On Natural Partial Orders of IC-Abundant Semigroups

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Intentionl Jounl of Mthemtics nd Computtionl Science Vol. No. 05 pp. 5-9 http://www.publicsciencefmewok.

IC-bundnt semigoups. Afte giving some popeties nd chcteiztions of ntul ptil odes on bundnt semigoups we conside IC-bundnt semigoups.

1 Intentionl Jounl of Mthemtics nd Computtionl Science Vol. No. 05 pp On Ntul Ptil Odes of IC-Abundnt Semigoups Chunhu Li Bogen Xu School of Science Est Chin Jiotong Univesity Nnchng Jingxi Chin Abstct In this ppe we will investigte the ntul ptil odes on IC-bundnt semigoups. Afte giving some popeties nd chcteiztions of ntul ptil odes on bundnt semigoups we conside IC-bundnt semigoups. We pove tht n IC-bundnt semigoup is loclly mple if nd only if the ntul ptil ode on the semigoup is comptible with the multipliction. Keywods Abundnt Semigoup Ntul Ptil Ode IC-Abundnt Semigoup Loclly Ample eceived: Mch 4 05 / Accepted: Mch 0 05 / Published online: Mch 3 05 The Authos. Published by Ameicn Institute of Science. This Open Access ticle is unde the CC BY-NC license. Intoduction The concepts of the ntul ptil odes on egul semigoup wee intoduced by Nmbooipd [3] in 980. As geneliztion of egul semigoups in the nge of bundnt semigoups El-Qllli nd Fountin [ ] intoduced bundnt semigoups. Afte tht vious clsses of bundnt semigoups e eseched (see [ 6-4] ). In 987 Lwson defined ntul ptil odes on bundnt semigoups nd extend Nmbooipd s esults. The eltion semigoup S by the ule tht elements b is defined on b if nd only if the of Se elted by Geen s eltion in some ovesemigoup of S. The eltion L is defined dully. In this ppe we shll study the ntul ptil odes on IC-bundnt semigoups by using the notion of mjoiztion of the eltions nd L. Some popeties nd constuctions on IC-bundnt semigoups will be descibed in tems of its ntul ptil ode. We shll poceed s follows: section povides some known esults. In section 3 we give some chcteiztions of the ntul ptil odes on bundnt semigoups. The lst section we conside the ntul ptil odes on IC-bundnt semigoups.. Peliminies Thoughout this ppe we shll use the notions nd nottions of [4-5]. Hee we povide some known esults used epetedly in the sequel. At fist we ecll some bsic fcts bout the eltion L nd. Lemm II. [] Let S be semigoup nd b S. Then the () Lb( b); () fo ll bx = by( xb = yb). x y( x y) = = if nd only if As n esy but useful consequence of Lemm II. we hve Coolly II. [] LetS be semigoup nd e = e S.Then the () L e( e ); () = e( = e) nd fo ll x = y( x = y) Coesponding utho E-mil ddess: chunhuli66@63.com (Chunhu Li) bogenxu@63.com (Bogen Xu)

2 6 Chunhu Li nd Bogen Xu: On Ntul Ptil Odes of IC-Abundnt Semigoups implies bx = by( xb = yb). Evidently L is ight conguence while is left conguence. In n bity semigoup we hve. But fo egul elements b we get L L nd Lb( b ) if nd only if Lb( b ). Fo convenience we denote by [ ] typicl idempotent ( ) denotes the elted [ L elted ] to L clss ( clss ) contining. L And E( T ) denotes the set of idempotents of T ; e g( T ) denotes the set of egul elements of T. We denote by V( ) the set of ll inveses of. A semigoup S is clled bundnt if nd only if ech L clss nd ech clss contins t lest one idempotent. An bundnt semigoup S is clled qusi-dequte if its set of idempotents constitutes subsemigoup (i.e. its set of idempotents is bnd). Moeove qusi-dequte semigoup is clled dequte if its bnds of idempotents is semilttice (i.e. the idempotents commute). An bundnt semigoup S is clled mple if fo ll e = e S e = ( e) nd e = ( e). Follwing[] n bundnt semigoup S is clled idempotent-connected fo shot IC povided fo ech S nd fo some bijection θ : thee exists x = xθ fo ll < > < > such tht ( ) x < > whee < > [esp. < >] is the subsemigoup of S geneted by the set { y E( S): y = y = y} [esp. { y E( S): y y y} = = ]. In fct n mple semigoup is n IC-dequte semigoup nd vice ves. A semigoup S is clled loclly P-semigoup if fo ll ese is P-semigoup. An equivlence eltion ρ on S is clled ρ -unipotent if ech ρ -clss of S contins exctly one idempotent. Evidently n dequte semigoup is both nd - unipotent. It is well known tht L- unipotent [ esp. - unipotent ] egul semigoup is n othdox semigoup whose bnd of idempotents is ight [ esp. left ] egul bnd ( bnd B is left [esp. ight] egul bnd if fo y B xy = xyx [esp. xy = yxy ][ see4]). Definition II. [5] Let S be n bundnt semigoup. We define thee eltions on S s follows: () b nd thee exists n idempotent b such tht = fb; () b L L nd thee exists n idempotent l b such tht = be; e = e S. f e L (3) = i.e. b thee exist idempotents e f such tht = eb = bf. l Lemm II. [5] Let S be n bundnt semigoup nd b S. Then b if nd only if thee exists b e E( S) f E( S) such tht = eb nd e f. Lemm II.3 [ 5] Let S be n bundnt semigoup. ThenS is IC if nd only if = =. l Definition II. [ 4] Let ρ be n equivlence eltion on semigoup S T be subset of S. T is clled to stisfy ρ - mjoiztion if fo ny b c T b c nd bρ c implies tht b = c. 3. Popeties nd Chcteiztions The im of this section is to intoduce the ntul ptil on bundnt semigoups nd to give some popeties nd chcteiztions of the ntul ptils on such semigoups. PopositionIII. Let S be n bundnt semigoup nd b S. Then b if nd only if thee exist tht = xb = by nd x =. such Poof. We only pove the sufficiency pt. To see this let such tht xb by = = nd x =. Then = x = xby = y. Hence by Coolly II. = x. Futhemoe we get = = ( xb) = ( x) b = = mens tht b. = y nd y x E( S). Thus ( by) = b( y ). This LemmIII. Let S be n bundnt semigoup. If S stisfy mjoiztion then fo ny e E( S) ese is unipotent. Poof. Let f g E( ese). Then e f nd e g. If f L ( ese) g then f L ( S) g. Since S stisfy mjoiztion we hve f = g. So ese is unipotent. LemmIII. Let S be n bundnt semigoup. If S is unipotent then S stisfy mjoiztion. Poof. Let b c S such tht b c nd bl c. By the dul gument of Lemm II. thee exist e E( Lb) e E( Lc) such tht b = e c = e. Hence e LbLcLe. But S is unipotent we hve e = e. So b = e = e = tht is S stisfy c mjoiztion.

3 Intentionl Jounl of Mthemtics nd Computtionl Science Vol. No. 05 pp Poposition III. Let S be n bundnt semigoup. Then the () fo ny e E( S) ese is () fo ny e E( S) ese stisfy (3) E( S ) stisfy L- mjoiztion; unipotent; (4) e g( S ) stisfy L- mjoiztion. mjoiztion; Poof. () () It follows immeditely fom Lemm Ⅲ.. () (3) Suppose tht () holds. Let e f g E( S) nd e f e g f L g. Then f g E( ese) nd f Hence f = g nd so E( S ) stisfy L- mjoiztion. L ( ese) g. (3) (4) Assume tht E( S ) stisfy mjoiztion. Let b c e g( S) nd b c bl( S) c. Then thee exist e f E( S) such tht b = e = f. On the othe hnd since is egul we hve x V( ) such tht ( xb) xexf xef xb E( S). = = = Agin since xb = ( xx) b = ( x)( xb) = xe = ( xe) x = ( xb)( x) we hve tht xb x. Notice tht b = f = xf = ( xb) we get tht bl( S) xb. Similly we cn pove tht xc E( S) c = ( xc) xc x nd cl( S) xc. Hence xbl( S) xc. By the hypothesis E( S ) stisfy L- mjoiztion we hve tht xb = xc. Theefoe b = ( xb) = ( xc) = c. Tht is e g( S ) stisfy L- mjoiztion. (4) () Assume tht e g( S ) stisfy L- mjoiztion. Let f g E( ese) nd f L ( ese) g. Hence e f e g nd f L( S) g. By (4) we hve f = g nd so ese is unipotent. Theoem III. Let S be n bundnt semigoup stisfying the egulity condition. Then the following sttements e equivlent: () e g( S ) is comptible with espect to ; ()S is loclly dequte semigoup; (3) fo ny e E( ese) ese stisfies both -unipotent; (4) fo ny e E( ese) ese stisfies both mjoiztion; (5) E( S ) stisfies both L- nd - mjoiztion; nd nd (6) e g( S ) stisfies both L- nd - mjoiztion. - Poof. () () Suppose tht e g( S ) is comptible with espect to. Then e g( S ) is loclly invese semigoup (see [ 3 ]). Noticing tht fo ny e E( S) ese is n bundnt semigoup which implies tht E( ee g( S) e ) is semilttice nd e g( ese ) = ee g( S) e. We obseve tht E( ese ) is semilttice (since ee g( S) e is invese semigoup ). Thus ese is n dequte semigoup tht is S is loclly dequte semigoup. () (3) This is cle. (3) (4) (5) (6) Follow fom Poposition Ⅲ.. (6) () Suppose tht e g( S ) stisfies both nd - mjoiztion. By Poposition Ⅲ. we hve ese stisfies both nd -unipotent. But e ( ) g ese = ee g( S) e we hve tht e g( ese ) is egul semigoup stisfying both nd -unipotent. Futhemoe e g( ese ) is n invese semigoup. Theefoe e g( S ) is loclly invese semigoup nd so e g( S ) is comptible with espect to (see[ 3 ]). 4. Ntul Ptil Odes on IC-Abundnt Semigoups In this section we will conside the ntul ptil odes on IC-bundnt semigoups. Theoem IV. Let S be n IC-bundnt semigoup. Then the () S is ight comptible with espect to ; ()fo ny e f g E( S) e f nd e g fg = gfg; (3)fo ny e E( S) the egulity condition; (4) fo ny e E( S) egulity condition. ese stisfies both ese stisfies both mjoiztion nd unipotent nd the Poof. () () Let e f g E( S) e f nd e g. By () we hve tht eg fg. Hence g fg. By Poposition.7 of [5 ] we hve fg E( S). Theefoe fg = gfg. () (3) Let f g E( ese). Then e f nd e g. By () we hve tht fg = gfg. Hence ( fg) fgfg f( fg) fg E( S) = = = which implies tht ese stisfies the egulity condition. Next we show tht ese stisfies mjoiztion.

4 8 Chunhu Li nd Bogen Xu: On Ntul Ptil Odes of IC-Abundnt Semigoups Let b c ese such tht b c nd bl( S) c. By the dulity of Lemm II. thee exist e E( Lb) ese e E( Lc) ese such tht b = e c = e. Obviously e e e e e ee e e nd el( ese) e. Hence = so tht e = ee = eee = ee = e. Thus b = e = e = Theefoe ese stisfies c. (3) (4) It follows fom Poposition III.. mjoiztion. (4) () Suppose tht (4) holds. Then e g( ese ) is n unipotent semigoup. Hence E( ese ) is ight egul bnd. Let b c S nd b. By Lemm II. fo thee exists f E b e E ( ) ( ) such tht e f nd = eb. Hence c = ebc. Since fbc = bc by LemmII. we hve f( bc) = ( bc) E( bc ). It is esy to see tht ( bc) f E( bc ). Thus e( bc) f E( fsf) since e( bc) f E( fsf). x V( e( bc) f) fsf. We hve tht is xe( bc) f( bc) fxe( bc) f E( fsf) nd xe( bc) fl( fsf)( bc) fxe( bc) f xe bc fl fsf bc fxe bc f ( ) ( )( ) ( ). Tke Since E( fsf ) is ight egul bnd by ssumption we hve xe( bc) f = ( bc) fxe( bc) f. Multiplying the pio fomul on the ight by x we obtin tht x ( bc) = fx. Hence x = xe( bc) fx = xe nd so xe e( bc) fxe E( fsf). Notice tht xel( fsf) e( bc) fxe the fct tht fsf is by unipotent we hve tht xe = e( bc) fxe. If we multiply this equlity on the ight by ( bc) f we cn obtin tht xe( bc) f = e( bc) f. Hence e( bc) f = e( bc) fxe( bc) f = e( bc) fe( bc) f. Tht is e( bc) f E( fsf ). Note tht ( bc) f( bc) fe( bc) f E( fsf) e bc fl fsf bc fe bc f ( ) ( )( ) ( ) by (4) we obseve ( bc) fe( bc) f e( bc) = f. Hence e( bc) f ( bc) f. Since c = ebc = e( bc) fbc by Lemm II. we hve c bc. Agin since S is n IC-bundnt semigoup we get tht c bc. The poof is completed. Theoem IV. Let S be n IC-bundnt semigoup. Then the () S is comptible with espect to ; ()fo ny e f g E( S) e f nd e g fg = gf; (3) S is loclly mple semigoup; (4) fo ny e E( S) ese stisfies both -mjoiztion nd the egulity condition. Poof. It follows fom Theoem IV. nd its dul. 5. Conclusions nd In this ppe we investigte the ntul ptil odes on IC-bundnt semigoups nd give some popeties nd chcteiztions of ntul ptil odes on bundnt semigoups by using the notion of mjoiztion. We genelize nd stengthen the esults of Fountin on bundnt semigoups. Acknowledgements This wok is suppoted by the Ntionl Science Foundtion (Chin) (No. 608; No. 3604) the Ntul Science Foundtion of Jingxi Povince ( No. 0BAB008 ) the Science Foundtion of the Eduction Deptment of Jingxi Povince (No. GJJ438;No. KJLD067). efeences [] El-qllli A. & Fountin J. B. Idempotent-connected bundnt semigoups. Poc. oy. Soc. Edinbugh 9 pp [] Fountin J. B. Abundnt semigoups Poc. London Mth. Soc. 44 pp [3] Nmboopd K. SS. The ntul ptil ode on egul semigoup Poc. Edinbugh Mth. Soc. 3 pp [4] Petich M. Completely egul semigoups. New Yok Jhon Wiley nd Sons Inc [5] Lwson M. V. The ntul ptil ode on n bundnt semigoup Poc. Edinbugh Mth. Soc. 30 pp [6] Li C. Guo X. & Liu E. Good conguences on pefect ectngul bnds of dequte semigoups Advnces in Mthemtics 38 pp [7] Li C. & Liu E. Fuzzy good conguences on Left semi-pefect bundnt semigoups Comm. Algeb 39 pp [8] Li C. On fuzzy conguences of type B semigoup I Intentionl Mthemticl Foum 9 pp [9] Li C. Wng L. & Fn Z. The stuctue of bundnt semigoups with multiplictive type B tnsvesl Advnces in Mthemtics 43 pp

5 Intentionl Jounl of Mthemtics nd Computtionl Science Vol. No. 05 pp [0] Li C. & Wng L. On the tnsltionl hull of type B semigoup Semigoup Foum 8 pp [] Li C.& Wng L. A pope cove on ight type B semigoup Jounl of South Chin Noml Univesity 44 pp [] Li C. & Shu C. Some notes on pope type B semigoups Jounl of Nnchng Univesity 38 pp [3] Li C. Xu B. & Hung H. Fuzzy-t conguences on bundnt semigoups Fuzzystems nd Mthemtics 8 pp [4] Li C. & Xu B. Fuzzy good conguences on bundnt semigoups Fuzzystems nd Mthemtics pp

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g: