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 009.. 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
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
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
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.
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
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.
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 0.9 0. 0.7 0.} 0.8 0. 0.5 0.} 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 0.8 0. 0. 0.} 0.7 0. 0.6 0.} Hnc is fzzy soft -intio idl nd fzzy soft -idl ov S.
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.
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
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
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
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
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 0-0-05 in Indi. fncs. M.I. li. ng X. Y.i nd M. Shbi On som nw otions in soft st thoy Comt Mth. l. 57 57-55009.. M.I. li nd M. Shbi 009 Soft idls nd gnlizd fzzy idls in smigos. Nw Mth. Nt. Com. 5599-65009.. V. Chinndi nd K. lmozhi Intvl vld fzzy soft smigos nnmli Univsity scinc jonl 50 5-06.. Chinm nd Jiojkl On bi- -idls in -smigos songklnkin J. sci.tchnol. 9-007. 5. P.Dhn nd S.Comssn Chctiztion of gl -smigos thogh fzzy idls Inin jonl of fzzy systms 57-68007. 6. N.Khyol S.jos nd G. os not on bi-idl nd si- idls of smigos odd smigos 8 75-8997. 7. P. K. Mji. isws nd. oy Soft st thoy Comt. Mth. l. 5 555-5600. 8. D. Molodtsov Soft st thoy fist slts Comt. Mth. l 7 9-999. 9. Mhmmd km J. Kvikm nd zm in Khmis zzy soft - smigos l. Mth. Inf. Sci 8 99-90. 0. Mhmmd Ifn li Mhmmd Shbi nd K.P. Shm On soft idls ov smigos Sothst sin lltion of Mthmtics 595-6000.. Mnzz Nz Mhmmd Shbi nd Mhmmd Ifn li On zzy Soft Smigos Wold l. Sci 6-80.. D.. Pinc Willims t l zzy bi-idls in -smigos Hctt Jonl of Mthmtics nd Sttistics 8-5- 009. M. K. Sn nd N. K Sh On - smigo I ll. Clctt Mth. Soc. 78 80-86 986.. Sjit Km Sd On zzy Idls in -Smigos Intntionl Jonl of lgb 6 775-78 009 5.. Zdh zzy sts. Infom nd contol. 8 8-5965. 50