T-Rough Fuzzy Subgroups of Groups

Transcription

1 Journal of mahemaic and compuer cience 12 (2014), T-Rough Fuzzy Subgroup of Group Ehagh Hoeinpour Deparmen of Mahemaic, Sari Branch, Ilamic Azad Univeriy, Sari, Iran. Aricle hiory: Received July 2014 Acceped Augu 2014 Available online Augu 2014 Abrac The rough e heory wa inroduced by Pawlak in I wa propoed for preenaion equivalence relaion. Bu he concep of fuzzy e wa inroduced by Zadeh in In hi paper,he concep of he rough e,t-rough e,t-rough fuzzy e, T-rough fuzzy ubgroup, T-rough fuzzy ideal, and e-valued homomorphim of group will be given. A neceery and ufficien condiion for a fuzzy ubgroup(ideal) and fuzzy prime ideal of a group under a e-valued homomorphim o be a T-rough fuzzy ubgroup(ideal) and T-rough fuzzy prime ideal i aed. The purpoe of hi paper i o inroduce and dicu he concep of T-rough fuzzy group of group ha hoe have been proved in everal paper. Alo, we proved ha inerecion wo fuzzy ubgroup(ideal) of a e under a e-valued homomorphim i a T-rough fuzzy ubgroup of oher e. Keyword: Approximaion pace, T-rough e, T-rough fuzzy e, Fuzzy ubgroup, Fuzzy ideal, e-valued homomorphim. 1. Inroducion The noion of rough e wa inroduced by Z.Pawlak in he year 1982 [24-25]. Rough e heory, a new mahemaical approach o deal wih inexac, uncerain or vague. Knowledge, ha recenly received wide aenion on he reearch area in boh of he real-life applicaion and he heory ielf. I ha found pracical applicaion in many area uch a knowledge dicovery,machine learning, daa analyi, approximae claificaion, conflic analyi, and o on [21-28]. Rough e heory i a mahemaical framework for dealing wih uncerainy and o ome exen overlapping fuzzy e heory. The rough e heory approach i baed on indicernibiliy relaion and approximaion. The heory of rough e i an exenion of e heory. The concep of a fuzzy e wa inroduced by Zadeh [32], and i i now a rigorou area of reearch wih manifold applicaion ranging from engineering and compuer cience o medical diagnoi and ocial behavior udie. In paricular, ome reearcher [12,15] applied he noion of fuzzy e o ideal of a ring. The algebraic approach of rough e wa udied by ome auhor, for example, Q.M.Xiao and Z.L.Zhang [30] udied rough prime ideal and rough fuzzy prime ideal in emigroup. Baed on he

2 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), conrucive mehod, exenive reearch ha alo been carried ou o compare he heory of rough e wih oher heorie of uncerainy uch a fuzzy e and condiional even. Since Roenfeld [29], applied he noion of fuzzy e o algebra and inroduced he noion of fuzzy ubgroup, ince hen many reearcher are engaged in exending he concep of abrac algebra o he broader framework of he fuzzy eing. In 1982, Liu [17] defined and udied fuzzy ubring and fuzzy ideal of a ring. Reader in [4,9,10,11,14], will ge ome definiion and baic reul abou fuzzy algebra ha heir properie were carefully udied o a cerain exen. Rough fuzzy e and fuzzy rough e are alo udied by Nakamura [19], Nanda [20], Biwa [2,3], and by Banerjee and Pal [1]. Several reearch direcion have been uggeed on fuzzy rough e and rough fuzzy e. In hi paper, we inroduce he concep of he rough e, T-rough e, T-rough fuzz e,trough fuzzy ubgroup(ideal), T-rough fuzzy prime ideal and e-valued homomorphim in a group and give ome properie of uch ideal and hen exended ome heorem in which have been proved in [12,14]. 2. Preliminarie The following definiion and preliminarie are required in he equel of our work and hence preened in brief. Some of hem were in [21]. Suppoe ha U i a non-empy e. A pariion or claificaion of U i a family of non-empy ube of U uch ha each elemen of U i conained in exacly one elemen of. Recall ha an equivalence relaion on a e U i a reflexive, ymmeric, and raniive binary relaion on U. Eech pariion induce an equivalence relaion on U by eing. x y x and y are in he ame cla of. Converely, each equivalence relaion on U induce a pariion of U whoe clae have he form x y U x y. Definiion 2.1.[25]. A pair U, where U and i an equivalence relaion on U i called an approximaion pace. Definiion 2.2.[25]. For an approximaion pace U, by a rough approximaion in U, we mean a mapping Apr : P( U ) P( U ) P( U ) defined by for every X P( U ), Apr ( X ) ( Apr ( X ), Apr ( X )), where Apr ( X ) i called a lower rough approximaion of X in, rough approximaion of X in U,. Definiion 2.3. Given an approximaion pace U, a pair (, ) rough e in U, if ( A, B ) ( Apr( X ), Apr( X )) for ome X P( U ) Definiion 2.4.[15]. Le, 0,1. If x U. We define a x a x Apr ( X ) x U x X, Apr ( X ) x U x X. U wherea Apr ( X ) i called a upper A B in P( U ) P( U ) i called a. U be an approximaion pace. A ube fuzzy i a mapping from U o Apr ( )( x ) ( a), Apr ( )( x ) ( a) 187

3 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), They are called, repecively, he lower and he upper approximaion of he fuzzy ube. Apr ( ) ( Apr( ), Apr( )) i called a rough fuzzy e repec o if Apr ( ) Apr ( ). Definiion 2.5.[15]. Le be a fuzzy ube of G, 0,1. Then he e xg ( x) ; xg ( x) are called, repecively, -levele and -rong levele of he fuzzy e. Definiion 2.6.[ 29 ]. A fuzzy ube of a group G i called a fuzzy ubgroup if, for all xy, in G, (1) ( xy) ( x) ( y ); 1 (2) ( x ) ( x ). Definiion 2.7.[29 ]. A fuzzy ube of a group G i called a fuzzy ideal if, for all xy, in G, ( xy) ( x) ( y ) Definiion 2.8.[15 ]. A fuzzy ube of a group G i called a fuzzy prime ideal if, for all xy, in G, ( xy) ( x) or ( xy) ( y ) Definiion 2.9.[ 15,18 ]. Le and be wo fuzzy ube of G, hen produc i defined by ( )( z ) ( ( ) ( )) zxy x y for ome x, y G. 3. Se-valued homomorphim which induced by group Definiion 3.1. [ 5 ]. Le X and Y be wo non-empy e and B Y. Le T : X P ( Y ) be a evalued mapping where P ( Y ) upper invere of B under T are defined by denoe he e of all non-empy ube of Y. The lower invere and 1 T ( B ) x X T ( x ) B ; T ( B ) x X T ( x ) B. Definiion 3.2. [ 1 ]. Le X and Y be wo non-empy e and B Y. Le T : X P ( Y ) be a evalued mapping where P ( Y ) denoe he e of all non-empy ube of Y. 1 ( T ( B ), T ( B )) called T-rough e of G. Example 3.3. Le U, be an approximaion pace and T : U P ( U ) be a e-valued mapping where T ( x ), hen for any B U, T ( B ) Apr ( B ) and x T 1 ( B ) Apr ( B ). Definiion 3.4. [ 29 ]. Le G and G be wo group and T : G P ( G) be a e-valued mapping. T i called a e-valued homomorphim if for all x, y G, (1) T( xy) T( x) T( y ); (2) T x -1 a -1 a T x T x -1 ( ( )) ( ). Remark 3.5. Example 3.3 i a e-valued homomorphim. So a group homomorphim i a pecial cae of x y xy for all a e-valued homomorphim. Le be a complee congruence relaion, i.e., x, y G. Define : ( ) T G P G by T ( x ) x Furher, by Example 3.3, he rough e are T-rough e. for all xg, hen T i a e-valued homomorphim. i 188

4 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), Definiion 3.6.[12,13]. Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. Le be a fuzzy ube of G. For every xg, we define T ( ) and T T x a T x a at ( x ) at ( x ) 1 ( )( ) ( ) ; ( )( ) ( ). 1 ( ) are called, repecively, he T- rough lower and he T- rough upper fuzzy ube of G. If T ( ) and 1 1 T ( ) are fuzzy prime ideal, ( T ( ), T ( )) i aid o be T- rough fuzzy prime ideal of G. Propoiion 3.7. Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. Le and be wo fuzzy ube of G, hen following hold: (1) T ( ) T ( ) T ( ); (2) T ( ) T ( ) T ( ); (3) implie T ( ) T ( ) and T ( ) T ( ) T ( ); -1-1 T ( ) T ( ); (4) (5) T ( ) T ( ) T ( ). Proof. The proof i wih uing of definiion. Theorem 3.8. Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. If i a fuzzy ubgroup of G, hen T 1 ( ) i a fuzzy ubgroup of G. Proof. Le i a fuzzy ubgroup of G. Then we have for all x, y G, Hence 1 T ( )( xy) ( z) ( ab) z T ( xy) a T ( x), bt ( y) ( ( a) ( b)) a T ( x), b T ( y) T 1 ( )( xy) T 1 ( )( x) T 1 ( )( y ). And Therefore T x T x. ( ( a)) ( ( b)) a T ( x) b T ( y) 1 1 T ( )( x) T ( )( y) T x a a -1-1 a T x a T x T -1 x Theorem 3.9. Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. If i a fuzzy ubgroup of G, hen T ( ) i a fuzzy ubgroup of G. Proof. Le i a fuzzy ubgroup of G. Then we have for all x, y G,. 189

5 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), T ( )( xy) ( z) ( ab) z T ( xy) a T ( x), b T ( y) Hence T ( )( xy) T ( )( x) T ( )( y ). And -1 + Therefore T x T x. Corollary Le G and ( ( a) ( b)) a T ( x), b T ( y) ( ( a)) ( ( b)) a T ( x) b T ( y) T ( )( x) T ( )( y) T x a a -1-1 a T x a T x T x. G be wo group and T : G P ( G) be a e-valued homomorphim. If i a fuzzy ubgroup of G, hen 1 ( T ( ), T ( )) i a T- rough fuzzy ubgroup of G. Theorem Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. If i a fuzzy ideal of G, hen 1 ( T ( ), T ( )) i a T- rough fuzzy ideal of G. Proof. Le i a fuzzy ideal of G. Then we have for all x, y G, Hence T ( )( xy ) T ( )( x ) T ( )( y ). Alo 1 T ( )( xy ) ( z ) ( ab ) z T ( xy ) at ( x ), bt ( y ) ( ( a) ( b)) at ( x ), bt ( y ) ( ( a)) ( ( b)) at ( x ) bt ( y ) T 1 ( )( x ) T 1 ( )( y ). T ( )( xy ) ( z ) ( ab ) z T ( xy ) at ( x ), bt ( y ) ( ( a) ( b)) at ( x ), bt ( y ) ( ( a)) ( ( b)) at ( x ) bt ( y ) T ( )( x ) T ( )( y ). Therefore T ( )( xy ) T ( )( x ) T ( )( y ) Propoiion Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. Le and be wo fuzzy ube of G, hen following hold: 190

6 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), T ( ) T ( ) T ( ); (1) (2) T ( ) T ( ) T ( ). Proof.(1) For any x, y, z G, we have Therefore T 1 ( )( x ) ( )( a ) ( min( ( y ), ( z )) ) a T ( x) a T ( x) a yz ( min( ( a), ( a)) ) a T ( x) a yz (min ( a), ( a) ) ( )( a) a T ( x) a T ( x) T1 ( )( x ) T ( ) T ( ) T ( ) T ( ). (2). The proof i imilar o he proof (1). Lemma Le be a fuzzy ube of a group G. Then i a fuzzy ubgroup of G if and only if for all 0,1, if,, hen and are fuzzy ubgroup of G. 1 Proof. Aume i a fuzzy ubgroup of G. We how ha xy, x for all, xy. Since 1 i a fuzzy ubgroup of G, we have ( xy) ( x) ( y), ( x ) ( x). Since x, y, 1 hen ( x ), ( y ). I implie ha xy, x. Hence i a fuzzy ubgroup of G. Similarly, we can how i a fuzzy ubgroup of G, oo. Converely, le i a fuzzy ubgroup of G for all 0,1, and. Aume x y, x for all x, y G. I implie xy,. Since i a fuzzy ubgroup of G, hen 1 1 xy, x. Therefore ( xy) ( x) ( y), x ( x ). Hence i a fuzzy ubgroup of G. Lemma Le be a fuzzy ube of a group G. Then i a fuzzy ideal of G if and only if for all 0,1, if,, hen and are fuzzy ideal of G. Proof. The proof i imilar o he proof lemma Lemma Le be a fuzzy ube of a group G. Then i a fuzzy prime ideal of G if and only if for all 0,1, if,, hen and are fuzzy prime ideal of G. Proof. Aume i a fuzzy prime ideal of G. Aume. By lemma 3.14, i an ideal of G. Le x, y G, uch ha xy. Since i a fuzzy prime ideal of G, hen ( xy) ( x ) or 191

7 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), ( xy) ( y ). I implie x or y. Therefore i a prime ideal of G. Similarly, we can how i a prime ideal of G,oo. Converely, aume for all 0,1, if, hen i a prime ideal of L. Le x, y L, By lemma 3.14, i a fuzzy ideal of G. I implie ( xy) ( x ) and ( xy) ( y ). Le ( xy ). Thu xy. Since i a prime ideal of G, x or y. I mean ha ( x) ( xy ) or ( y) ( xy ). Hence ( xy) ( x ) or ( xy) ( y ). Therefore i a fuzzy prime ideal of G. Lemma3.16. Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. If i a fuzzy ideal of G, hen for any 0,1, (1) ( T ( )) T ( ); (2) 1 1 ( T ( )) T ( ); (3) ( T ( )) T ( ); 1 1 ( 4) ( T ( )) T ( ). Proof. (1). x ( T ( )) T ( )( x ) ( a) a T ( x ) (2). a T ( x ), ( a) T ( x ) x T ( ). 1 1 x ( T ( )) T ( )( x) ( a) a T ( x) a T ( x), ( a) 1 T ( x) x T ( ). (3) and (4) are imilar o (1) and (2). Theorem Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. If i a fuzzy prime ideal of G, hen T ( ) and T 1 ( ) are fuzzy prime ideal of G. Proof. Le be a fuzzy prime ideal of G. By lemma 3.15, for all 0,1, if, hen i a prime ideal of G. By lemma 3.16 and 3.15, for all 0,1, if ( T ( )),( T ( )) i a prime ideal of G. By lemma 3.15, T ( ) i a fuzzy prime ideal of G. Hence i a lower T-rough fuzzy prime ideal of G. Similarly, i a upper T-rough fuzzy prime ideal of G. Therefore T ( ) are fuzzy prime ideal of G. and T 1 ( ) 192

8 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), Propoiion LeG and G be wo group and T : G P ( G) be a e-valued homomorphim. Le and be wo fuzzy ubgroup( ideal) of G, hen i a upper T- rough fuzzy ubgroup(ideal) of G. Proof. Le x, y G, hen 1 T ( )( xy ) )( z ) ( )( ab ) z T ( xy) a T ( x), b T ( y) (min ( ab), ( ab) ) a T ( x), b T ( y) a b a b (min min ( ), ( ),min ( ), ( ) ) a T ( x), b T ( y) a a b b (min min ( ), ( ),min ( ), ( ) ) a T ( x), b T ( y) (min ( )( a),( )( b) ) a T ( x), b T ( y) ( ( )( a)) ( ( )( b)) a T ( x) b T ( y) 1 1 T ( )( x) T ( )( y) Therefore T ( )( xy) T ( )( x) T ( )( y ). Alo T x a a T x 1 a T x. at x Hence i a upper T- rough fuzzy ubgroup of G. Propoiion Le G and G be wo group and T : G P ( G) be a e-valued homomorphim. Le an be wo fuzzy ubgroup( ideal) of G, hen i a lower T- rough fuzzy ubgroup ( ideal) of G. Proof. The proof i imilar o he proof propoiion Concluion Fuzzy e heory and rough e heory ake ino accoun wo differen apec of uncerainy ha can be encounered in real-world problem in many field. Fuzzy e deal wih poibiliie uncerainy, conneced from ambiguiy of informaion. The combinaion of fuzzy e and rough e lead o variou model. Thi paper i inended o buil up a connecion beween rough e, fuzzy e and group heory. The noion of T- rough fuzzy ubgroup in a group i a generalizaion of he noion of fuzzy ubgroup in a group. Alo, uing he concep of relaion beween rough fuzzy ubgroup and level rough e. In hi paper, we ubiued a univere e by a group, and inroduced he e-valued homomorphim and T- rough fuzzy ubgroup, and T-rough fuzzy ideal and T-rough fuzzy prime ideal in a group baed on 193

9 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), definiion in [12,13,16,30,31]. We generalized ome idea preened by Davvaz [5,6]. Furher, we udied and inveigaed ome heir inereing properie of a e-valued homomorphim induced by a group homomorphim.we hope ha hi exended reearch many provide a powerful ool in approximae reaoning. Alo, we believe, hi paper offered here will urn ou o be more ueful in he heory and applicaion of rough e and fuzzy e. Acknowledgmen The auhor i highly graeful o referee for heir valuable commen and uggeion for improving he paper. Reference [1] M.Banerjee, and S.K. Pal, Roughne of a fuzzy e, Inform. Sci.93, (1996), [2] R.Biwa, On rough e and fuzzy rough e, Bull. Pol.Acad. Sci. Mah.42, (1994), [3] R. Biwa, On rough fuzzy e, Bull. Pol. Acad. Sci. Mah.42,(1994), [4] Z. Bonikowaki, Algebraic rucure of rough e, in: W.P. Ziarko (Ed.),Rough Se, Fuzzy Se and Knowledge Dicovery, Springer-Verlag, Berlin, (1995), [5] B. Davvaz, A hor noe on algebraic T-rough e,informaion Science, Vol.17, (2008), [6] B. Davvaz, Roughne in ring, Inform. Sci. 164,(2004), [7] D.Duboi, H. Prade, Rough fuzzy e and fuzzy rough e, In. J. General Sy. 17 (2-3),(1990), [8] D. Duboi, H. Prade, Two fold fuzzy e and rough e-ome iue in knowledge repreenaion, Fuzzy Se Sy. 23, (1987), [9] E.Ranjbar-Yanehari, M.Aghari-Larimi, union and inerecion fuzzy ubhypergroup, JMCS, Vol.5, No.2, (2012) [10] E. Hendukolaie, On fuzzy homomorphim beween hypernear-ring, JMCS, Vol.2, No.4, (2011) [11] E. Hendukolaie, M.Aliakbarnia.omran, Y.Naabi, On fuzzy iomorphim heorem of - hypernear-ring, JMCS, Vol.7, (2013) [12] S.B. Hoeini, N. Jafarzadeh, A. Gholami, T-rough Ideal and T-rough Fuzzy Ideal in a Semigrou, Advanced Maerial Reearch, Vol ,(2012), [13] S.B.Hoeini, N. Jafarzadeh, A. Gholami, Some Reul on T-rough (prime, primary) Ideal and T-rough Fuzzy (prime, primary) Ideal on Commuaive Ring,In.J.Conemp. Mah Scince, Vol.7,( (2012), [14] T.Iwinki, Algebraic approach o rough e, Bull. PolihAcad. Sci. Mah.35, (1987), [15] O. Kazanci, B. Davvaz, On he rucure of rough prime (primary) ideal and rough fuzzy prime (primary) ideal in commuaive ring, Informaion Science, 178, (2008), [16] N.Kuroki, Rough ideal in emigroup, Inform. Sci. 100, (1997), [17] W.J. Liu, Fuzzy invarian ubgroup and fuzzy ideal, Fuzzy Se Sy. 8,(1982) [18] J.N. Mordeon, M.S. Malik, Fuzzy Commuaive Algebra, World Publihing, Singapore,1998. [19] A.Nakamura, Fuzzy rough e, Noe on Muliple-valued Logic in Japan,9 (8),(1988), 1-8. [20] S. Nanda, Fuzzy rough e, Fuzzy Se and Syem, 45,(1992), [21] Z. Pawlak, Rough e, In. J. Inform. Compu. Sci. 11, (1982), [22] Z. Pawlak, Rough e- Theoreical Apec of Reaoning abou Daa, Kluwer Academic Publihing, Dordrech,1991 [23] Z. Pawlak, Rough e power e hierarchy, ICS PAS Rep.470, (1982). [24] Z. Pawlak, Rough e algebraic and opological approach, ICS PAS Rep. 482, (1982). [25] Z. Pawlak, Rough e and fuzzy e, Fuzzy Se and Syem.17, (1985), [26] Z. Pawlak, Some remark on rough e, Bull. Pol. Acad.Tech. 33, (1985). [27] Z. Pawlak, A. Skowron, Rough e and Boolean reaoning, Informaion Science, 177, (2007),

10 E. Hoeinpour / J. Mah. Compuer Sci. 12 (2014), [28] Z. Pawlak, A. Skowron, Rough e: ome exenion, Informaion Science, 177, (2007), [29] A. Roenfeild, Fuzzy Group, Journal of Mahemaical Analyi and Applicaion, 35, (1971) [30] Q.M. Xiao, Z.L. Zhang, Rough prime ideal and rough fuzzy prime ideal in emigroup, Informaion Science, 176, (2006), [31] S. Yamak, O. Kazanci, B. Davvaz, Generalized lower and upper approximaion in a ring,informaion Science 180, (2010) [32] L.A. Zadeh, Fuzzy e, Inform. Conrol 8, (1965)