MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 Mltesol fe ot reslts for two ybr rs rtlly orere etrc sce R A Rsw rr_rsw5@yooco I Mostf s6@yooco Dertet of Mtetcs Fclty of cece Asst Uversty Asst 756 Eyt Abstrct- I ts er we troce te oto of e wely ootoe roerty for two ybr rs ec of te cossts of lt-vle : sle vle : efe o rtlly orere etrc sce te we rove cocece coo fe ot teores for two ybr rs er fferet cotrctve cotos ese teores ete eerlze very recet reslts tt c be fo [] y oters AM Mtetcs bect Clssfcto : 7 75 79 7 55 Keywors: Metrc sce rtlly orere set fe ot of orer lt-vle e wely ootoe roerty cotble s w I INRODUCION I recet yers tere s bee row terest sty te estece of fe ots for cotrctve s stsfy ootoe roertes orere etrc sces s tre ws tte by R Rers [6] were tey etee te fos Bc cotrcto rcle rtlly orere sets wt soe lctos to tr eqtos e sty of fe ots for lt-vle cotrctos s te sorff etrc ws tte by Nler [5] wo etee te Bc cotrcto rcle fro sle vle to lt-vle Lter y tors eveloe te estece of fe ots for vros lt-vle cotrctos er fferet cotos For etls we refer te reer to [ 6 7 8 9 ] te refereces tere Fe ot teory of sc s s lctos cotrol teory cove otzto fferetl clso ecoocs G-Bsr Lst [5] troce te cocet of cole fe ot rove soe cole fe ot reslts er cert cotos colete etrc sce eowe wt rtl orer ey le ter reslts to sty te estece of qe solto for eroc bory vle roble ssocte wt frst orer orry fferetl eqto Lter Lst C rc [] estblse te estece of cole cocece cole coo fe ot reslts to eerlze te reslts [5] Be Btt [] ve followe te tecqe of Bsr Lst rove soe cole fe ot reslts for lt-vle s rtlly orere etrc sces For ts rose tey troce eerlze e ootoe roerty for set vle II PRELIMINARIE root ts er wll be ostve teer wll be o-etve teers Frterore X wll eote o-ety set X wll eote te roct sce X tes X X Uless oterwse stte "for ll " wll e "for ll " resectvely 56
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 Defto A etrc o X s : X X R stsfy for ll y z X : y y ; y ; y y ; z y y z e lst reqreet s clle te trle eqlty If s etrc o X we sy tt X s etrc sce Defto [8] A trle X s clle orere etrc sce ff X s etrc sce X s rtlly orere set Defto [5] Let X be orere etrc sce X s s to ve te seqetl ootoe roerty f t verfes te follow roertes: If s cres seqece wt te for ll N If y s ecres seqece wt y y te y y for ll N Lst C r c [] troce te follow cocets for two sle vle s F : X X X : X X efe o rtlly orere set X Defto [] A eleet y X X s s to be cole cocece ot of te s F f F y y F y ; cole coo fe ot of te s F f F y y y F y Defto 5 [] e F s te e -ootoe roerty f F y s ootoe o-ecres ts frst ret ootoe o-cres ts seco ret tt s for y y X X les F y F y y y X y y les F y F y If s te etty we obt te Bsr lst s oto of e ootoe roerty of te F Defto 6 [] e F re clle w cotble f F y F y weever F y y F y e Rolá et l [] etee te revos otos by ef cocece ot betwee two s y ber of vrbles F rtto A B of tt s A B A B be s fro to tself We wll eote A B : : A A B B A B : : A B B A If X s rtlly orere sce y X we wll se te follow otto y A y y B Isre by te bove otto we eow te roct sce X wt te follow orer: For X we sy Also we sy tt re corble f or 57
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 Defto 7 [] Let X be rtlly orere set F : X X : X X be s We sy tt F s te e -ootoe roerty f F s -ootoe o-ecres rets of A -ootoe ocres rets of B tt s for ll y z X ll y z F Defto 8 [] A ot y F z s clle X fe ot of te F f F ; cocece ot of te s F f F ; coo fe ot of te s F f F eore [] Let X be colete orere etrc sce Let be -tle of s fro to tself sc tt A B s ertto verfy tt A B f B Let F : X X : X X be two s sc tt F s te e -ootoe A B f A roerty o X F X X cotes wt F Asse tt tere ests [ verfy F F y y y y for wc y for ll ose eter F s cotos or X s te seqetl ootoe roerty If tere est X verfy F e F ve t lest oe -cocece ot Let X be etrc sce CB X be te clss of ll oety close boe sbsets of X For A B CB X let A B s Bs b A were A bb A f A X s s to be sorff etrc ce by Let F : X X te ower set of X be set vle e X X y F y s sbset of X Defto 9 [] Let X be rtlly orere set F : X X CB X be set vle F s s to be e ootoe f F s orer-reserv orer-revers y e y y y X ly for ll F y tere ests F y sc tt for ll v F y tere ests v F y sc tt v v Defto [] A ot y X X s s to be cole fe ot of te set vle F f F y y F y Defto [] Let F : X X CB X : X X be ybr r of s A eleet y X X s clle cole cocece ot of ybr r F f F y y F y ; cole coo fe ot of ybr r F f F y y y F y Le [5] Let A B CB X > e for every A tere est b B sc tt b A B 58
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 e for every A Le [5] Let A B CB X > tere est b B sc tt b A B I or recet er [8] we ve te follow eftos for ybr r of set vle wt vrble sle vle Defto [8] Let X be rtlly orere set F : X CB X : X X be s We sy tt F s te e -ootoe roerty f F s -ootoe o-ecres rets of A -ootoe o-cres rets of B tt s for y y z X we ve y z F y F z tt s for y eleet F y v F z we ve v Defto [8] A ot X s clle fe ot of te s F f F ; cocece ot of te s F f F ; coo fe ot of te s F f F I Gor et l [9] troce te cocet of te e wely cres roerty of s rove soe cole fe ot reslts Defto [9] Let X be rtlly orere set : X X X be s We sy tt r s te e wely ootoe roerty o X f for y y X y y y y y y y y y y y y y y y y y y I [] te tors obte trle cocece coo fe ot reslts for two ybr rs cosst of lt-vle sle vle s er two fferet cotrctve cotos eore [] Let be etrc sce : : be s sc tt For ll were 7 re o-etve rel bers sc tt If s colete sbset of X te ve trle cocece ot Frterore ve trle coo fe ot f oe of te follow cotos ols for soe : re w- cotble l l l s cotos t v w ; b f s - eotet; c s cotos t y z l l l Isre by te reslts of Rolá et l [] Gor et l [9] te revos reslt of Ktb et l [] ts er we estbls - cocece - coo fe ot teores for two ybr rs ec of te cossts of lt-vle wt - vrble sle vle er fferet cotrctve cotos by s te oto of e wely ootoe roerty Or reslts rove ete ll bove reslts 59
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 III MAIN REUL Frst of ll we ve te follow eftos Defto Let X be rtlly orere set : X CB X be s We sy tt r s te e wely ootoe roerty o X f for y X Defto e s : X CB X : X X re clle w cotble f weever were s te set of ll - cocece ots of Defto e s clle eotet t soe ot X f Now we stte or reslts eore Let X be orere etrc sce be tle of s fro to tself for wc fe e for fe Let : X CB X : X X be s sc tt ve te e wely ootoe roerty o X X X X or or X X s colete sbsce of X Asse tt tere est o-etve rel bers [ sc tt < for oly corble eleets X Also sse tt X s te seqetl ootoe roerty If tere est X wt or te ve cocece ot X Frterore ve coo fe ot f oe of te follow cotos ols for soe X : re w cotble l s cotos t ; b f s eotet; c s cotos t l Proof Coser Us te e wely ootocty for yels e for we ve 6
te t s for we ve For by s we et Also by for we ve 5 Cot ts wy we c costrct seqeces X for wc 6 7 If for ll te fro Ily tt for ll ece re cocece ots of rs resectvely o we sse tt > for soe wc ves tt < < Now we ly Le te cotrcto coto to c sy tt for tere est sc tt for ll MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 6
8 Now we terce te role of to et 9 A over ll 8 9 ts ves A fro ce < MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 6
< te > tt s < s fro we obt By slr wy s bove ly Le te cotrcto coto yel tt for tere est sc tt for ll ll Also we ve 5 A over ll 5 ts ves 6 7 A fro 6 7 8 Fro 8 9 MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 6
Cot ts rocess we obt By reetely se of trle eqlty for every N wt > we obt ] [ ] [ l l l l ce < we cocle tt re Ccy seqeces X wc s colete te tere est X sc tt s Us 7 v te roertes o X te Fro we et O t lt s we et wc yels to lrly we ve As we ve s s cocece ot for te rs ose tt ols by cto we c rove tt for ll ce re w cotble te we ve for ll MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 6
o t s If for soe te we et for ll o t s Also s s cotos t tese ve l l Now we rove tt re ootoe seqece ecres or cres for ll ce te wc les 5 e by 5 6 A by ootocty for we ve 7 Fro 7 8 Cot ts wy we c et 9 erefore for ec s ootoe seqece coveres to te Now we ly wt s ece Coseqetly MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 65
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 lrly ose tt b ols ce s eotet for soe we ve Flly sose tt c ols ce s cotos t l te ece l for soe l Corollry Let X be orere etrc sce : X CB X : X X be s sc tt ve te e wely ootoe roerty o X X X X or y z or y z for y y z X X s colete sbsce of X Asse 7 tt tere est o-etve rel bers 7 sc tt < y z v w y v z w for ll y z v w y z v w 6 y z v w X were y v z w If tere est y z X wt 5 7 z y z y [ y y y z z ] or z y z y z [ y z ] e ve trle cocece ot tt s tere est y z X sc tt F y z y F y z z F z y Moreover ve trle coo fe ot e y z X sc tt F y z y y F y z z z F z y f oe of te follow cotos ols for soe y z v w X : re w cotble l l y v l z w s cotos t v w ; b f y y z z s eotet; c s cotos t y z l l v y l w z Proof Coser A s te set of o bers B s te set of eve bers Defe : by 66
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 ece Corollry follows fro eore te efto of trle cocece coo fe ot follows rectly fro Defto Rer Corollry s te orere verso of eore 7 of Ktb et l [] Note tt eore s ot sefl ere becse er revos coce for we ve s ot A B or A B eore Let X be orere etrc sce be tle of s fro to tself Let : X CB X : X X be s sc tt ve te e wely ootoe roerty o X X X X or or X X s colete sbsce of X Asse tt tere ests o-etve rel ber < sc tt for X were Also sse tt X s te seqetl ootoe roerty If tere est X wt or te ve cocece ot X Frterore ve coo fe ot f oe of te cotos b or c of eore ols Proof As eore we be wt roerty for to costrct seqeces for ll wle If te Ily tt se te e wely ootoe ece re cocece ots of rs resectvely o we sse tt > > Now we ly Le te cotrcto coto to c sy tt for tere est sc tt for ll 67
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 If te 5 e erefore Also for 5 6 tere est sc tt for ll Us 6 7 we et for ll 8 Coser for soe s fro 8 For N wt > we ve [ [ ce < we cocle tt re Ccy seqeces X wc s colete te tere est X ] ] 7 9 sc tt s Us te seqetl roerty o X ll wll ve 68
By t lt s we et ece Also we ve At we obt s s cocece ot for te rs ose tt ols by eore we ve ece Also we c et erefore s ootoe seqece coveres to te Now we ly wt 5 s ece Coseqetly lrly If b or c we c rove te estece of coo fe ot s eore Corollry Asse se yotess of Corollry bt relce Coto wt oter oe For < < w v z y w v z y w z v y w v z y e we ve te se reslts Rer Corollry s te orere verso of eore 8 of Ktb et l [] MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 69
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 REFERENCE [] M Abbs L C rc B Dov c M A K: Cole cocece coo fe ot teores for ybr r of s Fe Pot eory Al : [] M Abbs M A K Reovc: Coo cole fe ot teores coe etrc sces for w- cotble s Al Mt Cot 7 95 [] M Ars J A: O ltvle cotrctos coe etrc sces wtot orlty e cetfc Worl Jor Artcle ID 86 es [] I Be A R Btt: Cole fe ots of set vle s rtlly orere etrc sces J Noler c Al 79-85 [5] G Bsr V Lst: Fe ot teores rtlly orere etrc sces lctos Noler Al 656 79-9 [6] Co: Fe ots for ltvle s b-etrc sces Al Mt c 59 6 97-9 [7] L C r c : Mlt-vle oler cotrcto s J Mt Al Al 7 9 76 7 [8] L C r c M Abbs B Dov c R t: Coo fzzy fe ot teores orere etrc sces Mt Cot Moel 5 77-7 [9] M E Gor E Abrtbr Y J Co M Reez: Cole coo fe ot teores for e wely ootoe s rtlly orere etrc sces Fe Pot eory Al 95 [] N ss A Alotb: Cole coceces for lt- vle cotrctos rtlly orere etrc sces Fe Pot eory Al : 8 [] A Kewcroe A Kewo: Coo fe ots for sle vle lt-vle s G etrc sces It Jorl Mt Al 56 775-79 [] M A Ktb J A M Abbs M Ars: rle cocece coo fe ot reslts for two rs of ybr s Abstrct Al Al Artcle ID 879 Pes [] V Lst L C r c : Cole fe ot teores for oler cotrctos rtlly orere etrc sces Noler Al 7 9-9 [] G L: s-cocece s-coo fe ot teores for two rs of set-vle ocotble s etrc sce J Noler c Al 55-6 [5] Nler Jr: Mlt-vle cotrcto s Pcfc J Mt 969 75-88 [6] A C M R M C B Rers A fe ot teore rtlly orere sets soe lctos to tr eqtos Proc Aer Mt oc 5 5 [7] K P R Ro G N V Ksore P R Bb rle cocece ot teores for lt - vle s rtlly orere etrc sces Uversl J Cott Mt 9- [8] R A Rsw I M : N-Cocece Coo N-fe Pot for ybr Pr of Ms Prtlly Orere Metrc ces J cetfc Res Re 5 6 - [9] B E Roes: A fe ot teore for ltvle o-self- Coettoes Mt Uv Crole 7 996 - [] A Rolá J Mrtíez- Moreo C Rolá: Mltesol fe ot teores rtlly orere colete etrc sces J Mt Al Al 96 56-55 7
MAYFEB Jorl of Mtetcs - IN 7-69 Vol 7 - Pes 56-7 [] R K A r: Coo fe ot reslt of ltvle slevle s rtlly orere etrc sce Av Pre Mt 977-8 7