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1 Inernaonal Journal of Mahemacal rchve-75), 06, 9-98 valable onlne hrough wwwjmanfo ISSN NOTE ON FUZZY WEKLY OMPLETELY PRIME - IDELS IN TERNRY SEMIGROUPS U NGI REDDY *, Dr G SHOBHLTH Research scholar, Dep of Mahemacs, SrKrshnadevaraya Unvers nanhapuramu, P), Inda 5500 Professor, Dep of Mahemacs, SrKrshnadevaraya Unvers nanhapuramu, P), Inda 5500 Receved On: 0-0-6; Revsed & cceped On: ) BSTRT Ideal heory n ernary semgroups s nroduced by FM Soson Sam Kumar Majumder nroduced and suded on fuzzy compleely prme deals n gamma semgroups In hs paper we proved some properes of prme -deals and fuzzy weakly compleely prme -deals n ernary semgroups I s prove ha If be a non-empy fuzzy subse of a ernary semgroup T, hen s a fuzzy ernary sub semgroup of f and only f s a fuzzy weakly compleely prme - deal n T Key words: Ternary semgroup, fuzzy se, Fuzzy lef laeral, rgh) deals, Fuzzy weakly compleely prme deals INTRODUTION The heory of ernary algebrac sysem was nroduced by DH Lehmer n 9 bu earler such srucures were suded by Kasner n90, who gave he dea of n-ary algebras Ternary semgroups are unversal algebras wh one assocave ernary operaon The oncep of quas-deals n semgroups was nroduced n 956 by O Senfeld D and Dewan suded quas-deals and bdeals n ernary semgroups The nroducon of fuzzy ses by L Zadeh fer N Kurok nroduced and suded he noon of fuzzy semgroups He also suded he concep of fuzzy b-deals 976) and fuzzy quas-deals98) of semgroups Shabr, Jun and Bano nroduced and suded he prme, srongly prme, semprme and rreducble fuzzy b-deals of semgroups They characerzed hose semgroups for whch each fuzzy b-deal s semprme and also characerzed hose semgroups for whch each fuzzy b-deal s srongly prme Several researchers conduced he researches on he generalzaons of he noons of fuzzy ses wh huge applcaons n compuer, logcs and many branches of pure and appled mahemacs hnram and Saelee n 00 suded he concep of fuzzy deals and fuzzy flers of ordered ernary semgroups BSI DEFINITIONS ND PRELIMINRIES Defnon: non-empy se T s sad o be ernary semgroup f here ess a ernary operaon : T T T T wren as a, b, c) a b c sasfes he followng deny a b c) d e = a b c d) e = a b c d e) for any a, b, c, d, e T Defnon: non-empy subse of a ernary semgroup T s called a ernary subsemgroup of T f Defnon: non-empy subse of a ernary semgroup T s called a lef rgh, laeral) deal n T f TT TT, TT ) Defnon: non-empy subse of a ernary semgroup T s called a wo sded deal of T f s boh lef and rgh deal n T orrespondng uhor: U Nag Reddy *, Research scholar, Dep of Mahemacs, SrKrshnadevaraya Unvers nanhapuramu, P), Inda 5500 Inernaonal Journal of Mahemacal rchve- 75), May 06 9

8-05-6) BSTRT Ideal heory n ernary semgroups s nroduced by FM Soson Sam Kumar Majumder nroduced and suded on fuzzy compleely prme deals n gamma semgroups In hs paper we proved some properes of prme

2 U Nag Reddy *, Dr G Shobhalaha / Noe On Fuzzy weakly ompleely Prme - Ideals n Ternary Semgroups / IJM- 75), May-06 5 Defnon: non-empy subse of a ernary semgroup T s called a deal n T f s lef, rgh and laeral deal n T 6 Defnon: Le T be a non-empy se fuzzy subse of a ernary semgroup T s a funcon : T [0,] 7 Defnon: Le be a fuzzy subse of a non-empy se T for any [0,], he subse = { T : } of T s called a level se of 8 Defnon: For any wo fuzzy subses and and denoed by and are fuzzy subses of T and defned as ) ) = ma{ ), } = ) and S for all T where denoes mamum or supremum and denoes mnmum or nfmum of a non-empy se T, he unon and he nersecon of 9 Defnon: Le, and are any hree fuzzy ses of a ernary semgroupt Then her fuzzy produc s defned by 0 Defnon: fuzzy se of a ernary semgroup T s called a fuzzy ernary subsemgroup of T f yz ) { } for all, Defnon: fuzzy se of a ernary semgroup T s called a fuzzy lef rgh, laeral) deal n T f,, ) for all, z T Defnon: fuzzy se of a ernary semgroup T s a fuzzy deal n T f s fuzzy lef, rgh and laeral deal n T Defnon: Le be a non-subse of a ernary semgroup T Then he characersc funcon of s defned by = 0 f f We denoe he characersc funcon T of T e, T = T hus T = for all T Defnon: subse of a ernary semgroup T s sad o be a prme deal n T f yz mples or y or z 5 Defnon: fuzzy deal of a ernary semgroup T s called a fuzzy weakly compleely prme deal n T f ) or y ) or z ) for all, 6 Defnon: fuzzy deal of a ernary semgroup T s called a fuzzy prme deal n T f nf ma,, z for all, { )} 7 Proposon: If and 5 are fuzzy subses of a non-empy se T hen ) ) = ) ) ) T T ) T T ) T T ) ) ) = ) ) v) ) = ) ) v) ) = ) = ) , IJM ll Rghs Reserved 9

a level se of 8 Defnon: For any wo fuzzy subses and and denoed by and are fuzzy subses of T and defned as ) ) = ma{ ), } = ) and S for all T where denoes mamum or supremum and denoes mnmum or nfmum

3 U Nag Reddy *, Dr G Shobhalaha / Noe On Fuzzy weakly ompleely Prme - Ideals n Ternary Semgroups / IJM- 75), May-06 8 Proposon: If and ) ) T T ) T T ) T T ) ) T ) T ) T T) T T ) ) T T ) ) T T ) T T ) are wo fuzzy subses of a non-empy se T hen Proof:) Le T If for any p, q, r T hen T T ) T T )) ) = 0 = ) T T ) ) If = for any p, q, r T hen ) T T ) ) = { ) T q) T )} = r { } { } = = { { }} = { { }} { { }} = { { T q) T r)}} { { T q) T r)}} = T T ) T T ) = T T ) T T )) ) T T ) ) T T ) T T )) T T ) T T T Therefore ) ) Smlarly we can prove ha ) T ) T ) T T) T T ) and ) T T ) ) T T ) T T ) RESULTS Theorem: Le be a non-empy fuzzy subse of a ernary semgroup T Then s a fuzzy ernary T T subsemgroup of f and only f s a fuzzy weakly compleely prme deal n T Proof: Le be a non-empy fuzzy subse of a ernary semgroup = / ssume ha s a fuzzy ernary subsemgroup of T and le, Then = mn{,, } mn{ ),, } ma{,, } ma{,, } ma{,, } e, ma{ ),, } Therefore ) or y ) or z ) Hence s a fuzzy weakly compleely prme deal n T 06, IJM ll Rghs Reserved 95

{ }} = { { T q) T r)}} { { T q) T r)}} = T T ) T T ) = T T ) T T )) ) T T ) ) T T ) T T )) T T ) T T T Therefore ) ) Smlarly we can prove ha ) T ) T ) T T) T T ) and ) T T ) ) T T ) T T ) RESULTS

4 U Nag Reddy *, Dr G Shobhalaha / Noe On Fuzzy weakly ompleely Prme - Ideals n Ternary Semgroups / IJM- 75), May-06 onversel assume ha s a fuzzy weakly compleely prme deal n Then we have ) or y ) or z ) / Therefore s a fuzzy ernary subsemgroup of T Theorem: Le { : I} be a famly of fuzzy weakly compleely prme deals n a ernary semgroup T Then s also a fuzzy weakly compleely prme deal n Proof: Le { : I} be a famly of fuzzy weakly compleely prme deals n a ernary semgroup T Then we have or or for all, y zz TT, I Then I or or T ma{ ),, } ma{ ),, } mn{ ),, } / / / / mn{,, } = T = nf{ : I} I nf{ : I} I nf{ : I} I nf{ : I} I Hence s fuzzy weakly ernary compleely prme deal n I T T T Theorem: Le be a ernary semgroup and be a non-empy fuzzy subse of Then he followng are equvalen ) s a fuzzy weakly compleely prme deal n T ) For any [0,], f s non-empy) s a prme deal n T Proof: Le be a fuzzy weakly compleely prme deal n a ernary semgroup T Then we have ) or y ) or z ) for all, Le [0,] be such ha s non-empy Le,, yz Then Snce s a fuzzy weakly compleely prme deal n a ernary semgroup T, so, we have ) or y ) or z ) for all, Then or or whch mples ha or y or z Hence s a prme -deal n T onversel le us suppose ha s a prme deal n a ernary semgroup T Le s non-empy Then yz yz) Snce s a prme deal n T, we have or y or z Then or or whch mples ha ) or y ) or z ) Hence s a fuzzy weakly compleely prme deal n a ernary semgroup T Theorem: Le be a non-empy subse of a ernary semgroup T and be he characersc funcon of Then s a lef deal of T f and only f s a fuzzy lef deal of T 06, IJM ll Rghs Reserved 96

I} be a famly of fuzzy weakly compleely prme deals n a ernary semgroup T Then we have or or for all, y zz TT, I Then I or or T ma{ ),, } ma{ ),, } mn{ ),, } / / / / mn{,, } = T = nf{ : I} I nf{ : I}

5 U Nag Reddy *, Dr G Shobhalaha / Noe On Fuzzy weakly ompleely Prme - Ideals n Ternary Semgroups / IJM- 75), May-06 Proof: Le be non-empy subse of a ernary semgroup T and be a characersc funcon of We assume ha s lef deal of T hen TT We have o prove ha s fuzzy lef deal of T e, Le a = yz for all, onsder TT yz) = T T ) yz) = T T q) r)) yz = = T T ) = ) = z) yz) Therefore s a fuzzy lef deal of T onversel suppose s a fuzzy lef deal n T Then yz) for all, z To prove ha s a lef deal n T e, TT Le, hen = T, T =, = T T onsder yz) = ) = T T ) = ) yz) T T yz) = T T ) yz) T T TT TT Therefore s a lef deal n a ernary semgroup T 5 Theorem: Le be a non-empy subse of a ernary semgroup T and be he characersc funcon of Then s a rgh deal laeral deal, deal) n T f and only f s a fuzzy rgh deal fuzzy laeral deal, fuzzy deal) n T Proof: Smlar o he proof of Theorem 6 Theorem: Le T be a ernary semgroup and be a non-empy subse of T Then he followng are equvalen ) s prme deal n a ernary semgroup T ) The characersc funcon of s a fuzzy weakly compleely prme deal n T Proof: Le be a prme deal n a ernary semgroup T and be he characersc funcon of Snce φ, so, s non-empy Le, Suppose yz Then = Snce s a prme deal n T, 06, IJM ll Rghs Reserved 97

onversel suppose s a fuzzy lef deal n T Then yz) for all, z To prove ha s a lef deal n T e, TT Le, hen = T, T =, = T T onsder yz) = ) = T T ) = ) yz) T T yz) = T T ) yz) T T TT TT Therefore s a lef

6 U Nag Reddy *, Dr G Shobhalaha / Noe On Fuzzy weakly ompleely Prme - Ideals n Ternary Semgroups / IJM- 75), May-06 or y or z whch mples ha ) or = or z ) = Hence or or Suppose yz Then = 0 Snce be a prme deal n T, or y or z whch mples ha = 0 or = 0 or = 0 Hence or onsequenly s a fuzzy weakly compleely prme deal n T onversel le he characersc funcon of s a fuzzy weakly compleely prme deal n T Then s a fuzzy deal n T By he heorem, s an deal n T Le, be such ha yz Then = Le f possble S and y T and z T Then = = = 0 whch mples < and < and < Ths conradcs our assumpon ha s a fuzzy weakly compleely prme deal of T Hence s prme deal n T REFERENES = D, VN and Dewan,S, noe on quas deals and b - deals n ernary semgroups, Inerne, JMah and Mahsc vol8, No 995) Dua,TK, Kar,S and MaBK, On deals n regular ernary semgroups, Dscuss MahGen lgebra ppl 8 008), 7-59 Kar, S and Palu Sarka, Fuzzy quas deals and fuzzy b- deals of ernary semgroups, nnals of Fuzzy Mahemacs and Informacs Volume, No, Ocober 0), pp 07- Kurok N, Fuzzy b-deals n Semgroups, ommen Mah Unv S Paul, 8: 979, 7-5 Lehmer, D H, ernary analogue of abelan groups, mer J Mah 5 9) Sam Kumar Majumder, Fuzzy compleely prme deals n gamma semgroups, SDU Jornal of Scnce E- Journal), 00, 5): Sson,FM, Ideal heory n ernary semgroups, Mahs Japan, 0 965), 68 8 Zadeh, L, Fuzzy ses, Inform onrol 8 965) 8-5 Source of suppor: Nl, onflc of neres: None Declared [opy rgh 06 Ths s an Open ccess arcle dsrbued under he erms of he Inernaonal Journal of Mahemacal rchve IJM), whch perms unresrced use, dsrbuon, and reproducon n any medum, provded he orgnal work s properly ced] 06, IJM ll Rghs Reserved 98

s a fuzzy deal n T By he heorem, s an deal n T Le, be such ha yz Then = Le f possble S and y T and z T Then = = = 0 whch mples < and < and < Ths conradcs our assumpon ha s a fuzzy weakly compleely

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Fourier Transforms and the -Adic Span of Periodic Binary Sequences IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 687 ; V 2 m+ where V s some neger and le f be he number of ; V 2 m+. Thus, From (5) M 2s l f V 2s 2 2s(m+) : (6) M 0 6 0 0qM 0 4 +6q