Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi

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Gül Karadeniz
7 yıl önce
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Joural of Egieerig ad Naural Scieces Mühedislik ve Fe Bilimleri Dergisi Sigma 004/ ON THE GENERALIZATION OF CARTESIAN PRODUCT OF FUZZY SUBGROUPS AND IDEALS Bayram Ali ERSOY * Deparme of Mahemaics,

1 Joural of Egieerig ad Naural Scieces Mühedislik ve Fe Bilimleri Dergisi Sigma 004/ ON THE GENERALIZATION OF CARTESIAN PRODUCT OF FUZZY SUBGROUPS AND IDEALS Bayram Ali ERSOY * Deparme of Mahemaics, Faculy of Ars ad Sciece, Yildiz Techical Uiversiy, Davupaşa-ISTANBUL, Geliş /Received: 0000 Kabul/Acceped: BULANIK ALT GRUPLARIN VE İDEALLERİN KARTEZYEN ÇARPIMLARININ GENELLEŞTİRİLMESİ ÖZET Bu çalışmada Malik ve Mordeso u makalesi geelleşirildi Yai farklı grupları (halkaları bulaık al gruplarıı (bulaık ideallerii karezye çarpımları iceledi G ve Gboşa farklı iki grup olmak üzere eğer ve G ve G ( R ve R birimli olmak zoruda olmaya değişmeli iki halka olmak üzere i bulaık al grupları(bulaık idealleri ise karezye çarpımları da G G ( R R i bulaık al grubudur (bulaık idealidir Yukarıdaki ifadesii ers yöleri de çalışılmışır Bu ifadeleri farklı grup (halka içi de geelleşirilmişir Aahar Sözcükler: Bulaık al küme, Bulaık Al grup, Seviye al grubu, Bulaık ideal, Seviye ideali, Bulaık bağıı, Karezye çarpım ABSTRACT I his work I geeralize Malik ad Mordeso s paper [] I aalysis he caresia produc of fuzzy subgroups (ideals of wo groups (wo commuaive rigs Rigs which have o ecessarily ideiy eleme Tha is; if ad σ are fuzzy subgroups (ideals of G ad G ( R ad R respecively he σ subgroup (ideal of G G ( R R Coversely he opposie direcio of he above saemes is sudied We geeralize he above saemes for differe Groups (Rigs Keywords: Fuzzy subse, Fuzzy subgroup, Level subgroup, Fuzzy ideal, Level ideal, Fuzzy relaio, Caresia produc INTRODUCTION The cocep of a fuzzy subse was iroduced by Zadeh[5] Fuzzy subgroup ad is impora properies were defied ad esablished by Rosefeld[] The may auhors have sudied abou i Afer his ime i was ecessary o defie fuzzy ideal of a rig The oio of a fuzzy ideal of a rig was iroduced by Liu [] Malik, Mordeso ad Mukherjee have sudied fuzzy ideals The * ersoya@yildizedur ; el: ( PDF creaed wih pdffacory Pro rial versio wwwpdffacorycom

2 B A Ersoy Sigma 004/ cocep of a fuzzy relaio o a se was iroduced by Zadeh[6] Bhaacharya ad Mukherjee have sudied fuzzy relaio o groups Malik ad Mordeso [] sudied fuzzy relaio o rigs Moreover Malik ad Mordeso have wrie very impora book for Fuzzy algebra which is Fuzzy Commuaive Algebra [4] I his paper G ( i =,,, is a group ad R ( i =,,, is a commuaive rig i A fuzzy relaio o R is he fuzzy subse of R R I our paper he caresia produc of wo ses G ad G ( R ad R is defied like ha: ( a, b,( a, b G G( R R ( a, b + ( a, b = ( a+ a, b+ b, ( a, b( a, b = ( aa, bb I geeralize Malik ad Mordeso s paper Tha is; if, are fuzzy subgroups (ideals of G ad G ( R ad R respecively he subgroup (ideal of G G ( R R Le, be fuzzy subses of G, G respecively such ha subgroup (ideal of G G ( R R The or is fuzzy subgroup (ideal of G or G ( R or R respecively Le such ha ( e ( e, i ad be fuzzy subses of R subgroup (ideal of G G ( R R If x G, y G = ( x ( ead ( y ( e ( x R, y R (0 = (0, ( x (0 ad ( y (0 he boh ad are fuzzy subgroups (ideals of G ad G ( R ad R Also I exed hese above heorems for differe Groups (Rigs Tha is if,,,, are fuzzy subgroups (ideals of G, G,, G ( R, R,, R respecively, he is fuzzy subgroup (ideal of G G G ( R, R,, R The I prove he opposie direcio of he previous saeme uder some codiios PRELIMINARIES I his secio, we review some basic defiiios ad resuls Defiiio : A fuzzy subse of o empy se S is a fucio : S [ 0,] Defiiio : A fuzzy subse of a group G is called a fuzzy subgroup of G if (i ( xy mi( ( x, ( y (ii for all xy, G ( x ( x If subgroup of G he ( x = ( x for all x G Defiiio : If subse of S, he for ay, he se = { x S ( x } Im is called he level subse of S wih respec o Theorem 4 ( [] : Le be fuzzy subse of G subgroup of G if ad oly if is a subgroup of G for Im Here, if subgroup of G, he is called a level subgroup of Defiiio 5 ([] : A fuzzy subse of a rig R is called a fuzzy lef (righ ideal of R if (i ( x y mi( ( x, ( y (ii for all x, y R ( xy ( y ( ( xy ( x 96 PDF creaed wih pdffacory Pro rial versio wwwpdffacorycom

3 (, O he Geeralizaio of Caresia Produc A fuzzy subse of R is called a fuzzy ideal of R if lef ad fuzzy righ ideal of R Defiiio 6 ( [5] : If subse of R, he for ay Im, he se = { x R ( x } is called he level subse of R wih respec o Theorem 7 []: Le be fuzzy subse of R ideal of R if ad oly if is a ideal of R for Im Here, if ideal of R, he is called a level ideal of Defiiio 8 ( [6] : A fuzzy relaio o R is he fuzzy subse of R R Defiiio 9 ( [] : Le ad σ be fuzzy subses of R The Caresia produc of ad σ is σ ( xy, = mi( ( x, σ( y for all xy, R FUZZY SUBGROUPS AND FUZZY IDEALS Now we will geeralize some heorems i [] Theorem : If ad are fuzzy subgroups of G ad G respecively, he is a fuzzy subgroup of G G Proof: Le ( a, b,( a, b G G (( a, b( a, b = ( aa, bb ad = mi( ( aa, ( bb mi( ( a, ( a, ( b, ( b mi(mi( ( a, ( b,mi( ( a, ( b = mi( ( a, b, ( a, b (( a, b = ( a, b = mi( ( a, ( b mi( ( a, ( b = a b Therefore subgroup of G G Theorem : If, are fuzzy ideals of R, Rrespecively, he is fuzzy ideal of R R Proof: = (0,0 mi( (0, (0 Le Im( he (0 ad (0 Thus ad are ideals of R ad R respecively Hece for all Im(, ( = is lef ideal of R R Because ( xy,,( z, ( ad ( ab, ( R, R we mus show ha ( x z, y ( ad ( xa, yb ( ( x z, y = mi( ( x z, ( y ad sice ad are ideals of R ad R respecively mi( ( x z, ( y he ( x z, y ( Sice ( xa, yb = mi( ( xa, ( yb ad ad are 97 PDF creaed wih pdffacory Pro rial versio wwwpdffacorycom

4 B A Ersoy Sigma 004/ ideals of R R R R of mi( ( xa, ( yb he ( xa, yb ( Hece ( is ideal Corollary i If,,,, are fuzzy subgroups of G, G,, G respecively, he is fuzzy subgroups of G G G ii If,,,, are fuzzy ideals of R, R,, R respecively, he is fuzzy ideal of R, R,, R Proof: Oe ca easily show by iducio mehod Theorem 4: Le, be fuzzy subses of G, Grespecively such ha subgroup of G G The or is fuzzy subgroup of G or G respecively Proof: We kow ha ( e, e = mi( ( e, ( e ( xy,, ( xy, G G The ( x ( e or ( y ( e If ( x ( e, he ( x ( e or ( y ( e Le ( x ( e The x G ( xe, = ( x xy, G ad ( xy = ( xy, e = (( xe, ( ye, mi( ( xe,, ( ye, = mi( ( x, ( y ( x = ( x, e = ( x, e = ( xe, mi ( xe, = ( x Therefore is fuzzy subgroup of G Now suppose ha ( x ( e is o rue for all x G If ( x > ( e x G, he ( y ( e y G Therefore ( e, y = ( y for all y G Similarly xy, G ad ( xy = ( e, xy = (( e, x( e, y mi( ( e, x, ( e, y = mi( ( x, ( y ( x = ( e, x = ( e, x = ( e, x mi ( e, x = ( x Hece is fuzzy subgroup of G Cosequely eiher or is fuzzy subgroup of G or G respecively 98 PDF creaed wih pdffacory Pro rial versio wwwpdffacorycom

5 O he Geeralizaio of Caresia Produc Theorem 5: Le, be fuzzy subses of R, R respecively such ha ideal of R R The or is fuzzy ideal of R or R respecively Proof: We kow ha (0,0 = mi( (0, (0 ( xy,, ( xy, R R The ( x (0 or ( y (0 If ( x (0, he ( x (0 or ( y (0 Le ( x (0 The x R ( x,0 = ( x xy, R ( x y = ( x y,0 ad = (( x,0 ( y,0 mi( ( x,0, ( y,0 = mi( ( x, ( y ( xy = ( xy,0 = (( x,0 ( y,0 = ( x,0 or ( y,0 mi ( x,0or mi ( y,0 = ( x or = ( x Therefore is fuzzy ideal of R Now suppose ha ( x (0 is o rue for all x R If ( x > (0 x R, he ( y (0 y R Therefore (0, y = ( y for all y G Similarly xy, R ad ( x y = (0, x y = ((0, x (0, y mi( (0, x, (0, y = mi( ( x, ( y ( xy = (0, xy = ((0, x(0, y (0, x, ( (0, y = mi( (0, ( x,( = mi( (0, ( y = ( x, ( = ( y Therefore is fuzzy ideal of R Corollary 6: Le,,,, be a similar fuzzy subses of G, G,, G ( R, R,, R of such ha is fuzzy subgroup (ideal of G G G ( R, R,, R The or or or or subgroups (ideals of G, G,, G ( R, R,, R respecively Corollary 7: Le ad be a similar fuzzy subses of G ad G ( R ad R of such ha is fuzzy subgroup (ideal of G G ( R R If x G, y G ( e = ( e ( x ( e ad ( x R, y R (0 = (0, ( x (0 ad ( y (0 he, subgroups (ideals of G ad G ( R ad R 99 PDF creaed wih pdffacory Pro rial versio wwwpdffacorycom

6 B A Ersoy Sigma 004/ Corollary 8: Le,,,, be a similar fuzzy subses of G, G,, G ( R, R,, R of such ha is fuzzy subgroup (ideal of G G G ( R R R If x G, x G,, x G ( e = ( e = ( e = = ( e, G, G,, G ( x ( e, ( x ( e, ( x ( e,, ( x ( e ( x R, x R,, x R (0 (0 (0 (0, ( x (0, subgroups (ideals of,,, 4 CONCLUSIONS = = = = ( x (0, ( x (0,, ( x (0 he,,,, G G G ( R, R,, R respecively Oe ca examie hese heorems i ay Rigs Tha is, i rue ha hese heorems are valid i o commuaive rigs wihou ideiy eleme REFERENCES [] Liu, WJ, Fuzzy ivaria subgroups ad fuzzy ideals, Fuzzy Ses ad Sysems, 8, - 9, 98 [] Rosefeld, A, Fuzzy groups, J Mah Aal Appl , 97 [] Malik DS ad Mordeso J N, Fuzzy relaios o rigs ad groups, Fuzzy Ses ad Sysems 4 7-, 99 [4] Malik DS ad Mordeso J N, Fuzzy Commuaive Algebra, World scieificpublishig, 998 [5] Zadeh L A, Fuzzy ses, Iform Corol 8 8-5, 965 [6] Zadeh L A, Similariy relaios ad fuzzy orderig, Iform Sci, 77-0, PDF creaed wih pdffacory Pro rial versio wwwpdffacorycom

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