounded lattce. An operaton u : L t s an assocatve symmetrc aggregaton functon whch has a neutral element e L. L s called a unnorm on L f III. Matrx-Norm Aggregaton Operators Let M M ([0,]) be the set of all n k matrces wth entres from [0,]. Defne the bnary nk relaton on M as, for A [ a ] and B [ b ] n M, A B f and only f a b for every,,..., n and,,..., k. Let 0 denotes an n k matrx wth all entres zero and denotes an n k matrx wth all entres one. Then M,, 0, s a bounded lattce wth the least element 0 and the greatest element. Defnton 3. Consder the bounded lattce M,, 0,. The bnary operaton U : M M s called a matrxnorm aggrtegaton operators(smply, matrx-norm) on M f t satsfes the followng axoms; for all A, B, C, D M, (U) Commutatvty: U( A, B) U( B, A) (U) Monotoncty: U( A, B) U( C, D) f A C and B D (U3) Assocatvty: U( A, U( B, C)) U( U( A, B), C) (U4) There exst some elements M called the neutral matrx such that for all AM, U( A, ) A. Note that matrx-norms aggregaton operators are unnorm aggregaton operators on the bounded lattce M,, 0,. An element AM s sad to be dempotent f U( A, A) Aand a matrx norm U s called dempotent f U( A, A) Afor all AM. Theorem 3. For any matrx-norm U, ts neutral matrx s unque. Proof. Suppose that and are two neutral matrces of the matrx-norm U. Then, from the property of neutral matrx, U(, ) and U(, ). Then, from commutatvty, and hence neutral matrx s unque. Theorem 3. Neutral matrx of a matrx-norm U s dempotent. Proof. From the property of neutral matrx, U(, ) and hence neutral matrx s dempotent. Natonal Conference On Dscrete Mathematcs & Computng (NCDMC-07) 3 Page
Theorem 3.3 Assume that U s a matrx-norm wth neutral matrx. Then (a) For any AM and all B, we get U( A, B) A. (b) For any AM and all B, we get U( A, B) A. Proof. (a) From the property of neutral matrx U( A, ) U( A, B) U( A, ). Therefore U( A, B) A. (b) From the property of neutral matrx U( A, ) Therefore U( A, B) A. Matrx-Norm Aggregaton Operators A. If B then, from monotoncty, A. If B then from monotoncty, U( A, B) U( A, ). Theorem 3.4 Assume that U s a matrx-norm wth neutral matrx. Then (a) U( A, 0) 0 for all A (b) U( A, ) for all A Proof. (a) For A, U( A, 0) U(, 0) 0. Therefore U( A, 0) 0. (b) For A, U( A, ) U(, ). Therefore U( A, ). The followng two corollares are easy to prove. Corrollary 3.5 If 0, U( A, ) for all AM. Ths s the case of multple t-conorm. Corrollary 3.6 If, U( A, 0) 0 for all AM. Ths s the case of multple t-norm. Theorem 3.7 If U s a matrx-norm then U( 0, 0) 0 and U(, ). Proof. Snce 0, from monotoncty, U( 0, 0) U(, 0 ). From the property of neutral matrx, U(, 0) 0 and therefore U( 0, 0) 0. Snce U( 0, 0) 0, we have U( 0, 0) 0. Smlarly, snce, from monotoncty, U(, ) U(, ). From the property of neutral matrx U(, ) and therefore U(, ). Snce U(, ), we have U(, ). Defnton 3. Let u be unnorm aggregaton operator. Defne the bnary operaton U on M as follows; A [ a ] and B [ b ] n M are mapped to C [ c ] n M by c u( a, b ) for every,,..., n and,,..., k. Then U s a matrx norm on M, called the matrx norm nduced by unnorm u. Matrx-norm nduced by the unnorm u ( a, b) mn( a, b) s the multple t-norm wth neutral matrx and matrx-norm nduced by the unnorm u ( a, b) max( a, b) s the multple t-conorm wth neutral matrx 0. Defnton 3.3 Consder the matrx-norms U and U. Then U s dstrbutve over U f t satsfes U ( A, U ( B, C)) U ( U ( A, B), U ( A, C)) for all A, B, C M. Theorem 3.8 Let U and U be two matrx-norms wth neutral matrces and, respectvely. Suppose that U s dstrbutve over U. Then U (, ) (a) (b) If U(, ), then U s dempotent. (c) If and U ( 0, ), then U ( 0, ). (d) If and U ( 0, ) 0, then U ( 0, ) 0. Proof. (a) Takng A C and B n equaton (), we get U( U(, ), U(, )) U(, U(, )), whch gves U(, U(, )) U(, ). Therefore U (, ). Natonal Conference On Dscrete Mathematcs & Computng (NCDMC-07) 3 Page
Natonal Conference On Dscrete Mathematcs & Computng (NCDMC-07) Matrx-Norm Aggregaton Operators (b) Let AM. Takng B C n equaton (), we get U( U( A, ), U( A, )) U( A, U(, )), whch gves U( A, A) U( A, ). Therefore U ( A, A) A and hence U s dempotent. (c) Takng A 0, B, C n equaton (), we get U( U( 0, ), U( 0, )) U( 0, U(, )). Snce, we have U(, ) U(, ). Therefore U ( U ( 0, ), U ( 0, )) U ( 0, U (, )) U ( 0, U (, )) U ( 0, ), whch gves U ( 0, ). Therefore U ( 0, ). (c) Takng A 0, B, C n equaton (), we get U( U( 0, ), U( 0, )) U( 0, U(, )). Snce, we have U(, ) U(, ). Therefore U ( U ( 0, ), U ( 0, )) U ( 0, U (, )) U ( 0, U (, )) U ( 0, ) 0, whch gves U ( 0, ) 0. Therefore U ( 0, ) 0. IV. Concluson Unnorms are mportant generalzatons of t-norms and t-conorms, havng a neutral element lyng anywhere n the unt nterval. It has found many applcatrons n varous felds. After unnorms on unt nterval, unnorms on bounded lattces have recently become a challengng study obect. Ths paper extended ths study of unnorms on the bounded lattce M and dscussed some of ts propertes. Unnorms on the bounded lattce M are called matrx-norm aggregaton operators and they are the generalzatons of multple t-norms and t- conorms. Also, matrx-norms nduced by unnorms are developed and ther propertes are nvestgated. Acknowledgements The frst author acknowledges the fnancal assstance gven by the Mnstry of Human Resource Development, Government of Inda and Natonal Insttute of Technology, Calcut, throughout the preparaton of ths paper. References []. Zadeh, Lotf A. "Fuzzy sets." Informaton and control 8(3), 965, 338-353. []. Gau, Wen-Lung, and Danel J. Buehrer. "Vague sets." I transactons on systems, man, and cybernetcs 3(),993, 60-64. [3]. Atanassov, Krassmr T. "Intutonstc fuzzy sets." Fuzzy sets and Systems 0(),986, 87-96. [4]. Sebastan, Sabu, and T. V. Ramakrshnan. "Mult-fuzzy sets." Internatonal Mathematcal Forum. 5(50), 00. [5]. Yager, Ronald R. "On the theory of bags." Internatonal Journal of General System 3(), 986, 3-37. [6]. Vad, Shna, Sunl Jacob John and Antha Sara Thomas. " Multple sets. " Journal of new results n scences 9, 05, 8-7. [7]. Vad, Shna, Sunl Jacob John and Antha sara Thomas. " Multple sets: a unfed approach towards modellng vagueness and multplcty. " Journal of new theory,, 06, 9-53. [8]. R. Yager, On the general class of fuzzy connectves, Fuzzy Sets Syst. 4, 980, 35 4. [9]. J. Domb, A general class of fuzzy operators, A De Morgan s class of fuzzy operators and fuzzness nduced by fuzzy operators, Fuzzy Sets Syst, 8, 98, 49 63. [0]. S. Weber, A general concept of fuzzy connectves, negatons and mplcatons based on t Norms and t Co norms, Fuzzy Sets Syst,, 983, 5 34. []. Deschrver, Glad, Chrs Cornels, and tenne. Kerre. "On the representaton of ntutonstc fuzzy t-norms and t- conorms." I transactons on fuzzy systems, (), 004, 45-6. []. Sebastan, Sabu, and T. V. Ramakrshnan. "Mult-fuzzy sets: an extenson of fuzzy sets." Fuzzy Informaton and ngneerng, 3(), 0, 35-43. [3]. Vad, Shna and Sunl Jacob John. " t-norms on multple sets. " presented n Natonal Semnar on Dscrete Mathematcs and ts Applcatons ( th & th August 04), held at CAS College, Kannur, Kerala, Inda. [4]. Vad, Shna and Sunl Jacob John. " On Multple t-norms and t conorms. " Materals Today: Proceedngs. Submtted. [5]. Yager, Ronald R., and Alexander Rybalov. "Unnorm aggregaton operators." Fuzzy sets and systems, 80(), 996, -0. [6]. Fodor, János C., Ronald R. Yager, and Alexander Rybalov. "Structure of unnorms." Internatonal Journal of Uncertanty, Fuzzness and Knowledge-Based Systems, 5(04), 997, 4-47. [7]. De Baets, Bernard. "Idempotent unnorms." uropean Journal of Operatonal Research, 8(3), 999, 63-64. [8]. L, Yong-Mng, and Zhong-Ke Sh. "Remarks on unnorm aggregaton operators." Fuzzy Sets and Systems, 4(3), 000, 377-380. [9]. Yager, Ronald R. "Unnorms n fuzzy systems modelng." Fuzzy Sets and Systems, (), 00, 67-75. 33 Page
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