TERNARY SEMIHYPERGROUPS IN TERMS OF BIPOLAR-VALUED FUZZY SETS

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1 IJRR 6 Jul wwwarpapresscom/volumes/vol6issue/ijrr_6 6pf TERNRY EMIHYPERGROUP IN TERM OF IPOLR-VLUED FUZZY ET Ibsam Masmal Deparmen of Mahemacs College of cence Jaan Unvers Jaan Kngom of aua raba E-mal: bsam4@homalcom TRCT Ths paper represens he concep of bpolar-value fu ernar subsemhpergroups lef hpereals rgh hpereals laeral hpereals hpereals of ernar semhpergroups Kewors: Ternar semhpergroups; Hpereals; polar-value fu ses; polar-value fu lef rgh laeral hpereals M Classfcaon: N N5 M7 INTRODUCTION Hpersrucure heor was nrouce n 94 when F Mar [8] efne hpergroups began o anale her properes apple hem o groups In he followng ecaes nowaas a number of fferen hpersrucures are wel sue from he heorecal pon of vew for her applcaons o man subjecs of pure apple mahemacs b man mahemacans Nowaas hpersrucures have a lo of applcaons o several omans of mahemacs compuer scence he are sue n man counres of he worl The neresng fac n he hper srucure of an algebrac srucure s ha n a classcal algebrac srucure he composon of wo elemens s an elemen whle n an algebrac hpersrucure he composon of wo elemens s a se number of algebras have conrbue a lo of papers n hs recon several books have been wren on hpersrucure heor see [] [] [] [] recen book on hpersrucures [] pons ou on her applcaons n rough se heor crpograph cong heor auomaa probabl algebrac geomer laces bnar relaons graphs hpergraphs noher book [] s evoe especall o he su of hperrng heor n he smlar wa man more n he su of hpernearrng We hope o su he conceps n case of hpersemrngs everal kns of hperrngs are nrouce anale ll now The volume ens wh an oulne of applcaons n chemsr phscs analng several specal kns of hpersrucures: e -hpersrucures ransposon hpergroups In 9 Lehmer nrouce he concep of a ernar semgroups [6] non-emp se X s calle a ernar semgroup f here ess a ernar operaon X X X X ; wren as sasfng he followng en for an 4 5 ] 45] [ [ 4 ] 5] [ [ 45]] X n semgroup can be reuce o a ernar semgroup However anach showe ha a ernar semgroup oes no necessarl reuce o a semgroup For hs we can conser some general eamples as below T { } T { } are ernar semgroups wh ero whou ero elemen whle T T are no a semgroup wh ero or whou ero elemen uner comple mulplcaon noher general eample s he se of negave negers Z or se of negave negers wh ero elemen Z Los showe ha ever ernar semgroup can be embee no a semgroup [7] whereas orowec e al [6] have gven a connecon beween ernar semgroups some ornar semgroups Furher Duek Mukhn [7] have shown he creron ha an ornar semgroup nuces a ernar semgroup Hla Naka [9 ] worke ou on ernar semhpergroups nrouce some properes of hpereals n ernar semhpergroups also see [] The concep of a fu se nrouce b Zaeh n hs classc paper [5] proves a naural framework for generalng some of he noons of classcal algebrac srucures also see [6] Fu semgroups have been frs consere b Kurok [] fer he nroucon of he concep of fu ses b Zaeh several researches conuce he researches on he generalaons of he noons of fu ses wh huge applcaons n compuer logcs man branches of pure apple mahemacs Fu se heor has been shown o be an useful ool o escrbe suaons n whch he aa are mprecse or vague Fu ses hle such suaons b arbung a egree o whch a ceran objec belongs o a se In 97 Rosenfel [] efne he concep of fu group nce 5

2 IJRR 6 Jul Masmal polar-value Fu es hen man papers have been publshe n he fel of fu algebra Recenl fu se heor has been well evelope n he cone of hperalgebrac srucure heor recen book [] conans an wealh of applcaons In [5] Davva nrouce he concep of fu hpereals n a semhpergroup also see [4 8 9] Yaqoob ohers [4] nrouce he concep of rough fu hpereals n ernar semhpergroups several papers are wren on fu ses n several algebrac hpersrucures The relaonshps beween he fu ses algebrac hpersrucures have been consere b Corsn Davva Leoreanu Zhan Zahe mer Crsea man oher researchers There are several kns of fu se eensons n he fu se heor for eample nuonsc fu ses nervalvalue fu ses vague ses ec polar-value fu se s anoher eenson of fu se whose membershp egree range s fferen from he above eensons Lee [] nrouce he noon of bpolar-value fu ses polar-value fu ses are an eenson of fu ses whose membershp egree range s enlarge from he nerval [] o [ ] In a bpolar-value fu se he membershp egree ncae ha elemens are rrelevan o he corresponng proper he membershp egrees on ] assgn ha elemens somewha sasf he proper he membershp egrees on [ assgn ha elemens somewha sasf he mplc counerproper [ 4] In [] Jun park apple he noon of bpolar-value fu ses o CH-algebras The nrouce he concep of bpolar fu subalgebras bpolar fu eals of a CH-algebra Lee [5] apple he noon of bpolar fu subalgebras bpolar fu eals of CK/CI-algebras lso some resuls on bpolar-value fu CK/CIalgebras are nrouce b ae n [] In hs paper we su he concep of bpolar-value fu ernar subsemhpergroups lef hpereals rgh hpereals laeral hpereals hpereals of ernar semhpergroups TERNRY EMIHYPERGROUP In hs secon we wll presen some basc efnons of ernar semhpergroups map : H H H s calle hperoperaon or jon operaon on he se H where H s a non-emp se H H \ { } enoes he se of all non-emp subses of H hpergroupo s a se H wh ogeher a bnar hperoperaon Defnon hpergroupo H whch s assocave ha s s calle a semhpergroup Le be wo non-emp subses of H Then we efne a b a b a a a a Defnon map f : H H H H s calle ernar hperoperaon on he se H where H s a non-emp se H H \ { } enoes he se of all non-emp subses of H Defnon ernar hpergroupo s calle he par H f where f s a ernar hperoperaon on he se H Defnon 4 ernar hpergroupo f s calle a ernar semhpergroup f for all a a a we have f f a a a a4 a5 f a f a a a4 a5 f a a f a a4 a5 Defnon 5 Le f be a ernar semhpergroup Then s calle a ernar hpergroup f for all a b c here es such ha: c f a b f a b f a b 5 5

3 IJRR 6 Jul Masmal polar-value Fu es Defnon 6 Le f be a ernar semhpergroup T a non-emp subse of Then T s calle a subsemhpergroup of f onl f f T T T T Defnon 7 non-emp subse I of a ernar semhpergroup s calle a lef rgh laeral hpereal of f f I I f I I f I I Eample [4] Le { a b c e g} f for all where s efne b he able: a b c e a a b c e b { a b} b { c } { e g} c c c c { c } { c } { c } { e g} g g g g g f s a ernar semhpergroup Clearl { c } { c e g} are laeral hpereals Then e e g c e g { e g} g { c } of In wha follows le enoe a ernar semhpergroup unless oherwse specfe For smplc we wre f a b c as abc Defnon 8 Le be a ernar semhpergroup non-emp subse T of s calle prme subse of f for all T mples T or T or T ernar subsemhpergroup T of s calle prme ernar subsemhpergroup of f T s a prme subse of Prme lef hpereals prme rgh hpereals prme laeral hpereals prme hpereals of are efne analogousl IPOLR-VLUED FUZZY HYPERIDEL OF TERNRY EMIHYPERGROUP Frs we wll recall he concep of bpolar-value fu ses Defnon [4] Le X be a nonemp se bpolar-value fu subse VF-subse n shor of X s an objec havng he form : X Where : X ] : X ] The posve membershp egree corresponng o a bpolar-value fu se : X egree enoes he sasfacon egree of an elemen o he proper he negave membershp enoes he sasfacon egree of o some mplc couner proper of : X bpolar-value fu se : X For he sake of smplc we shall use he smbol for he 5

4 IJRR 6 Jul Masmal polar-value Fu es 5 Defnon be wo VF-subses of a ernar semhpergroup The smbol wll means he followng forall forall The smbol wll means he followng forall forall Defnon Le be hree VF-subses of a ernar semhpergroup Then her prouc s efne b : where oherwse f }} sup{mn{ oherwse f }} nf {ma{ for some for all Defnon 4 Le be a ernar semhpergroup VF-subse of s calle a VF-ernar subsemhpergroup of f } ma{ } sup mn{ nf for all a VF-lef hpereal of f sup nf for all a VF-rgh hpereal of f sup nf for all 4 a VF-laeral hpereal of f sup nf for all 5 a VF-hpereal of f

5 IJRR 6 Jul Masmal polar-value Fu es for all nf ma{ } sup mn{ } Eample Le { a b c } f for all where s efne b he able: c ccccc abc aa{ a b} a{ a b} bbbbbca{ a b} c{ c } bb Then f s a ernar semhpergroup Defne a bpolar-value fu subse n as follows: ll8f 5f { a b}f { c } ll 9f 7f { a b} f { c } roune calculaons can be seen ha he VF-subse s a VF-hpereal of Theorem 5 If { } s a faml of VF-ernar subsemhpergroups VF-lef hpereals VF-rgh Then hpereals VF-laeral hpereals VF-hpereals of s a VF-ernar subsemhpergroup VF-lef hpereal VF-rgh hpereal VF-laeral hpereal VF-hpereal of where nf : sup : Proof Conser { } s a faml of VF-ernar subsemhpergroups of Le H we have nf nf mn mn sup sup ma ma Hence hs shows ha Then for ever s a VF-ernar subsemhpergroups of The oher cases can be seen n a smlar wa Theorem 6 If { s a faml of VF-ernar subsemhpergroups VF-lef hpereals VF-rgh } 54

6 IJRR 6 Jul Masmal polar-value Fu es 55 hpereals VF-laeral hpereals VF-hpereals of Then s a VF-ernar subsemhpergroup VF-lef hpereal VF-rgh hpereal VF-laeral hpereal VF-hpereal of where : sup : nf Proof The proof s smlar o he proof of Theorem 5 Theorem 7 Le be a VF-rgh hpereal a VF-laeral hpereal a VF-lef hpereal of a ernar semhpergroup Then Proof Le be a VF-rgh hpereal a VF-laeral hpereal a VF-lef hpereal of a ernar semhpergroup If here o no es such ha hen If here es such ha hen }} sup{mn{ }} nf nf sup{mn{nf } mn{ }} nf {ma{ }} sup sup nf {ma{sup } ma{ Thus Le us conser

7 IJRR 6 Jul Masmal polar-value Fu es : forall n be a VF-subse of a ernar semhpergroup VF-subse such ha wll be carre ou n operaons wh a wll be use n collaboraon wh respecvel Theorem 8 Le be a VF-rgh hpereal a ernar semhpergroup Then a VF-lef hpereal of Proof Le ernar semhpergroup Le If here es be a VF-rgh hpereal such ha hen a VF-lef hpereal of a a If here o no es such ha hen sup{mn{ }} sup{mn{ sup{mn{ sup mn nf mn{ }} }} nf } nf {ma{ }} nf {ma{ nf {ma{ }} nf ma sup sup ma{ } Thus }} 56

8 IJRR 6 Jul Masmal polar-value Fu es Theorem 9 VF-subse of f onl f of a ernar semhpergroup s a VF-ernar subsemhpergroup Proof uppose hen If here es of a ernar subsemhpergroup If here o no es such ha such ha hen sup{mn{ }} supnf nf {ma{ }} nf sup Then for all we have such ha Hence Conversel assume ha nf nf a b nf mn{ c } abc mn{ a b c} sup sup a b sup ma{ c } abc ma{ a b c} Hence s a VF-ernar subsemhpergroup Theorem VF-subse of a ernar semhpergroup s a VF-lef hpereal VF-rgh hpereal VF-laeral hpereal of f onl f Proof Le such ha be a VF-lef hpereal of Le us suppose ha here es Then snce s a VF-lef hpereal of we have sup[mn{ }] sup[mn{ }] sup 57

9 IJRR 6 Jul Masmal polar-value Fu es nf [ma{ }] nf [ma{ }] nf In case of s a VF-lef hpereal of nf r r o n parcular a nf a es sup r r a for all a Hence sup a a a Thus a a a a If here o no such ha a hen a a a a Hence we ge Conversel le a Then a a sup a a a Consequenl We have nf a a sup[mn{ }] a mn{ } mn{ } a nf [ma{ }] a ma{ } ma{ } nf a sup a a a Hence s a VF-lef hpereal of The oher case can be seen n a smlar wa Proposon The prouc of hree VF-lef hpereals VF-rgh hpereals of a ernar semhpegroup s agan a VF-lef hpereal VF-rgh hpereal of Proof Le ernar semhpergroup hen b Theorem Ths complees he proof The oher case can be seen n a smlar wa be hree VF-lef hpereals of a Theorem Le be a ernar semhpergroup a non-emp subse of The followng saemens hol rue: s a ernar subsemhpergroup of f onl f s a VF-ernar 58

10 IJRR 6 Jul Masmal polar-value Fu es subsemhpergroup of s a lef hpereal rgh hpereal laeral hpereal hpereal of f onl f s a VF-lef hpereal VF-rgh hpereal VF-laeral hpereal VF-hpereal of Proof Le us assume ha s a ernar subsemhpergroup of Le Case nce s a ernar subsemhpergroup of we have Then nf sup mn{ } ma{ } Case or or Thus or or lso or or mn{ } nf ma{ Therefore } sup Therefore Conversel le We have nce nf sup s a VF-ernar subsemhpergroup of so mn{ } ma{ } Hence Le us assume ha s a lef hpereal of Le Case nce s a lef hpereal of hen Therefore Case Conversel le Thus nf nf sup Then We have nf nce nf sup Hence sup s a fu lef hpereal of sup The remanng pars can be seen n smlarl wa For an ] s ] Le be a VF-se n he se 59

11 IJRR 6 Jul Masmal polar-value Fu es s calle he VF-level se of U ; s { : s} Theorem Le be a VF-subse of a ernar semhpergroup The followng saemens hol rue: s a VF-ernar subsemhpergroup of f onl f for all ] s ] he se U ; s s eher emp or a ernar subsemhpergroup of s a VF-lef hpereal VF-rgh hpereal VF-laeral hpereal VF-hpereal of f onl f for all ] s ] he se U ; s s eher emp or a lef hpereal rgh hpereal laeral hpereal hpereal of Proof Le us assume ha s a VF-ernar subsemhpergroup of Le ] s ] such ha U ; s Le U ; s o s Thus mn{ } ma nce s a VF-ernar subsemhpergroup of Hence U ; s Conversel le nf h sup h h s h s Le we ake mn{ } s ma{ } Then s U ; s nce U ; s s a ernar subsemhpergroup of U ; s Thus Le us assume ha nf h mn{ } h sup h s ma{ } h Thus s a VF-lef hpereal of Le ] s ] such ha U ; s Le U ; s Thus Therefore U ; s Conversel le nf h sup h s h h Le we ake s Thus U ; s hs mples U ; s assumpon we have U ; s s a lef hpereal of o U ; s Therefore nf h sup h s h h Thus nf h sup h h The reman pars can be prove n a smlar wa h 6

12 IJRR 6 Jul Masmal polar-value Fu es Defnon 4 VF-subse of s calle a prme VF-subse of f nf ma{ } sup mn{ } for all VF-ernar subsemhpergroup of s calle a prme VF-ernar subsemhpergroup of f s a prme VF-subse of Prme VF-lef hpereals prme VF-rgh hpereals prme VF-laeral hpereals prme VF-hpereals of are efne analogousl Theorem 5 Le be a ernar semhpergroup a non-emp subse of The followng saemens hol rue: s a prme subse of f onl f s a prme VF-subse of s a prme ernar subsemhpergroup prme lef hpereal prme rgh hpereal prme laeral hpereal prme hpereal of f onl f s a prme VF-ernar subsemhpergroup prme VF-lef hpereal prme VF-rgh hpereal prme VF-laeral hpereal prme VF-hpereal of Proof Le us assume ha s a prme subse of Le Case nce s prme or or Thus ma{ mn{ } nf } sup Case Thus nf sup ma{ mn{ } } Conversel le such ha Thus for all nce s prme ma{ } hs mples or or lso mn{ } hs mples or or Hence or or I follows from Theorem Theorem 6 Le be a ernar semhpergroup be a VF-subse of The followng saemens hol rue: s prme VF-subse of f onl f for all ] s ] he se U ; s 6

13 IJRR 6 Jul Masmal polar-value Fu es s eher emp or a prme subse of s a prme VF-ernar subsemhpergroup prme VF-lef hpereal prme VF-rgh hpereal prme VF-laeral hpereal prme VF-hpereal of f onl f for all ] s ] he se U ; s s eher emp or a prme ernar subsemhpergroup prme lef hpereal prme rgh hpereal prme laeral hpereal prme hpereal of Proof Le us assume ha s a prme VF-subse of Le ] s ] Le us suppose ha U ; s Le such ha U ; s Thus nce nf h sup h h h s s prme or or also s or s or s Ths mples U ; s or U ; s or U ; s Conversel le Le we ake nf h s sup h Then U ; s nce h U ; s s prme U ; s or U ; s or U ; s Then or or also s or s or s Hence I follows from Theorem 5 h ma{ } nf h h mn{ } s sup h REFERENCE [] P Corsn V Leoreanu pplcaons of hpersrucure heor Kluwer caemc Publcaons [] P Corsn Prolegomena of hpergroup heor econ eon van eor 99 [] Davva VL Foea Hperrng heor applcaons Inernaonal caemc Press U 7 [4] Davva Fu hpereals n ernar semhperrngs Iran J Fu s [5] Davva Fu hpereals n semhpergroups Ialan J Pure ppl Mah [6] orowec W Duek Duplj -elemen represenaon of ernar groups Commun lgebra [7] W Duek VV Mukhn On n-ar semgroups wh ajon neural elemen Quasgroups Rela sem [8] Davva shor noe on nuonssc fu ernar plosubgroups Trens ppl c Res [9] O Dehkor Davva srong regular relaon on -semhperrngs J c IR Iran [] K Hla Davva K Naka On quas-hpereals n semhpergroups Commun lgebra [] Y Jun CH Park Flers of CH-lgebras base on b-polar-value fu ses In Mah Forum [] N Kurok Fu b-eals n semgroups Commen Mah Unv Paul [] KM Lee -polar-value fu ses her operaons Proc In Conf Inellgen Technologes angkok Thal 7-- [4] KM Lee Comparson of nerval-value fu ses nuonsc fu ses b-polar-value fu ses J Fu Logc Inel s [5] KJ Lee -polar fu subalgebras b-polar fu eals of CK/CI-algebras ull Malas Mah c oc h 6

14 IJRR 6 Jul Masmal polar-value Fu es [6] DH Lehmer ernar analogue of abelan groups mer J Mah [7] J Los On he eenng of moel I Funamena Mahemacae [8] F Mar ur une generalaon e la noon e groupe 8 em congres Mah cnaves ockholm [9] K Naka K Hla On some specal classes of hpereals n ernar semhpergroups Ulas Mahemaca accepe [] K Naka K Hla ome properes of hpereals n ernar semhpergroups accepe n Mahemaca lovaca [] Rosenfel Fu groups J Mah nal ppl [] ae polar-value fu CK/CI-algebras Worl ppl c J [] T Vougoukls Hpersrucures her represenaons Haronc Press Flora 994 [4] N Yaqoob M slam K Hla Rough fu hpereals n ernar semhpergroups v Fu s 9 pages [5] L Zaeh Fu ses Inform Conrol [6] L Zaeh The concep of a lngusc varable s applcaon o appromae reasonng I Inform c

Fuzzy derivations KU-ideals on KU-algebras BY

Fuzzy derivations KU-ideals on KU-algebras BY Fu ervaons KU-eals on KU-algebras BY Sam M.Mosaa, Ahme Ab-elaem 2 sammosaa@ahoo.com ahmeabelaem88@ahoo.com,2deparmen o mahemacs -Facul o Eucaon -An Shams Unvers Ro, Caro, Egp Absrac. In hs manuscrp, we