Compositions of Fuzzy T -Ideals in Ternary -Semi ring

Transcription

1 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces Copositios of Fuy T -Ideals i Terary -Sei rig RevathiK, 2, SudarayyaP 3, Madhusudhaa RaoD 4, Siva PrasadP 5 Research Scholar, Departet of Matheatics, GITAM Uiversity, Visakhapata, AP Idia 2 Asst Professor, Departet of Matheatics, Adikavi Naaya Uiversity, Rajahudry, AP Eail:revathiditakurthi@gailco 3 Departet of Matheatics, GITAM Uiversity, Visakhapata, AP Idia Eail:psudarayya@gailco 4 Departet of Matheatics, VSR & NVR College, Teali, AP Idia Eail: draths@gailco 5 Departet of BSH, VFSTR S Uiversity,Vadlaudi, Gutur, AP Idia Eail:pusapatisivaprasad@gailco Abstract The purpose of this paper is to itroduce differet types of operatios o fuy TΓ-ideals of terary Γ-seirigs ad to prove subsequetly these operatios give rise to differet structures o soe classes of fuy TΓ-ideals of terary Γ-seirigs A characteriatio of a regular terary Γ-seirig has also bee obtaied i ters of fuy subsets Keywords--- Terary Γ-Seirig, Regular terary Γ-seirig, left fiy TΓ-Ideal, right fiy TΓ-Ideal, lateral fiy TΓ-Ideal, fiy TΓ-Ideal Matheatics Subject Classificatio[2000: 6Y60, 6Y99, 03E72 I INTRODUCTION The theory of fuy sets was first ispired by Zadeh [9 Fuy set theory has bee developed i ay directios by ay scholars ad has evoked great iterest aog atheaticias workig i differet fields of atheatics Fuy ideals i rigs were itroduced by Liu [5 ad it has bee studied by several authors Ju [2 ad Ki ad Park [4 have also studied fuy ideals i seirigs I the year 2007, [6 we have itroduced the otios of fuy ideals ad fuy quasi-ideals i terary seirigs I the year 205, SajaiLavaya ad MadhusudhaaRao[6, 7, 8 itroduced the otio of terary Γ-Seirigs 2 PRELIMINARIES Defiitio 2: Let T ad Γ be two additive coutative seigroups x,, x,, x ) x x x appigfrot Γ T Γ TtoTwhichaps ( i) [[a b cγd e = [a [b c d e = [a b [c d e ii)[(a + b) c d = [a c d + [b c d iii) [a (b + c)βd = [a b d + [a c d iv) [a b (c + d) = [a b c + [a b d for all a, b, c, d T ad,,, Γ T is said to be a Terary -seirigif there exist a satisfyig the coditios : Defiitio 22: A eleet 0of a terary Γ-seirig T is said to be a absorbig ero of T provided 0 + x = x = x + 0ad 0 a b = a 0βb = a b 0 = 0 a, b, x T ad, Γ Defiitio 23: Let T be terary Γ-seirig A o epty subset S is said to be a terary -sub seirig of T if S is a additive sub sei group of T ad a b cs for all a,b,cs ad, Γ Defiitio 24: A oepty subset A of a terary Γ-seirig T is said to be left terary -ideal or siply left T -ideal of T if () a, b A iplies a + b A (2) b, ct, a A,, Γ iplies b c a A Defiitio 25: A oepty subset of a terary Γ-seirig T is said to be a lateral terary -ideal or siply lateral T -ideal of T if () a, b A a + b A (2) b, c T, a A,, Γ b a c A Defiitio 26: A oepty subset A of a terary Γ-seirig T is a right terary -ideal or siply right T -ideal of T if () a, b A a + b A (2) b, c T, a A,, Γ a b c A 35

2 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces Defiitio27:A o epty subset A of a terary Γ-seirig T is said to be terary -ideal or siply T -ideal of T if () a, b A a + b A (2) b, c T, a A,, Γ b c aa, b a c A, a b c A For ore o preliiaries we ay refer to the refereces ad their refereces 3 Operatios o fuy T -ideals: Defiitio 3: A o-epty fuy subset μ of a terary Γ-seirig Tis called a fuy left(lateral, right)terary -ideal or siply fuy left T -ideal of T if (i) (x + y ) i{μ(x), μ(y)} (ii) μ(xγy ) μ()[μ(xγy ) μ(y), μ(xγy ) μ(x) x, y, T, γ, Γ Defiitio32: A o-epty fuy subset μ of a terary Γ-seirig T is called a fuy -ideal of T if (i) (x + y ) i{μ(x), μ(y)} (ii) ( x y ) ( x) ( y) ( ) for ay x, y, T ad, Note33:A o-epty fuy subset μ of a terary Γ-sei rig T is called a fuy -ideal of T if it is a fuy left TΓ- ideal, a fuy right TΓ-ideal ad a fuy lateral TΓ-ideal of T Note 34: A fuy T -ideal of a terary Γ-sei rig T is a o-epty fuy subset of T which is a fuy left TΓ-ideal, fuy lateral TΓ-ideal ad fuy right TΓ-ideal of T Throughout this thesis uless otherwise etioed T deotes a terary Γ- seirig with uities ad FLTΓI(T),FMTΓI(T), FRTΓI(T) ad FTΓI(T) deotes respectively the set of all fuy left TΓ-ideals, the set of all fuy lateral TΓ-ideals, the set of all fuy right TΓ-ideals ad the set of all fuy TΓ-ideals of the terary Γ-sei rig T Also we cosider that (0) = for a fuy left TΓ- ideal(fuy lateral TΓ-ideal, fuy right TΓ-ideal ad fuy TΓ-ideal) of a terary Γ-sei rig T Defiitio 35: Let T be a terary Γ-seirig ad, 2, 3 FLT I( T) [FMTΓI(T), FRTΓI(T) FTΓI(T) The the 2 su ad the terary product 23 ad copositio o 2o 3 of, 2 3 are defied as follows: Sup {i[ ( u), 2 ( v): u, vt ( 2)( x) { xuv 0 if for ay u, v T, uv x sup [i{ ( ), 2( ), 3( )}:,, ;, ( )( ) { x u v u v w u v wt x w 0, if for ay u, v, wt ad for ay,, x u v w = { [{ ( u) ( v) 3( w)}: u, v, wt;, x u v w 2 0, if for ay u, v, wt ad for ay,, x u v w ( o o )( x) { sup [ i{ ( ui ), 2( vi ), 3( wi )}: ui, vi, wi T; i, i x u i iivi iw i i 0, if for ay u, v, w T;,, x u v w i = { i [{ ( u ) ( v ) ( w )}: u, v, w T;, x u v w i 2 i 3 i 0, if for ay u, v, w T;,, x u v w i 36

3 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces Sice T cotais 0, i the above defiitio the case x u + v for ay u, v T does ot arise Siilarly, sice T cotais left, lateral, right uity, the case x u v w for ay u, v, w T;, does ot arise I case of i product of, 2, 3 if T has left, lateral ad right uity, the the case x u v w for ay u, v, w T ad, Γ does ot arise ie i other words there are u, v, w T ad, Γ such that x = u v w Theore 36: I a terary -seirig T the followig are equivalet (i) is a fuy left(lateral, right)t -ideals of T (ii) + μ ad,(, )where is the characteristic fuctio of T Proof : Let μ be a fuy left TΓ-ideal of T Let a T Suppose there exist x, y, T ad, Γ such that a = x y The, sice is a fuy left TΓ-ideal of T, we have ( ) x y sup [i[ ( p), ( q): p, q T x y pq sup [i[ ( u s), ( v t) : u, v, s, t T xuv, yst sup [i[i[ ( u), ( s),i[ ( v), ( t): u, v, s, t T xuv, yst = sup [i[i[ ( u), ( v),i[ ( s), ( t): u, v, s, t T xuv, yst =i[ sup [i[ ( u), ( v), sup [i[ ( s), ( t): u, v, s, t T xuv yst =i[( )( x),( )( y) i[i[ ( x), ( x),i[ ( y), ( y) =i[i[, ( x),i[, ( y) i[i[ ( x), ( y) =i[ ( x), ( y) Therefore χ + μ μ Ad ( )( a) sup [i{ ( x), ( y), ( )} ax y sup [i{,, ( )} ax y sup { ( )} ( ) ax y Now sice is a fuy left TΓ-ideal, ( x y ) ( ) x, y, T ad, So i particular, ( ) ( a) for all a = x y Hece sup a x y ( ) ( a) Thus ( a) ( )( a) If there do ot exist x, y, T,, such that a = x y the ( )( a) 0 ( a) Therefore Coversely, suppose that Let x, y, T ad, Γ such that a = x y The ( x y ) ( a) ( )( a) Now ( )( a) sup [i{ ( x), ( y), ( )} ax y sup [i{,, ( )} ax y sup { ( )} ( ) ax y Hece ( x y ) ( ) x, y T ad Therefore is a fuy left TΓ-ideal of T Siilarly, we ca prove the reaiig parts of the stateet Theore 37: I a -sei group S the followig are equivalet (i) is a fuy T -ideals of S (ii) + ad, ad, where is the characteristic fuctio of T Proof : By usig the theores 36, we ca fid the proof of the theore easily Theore 38: Let, 2 FLT I(T)[FMT I(T), FRT I(T), FT I(T) The + 2 FLT I(T)[FMT I(T), FRT I(T), FT I(T) 37

4 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces ( )(0) [ [ ( u), ( v): u, v T [ (0), 2(0): u, vt 0 Thus Proof: 2 2 0uv 2 is oepty ad ( 2)(0) Let x, y, T ad, Γ The ( )( x y) [ [ ( x), ( y): p, q T 2 2 xy pq [ [ ( u s), ( v t): u, s, v, t T 2 x yuv, yst [ [ [ ( u), ( s), [ ( v), ( t): u, s, v, t T 2 2 x yuv, yst [ [ [ ( u), ( v), [ ( s), ( t): u, s, v, t T = 2 2 x yuv, yst [ [ [ ( u), ( v), [ ( s), ( t): u, s, v, t T = 2 2 xyuv yst [( )( x),( )( y) = 2 2 ( )( x y ) [ [ ( p), ( q) Agai 2 2 x y pq [ [ ( x yu), ( x yv) 2 uv [Sice x y x y ( u v) x yu x yv [ [ ( u), ( v) ( )( ) 2 2 uv Hece, 2 FLT I( T) Siilarly, oe ca prove the reaiig parts Theore 39: Let, 2, 3 FLT I(T)[FMT I(T), FRT I(T), FT I(T) The (i) 2 = 2 (ii) ( 2 ) + 3 = ( ) (iii) = = (iv) (v) 2 ad (vi) Proof: (i) 2 2 where is a fuy T -ideal of T, defied by ( )( x) [ [ ( u), ( v): u, v T xuv Therefore, 2 = 2 (ii) Let x T [ [ ( v), ( u): u, v T = ( 2 )( x) = 2 xuv [( ) ( x) [ [( )( u), ( v): u, v T xuv = xuv u pq ( x) { if x0 0 if x 0 [ [ [ [ ( p) ( q): p, qt, ( v): u, v T [ [ [ ( p) ( q), ( v) = xuv u pq 38

5 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces Siilarly, we ca deduce that [ [ ( p) ( q), ( v) = x pqv Therefore, ( 2 ) + 3 = ( ) (iii) for ay x T, [ ( )( x) = [ [ ( p) 2( q), 3( v) x pqv ( )( x) [ [ ( u), ( v): u, v T = [ (0), ( x) ( x) xuv Thus, = ad fro (i) = = ( )( x) [ [ ( u), ( v): u, v T (iv) Let x T The xuv ( u v) ( x) Hece = xuv ( x) [ (0), ( x) [ [ ( u), ( v): u, v T = ( )( x) Agai 0 xuv Therefore, Cosequetly, ( )( x) [ [ ( u), ( v): u, v T (v) Let x T The xuv [ ( u), (0) = ( x) > Therefore, 2, (vi) 2 x T The 3 3 Hece ( )( x) [ [ ( u), ( v): u, v T xuv [ [ ( u), ( v): u, v T ( )( x) xuv Theore 30: Let, 2, 3 FLT I(T)[FMT I(T), FRT I(T), FT I(T) The o 2 o 3 FLT I(T)[FMT I(T), FRT I(T), FT I(T) o o (0) Proof: Sice [ [ [ ( u ), ( v ), ( w ): u, v, w T,, Z = 0 uiiviiwi i i 2 i i i i [ (0), 2(0), 3(0) 0 (sice (0) 2(0) 3(0) o o is o epty ad ( o 2o 3 )(0) = Therefore, Now, for x, y, T, o o )(x + y) ( [ [ [ ( u ), ( v ), ( w ): u, v, w T,, Z = x y uiiviiwi i i 2 i i i i 39

6 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces [ i [ [ [ ( u ), ( v ), ( w ), [ ( p ), ( q ), ( r ),for x u v w, k l l k i 2 i 3 i k 2 k 3 k i y p q r, u, v, w, p, q, r T,,,,,, l Z k k k k k i i i k k k i i k k [ [ [ [ ( ui ), 2( vi ), 3( wi ): ui, vi, wi T, i, i, Z, x u v w i i l [ [ [ ( pk ), 2( qk ), 3( rk ): pk, qk, rk T, k, k, l Z y p q r k l k k k k k k [( o o )( x),( o o )( y) Now, ( o o )( x y ) [ [ ( u ), ( v ), ( w ): x y uiivi iwi i i 2 i 3 i i u, v, w T,,, Z [ [ [ ( x y r ), ( s ), ( t ) r s t j j j j j j j 2 j 3 j j [ [ [ ( r ), ( s ), ( t ) ( o o )( ) r s t j j j j j j j 2 j 3 j j Hece, o 2o3 FLT I(T) Siilarly, we ca prove the reaiig results Theore 3: Let, 2, 3 FLT I(T)[FMT I(T), FRT I(T), FT I(T) The o o Proof: If for ay u, v, w T ad for ay, Γ, u v w x the For ay x T, x uiiviiwi i i 2 i 3 i i o o ( o o )( x) [ [ ( u ), ( v ), ( w ): o o Thus, ui, vi, wi T, i, i, Z [ ( u), ( v), ( w) ( )( x) x u v w Theore 32: Let be a fuy left T -ideal, 2 be a fuy lateral T -ideal ad 3be a fuy right T -ideal of a terary -seirig T The Proof: Let be a fuy left TΓ-ideal ad 2 be a fuy lateral TΓ-ideal ad 3 be a fuy right TΓ-ideal of a terary -seirig T Let x T Suppose there exist u, v, w T ad, such that x u v w 40

7 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces ( )( x) Sup i{ ( u), ( v), ( w)} The xu v w xu v w 2 3 Sup [i{ ( u v w), ( u v w), ( u v w)} i{ ( x), ( x), ( x)} ( )( x) u v w T such that x = u v w Suppose there do ot exist,, The ( )( x) = 0 ( 2 3)( x) Therefore, Theore 33: Let T be a ultiplicatively regular terary -seirig ad, ad 3 be three fuy subsets of T The 2 3 Γ 2 Γ 3 Proof : Let c T Sice T is ultiplicatively regular, the there exists two eleetsx, y T ad, γ 2, 3, 4 Γ such that c = c x 2 c 3 y 4 c Now, ( )( c) Sup [i{ ( u), ( v), ( w)}: u, v, wt,, cuv w i{ ( c x c), ( c), ( c)} [sice c = c x 2 c 3 y 4 c = c x 2 c 3 y 4 c x 2 c 3 y 4 c ( c) Therefore, Defiitio 34: A eleet a of a terary Γ-seirig T is said to be terary ultiplicatively regular if there exist x, y T ad,,, such that a x a y a = a Theore 35: A terary -seirig T is ultiplicatively regular if ad oly if A B C = A B C for all left T -ideals A, for all lateral T -ideals B ad for all right T -ideals C of T Theore 36: I a terary -seirig T the followig are equivalet (i) T is ultiplicatively regular for every fuy left T -ideal (ii) T -ideal 3of T Γ ad every fuy lateral T -ideal 2 ad every fuy right Agai by theore 32, Proof: Let T be a regular terary Γ-seirig The by theore 33, Hece Coversely, let T be a terary Γ-seirig ad for every fuy left TΓ-ideal, every fuy lateral TΓ-ideal 2 ad every fuy Let L, M ad R be respectively a left TΓ-ideal, a lateral TΓ-ideal ad a right TΓ-ideal right TΓ-ideal of T, of T ad x L M R The x L, x M ad x R Hece L( x) M ( x) R( x) (where L( x ), ( x) M ad ( x) R are respectively the characteristic fuctio of L, M ad R) Thus L M R ( x) i{ L( x), M ( x), R ( x)} Sice L is a fuy left TΓ-ideal of T, M is a fuy lateral TΓ-ideal of T ad R is a fuy right TΓ-ideal of T Therefore by hypothesis, Hece ( )( ) L M R L M R L M R x ie, sup xu v w[i{ L ( u), M ( v), R ( w)}: u, v, wt;, This iplies that there exist soe r, s,t T ad, such that x = r s 2 t ad ( r) ( s) ( t) Hece r L, s M ad t R Therefore x LΓMΓR Thus L M 2 L M R R LΓMΓR Also LΓMΓR L M R Hece LΓMΓR = L M R Cosequetly, by theore 35, the terary Γ-seirig T is ultiplicatively regular,, FT I( T) The , 2, 3 Theore 37: Let Proof : By the theore 3, Γ 2 Γ 3 o 2 o 3 For ay x T, if ( o 2 o 3 )(x) = 0, the obviously o 2 o Now for ay x T, ( o o )( x) [ [ ( u ), ( v ), ( w ): x uiiviiwi i i 2 i 3 i i u, v, w T,,, Z 4

8 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces [ [ ( u v w ), ( u v w ), ( u v w ): x ui iviiwi i 2 3 i u, v, w T,,, Z [ (x), 2(x), 3(x) = ( 2 3 )(x) Therefore, o 2 o Agai ( 2 3 )(x) = [ (x), 2(x), 3(x) (x) Thus 2 3 Siilarly, it ca be show that ad Hece the theore Theore 38: Let, 2, 3, 4 FLT I(T)[FMT I(T), FRT I(T), FT I(T) The Γ 2 Γ 3 4 if ad oly if o 2 o 3 4 Proof: Sice Γ 2 Γ 3 o 2 o 3 it follows that o 2 o 3 4 iplies that Γ 2 Γ 3 4 Assue that Γ 2 Γ 3 4 u Let x T ad x = i iviiwi u, i, vi, wi T, i, i, Z The i ( x) ( u v w ) [ ( u v w ), ( u v w ),, ( u v w ) i [( )( u v w ),( )( u v w ),( )( u v w ) [ ( ( u ), ( v ), ( w ), ( ( u ), ( v ), ( w ), ( ( u ), ( v ), ( w ) ( x) [ [ [ ( u ), ( v ), ( w ) 4 x ui iviiwi i i 2 i 3 i i = ( o 2 o 3 )(x) Thus o 2 o 3 4 Theore 39: Let, 2, 3, 4 FLT I(T)[FMT I(T), FRT I(T), FT I(T) The ( i) ( o ) o o( o ) ( ii) o o o o ( iii) o o o o o o If T is a coutative terary -Seirig ( iv) eoeo where eflti ( T) is defiied by e( x) for all xt [respectively, eo oe =, oeoe =, eoeo = eo oe = oeoe = Proof: (i): Proof of (i) follows fro the defiitio (ii) : 2 Now ( o o )( x) [ [ ( u ), ( v ), ( w ): 3 4 x uiivi iwi i i 3 i 4 i i ui, vi, wi T, i, i, Z [ [ [ 2( u), 3( v ), 4( wi ) ( 2o3o4)( x) i Thus, o o o o ( o o )( x) [ [ ( u ), ( v ), ( w ): (iii): x uiivi iwi i i 2 i 3 i i u, v, w T,,, Z 42

9 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces [ [ 2( vi ), 3( wi ), ( ui ) x v w u i i 2 3 ( o o )( x) Therefore, ( o o )( x) ( o o )( x) o o o o Siilarly, we ca show that o o o o o o o o o o (iv): As T is with left uity e i L, which is defied by ei ieii x x i For every x T we have, ( eoeo )( x) [ [ e( u ), e( v ), ( w ): x uiivi iwi i i i i i ui, vi, wi T, i, i, Z [ [ [ e, e, ( w ) [ [ ( w ) [ [ ( u v w ) i i i i i ( ui ivi iwi ) ( x) Therefore, (e oeo ) i Agai ( eoeo )( x) [ [ e( u ), e( v ), ( w ): x uiiviiwi i i i i i ui, vi, wi T, i, i, Z [ [ [ e, e, ( wi ) ( x) i So eoeo ad hece eoeo The followig theore shows that terary ultiplicatio is distributive over additio fro three sides Theore 320: Let, 2, 3 FLTΓI(T)[respectively, FMTΓI(T), FRTΓI(T), FTΓI(T The ( i) o( ) o o o o o ( ii) ( ) o o o o o o ( iii) o o( ) o o o o Proof: Let x T be arbitrary The ( o( ) o )( x) [ [ [ ( u ),( )( v ), ( w ) 4 x uiivi iwi i i 2 3 i 4 i i : u i, vi, wi T, i, i, Z [ [ [ ( ui ), [ [ 2( ri ), 3( si ), 4( wi ) i 43

10 Iteratioal Joural of Advaced i Maageet, Techology ad Egieerig Scieces [ [ [ ( vi ), 2( vi ), 3( si ), 4( wi ) x ( u r w u s w ) i i [ [ [ [ ( p j ), 2( q j ), 4( t j ), x p q t p q t j j j j j i k k k k k j k [ [ ( p k ), 3( q k ), 4( t k ) k [ [( o o )( u),( o o )( v): u p q t, v p q t (( o o ) ( o o ))( x) Thus o( ) o o o o o j j j j i k k k k k j k Sice , therefore, o 2 o 4 o( 2 + 3)o 4 Siilarly, o 3 o 4 o( 2 + 3)o 4 Thus ( o 2 o 4 ) + ( o 3 o 4 ) o( 2 + 3)o 4 + o( 2 + 3)o 4 = o( 2 + 3)o 4 Hece we coclude that o( 2 + 3)o 4 = o 2 o 4 + o 3 o 4 Proof of (ii) ad (iii) follows siilarly Theore 32: Let, 2, 3 be three fuy left T -Ideals (fuy lateral T -ideals, fuy right T -ideals, fuy T -ideals) of a terary -seirig T The + 2 is the uique iial eleet of te faily of all fuy left T -Ideals (fuy lateral T -ideals, fuy right T -ideals, fuy T -ideals) of a terary -seirig T cotaiig, 2ad 2 3 is the uique axial eleet of the faily of all fuy left T -Ideals (fuy lateral T -ideals, fuy right T -ideals, fuy T -ideals) of a terary -seirig T cotaiig, 2, 3 Proof: Let, 2, 3 FLTΓI(T) The by theore 220(v),, Suppose ψ ad 2 ψ where, ψ FLTΓI(T) Now for ay x T, ( 2)( x) [ [ ( y), 2( ): y, T x y [ [ ( y), ( ) [ ( x y) ( x) Thus Agai,, 2 Let us suppose that ϕ FLTΓI(T) be such that ϕ, ϕ 2 ad ϕ 3 ( )( x) [ ( x), ( x), ( x) [ ( x), ( x), ( x) ( x) The for ay x T, Thus ϕ 2 3 Uiqueess of + 2ad 2 3 with the stated properties are obvious Proofs of other cases follow siilarly CONCLUSION Our ai purpose i this paper is to itroduce the operatios o fuy TΓ-ideal i terary Γ-seirigs We give soe characteriatios of fuy TΓ-ideals ACKNOWLEDGEMENTS The authors are deeply grateful to the referees for the valuable suggestios which lead to a iproveet of the paper ad the authors would like to thak the experts who have cotributed towards preparatio ad developet of the paper 44

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