be two non-empty sets. Then S is called a semigroup if it satisfies the conditions

Transcription

1 UZZY SOT GMM EGU SEMIGOUPS V. Chinndi* & K. lmozhi** * ssocit Pofsso Dtmnt of Mthmtics nnmli Univsity nnmling Tmilnd ** Dtmnt of Mthmtics nnmli Univsity nnmling Tmilnd bstct: In this w hv discssd bot th fzzy soft -gl nd -int gl smigos nd thi otis. Ind Tms: Soft smigos soft -idls -gl smigos -int gl smigos soft -gl smigos & fzzy soft - int gl smigos.. Intodction: Th conct of fzzy st ws intodcd by Zdh [5] in 965. Sn nd Sh [] dfind th -smigo in 986. Soft st thoy oosd by Molotsov [8] in 999. Soft st thoy hs bn lid in mny filds. Mji t l [7] wokd on soft st thoy nd fzzy soft st thoy. li t l [] intodcd nw otions on soft sts. Chinm nd Jiojkl [] dfind th bi-idls in -smigos in 007. Pinc t l [] sntd th fzzy - bi- idls in -smigos in 009. Khyol t l [6] inititd th bi-idls nd si idls of smigos nd odd smigos. Dhn t l [5] stdid th chctiztion of gl -smigos thogh fzzy idls in 007. Chinndi t l [] wokd on intvl vld fzzy soft smigos nd h stdid -gl smigos in fzzy idls in lgbic stcts. Mhmmd Ifn li [0] stdid soft idls ov smigos. Sjit Km Sd [] discssd th otis of fzzy idls in -smigos. Mhmmd km t l [9] discssd zzy soft smigos. Mnzz Nz t l [] wokd fzzy soft si- idls on fzzy soft smigos li t l [] stdid soft idls nd gnlizd fzzy idls in smigos in Pliminis: Dfinition. []: t S b c...} nd...} b two non-mty sts. Thn S is clld smigo if it stisfis th conditions i b S c b b c S nd. Dfinition. [5]: smigo S is clld gl if fo ch lmnt S th ists S nd sch tht. Dfinition. [8]: t U b th nivsl st E b th st of mts P U dnot th ow st of U nd b non-mty sbst of E. i is clld soft st ov U wh is ming givn by : P U. Dfinition. [7]: t b two soft sts ov common nivs U thn ND dnotd by is dfind s wh H G Dfinition.5 [7]: t b two soft sts ov common nivs U thn O dnotd by is dfind s wh H G. Dfinition.6 []: Th tndd nion of two fzzy soft sts ov common nivs U is fzzy soft st dnotd by G dfind s wh C c C. 8

2 if c H G if c G if c. Dfinition.7 []:Th tndd intsction of two fzzy soft sts ov common nivs U is fzzy soft st dnotd by G dfind s wh C c C. if c H G if c G if c. Dfinition.8 []: t b two soft sts ov common nivs U sch tht. Th stictd intsction of nd G is dnotd by nd dfind s wh C cc wh H G. Dfinition.9 []: Th stictd odct H of two fzzy soft sts nd G ov smigo S is dfind s H wh C by H G c C. Dfinition.0 []: soft st is clld soft smigo ov S if. Clly soft st ov smigo S is soft smigo if nd only if is sbsmigo of S. Dfinition. [0]: soft smigo ov smigo S is clld soft gl smigo if fo ch is gl.. Dfinition. [5]: t X b non-mty st. fzzy sbst of X is fnction fom X into th closd nit intvl [0 ]. Th st of ll fzzy sbsts of X is clld fzzy ow st of X nd is dnotd by P X.. Dfinition. [9]: fzzy soft st of smigo S thn is clld fzzy soft sbsmigo of S if y min y} y S. Dfinition. [9]: fzzy soft st of smigo S is clld fzzy soft lftight idl of S if y y y y S nd Dfinition.5 [9]: fzzy soft sbsmigo of smigo S is clld fzzy soft idl of S if zy m y} y z S. Dfinition.6 [9]: fzzy soft sbsmigo of smigo S is clld fzzy soft bi-idl of S if zy min y} y z S. Dfinition.7 [9]: fzzy soft sbsmigo of smigo S is clld fzzy soft intio idl of S if zy z y z S nd. Dfinition.8 []: fzzy soft ov S is sid to b fzzy soft si idl of S if is fzzy si idl of S.. Dfinition.9 []: fzzy soft ov S is sid to b fzzy soft gnlizd biidl of S if is fzzy gnlizd bi- idl of S.. zzy Soft Gmm gl Smigos: In This sction S dnots th soft gl smigo. 9

3 Dfinition.: soft smigo ov smigo S is clld soft gl smigo if fo ch is gl. Eml.: S } nd } wh is dfind on S with th following cyly tbl: Tbl- E } nd } } } }. Hnc S is soft gl smigo. Thom.: t b two fzzy soft idl bi-idl intio idl ov soft gl smigo S thn is fzzy soft idl bi-idl intio idl ov soft gl smigo S. Poof: t b two fzzy soft idl ov soft gl smigo S. Now w dfind wh C nd H G C. y y H min minm mmin m m G y H G G y} H G y}m }min y} G G y} y G G y} y}} Hnc is fzzy soft idl ov soft gl smigo S. Thom.: t b two fzzy soft idl bi-idl intio idl ov soft gl smigo S thn is fzzy soft idl bi-idl intio idl ov soft gl smigo S. Poof: Th oof is stightfowd. Thom.5: t b two fzzy soft -idl of Sthn is fzzy soft idl ov S. Poof: t b two fzzy soft idl ov S thn wh C c C if c H G if c G if c. t s t S nd. iif c 0

4 st st H iiif c st H m m G m m H st G H s s s s H G H t} t} t} t} iiiif c thn H min G } G } Now vify tht H s t m H s H t} s t S nd c C. Ths st m s }. Hnc is fzzy soft H H H t idl ov S. Thom.6: t b two fzzy soft idl ov S is fzzy soft idl ov S. Poof: Th oof is stightfowd. Thom.7: t b two fzzy soft sts of soft gl smigo S. nd two non-mty sbsts of S. i Poof: t S if thn nd. w hv min } min00} 0 Sos thn nd min } min00} 0 t S if thn th ists nd sch tht s min d} cd min min} } So sinc.sos thn nd.

5 min } min00} 0 Hnc Thom.8: t b soft sbst of smigo S b soft sbsmigo of S if nd only if is fzzy soft sbsmigo of S. Poof: t b soft smigo of S if 0 if. t b S if min } thn b nd 0 this imlis tht b sinc is sbsmigo of S b nd hnc which is contdiction. Ths min } b nd. Convsly ssm tht is soft sbsmigo of S. t b thn } b fzzy soft sbsmigo. Now min this imlis tht nd hnc b. Thfo is soft sbsmigo of S. Thom.9: t b soft bi-idl of S if nd only if chctistic fnction is fzzy soft bi-idl of S. Poof: ssm tht b soft bi-idl of S by thom.7 whv S S hnc is fzzy soft bi-idl of S. Convsly ssm tht is fzzy soft bi-idl of S S by thom.8 it is cl tht is soft sbsmigo of S. t S sch tht S thn is fzzy soft bi-idl w hv S which imlis tht nd hnc S S. Hnc is soft bi-idl of S. Thom.0: t b soft idl of S if nd only if chctistic fnction is fzzy soft idl of S. Poof: Th oof is stightfowd. Th following thom is ltion btwn soft st nd fzzy st. Thom.: t b soft sbst of smigo S thn is soft - si idl of S if nd only if chctistic fnction is fzzy soft -si idl of S. Poof: Sos is soft si idl of S nd b th chctistic fnction of S lt S. If thn f. If thn S E S E. Csi: t S E S E. If b thn. Thn

6 min smin }} smin v}}} b v 0. Thfo. Cs ii: t S E S E. If v thn v. Thn min smin }} smin v}}} b v 0. Thfo. Cs iii: t S E S E. If b thn thn nd if v thn v. Thn min smin }} smin v}}} b v 0. Thfo. Hnc is fzzy soft si idl of S. Convsly sos tht is fzzy soft si idl of S. t thn th ists s t S y z nd sch tht ys tz. s min b min y min} s} }} Similly. Sinc. min } min} Ths nd hnc S E S E.Thfo is soft -si idl of S. Thom.: Th following conditions ivlnt i Evy soft -bi-idl is soft -idl of S.. ii Evy fzzy soft -bi-idl of S is fzzy soft -idl of S. Poof: ssm tht condition i holds lt b ny fzzy soft -idl of S. t b ny fzzy soft -bi-idl of S nd S sinc th st S E is soft -bi-idl of S by th ssmtion is soft -ight idl of S is soft -gl w hv th ists S sch tht sinc is fzzy soft -bi idl of S.

7 min. Hnc is fzzy soft -ight idl of S. Similly is fzzy soft -lft idl of S. Thfo is fzzy soft - idl of S. Hnc i. Convsly ssm tht ii holds. t b soft -idl of S by thom.9 th chctistic fnction is fzzy soft -bi-idl of S. Hnc by ssmtion is fzzy soft -idl of S ths by thom.0 is soft -idl of S. Hnc i Th following mls shows tht fzzy soft -idl nd fzzy soft -bi-idl of S. Emls.: t S } nd } in th tbl.. t E } } thn is fzzy soft st dfind s } } Hnc is fzzy soft -bi-idl nd fzzy soft -idl ov S. Thom.: Th following conditions ivlnt. i is fzzy soft -idl of S. ii is fzzy soft -intio idl of S. Poof: t is fzzy soft -idl of S. W hv sinc is - lft idl of S. sinc is - ight idl of S. Hnc S nd. Convsly ssm tht is fzzy soft - intio idl of S. t S sinc S is soft - gl smigos th ists ys sch tht nd y nd. Ths w hv y y nd Hnc ovd. Th following ml shows tht fzzy soft - idl nd - intio idl of S. Emls.5: t S } nd } in th tbl.. t E } } thn is fzzy soft st dfind s } } Hnc is fzzy soft -intio idl nd fzzy soft -idl ov S.

8 5 Thom.6: Evy fzzy soft -si idls fzzy soft -bi-idl of S. Poof: t is fzzy soft - si idl of S. It is sfficint to ov tht S } min nd. sinc is fzzy soft -si idl of S. } min } min } min s v v } min } min s v v Hnc. } min Thom.7: In soft -gl smigo S thn fzzy soft -si-idls nd fzzy soft -bi-idls coincid. Poof: It is mins to ov tht vy fzzy soft -bi-idls fzzy soft -siidls if b fzzy soft -bi-idl of S thn.... i t S sos tht. } min Now sos tht thn th ists y S y sch tht } min y tht is... ii now w ov tht so tion i is stisfid. Sinc } min s. Pov tht } min S sinc S is soft -gl th ists S sch tht. y Thn sinc is fzzy soft - bi-idl w hv }. min y If } min thn which is imossibl by tion ii ths } min thn. } min Thom.8: Th following conditions ivlnt i is -gl.

9 fo vy soft -bi-idl nd vy soft -lft idl of S. Poof: i y dfinition K wh K is fnction K : P K. Now K wh K is fnction to PS sch tht K. w sos tht sinc nd is gl th ists S sch tht.now nd hnc. This shows tht. i.sos tht nd is fnction fom to P dfind by nd is fnction fom to P dfind by. Thn is soft -bi-idl nd is soft -lft idl ov by hyothsis. Thfo is soft -gl. Thom.9: Evy fzzy soft -gnlizd bi-idl is fzzy soft -bi-idl of S. Poof: t b ny fzzy soft -gnlizd bi-idl of S nd lt S sinc S is soft -gl th ists S sch tht. w hv min }. This imlis tht is fzzy soft -sbsmigo of S nd so is fzzy soft - bi-idl of S. Thom.0: Th following conditions ivlnt i is -gl. fo vy fzzy soft -gnlizd biidl nd vy fzzy soft -lft idl of S. iii fo vy fzzy soft - bi-idl nd vy fzzy soft -lft idl of S. Poof: i.y dfinition N wh N is fnction N : P N. t b fzzy soft gnlizd -bi-idl nd b fzzy soft - lft idl of S. Sinc is soft - gl lt th ists S sch tht. s min } min } min } Which imlis tht. iii. y thom.9. 6

10 iii i. lt b fzzy soft -bi-idl nd b fzzy soft - lft idl of S lt by thom.0 nd thom.9 is fzzy soft - lft idl nd is fzzy soft - bi-idl of S nd N wh N : P nd N. Thn by hyothsis. w hv min } min}. sinc is fzzy soft sbst of S w hv. S s min min } } min }. Hnc ths s min } which imlis tht nd it follows tht thn. Thfo by thom.8 hnc is soft -gl smigo. Thom.: Th following conditions ivlnt i is lft -gl. I I fo vy fzzy soft -idl I nd fzzy soft -bi-idl of S. Poof: i y dfinition I K wh K is fnction K : P K I. t b soft lft gl nd thn th ists S nd sch tht. t I b fzzy soft idl nd b fzzy soft bi-idl of S. s min } min } min } Hnc I I i sos I nd fzzy soft idl nd fzzy soft biidl of S sch tht I I. t I b ny soft idl nd b ny soft bi-idl of S I by thom.0 I is fzzy soft idl of S nd by thom.9 is fzzy soft bi- idl of S. Now by thom.7 w hv I I nd hnc 7

11 I I I I by thom.7 It follows tht I nd hnc I I. Hnc is soft lft -gl.. zzy Soft Gmm Int gl Smigos: In this sction S dnots th soft int gl smigo. Dfinition.: soft smigo ov smigo S is clld soft int gl smigo if fo ch is int gl. Eml.: S } nd } wh is dfind on S with th following cyly tbl: Tbl- E } nd } } } } Hnc S is soft int gl smigo. Thom.: t b two fzzy soft idl bi-idl intio idl ov soft int gl smigo S thn is fzzy soft idl biidl intio idl ov soft int gl smigo S. Poof: Th oof is Stightfowd Thom.: t b two fzzy soft idl bi-idl intio idl ov soft int gl smigo S thn is fzzy soft idl biidl intio idl ov soft int gl smigo S. Poof: Th oof is Stightfowd Thom.5: Th following conditions ivlnt. i is fzzy soft -idl of S. ii is fzzy soft -intio idl of S. Poof: t is fzzy soft -idl of S. W hv sinc is - lft idl of S. sinc is - ight idl of S. Hnc S nd. Convsly ssm tht is fzzy soft - intio idl of S. t S sinc S is soft - int gl smigos th ists y vs sch tht y nd 8

12 v nd. Ths w hv y nd y v v Hnc ovd. Thom.6: Th following conditions ivlnt i is -int gl. fo vy soft -lft idl nd vy soft -ight idl of S. Poof: i y dfinition K wh K is fnction K : P K. w hv K wh K is fnction to PS sch tht K. w sos tht sinc nd is int gl th ists S sch tht.now nd hnc This shows tht i.sos tht nd N is fnction fom to P dfind by nd is fnction fom to P dfind by. Thn is soft -lft idl nd is soft -ight idl ov by hyothsis. Thfo is soft -int gl. Thom.7: Th following conditions ivlnt i is lft int gl. fo vy fzzy soft lft idl nd fzzy soft ight idl of S. Poof: i y dfinition M wh M is fnction M : P M. M wh M is fnction P S sch tht M t b soft int gl nd thn th ists S nd sch tht. s min } min } min } Hnc i sos nd fzzy soft lft idl nd fzzy soft 9

13 ight idl of S sch tht nd by thom.0 is fzzy soft lft idl of S nd is fzzy soft ight idl of S. Now by thom.7 w hv nd hnc by thom.7 It follows tht nd hnc Th bov thom hnc is lft int gl. cknowldgmnt: Th sch of th scond tho is tilly sotd by UGC- S gnt:.5-/0-5s/7-5/009s dtd in Indi. fncs. M.I. li. ng X. Y.i nd M. Shbi On som nw otions in soft st thoy Comt Mth. l M.I. li nd M. Shbi 009 Soft idls nd gnlizd fzzy idls in smigos. Nw Mth. Nt. Com V. Chinndi nd K. lmozhi Intvl vld fzzy soft smigos nnmli Univsity scinc jonl Chinm nd Jiojkl On bi- -idls in -smigos songklnkin J. sci.tchnol P.Dhn nd S.Comssn Chctiztion of gl -smigos thogh fzzy idls Inin jonl of fzzy systms N.Khyol S.jos nd G. os not on bi-idl nd si- idls of smigos odd smigos P. K. Mji. isws nd. oy Soft st thoy Comt. Mth. l D. Molodtsov Soft st thoy fist slts Comt. Mth. l Mhmmd km J. Kvikm nd zm in Khmis zzy soft - smigos l. Mth. Inf. Sci Mhmmd Ifn li Mhmmd Shbi nd K.P. Shm On soft idls ov smigos Sothst sin lltion of Mthmtics Mnzz Nz Mhmmd Shbi nd Mhmmd Ifn li On zzy Soft Smigos Wold l. Sci D.. Pinc Willims t l zzy bi-idls in -smigos Hctt Jonl of Mthmtics nd Sttistics M. K. Sn nd N. K Sh On - smigo I ll. Clctt Mth. Soc Sjit Km Sd On zzy Idls in -Smigos Intntionl Jonl of lgb Zdh zzy sts. Infom nd contol