Classifications of Ordered Semigroups in Terms of Bipolar Fuzzy Bi-Ideals

Transcription

1 J l Eviro iol Sci TeRoad Publicaio ISSN: Joural o lied Eviromeal ad iological Scieces wwweroadcom Classiicaios o Ordered Semigrous i Terms o iolar Fu i-ideals Tariq Mahmood M Ibrar sghar Kha * Hidaa Ullah Kha3 ad Faima bbas 4 Dearme o Elecroic Egieerig Uiversi o Egieerig ad Techolog Taila Sub Camus Chakwal Pakisa Dearme o Mahemaics bdul Wali Kha Uiversi Marda Marda Khber Pakhukhwa Pakisa 3 Dearme o Mahemaics Uiversi o Malakad a Chakdara Disric Dir L Khber Pakhukhwa Pakisa 4 Dearme o Mahemaics Gomal Uiversi D I Kha Khber Pakhukhwa Pakisa STRCT Received: Ma 07 cceed: Jul iolar u se is a eesio o u se I biolar u se he rage o he membershi ucio is [ ] whereas i u se i is [ 0] We emloed he coce o biolar u se heor i he srucure o ordered semigrou ad iroduced a geeraliaio o biolar u bi-ideals This geeralied orm o biolar u bi-ideals is called -biolar u bi-ideals o ordered semigrous We obaied some ieresig characeriaio resuls o ordered semigrou i erms o his ew coce KEYWORDS: iolar u bi-ideals biolar u oi α β -biolar u bi-ideals osiive -cu egaive s-cu -cu INTRODUCTION The lis o abbreviaioha are used requel i his aer are give i see Table Table : NMES ND REVITIONS Name bbreviaio iolar u bi-ideals FI iolar u se FS Fu i-ideals FI Geeralised Fu i-ideals GFI Ordered Semigrous OS I ad ol i i The se o all biolar u subses o S FS Ordered iolar Fu Poi OFP iolar u bi-ideals α β -FI -biolar u geeralied bi-ideal -FGI -biolar u bi-ideal -FI -biolar u bi-ideal -FI iolar u le righ ideal iolar u ideal FL R I FI Zhag Zhag urher geeralied he heor o u se Zadeh 965 ad deied FS I FS he rage o he membershi ucio is [ ] raher ha [ 0] as i case o u se I FS we sli he rage [ ] io 0] 0 ad [ 0 I he membershi degree o a eleme sa belogo he ierval 0] he i idicaeha o some ee saisiehe roer whereas i he membershi Corresodig uhor: sghar Kha Dearme o Mahemaics bdul Wali Kha Uiversi Marda Marda Khber Pakhukhwa Pakisa ahar4se@ahoocom 34

2 Mahmood e al 07 degree is equal o 0 he we sa ha is irreleva o he corresodig roer O he oher had ad he membershi degree o belogo [ 0 he saisiehe imlici couer roer o a cerai degree lhough FS look like similar o iuiioisic u ses bu he are oall diere rom each oher as meioed i Lee 004 Kuroki Kuroki wahe irs o emloed he heor o u se i he srucure o semigrous I haka ad Das 99 a geeralied orm o u subgrou Roseeld 97 is reseed I addiio he geeraliaio o FI i semigrous is give i Kaaci ad Yamak 008 Furher Ju e al Ju e al 005 geeralied u sub-algebra i CK/CI algebra ad reseed he idea o α β -u sub-algebra where α ad β are i { q q q} ad α q I his regard or urher sud i diere braches o algebra see Crise 00 Davva 00 Kaaci 008 Shabir 00 Ma 008 FS i a se S is deoed b: { : S [ 0] : S [0] S} where ad are called he egaive membershi ad osiive membershi maigs resecivel I or a eleme i S we have 0 0 he has a osiive saisacio o a FS i S O oher had i 0 0 he saisiehe couer roer o a FS i S u o some ee There is a ossibili or a eleme s such ha 0 For shor wrie S; o rerese a FS This sud aimo iroduce a more geeralied orm o FI o OS ad ivesigae several characeriaio o OS i erms o his ew idea We urher discuss relaios bewee -FI -FI ad a q -FI PRELIMINRIES Le S be a o-em se deoes oeraio o mulilicaio ad deoes arial order relaio I S saihe ollowig codiios: S is a semigrou S is a arial ordered se 3 a b S a b a b ad a b The S is called a OS ad is deoed b S I φ S ad he we deoe ad deie he ollowig: ]: { S : h } { ab : a adb } { S S } ] ] ] ] ] ] ]] ] The characerisic ucio o is deoed b χ S χ χ where χ ad i i χ χ 0 i 0 i I he is called a subsemigrou subsemigrou o S is a bi-ideal i: S or all S χ are deied as: The egaive s -cu is deied ad deoed as N ; s : { S s} whereahe osiive -cu is rereseed b P ; : { S } or a FS S; We rerese ad deie he - cu o as C ; N ; s P ; 35

3 J l Eviro iol Sci For wo FSs S; ad g S; g g ad or all S we sa ha: i g i g ad g ii g i g ad g iii g S; g g g S; g g S he FS 0 S;0 0 S; 0 0 ad S; ad S; g g iv For all 0 The roduc o ad are deied as ollows: g is deied ad deoed as: o g S; og og where { g } i φ o g 0 i φ ad { g } i φ o g 0 i φ The mulilicaio o is well deied ad associaive ad clearl F S o is a ordered semigrou Deiiio Shabir 03: FS S; i S is called a FL R I o S i: ad ad ad or all S FS is called FI o S i i is boh FLI ad FRI Deiiio Shabir 03: FS is called FI o S i: ad } ad { } 3 } ad { } or all Theorem 3 Le be a FS i S The is a FI i φ s 0 0] Proosiio 4 The ollowig hold or i i ad ol i ii χ χ χ χ χ S C ; is a bi-ideal o S or all iii χ o χ χ ] Proo i I is sraighorward o rove ad hus omied ii Firs we cosider φ The clearl χ χ χ O he oher had i φ he we le ad hereore χ χ χ ad Hece χ χ χ χ χ χ iii Now we rove χ o χ χ ] For his i is eough o show ha χ o χ χ ] ad χ o χ χ Le S : I ] he χ χ ad ab or some a ad ] b Hece ] a b ad hereore φ ad we have ] 36

4 Mahmood e al 07 ad χ oχ χ oχ { χ χ } { χ a χ b} { χ { χ a χ χ } b} I ollowha χ χ oχ ad χ χ oχ ] ] 3 Geeralied FI o OS The idea o FI is geeralied o iroduce he oio o α β -FI i OS i his secio I s 0 0] he a FS S; o he orm: s i ] i ] : : 0 i ] 0 i ] is called a OFP wih ad s are he value ad suor o he OFP ad is deoed b OFP is said o belogo deoed as + > he is said o be quasi-coicidece wih deoed as he we wrie α does o hold s i s ad I ad q I s or s q q we mea s ad q We wrie α i he relaio Le 0 5 ad 0 5 or all S s 0 0] such ha q The s ad +> I ollowha > + s ad + < + ha is < 0 5 ad > 0 5 I ollowha } φ + From he above discussio we coclude ha he case α q will be omied i his sud { q Deiiio 3 FS i S is called α β -biolar u le righ ideal o S where α q i he ollowig codiios hold or all s 0 0] ad S : I he α imlies β s α β α β Deiiio 3 FS i S is called α β -FI he ollowig codiios hold or all s 0 0] ad : I he α β s α ad s α { s s } { } β 3 s α ad s α { s s } { } β Theorem 33 Le be a FS i S The is a FI i ad ol i or all s 0 0] ad he ollowig codiios hold simulaeousl: I he s ad s { s s } { } 3 ad Proo Suose ha is a FI I S such ha ad he s ad Deiiio we have ad This imlieha s ad ad so I s ad s he s ad s Deiiio we 37

5 J l Eviro iol Sci have { { s s } ad { } { } i ollowha { s s } { } I such ha s ad s he s ad s Deiiio 3 we have s s } his imlies s } { { s } { ad } { } Coversel le S such ha I s ad he ad hece b we have s This imlies s ad Hece we have I ad he b we have { } { } This imlieha ad { } Le such ha ad he b 3 we have { } { } I ollowha { } Proosiio 34 Ibrar e al 06 FS i S is a -FGI i he ollowig codiios hold simulaeousl or all : 05} ad { 05} { 05} ad { 05} Theorem 35 FS i S is a -FI i he ollowig codiios hold simulaeousl or all : 0 5 ad } ad { 05} 3 05} ad { 05} Proo The roo ollows rom Proosiio 34 Theorem 36 Ever -FI is -FI Proo I ca be roved easil The below give eamle showha ever -FI is o -FI Eamle 37 Le S { a b c d e} be a se wih he ollowig mulilicaio able ad order relaio Table a b c d e a a d a d d b a b a d d c a d c d e d a d a d d e a d c d e : { a a a c a d a e b b c c c e d d e e} The S is a OS Le be a FS i S deied b Table below: 38

6 Mahmood e al 07 Table S a b c d e The is -FI bu o a -FI because ae d { 07 06} {07055} Theorem 38 Ever q -FI is a -FI Proo Le be a q -FI o S Le a e ad bu S such ha ad s 0 0] I he ad hece b hohesis we have Ne we cosider s s 0 0] ad s s The q s q ad hece b hohesihis imlies s s } { gai we le such ha s ad s { } imlies s ad q s hohesis q { s s } { } Hece S; is a -FI Theorem 39 I is a oero -FI o S he he se S { S 0} { S 0} is a bi-ideal o S o Proo Le S such ha ad The 0 ad 0 Hece < 0 ad So > 0 Suose ha 0 or 0 Sice ad > or < This imlieha which is a coradicio lso + or + This imlieha q which is a coradicio Thereore 0 ad 0 Hece I o S o S he ad 0 So < 0 < 0 > 0 ad > 0 Suose ha he 0 or 0 Clearl ad S o Sice 0> or 0< This imlies a coradicio lso + or + This imlieha q agai a coradicio Thereore 0 ad 0 Hece S o Le such ha The o ad 0 So < 0 < 0 > 0 ad > 0 I S he 0 o or 0 Clearl ad Sice 0> or 0< This imlies a coradicio lso + or + This imlieha q a coradicio Thereore S o Theorem 30 Le be a The is a -FI Proo Suose be a -FI o S Le S be such ha ad -FI o S ad or all S or s 0 0] he s ad 39

7 J l Eviro iol Sci Theorem 35 we have 0 5 s ad 0 5 ad so s Le S ad s s 0 0] be such ha s ad s he s s ad Theorem 35 we have 05} s s } ad { 05} { } which imlies { s s } { } Le ad s s 0 0] such ha s ad s he s s ad Theorem 35 3 we have 05} s s } ad { 05} { } I which i ollows ha { s s } { } Hece is a -FI Theorem 3 Le I be a bi-ideals o S ad a FS i S such ha i 0 or all S \ I ii 05] [05] or all I The is q -FI o S Proo Le S such ha ad s 0 0] Le q he ad +> This imlieha I Sice I is bi-ideals o S so I I order o check we cosider he ollowig our cases: s 05 ad 0 5 s < 0 5 ad> s < 05 ad s 0 5 ad> 0 5 The irs case iduces 0 5 s ad 0 5 ad hece s The secod case imlies ha ad +> i ollowha q Sice q so case 3 ad 4 do o occur Cosequel Le S such ha s q ad q s or s s 0 0] he +s < +s < + > ad + > ad so I Sice I is a bi-ideals o S hereore I I order o check s s { we cosider he ollowig our cases: } { } s 5 ad < 05 ad > < 05 ad 05 ad > s s s 0 5 The irs case iduces 05 s ad 05 ad so s s The secod case imlieha + s < ad + > ha is s s q Sice q s ad s q so cases 3 ad 4 do o occur Cosequel s s Le such ha s q ad q s or s s 0 0] he +s < +s < + > ad + > i which i ollowha I Sice I is bi-ideals o S hereore we have I Now o show ha s 5 ad < 05 ad > < 05 ad 05 ad > s s s 0 5 s s we cosider he ollowig our cases: 40

8 Mahmood e al 07 The irs case iduces 05 s s ad 05 ad so s s The secod case imlieha + s s < ad + > ad hereore s s q Sice q s ad q ad hereore boh Case 3 ad Case 4 do o occur Hece { s s { } s } I geeral ever -FI ma o be q -FI as show i he ollowig eamle Eamle 3 Cosider he OS o Eamle 37 ad deie FS i S as i Table 3 S Table 3 a b c d e a e The is a -FI o S bu o a q -FI because 0507 q ad q bu ae d I he ollowig heorem we rovide a codiio or a -FI o be q -FI Theorem 33 Ever -FI is a q -FI o S or codiio ha s ] Proo Le be a -FI o S ad S such ha ad q or s ] The ad +> ha is < s s ad > ad so Sice is a -FI o S Thereore Le S such ha s q ad q s or some s s ] he +s < +s < + > ad + > This imlieha < s s < s s > ad > Hece s ad s ad sice is a -FI o S hereore s s Le such ha s q ad q s or s s ] he +s < +s < + > ad + > This imlieha < s s < s s > ad > ha is s ad s Sice is a -FI o S hereore s s ad hece is q -FI o S REFERENCES [] haka S K ad P Das 99 O he deiiio o a u subgrou Fu Ses ad Ssems Vol 5: 35-4 [] Crise I ad Davva 00 aassov's iuiioisic u grade o hergrous Iorm Sci Vol 80: [3] Davva M Fahi ad R Salleh 00 Fu herrigs Hv-rigs based o u uiversal ses Iorm Sci Vol 80: [4] Ibrar M Kha ad Davva 06 Characeriaios o regular ordered semigrou i erms o α β -biolar u geeralied bi-ideals Iellige ad u ssems submied 06 [5] Ju Y 005 O α β -u subalgebras o CK / CI-algebras ull Korea Mah Soc Vol 4 4: [6] Kaaci O ad Davva 008 O he srucure o rough rime rimar ideals ad rough u rime rimar ideals i commuaive rigs Iorm Sci Vol 78: [7] Kaaci O ad S Yamak 008 Geeralied u bi-ideals o semigrous So Comu Vol : 9-4 4

9 J l Eviro iol Sci [8] Kuroki N 979 Fu bi-ideals i semigrous Comme Mah Uiv S Pauli Vol 8 : 7- [9] Kuroki N 98 O u ideals ad u bi-ideals i semigrous Fu Ses Ssem Vol 5: 03-5 [0] Kuroki N 99 O u semigrous Iorm Sci Vol 53: [] Lee K M 004 Comariso o ierval-valued u ses iuiioisic u ses ad biolarvalued u ses J Fu Logic Iellige Ssems Vol 4 : 5-9 [] Ma X J Zha ad Davva 008 Some kids o -ierval-valued u ideals o CI-algebras Iorm Sci Vol 78: [3] Roseeld 97 Fu grous J Mah al l Vol 35: 5-57 [4] Shabir M Y Ju ad Y Nawa 00 Characeriaios o regular semigrous b α β - u ideals Comu Mah l Vol 59: 6-75 [5] Zadeh L 965 Fu ses Iormaio ad Corol Vol 8: [6] Zhag W R 994 iolar u ses ad relaios: a comuaioal ramework or cogiive modelig ad muliage decisio aalsis Proc o IEEE Co [7] Zhag W R 998 iolar u ses Proc o IEEE