Comments
Description
Transcript
Dimension of Orders
Title On the Dimension of Orders Author(s) Hiraguchi, Toshio Citation The science reports of the Kanazawa University=金沢大学理科報告, 4(4): 1-20 Issue Date 1955-10 Type Departmental Bulletin Paper Text version publisher URL http://hdl.handle.net/2297/33759 Right *KURAに登録されているコンテンツの著作権は,執筆者,出版社(学協会)などが有します。 *KURAに登録されているコンテンツの利用については,著作権法に規定されている私的使用や引用などの範囲内で行ってください。 *著作権法に規定されている私的使用や引用などの範囲を超える利用を行う場合には,著作権者の許諾を得てください。ただし,著作権者 から著作権等管理事業者(学術著作権協会,日本著作出版権管理システムなど)に権利委託されているコンテンツの利用手続については ,各著作権等管理事業者に確認してください。 http://dspace.lib.kanazawa-u.ac.jp/dspace/ T h eS e i e n c eR e p o r t円 。 t h eK a n a % a w aUn C l ' s i t y ,VoLIV N, け 1,p p .l . . 2 0,O c t o r h e r,1955, 刊 ‘ Onthe Dimel1siou of Orders By T o s i oHIRAGUTI ( R e c e i v e dS e p t e m b e rH,1955) 1 . I ntroduction. Thep r e s e n tpaperi sasummaryo ft h ea u t h o r ' s two p r e v i o u so n e s[ 1 J,[ 2 J,r e v i s e c l and s u p p l e m e n t e d . Throughout t h es e q u e lt h e term “ o r d e r i su s e dI np l a c eo f the 円 term“p a r t i a lo r d e r " . I tseemst ot h ea u t h o rt h a t,s of a ra st h el i n e a re x t e n s i o nand t h ed i m e n s i o no fo r d e r sa r ec o n c e r n e d,i ti sc o n v e n i e n tt oc o n s i d e rano r d e ra sas u b s e t o fa C a r t e s i a np r o d u c t . Thuswehavet h ef o l l o w i n gd e f i n i t i o n s . e f i n e d on a s e t A we meana s u b s e tP o ft h ep r o d u c t A xA 1 . 1' . By an order d whichs a t i s f i e st h ef o l l o w i n gc o n d i t i o n s : 0 1 : ( ( x, x)i x, ==A}cP, 0 2 : ( x, y) c ==Pand (y, X)EP imply x=y, 0 3 : ( X, y)E 三P a nd (y, Z)EPiml りl y( x, z )εP. Byt h eabovec 1e f i n i t i o n{(x, x )I x, ==A}i si t s e l f an o r d e r onA,whichi ss a i danull o r d e r . By a l i n e a r order d e f i n e d on a s e t A we mean an o r d e r L whichs a t i s f i e s the c o n d i t i o n 0 4 : Fore v e r yx, yEA,e i t h e r( X, y)E L o r (y, X)EL . By t h edomaino fano r d e rwemeant h es e tonwhichthe o r d e ri sd e r r n e d . By an ordereds e tA(P) wemeanas e tA c o n s i d e r e dt o g e t h e rwithano r d e rP c 1i f i n e don i t . Wes h a 1 1u s et h eu s u a lt e r m i n o l o g i e si nt h et h e o r yo ft h eo r c 1e r e ds e t s,accompanied by "(P)" which may b ei n t e r p r e t e di nt h eo b v i o u sf a s h i o n . Forexample: な andy a r ecomparable(P)ηmeanst h a te i t h e r( X, y)EPo r(y, x)εP;X andy a r eincomprable (P)"meanst h a tn e i t h e r( x, y )己 P nor( y, x )E P,whichwi 1 lbea b b r e v i a t e dby乃和 (P)"; “ xprocedes(P)y . " meanst h a t( x, y )巴 P,b u tXヂy,which wiUb ea b b r e v i a t e db y“ Zく y (P)"; “ αisamaximal(P) elemento fA"meanst h a taく x(P) f o rnoe l e m e n t xEA; “ Bi sal i n e a r ( P )s u b s e to fA"meanst h a tB i sas u b s e to fA whichi sl i n e a r l yo r d e r e c 1 byt h eo r d e rP,e t c . 1 .2 . LetP b eano r d e ronas e tA andB as u b s e to fA. ThenP(B)={(x, YI(x, y) E P andx, yEB}i sano r d e rd e f i n e dont h es e tB whichi ss a i das u b o r d e ro fP onB. Thes u b o r d e ro fP onas e to fs i n g l eelement b w i l lbec l e n o t e c lbyP(b) ors i m p l y by b . WhenP(B) i sa l i n e a ro r d e ri ti ss a i dal i n e a rs u b o r d e randB a l i n e a r ( P )s u b s e ! o fA . 2 T . HIRAGUTI 1 . 3 . Byane x t e n s i o no fano r d e rP wemeanano r d e rQ d e f i n e dont h edomaino f P such thatP長 Q . An e x t e n s i o nL o f an o r d e rP i ss a i da l i n e a re : : . t e s i o no fP p r o v i d e dt h a tL i sa l i n e a ro r d e r . SES}be a system o fo r d e r sd e f i n e d on af i x e ds e tA. Thei n t e r s e c t i o n Let {PsI 什 SES Ps i sa l s o an o r d e r on A and eachPs i s an e x t e n t i o no fi t . The u n i o n USES PSi sn o t always an o r d e r,b u ti fe i t h e rPs~Ps' o rP S I長 Psf o revery theni ti sano r d e ronA andane x t e n s i o no feachPS' ES, 1 . 4 . Let{As i SES}beasystemo fp a i r w i s ed i s j o i n ts e t s,PSano r d e rd i f i n e donAs f o reachSESandQ ano r d e rd e f i n e dont h es e t5 . Then Us Es P, u{ (x"xs ' )! υ X s, EAs'ands くs ' ( Q ) } i sano r d e rd e f i n e dont h es e t Us ES A" whichi ss a i dt h eo r d i n a lsumo ft h esystem SES}a c c o r d i n gt ot h eo r d e rQ anddenotedby o fo r d e r s{PsI P s• 1 np a r t i c u l a r s the n u l l o r d e r on 5,t h e sum c o i n c i d e s with Us巳 sPs and i ss a i dt h e when Q i Theo r d i n a lsumo f五n i t enumbero fsummands c a r d i n a lsumanddenoted I . :s ES Al, A2' ・ ・, An according t ot h eo r d e ro ft h ei n d e xnumbersw i l lbedenoted Al十 A2→ . ..十 α beasystemo fo r d e r sPs dennεdona s e tA f o reachsE5,and L5. Let{Ps! F the s e to fa l lmappings o fS I n t o UsESAs such t h a t EAs f o r every sE5. Then {(f,f)[fEF}U { I f , gEFand (f( s ) , g(S))EPsforv e r y i sanorderd e f i n e dont h es e tF whichi ss a i dt h ec a r d i幻a l1 うr odudo ft h esystemand denotedby n 1 np a r t i c u l a r,正 ps P f o revery sE5, F becomes t h es e to f sE' S ニ n t oP. 1 nt h i sc a s et h ep r o d u c t 18 d e n o t ec 1 a l lmappingso fS i ps. I fa l l a r e i s o m o r p h i ct o af i x e do r d e rP,thent h ep r o d u c ti si s o m o r p h i ct ops。 1 . 6 . Let{Ps'sES},As andF meanthesamea si n1 .5andl e t W beaw e l l o r d e r .Then d e f i n e donthes e tS I f , g巴 F and り くg(σ) ( P a ) t h el e a s t ( W )e l e m e n t(f s u c ht h a tf( σ)ヂ σ)} i s an o r d e rd e f i n e d on F whichi ss a i dt h eo r d i n a lp r o d u c to ft h esystem{PsI SES} a c c o r d i n gt ot h ew e l l o r d e r W andd e n o t e dby f]wCs)ps' { u{ 2“ LinearExtensions of anOrder. Thef o l l o w i n gtheoremhasbeena l r e a d yknown [ 3 J . e tP b eano r d e rd e f i n e d0幻 α s e tA anda, banyt w oincom ρa r 2 . 1 . THEOREM. L a b l e ( P )e l e m e n t s of A,t h e nt h e r ee s i s t sal i n e a re x t e n s i o n EL anda l i n e a re x t e n s i o nL2s u c ht hαt( b, α) 2 ・ s u c ht h a t( αふ)EL1 Thistheorem w i l lbeg e n e r a l i z e da sf o l l o w s : 22 THEOREM. L e tP b e ano r d e rd e f i n e donas e tA andB anull-ordered(P) 白 Ont h eDimensiono lOrders 3 s u b s e t of A,的 t h e ω ? 幻zl h e ω r eι ? ι e X 工 1 d 叫x t μ 6 ω削 n t 岱s i o nL ofP 幻 s l t 叱 c ht 的 h α ωtL(B)~豆二 L. Proo f .Forevery (a, b )E三L(B) p u t P α b={( 二 τ, y)ixフYI=Aand ( x, a ), ( b, y )1三 P}, t h e 1 1Qニ pu(U(刊 )三 &(B) Pa b )i san e x t e n s i o no fP s u c ht h a tL(B)~ (,ì. Let~[ b巴 the s e to fa l le x t e n s i o n sQ o fP s u c ht h a tL(B)亘 ( , i . Byt h ef a c tj u s tmentioned2 !ヂ . ( ) and fWト i i fs p e c i f i e dby t h es u b s e t心 o 心 ={(Q , Q ' )I Q, Q'E2f andQC 二Q'} i sa 1 1o r d e rd e n 1 1 e don9 f . L et~ b eanyl i n e a rs u b o r d e ro f心 andZ l 'i t sdomain. By the remark s t a t ec 1a tt h e end o f1 .3, U(JE~} Q i s ano r d e r0 1 1A an1 ci sane x t e n s i o no f eachQE2,i . e .i ti sa 1 1upperbound(心) o fI B . Therefore,by t h eZ o r n ' s lemma,there e x i s t sa 1 1 element L i n 心 which has 1 1 0p r o p e re x t e n s i o 1 1 . Li sn e c e s s a r i l yal i n e a r o rc 1 e ronA. ThusL i sa l i n e a re x t e n s i o no fP whichc o n t a i n sL(B)a sa s u b o r1 c e r . I i n e a re x t e n s i o no fano r d e rP w i t hr e s p e c tt oane l e m e n ta wemean a Byaright l l i n e a re x t e n s i o nL s a t i s f i y i n gt h ec o n d i t i o n ( α i f aOx(P),t h e n( a , x )εL . Duallyal e f tl i n e a re x t e n s i o nL 'w ithr e s p e c tt oai sd e f i n e dbyt h ec o n d i t i o n 〔 戸 ) i f αチ x(P),then(x , a ) l = = L ヘ Wehavet h ef o l l o w i n gt h e o r e m . v e r yo r d e rP dξi 1nedonαs e tA α1'ldfo re v e r ye leme1'ltα己 三 A, t h e r e THEOREM. Fore a re x t e n s i o1'l with r e s ρe c tt oα . e x i s t ar i g h tl i n e a re x t e n t i o nanda l e f tl i1'le Prool .Put Alニ [ x :x己 A a1'ld ( x, α)ε P} , A:! ニ A~-A.l , {xIxl=Aand、(a, X)EP}, Aι 1 A --. f 1 g , andl e tLi b eal i n e a re x t e n s i o no fP(A;) f o reach i=1, 2, 3, 4 . Thent h eo r d i n a l sum A;1ニ 二二 (*) L=L]十 L2andL '=L. l+L3 a r er i g h tandl e f tl i n e a re x t e n s i o n so fP w i t hr e s p e c tt oα r e s p e c t i v e l y . One s e e se a s l l yt h a te v e r yr i g h tl i n e a re x t e n s i o nL and everyl e f tl i n e a re x t e n s i o n L 'w i t h1'e s p e c tt oane l e m e n t ahavet h eforms (ホ) 1'e s p e c t i v e l y . 1 'e x t e n s i o no fag i v e no 1 'd e1' whichi s1'i g h t( l e f t ) withr e s p e c tt oeveryelement Al i n e a 1 'd e1' i ss a i dright ( le f t ) withr e s ρe c tt ot h es u b s e t . o fas u b s e to ft h edomaino ft h eo 2 . 4 . THEOREl¥1. LetP b eanorderd e f i n e d0幻 αs e tA _ andB αs u b s e t ofA.There e x i s t s a right ( le f t )l i n e a re x t e n s i o1'l ofP withr e s p e c tt oB,i fandonly ifB i sa u b s e tofA . l i n eαr(P) s Proo f .LetL bea right( le f t )l i n e a re x t e n s i o no fP withr e s p e c tt oB,andassume t h a tb φ'(P) for some b,b'(三 B. Then we have both (b, b')EL a nd( b ', b}ccL ,hence ' which c o n t r ac 1 i c t sb 併' ( P ) . The1'e f o r ee i 七h e1' ( b, b ' )正 二P 0 1 ' ( b ' ,b)EP,i . e .B i sa bニ b l i n e a r ( P )s u b s e to f A. C o n v e r s e l yl e tB i s al i n e a r ( P )s u b s e to fA,ands p l i tt h es e tA t ot h ef O l l o w i n g: 3 p a i r 可.v i s ed i sj o i n ts u b s e t s : T .HIRAGUTI A Al-= E PforαI I bEB}-B, Ao={x! εP for幻obEB}, A2= A… (AlUAg). L e tL1andLgb eanyl i n e a re x t e n t i o no ft h es u b o r d e r sP(A1)andP(Ag)r e s p e c t i v e l y andJ e t t h a to fP(A2) c o n s t r u c t e di nt h ef o l l o w i n gmanner. For each element xEA2p u tB ぉ二 {b!bEB and ( x, b )εP(A2)},and d e f i n eab i n a r y r e l a t i o n"-"onA2byw r i t i n gx-y託 ando n l yi fBx=By. Ther e l a t i o nt h u sd e f i n e di s Sd e v i d e di n t oc l a s s e s{A I C 'EX},t h es e to f ane q u i v a l e n tr e l a t i o nbywhicht h es e tA2 1 , e i n gas u b s e to fA Asi se a s i l yv e r i f i e dt h es u b s e t { ( C ' , ヲ )I C ' , 哲 巴X r e p r e s e n t a t i v e sX b and Bc B o fX xX i sal i n e a ro r d e rd e f i n e d on X . Now l e t L, be any l i n e a r Q= 2• 長 7 j } e x t e n s i o no fP(Ac) f o reachC 'E XandL2=2 : .Q ( x )Lc t h eo r d i n a lsumo fL c ' sa c c o r d i n g t ot h el i n e a ro r d e rQ onX.ThenL2i sal i n e a re x t e n s i o no fP(A心 。 t h a t S i n c ei ti se v i d e n t l Sa l i n e a ro r d e ronA2' i t remains o n l yt oshowt h a tP(A2)亘L2・ Buts i n c e 2 : .Q 2 'i ts u i f i c e st oshowt h a tP(A, , )亘 2 : .Q(X) P(Ac ) .L et ( X lP(Ar ; )長 L EP(A2). I f x, YEAcf o racEX , then(x, y)EP(AI , ' )長工:QP(Ac ),I fxEAc,yε f o rd i s t i n c tC ' , ザ 巴 X, t h e nwehaven e c e s s a r i l y (ご々)EQ. Foro t h e r w i s et h e r ee x i s t saneJement b巴 B s u c h t h a tbEB ηbutb$B ; r ・yEAq i m p l i e s By B7 jヨb ,hence (y, b)EP(A2) byt hεdennition τ y . o fB t o g e t h e rwith ( 払 y)EP(A2),i m p l i e s(x, y)εP(A!),andh ence But m p l i e sBc=Bxヨ bwhichc o n t r a d i c t sb倍Bc. Byt h ed e n n i t i o n o nt h eo t h e rhandx巴 Aci 1 .4 ), ( 乙 平 )EQi m p l i e s(x, y)E2 : .Q(x)P(Ac ) . ThuswehaveP(A2)亘 o ft h eo r d i n a lsum ( 戸QCβ P(Ac ) . Nowl e tL Ll十 L2トL3bet h eo r d i n a lsumo fLl, L2andL , s ; thenL I sar i g h tl i n e a r e x t e n s i o no fP withr e s p e c tt oB . S i n c ei ti se v i d e n tt h a tL i sal i n e a ro r d e rd e f i n e d 二一 tremainso n l yt oshowt h a tP~ L. o nA,i Buts i n c ewehave 工:P(A;) P(Al)1 -P(A2 )1 -P(AI)長 Ll+L 1 -L' J , 2i ts u f f i c e st oshowt h a tP CL ;P(A , ) .L et( x, y)εP. I fx, yEAif o rsomeム(x, y)εP(A 〉 也 ニ ニ ニ 豆工:P(Ai). I fxEAi,yEAif o ri キj,wehaven e c e s s a r i l yi くj ,t h e n c e (x, y)巴2 : .P(Ai). I nf a c t : I fxEAaandyEAluA2,then (x, b )E P f o r nobEB and(y, b )E Pf o rsome bEB. But l a t t e r,t o g e t h e r with (X, y)EP,i m p l i e s (x, b)EP which c o n t r a d i c t st h e f o r m e r . I fxEA2andyEA1,then (x, b )三 P f o rsomebEBand(yt)E Pf o ra l lb E 三B . Butt h el a t t e r,t o g e t h e rwith (x, y)己 p,i m p l i e s( X, b)EPf o ra l lb己 B whichc o n t r a d i c t s t h ef o r m e r . Hence wehavei くj , I no r d e rt o show t h a tLi sr i g h t with r e s p e c tt o B,i ti ss u出 c i e n tt oshowt h a t ( b , x )ELwhenever (ふ的手 P f o ranyf i x e dbEB. But (X, b ) E 三P i m p l i e s xEA2リ Ag・ I f XEAH,e v i d e n t l y( b, x )ELs i n c ebEB長 ・ I fxEA2,wehaven e c e s s a r i l yXE 三Ac ,b Aflニ {XI( b, X)EPfor nobEB}, 刀 。 theDimension0/Orders A':~ ,~{ 叫 Cb , x);=, P v u f υ ,ra l lf l ' " " ,B } B , A ' : ! A--( A '1UA'a) , and l e tL '1,L ' a be any l i n e a re x t e n s i o n so fP ( A '1) and P(A'g) r e s p e c t i v e l y . Put 二 B1 ={bjbEBand ( b, x ) ; : = : P ( A ' : ! ) }f o reachXEA'2・ Thes e t A'2 w i l l be d i v i d e di n t o h ee q u i v a l e n tr e l a t i o n"-ビl e f i n e dbyp u t t i n gx~y i fa n do n l y c l a s s e s{A'ciごEX'}byt ♂ i f B'. r = B ' 1 I ' ThenQ'{(c , l ) ) 乙1 )εX' a n c l B'c 三 B'η }i s al i n e a ro r c l e ronX'. L e t Dc 1 命 be any l i n e a re x t el1s i ol1 o f P(A'I , )a n c lp u t L'2~~LQICC) D, ; t t hen L ' 2i s al i n e a r e x t e n s i o no fP(A'2) a n c lL ' L ' l→L '2+L '3 i sa l e f tl i n e a re x t e n s i o no fP withr e s p e c t 二 . t oB LetB a n c lB' betwos u b s e t so ft h ec l o m a i no fano r d e rP. B i ss a i c lo r d e r d i s j o i n t (P) u ρwards (doumwards) t oB 'p r o v i c l e c lt h a t( B 'xB)円 p=o( ( B B')円 P=O). 羽Then Ba n c lB' a r eo r c l e r c l i s j o i n tu p w a r c l st oeacho t h e r,t h e ya r es i m p l ys a i c lo r d e r d i s j o i n . t (P). Weh avethef o l l o w i n 只t heorema n c lthec o r o l l a r y . L e tP b eano r d e rdennedonas e tA andB andB 't 1 イ)0 l i n e a rぐP) s u b s e t sofA s u c ht h a tβ i so r d e r d i s j o i n t (P) up ω αr ds ( d oωnωαr d s )t o Bぺ Then 2 . 5 . THEOREM. i ol1 ofP ωh i c hi sb o t hr i g h t( le f t )w i t hr e s ) うe c tt oB a刀d t h e r ee x i s t sal i n e a re x t el1s l e f t( r i g ht )w i t hr e s ρe c tt oB ' . P r o o f . Let A . ;(i=1, 2, . ' 3 )a n c lA ' j(j=1, 2, B )b et h e same p a r t i t i o n so fA a si n the n c lp u tAij=A 1A l j . Then c o n s i c l e r i n gt h ec o n c l i t i o n that p r o o fo ft h el a s ttheorem,a (B、〆 B)np土仇 oneseese a s i l yt h a t 乱 A12~Alg A22~A2:FO Hencewehave cA21, Al=A1 ] , A2Ag A:nuAH2UA a s . A' IコAlluA21uA:n, A'2 A : l 2, A'a=A a g . Let Ll Lll,L:n a n c lL ' Hcc Lm: b eanylineaγe三t e n s i o n so fP(A1)P(All),P(AmJ a n c l P(A'a) PCAgg) r e s p e c t i v e l ya n c lL : ! L立1,L ' : ! LW2t h er i g h ta n c ll e f tl i n e a r 1 )a n c l P(A'2), P(A32) r e s p e c t i v e l yc o n s t r u c t e c li n the e x t e n s i o n so f P(A2)ニ P(A2 て 二 二 samemannersa si nt h ep r o o fo ft h el a s tt h e o r e m . Thena si se a s i l ys e e n Lg=Lal+L日立ト L日日 a n c l L ' 1=LllトL21+L:n a r el i n e a re x r e s i o n so fP(A : l )a n c lP(A, , )respectively. Nowput I ,cLn十L 21トLg l+Lfl2+L : m, thenwehave L=L1j L2+L , ? =D] トL '2+L' 日 which shows t h a tL i sr i g h tw i t hr e s p e c tt oB a n c ll e f t with r e s p e c ttoB' The r o v e c lc l u a l l y . remainingp a r to ft h etheoremmaybep 2 . 6 . COROLLARY. L e tP b eano r d e rd i f i n e donas e tA ωz dB al1d B' linear(P) 圃 s u b s e t sofA whicha r eo r d e r d i s i . o i n t C P ) . Thent h e r ee x i s t sal i n e a re x t e n s i o nwhich うe c tt oB andl e f tωi t hr e s ρe c it oB ',anda l il1e a rc x t e n s i o n which i sr i g h twithr e S 1 i s1 c f twithr e s ρe c tt oB andr i g h tωi t hr e s p e c tt oB ' T .HJRAGUTI 6 3 . Dimensiono fOrders. e a l i z e ro fano r d e rP wemeanas y s t e m{L , I SES}o fl i n e a re x t e n s i o n s 3 . 1 . Byar o fP s u c ht h a t nsL , =P,and by a minimalr e a l i z e r ar e a l i z e r{Lt:t ιT}s u c ht h a t TI o fi t si n d e x s e ti sl e s st h a nt h a to fe v e r yr e a l i z e ro fP . Byt h e t h ec a r d i n a l i t yI dimension o f an o r d e rP we mean t h ec a r d i n a l i t yo ft h ei n d e x s e to fi t sminimal r e a l i z e r . The d i m e n s i o no f aI 1o r d e rP i sd e n o t e d by D[P]. I ti se v i d e n tt h a ta system{L , I SES}o fl i n e a re x t e n s i o n so f an o r d e rP i sar e a l i z e ro fP,i fando n l yi f ,f o r any i n c o m p a r a b l e ( P )e l e m e n t s xandy,s , S'ESs u c ht h a t( x, y)EL , t h e r ee x i s t s and(y, x)εL " . By t h i s remarkandt h e theorem 2 . 1,e v e r yo r d e rhasar e a l z e rand hencet h ed i m e n s i o n . E v i d e n t l yt h ed i m e n s i o no fal i n e a ro r d e ri s 1andt h a to fan u l l o r d e ri s2 . The f o l l o w i n g theorem g i v e s an e s t i m a t i o no fthed i m e n s i o no fo r d e r sd i f i n e dona f i x e ds e tA. The dimension 0 /anorderP d o e sn o te x c e e dt h ec a r d i n a l i t y0 / i t sdomainA,i . e .D[P]三I A I . 3 . 2 . THEOREM. P r o o / . Let Lx,f o r each e l e m e n t xEA,ar i g h tl i n e a re x t e n s i o no fP w i t hr e s p e c t t ox . Then t h e system{L xEA}i sar e a l i z e ro fP,s i n c e( x, y )巴 Lyand( y, x )εLy yI f o ranyincompar able(P)e l e m e n t sxandy . The e s t i m a t i o ng i v e n above i sn o tt h es h a r p e s t . Thes h a r p e s tcnew i l lb eg i v e ni n s e c t i o n8 . Thel a s ttheoremw i l lb eg e n e r a l i z e da sf o l l o w s . 3 . 3 . THEOREM. L e tP b eano r d e rd e f i n e donas e tA and{A, I SES}b eas y s t e m 0 /pairwised i s j o i n tl i n e a r ( P )s u b s e t s0 /A. ThenD[P]<IA-UsA, i+I S1 . P r o o / . Let L f o r each e l e m e n t xEA-UsA" a r i g h tl i n e a re x t e n s lOn o fP w i t h o reachelementSES,al e f tl i n e a re x t e n s i G no fP w i t hr e s p e c t r e s p e c tt o x and L" f t oA. , Thenthesystem{LxlxEA-~nsA , }u{L, :SES} i sanr e a l i z e ro fP . 口 L e tP b ean o r d e rd e f i n e d on as e tA and{A,,!SES}as y s t e m 0 / 1 うa i r w i s ed i s j o i n tl i n e a r ( P )s u b s e t0 /A satisfyingA=UsA. , ThenD[P]主I S i・ 3 . 4 . COROLLARY. 4 . Dimension(lfSuborders. 4 .1 . L et P be ano r d e rd e f i n e donas e tA andB as u b s e to fA. ThenD[P(B)] 手D[P] whereP(B) i st h es u b o r d e ro fP onB . Thus i f some e l e m e n t sa r ed e l e t e d fromt h edomaino fano r d e r,t h ed i m e n s i o nd i m i n i s h e si ng e n e r a l . I nt h i ss e c t i o nt h e amount o ft h ed i m i n u t i o nc a u s e d by t h ed e l e t i o no fe l e m e n t sw i l lb ee s t i m a t e d . We s h a l lb e g i nwithshowingt h a tt h ed i m i n u t i o nc a u s e dbyd e l e t i n gane l e m e n ti sa tmost1 . 4 . 2 . THEOREM. L e tP b e an o r d e rd e f i n e d on a s e tA andaane l e m e n t0 /A. ThenD[P]三D[P(A-a)J+l. lsES}be a minimal r e a l i z e ro fP(A-a),ands p l i tt h es e tA-at o P r o o / . L e t { L ', t h ef o l l o w i n gs u b s e t s : Ont h eDime 幻' s i o nofOrders 同 i ,( X, α) 三 ,P}, A1=xr;=A--a Ago {xI x ; : : : : A a フ ( a, x), =P} , τ A2=(A-a)--(A1uA; c : ) . Takeane l e m e n t σ ; : : : :S a n c l f i xi t . Put L20 0 L 'η(A2uA日 〉 司 L'1=α+L'η(A;l ) ' L1 D σ (Al): →仏 二 L白 c = L 'σ(AluA2), ThenL=L11 -L2a n c lL ネ c~L日-+ L 4 .arer i g h ta n c ll e f tl i n e a re x t e n s i o n so fP withr e s p e c t t oα r e s p e c t i v e l y . = : =S(J p u t Foreachsr Ad={X x ; : : : : Aand (X' , xly~L's forsomexlr : = :Al}, A.dニー (A- -As 1・ , Then L =UsL 九 : ;{ ( a , a)}u{ (x, a ) xr = : =A, } }U { ( a , x)IXEAs2} i sa J i n e a re x t e n s i o no fP a n c lt h e system{ L "I s ; : : : :S-σ }lJ{L, L*} i sar e a l i z e ro f P. Hencewehave r 斥 D[PJ三IS-O' +2=! S I+l=D[P(A-a)Jー トL Thel a s ttheoremshowst h a ti fnelementsa r ec l e l e t e d from t h ec l o m a i no fanorder ng e n e r a l,byn . But when the c l e l e t e c le l e m e n t ss a t i s f ya t h ec l i m e n s i o nc l i m i n i s h e s,i p a r t i c u l a rc o n d i t i o nt h ec l i m i n u t i o nw i l lbe l e s s e n d . 4 . 3 . THEOREM. L e tP b e an order d e f i n e d on a s e t A and a a minimal(P) faチ b(P),t h e n D[Pl三D[P']ト ー1 . P' element and b a maximalCP) e l e m e n t ofJL I b } . beingt h es u b o r d e rofP ont h es e t A' A-{α, と P r o o f . S p l i tt h es e tA 't ot h ef o l l o w i n宗 s u b s e t s : Al= . { x !x l = = A ',xゆ α( P)and ( x, b )竺 P}, A2~' { x :x = = : A ',xチ α(P) αndx併 b(P)}, Ag={xIx : = = A ',( α, x ), ==P仰 1 . dxφCP)}, a n c ll e tL ;b e al i n e a re x t e n s i o n0: 1 ' P( A, )f o reachi~ 1, 2, 3 .Then L~Ll ト b+L2 ト α :-L3 i sal i n e a re x t e n s i o no fP which i sl e f t withr e s p e c tt o aa n c lr i g h tw i t hr e s p e c tto b . Nowl e t{ L ' sI S ' e 三S }b eam i n il11a lr e a l i z e ro fP ' . Then f o reachs ぞ S ,Lsc =a+L ' s斗 b sr i g h twithr e s p e c tt o aa n c ll e f tw i t hr e s p e c tt o b, i sa l i n e a re x t e n s i o no fP whichi ヲ a n c lt h esystem{ L si SES}uL i sa r e a l i z e ro fP. Thereforewe have D[PJ<D[P'J十 1 . I ti s noteworthy t h a tt h ec l i l 1 1I n u t i o no ft h ec l i m e n s i o nc a u s e c l by d e l e t i n gal i n e a r ,whateverthec a r c l i n a l i t yl11ayb鳥 i sa tmost2 ,asthe: 1 ' o l l o w i n gtheorems h o w s . s u b s e t 4 . 4 . THEOREM. L e tP b e an orderd e f i n e donas e t A andC α linear(P) S 1 1 .b s e t ofA. ThenD[PJ~D[P(A-C)J 斗 2. Proo f . Let {L's s==:S}be a l11inimalrealizerofP(A-C). Foreachelementc==:C, 'i p u tUc {11.'U, = =A-C and( u , c )ιP}. O b v i o u s l yU/ , ; ; ;Uc f( c, c ' ) = = : P ( C ) . Forg i v e nSES ニ T .HIRAGUTI 8 and CEιput e i t h e r Asc τ :xEA~C and ιL ' s for some uζ U o rA s c c} 二 ο a c c o r d i n ga s UcヂD O r Uc 0,andp u tA*sc --Aw Then Ls= L 'suP(C)u{(克丸, ο C)icECα 削?幻 z d克 εA s c }リ { ( 化C , 幻 X) ¥ C三 C ω αndx巴 A木 ヘs i sal i n e a re x t e n s i o no fP. Now l e t Ll b ear i g h tl i n e a re x t e n s i o no fP withr e s p e c t 二 二 刊C e f tonewithr e s p e c tt oC . Then t h e system { L s 'S E C C S }u{Ll, Lui sa t oC andL2a l S Iィ 2=D[P(A-C)J+2. r e a l i z e ro fP. HencewehaveD[PJ三 i Tov e r i f yt h a tL ss a t i s f i e st h ec o n d i t i o n s0 1,0 2,0 4i sn o th a r d . To v e r i f yt h a ti t s a t i s f i e s0 3l e t (x, y),(y, z)ELs ' Therea r ethef o l l o w i n g8c a s e s : ( 1 ) x, y, zEA-C. Then ( x, y ),(y, z)EL'"hence ( x, z)EL's長 L s . ( 2 ) x, y, ZEG . Then (x, y ),(y, X)EP(C),hence( X, Z)EP(C)亘L s . ( : 3 ) x , yEA~C andZEC Then ( X, y)EL'sandyEA szthat i s (y, u)ELsf o r some 吻 UE グ x, u)EL'sf o rsomeUEU h a ti sXE 三A s z . Hencewehave ( x, z)ELs ' Therefore ( zt ( 4 ) x, zEA-CandyEC Then ( x, z )εL ' s and xEAs'l t h a ti s( x, u)EL ,f o rsome ' 幽 UEU ' I a ndzEAヘ 1 t h a ti s( u, z)EL ,f o ra l luE Hencewehave ( x, z)EL . , ' ( 5 ) y, zEA-CandXEC. Then ( y, Z)EL' sandyEAへ 必 固 Hence when U a 勾J ,( u, y ) ヘ ELsf o ra l luιwhichi m p l i e s( u, z )EL ' sf o ra l lu三 Uxt h a ti szEA ZE三 A~C=A へ x A⑤ When Ux s i n c eAsx-e . Thusi ne i t h e rc a s eweh a v e ε L s . EL' ,f o r some uE ( 6 ) x c三A-Cand y, ZEG . Then xEAsv t h a ti s hence ( ) , 手 ( ; ) , EL'sf o r someuEUzwhichi m p l i e s (x , z )巴 L s . f ( 7 ) yEA-Ca ndX, Zさ三 C . ThenyEAぺ andyEAszt h a ti s( u, y)EL o ra l luEU x S f and(y, μ)EL 二f o rsomeUEU . Assumingt h a t z 長 Ux EP(C) and克ヂ ZwehaveUz whichi si m p o s s i b l e . Hence ( X, Z)EP(C)長 L s・ ( 8 ) zEA-Ca ndX, yEC ThenU, 長U ! リ a ndzEAネ ザ When Ux ( ) ,Uyy()andwe 守L have ( u, z )EL ' sf o ra l l uEUy ,a f o r t i o r if o ra l l uEUx . Therefore zEAネ 目 。 When ω ヘ ο ,e v i d e n t l yzEA x .s i n c e O . Hence i ne i t h e rc a s ewehave ( x, z)ELs ' I no r d e rt ov e r i f yt h a tL si sane x t e n s i o no fP,l e t(X, y)EP. Therea r et h ef o l l o w i n g t h r e ec a s e s : 二 ( 1 ) x, yEA--C. Then ( x, y )εP(A-C)長L日 五L s . ( 2 ) XE 三A-Ca ndyEC. ThenXEU",hencexEAs! l w hichi m p l i e s( x, y)ELs ' ( 3 ) yEA-Ca ndXEC. I fUx O , 守L E Pf o ra l lUEUxwhichi m p l i e syEA*口・ I f Ux='O ,e v i d e n t l yyEAヘ x ' Hencei ne i t h e rc a s e( x, y )EL s . 45 . THEOREM. LetP b eanorderd e f i n e donas e tA andClandC2 b etwol i n e a r 缶 (P) s u b s e t sofA whicha r eo r d e r d i s j o i n t ( P ) . Then [PJ主D[P(A-CI-C2)]ート乙 Proo f . Let{L"sisES}beaminimalrealizerufP(A-CI-C2). LetL's bealinear e x t e n s i o no fP(A-C1) c o n s t r u c t e dfromL "si nt h esamemannera sL swasc o n s t r u c t e d 's i nt h ep r o o fo ft h l e theorem 4 . 4 . Thenl e tL s beal i n e a re x t e n s i o no fP from L c o n s t r u c t e dfromL's i nl i k emanner. Ont h eo t h e r hand s i n c eC1 andC2 a r eo r d e r d i s j o i n t (C) t h e r ee x i s t sal i n e a re x t e n s i o nb o t hl e f t with r e s p e c tt o Clandr i g h tw i t h r e s p e c tt o C2,and a l i n e a re x t e n s i o nb o t hr i g h t with r e s p e c tt o Cl andl e f tw i t h tb eLlandL2r e s p e c t i v e l y, Thent h esystem{Ls'sES}u{L1, L2} r e s p e c tt o C2. Leti Ont h eD u n e J t s i o nofOrders 9 C 2 ) ]ート乙 i sa r e a l i z e ro fP. HencewehaveD[P]三cD[P(A Cl- 4 . 6 . LetP bean o r d e rd e i i n e d on a s e t A and a, b two d i s t i n c te l e m e n t so f A. When ( a, b)EP ,( a , x)EPa 1 1 仁1( x, b)EPf o rnoxEA,i ti ss a i dt h a tbc o v e r s ( P )αora ィt i v e ( P ) and d e n o t e d by α ( :b)EP. When α ( :b)EP,t h ep a i ro f and b a r ec o n s e c z n t s a andbi sd e n o t e d by α (: b ) . Ap a i rα (: b )i ss a i do frankn i f c o n s e c u t i v ee l el11e YEA s u c hthat( x: b )εP,α ( :y)EPandx併 > ) I( P ) . t h e r ee x i s tn p a i rso felement sx, Wehavet h ef o l l o w i n gt h e o r el11. 4 . 7 . THEOREM. LetP b eanorderd e f i n e donas e tA and ( α . 'b ) ap a ir0 /consecu,t i v ee l e m e n t s0 /rank0or1 . ThenwehaveD[P(A-a-b)]十 I 圃 P r o o / . Let .¥f={L'sisE5} b eam i n il11a lr e a l i z e ro fP(A-a--b). Whenα (: b )i so f h o o s ea r b i t r a r i l yanelement o f) 1 " a ndl e ti tbeL二 . Whenα (: b )i so frank 1 rank0c t h e r ee x i s t sas i 昭 l ep a i rx0 andY os u c ht h a t( ι: b ),( α : J ω E P and xφ > ) Io ( P ) . Hence t h e r ee x i s t sanelemento fs rwhichcontains (xo>Yo)asanelement. Leti tbea l s oL'σS p l i tt h es e tA'=A--a-bt ot h ef o l l o w i n gf i v ed i s j o i n ts u b s e t s : xEA'and ( x, a)EP}, Al {x. ニ A2={xlxE 三A ',xチ α(P)and ( x, b )を 三 P} , A:lごご {xIXEA',xチ α(P)andx併(P)}, ふ'し Dl u A4一 { x :xEA ' , ( a, x)EPand xチ b(P)}, A" {xI xEA' and ( b , X)EP}. A12: l=AlU A2U Aぉ L12sc c L ' σ (A12日 ) , A : 1 4 ,,=A日 UA4 ,uA" , L;w;~L' σ (A 舟 4 ,,), and Li=L'σ (Ai ) f o r i=, 12, . 3 4, 5 . Then L =Ll十 α[ L2トb [ LB sar i g h tl i n e a re x t e n s i o no f P withr e s p e c tt o{ ムb } 4 5i 寸 ! L4+bィL"i sal e f tl i n e a re x t e n s i o no fP withr e s p e c tt o{ a , b } . andLネ コ L1 2 3 a Nowl e tL"f o rsE5σ,b et h el i n e a re x t e n s i o no fP c o n s t r u c t e dfromL 'sinthe .L 1f o rt h ec a s e where C= α (: b ),then samemannera si nt h ep r o o fo ft h etheorem4 , L*}i sa r e a l i z e ro fP.Hencewehave t h esystem{Ls SE5-a}u{L D[PJ三 5 σ . 1 ト2ニ 151+1ニ D[P(A-a-b)]+1 1 5 . D-reducible Orders. 5 .1. Let P be an o r d e rd e f i n e d on a s e tA . A s u b s e tBo fA i ss a i dt obed removablep r o v i d e dP[D(A-B)]=D[P]. Ano r d e rP i ss a i dt ob ed r e d u c i b l ei fthere e x i s t sa tl e a s ta d-removableelementi ni t sd o m a i n . One s t i m a t i n gt h ed i m e n s i o no fa g i v e no r d e r,i e ti fi t tw i l lb eo f t e nc o n v e n i e n tt od e l e t ebeforhandt h edremvables e x i s t s . As a c r i t e r i o nf h ef o l l o w i n g very o rt h ed r e m o v a b i l t yo fa s u b s e t we have t comprehensivetheorem. ← T.HIRAGUTI 1 0 5 . 2 . Let P b e an order d e f i n e d on a s e t A and B as u b s e t THEOREM*. A s a t i s f y i n gt h efollowingc o n d i t i o n s : 10 /ft h e r ee x i s t sanelemenl xEA~B 5 u c ht h a t( x, b)EP for an element bEB, t h e n( x, b )ιP fora l lelementsbEB. 20 /f t h e r ee x i s t s an element xEA~B s u c ht h a t( b , X)EPforanelementbEB, t h e n( b, X)EP fora l l elementsbEB. ThenB i sd-remvablee x c ψta tmostan element b c h o s e na r b i t r a r i l y ),provided o( D[P(A')]孟 D[P(B-bo)J ωh e r eAニ A-(B-bo ) ー P r o o f . Let s !sE5} b e a mimimal r e a l i z e ro fP(A') and ニ { L 't tE 1 ) S i n c eD[P(A'江主 D[P(B-bo)J meanst h a t[ 5 i 三ITi,theree x i s t sa t h a to fP(B-bo ー n t o mappIngo f ~j' 1 o Ls=L ' , 圃 Leti tbef .Then 'xEA',yE三B--b 。αnd CX, bo)EL's} {(x, y ) xEB~bo , yEA'and フ EL', , } s)u{(ぁ リ 1 i sal i n e a re x t e n s i o no fP f o reachSE5 and t h e system 町二 1 SE i sar e a l i z e ro f p HencewehaveD[PJ=D[P(A.')J. 園 I no r d e rt ov e r i f yt h a tL ss a t i s f i e s0 3l e t τherearet h ef o l l o w i n g8c a s e s : 1 ) x, y, zEA', 2 ) x , y , zEB~bO' B ) x, yEA'; ZEB~bm 4 ) ;xEB~b", 5 ) x, zEA'; yEB-b 6 ) XEA';y , z ι B~bO) ラ O) 7 ) 8 ) ZEA'; x, YEB-bo ' x , zEB~bO) I f1 )o r e v i d e n t l y( x, z)EL . , I f3 ),we have (xTo)EL'swhichi m p l i e s ELs. I f EL's and (y, bJε1 : " hence wehave (y, z ), εL ' s hence EL' , whichi m p l i e s( x, z)EL 'I f wehave ( x, b )( bOfz)EL'"hence ( X, Z)EL 's 長 L. ,I f s o ( x, y)EL m p l i e s EL 'swhichi m p l i e s( X, Z)EL 's 'I f wehavey=bos i n c e( b o, Y ), si (y, b )εL ' Hence ( 克 点 。 )EL シ and (bmZ)ELs which impliy EL's亘 園 I f8 ), o (y, z)EL m p l i e s si EL ' . , whichi m p l i e s( x, z)ELs ・ That L ss a t i s f i e st h ec o n d i t i o n s0 1,0 2and0 4maybe al i n e a ro r d e ronA . .i s v e r i f i ε d . T h e r e f o r eL I no r d e rt ov e r i f yt h a tL si sane x t e n s i o n,l e t( X, y)EP. Therea r et h ef o l l o w i n 広4 c a s e s : 2 ) XEA ' , yEβ bM 1 ) X, yEA ' , 3 ) yEA',XEB I f 4 ) x , yEB~Bo ・ e v i d e n t l y( x, y)EL'sC ; ; ;L " . I f2 ) byt h ec o n d i t i o n1,n e c e s s a r i l y( X, bo)EP. Hence 0 ヲ have we, (丸 bo)E L'~ which i m p l i e sC X, Y)ELs. I fB ),we have (bMy)EPfo 二L ' s which i m p l i e s ( 丸 I f4) ヲ (x , y)Ef( L' J~Ls' 1 no r d e rt ov e r i f yt h a t佼 i sa問 a l i z e ro fP,l e txoy(P). I fX, yEA ' , s i n c ex骨 ( P ( A ) ), ( x, y )巴 L F S 4 二Lsa n d ε L 午長L s 'f o rsomes , s'E5. I fxEA' and yEB bパ n e c e s s a r i l yxゆ *(6.4) (6.5) (6o )( o . 8 ),( o .り )a n c l( o . 1 0 )i n[ 1 1a r ea l ls p e c i a lc a s e so ft h i st h e o a m . 亀 ヲ 守 wehave byt h ec o n d i t i o n s1 and2 ぺ Thereforet h e r ee x i s t5, S 'ES s u c ht h a t 0 ‘ On t h eD i m e n S l : 0 1 i0 1Orders 1 1 C x, b u ) e = I :sand ( b o, y) r = I :s ence (x, y)e =Lsand (y, x)巴 Ll , ・Ifx, ye =B-ba> t h e r eef{i s t 行 h t , t 'E Tsucht h a t (x, y)EL't and (y, x)e = cL ' t l . But s i n c et h e r ee x i s ts , s"E5 s u c ht h a t L't=f(L'J andL ' t l = f C L ' t h e r ee x i s ts , s'E5aucht h a t (x, y)EL ,and (y, x)EL". 0 a I t must be n o t i c e dt h a tt h es e tB s a t i s f y i n gt h ec o n c l i t i o n s1 nd20 i sn o talways removablea saw h o l e . 6 . Dimension o f theSumof Orders. 6 . 1 . THEOREM. Let Q b e an order d e f i n e d on a s e t 5,{AsIsE5}α systemof ρazrωi s ed i s j o i n ts e t s,As an order d e f i n e donP andσ anelemeniof5 suchthat S ]と D[PsJlora l l sE5. Then D[Pσ LQ (s)P, J=Max (D[PoJ,D[QJ ) . D[QJ亘D[, Proof. L et B b eas e to fr e p r e s e n t s t i v e ss l e c t e dfromeachs e tA s . Thens u b o r d e r LQ (心 PS ont h es e tB i si s o m o r p h i ct oQ. T h e r f o r eD[QJ主 D[, LQ C s ) ps ] ' o f, Let S tニ{Ltt ET} b e a minimal r e a l i z e ro f Q and S P . , ニ {LtCs)!t ( s )ET, }t h a to fP, f o reachsE5. S i n c eD[Pr;J 与D[Ps J meansITa与!1 ' . , 1,t h e r ee x i s t samappingfso f ' ls . L et { I . , ;s 巴5 }b e a system o fs u c h mappings where, . 1 may b巴 chosen S l 'o n t oi 1 ! u tf o rσleti t七et h ei c l e n t i c a lmapping. For each t (σ〉εTa and a r b i t r a r i l yf o r5ヂ σ,b e tLt, t ( aニ エr , t ! J L f C a ) ),b et h esumo fI . s ( L t Cσ ) ) ' 8a c c o r d i n gt ot h el i n e a r f o reach tETl . Then L t t c,,) i s al i n e a ro r d e rc l e f i n e don UsAs a n c lal i n e a re x t e n s i o n o r c l e r Lt on5 nf a c t ,c o n s i d e r i n g, . 1(Lt ( σ ) )ニム ωforsomet(s)ET"Ltω2Ps andムヨ Q, o f ~Q(FJ)Ps' 1 wehave Lf s I s ( L f cσ))U{(X, y ) 1広 三 . A , , , ye = . A , , , and ( S, 5')ELt} 司氏。=しI コL JsPsu{(x, y) X εAs> yEAs ,and ( s 〆)EQ} i ニミ Q ( S ) ps 圃 I f D[Pσ ]乙D[Q],t h e r ee x i s t s a mapping o fTσonto T. Let i t be ヂ ラ t h e ns e [ニ { L r p C t Cσ ) )f (σ )t σ ()ETo}i sa r e a l i z e ro f, LQ ( s ) p , , ,andwehaveD[, LQ く の PJ=D[P]. I f D[Pσ ]く D[QJ,t h e r ee x i s t s a mapping o fT o n t o Tσ Let i t be ' 1 /,t h e n the , ψω I tET} i sar e a l i z e ro f, LQ く のP" a ndwehaveD[, LQ C s ) P s Jニ D[QJ. system 舵II={Lf I no r d e rt ov e r i f yt h a t1 . ¥ ' 1i sar e a l i z e ro f, LQ (め P" l e tx> 1 y(P). I fx, yE. A "f o rsome ,thenx> 1 y(P , ) . Hence t h e r ee 丈i s tt ( s ),t '( s )巴 T,s u c ht h a t (x, y)EL心 Jニf, CL ( σ ) ) sE5 t ) a n d x)ELt, ( s )ニfs CLt'(σ ) ) . Therefore ( ι y)ELHC ( y , x ) E L t . f / C σ ) f o r a l l t E T . 1 n and (y, σ p a r t i c u l a r (x, y ) E L r p C t (σ ) )( σ ) and (y, x)EL r p C t ' Cσ ) ), t ( σ ・ 〕 I f xEAs a n c l y( e =A" f o rs イ 可5 , ' f t h e ns ピ チ (Q). Hencethereexistt , t '巴 T s u c ht h a t( 5, 5 ' )巴 L t and ( s ', s)EL, ・ L e t t=ヂ ( t (σ ) )a n c lt 'ニヂ(t' ( σ ) ) . C o n s i d e r i n gt h a tt h ec l o m a i n so ff s ( L t (σ ) ) andf , (Lf ω)areAs andA s 'r e s p e c t i v e l y ,we have (x, y )εL r p(tc σ ) ) ' f C引 and (y, x)巴 L c p C t l Cη))及川σ ・ 〉 That 佼I Ii s フ , 圃 吋 ar e a l i z e ro f, LQ ( S ) psi sv e r i 良e ds i m i l a r l y . Thef o l l o w i n gtheoremi sanimmediater e s u l to ft h etheorem6 . 1 . eanorderd e t i n e donA =UsAs which i s decomtosablet o 6 . 2 . THEOREM. Let P b asum, LQ (8)P" Q andP,beingordersde花nedon5 andAsr e s t e c t i v e l y,αndσb ean element of 5 s u c ht h a t D[Pσ ]乙D[Ps J fora l l sE5. Then A-Aσ i s d-removable T .HlRAGUTI 1 2 ρr o v i d e dD[Pσ ]乙D[QJ,andA-{f ( s ) SES}i sd r e m o v a b l eρr o v i d e d D[PaJ>D[QJ, fb e i n gafunctionwhichs e l釘 t so n ee l e m e n tfrome a c hs e tA s . 1 h ef o l l o w i n gc o r o l l a r y . Asap a r t i c u l a rc a s eo f6 . 2,wehavet 6 . 3 . COROLLARY. L e tP b eano r d e rd i f i n e donas e tA. lfai sane l e m e n tofA う a r a b l e ( P )t oe a c he l e m e n t of A,t h e nai sd r e m o v a b l e . ln 1 り a r t i c u lar,t h e c o m l g r e a t e s t ( P )e l e m e n tandt h el e a s t ( P )e l e m e n ta r ed r e m o v a b l e . 7 . SomeExamples. 1 nt h i ss e c t i o nsomep a r t i c u l a ro r d e r sw i l lbes t u d i e d . L e t X={xiliEl},Y=:{y IiEI }b et w od i s j o i n ts e t swhere1i st h es e tofa l l 7 .1 i n t e g e r s,andl e tW b et h eo r d e ronXしJY s p e c i f i e db y 色 , ;X i )1i εI }U{ ( Y i ' Y i )IiEl} W=={(X , ;Y βliEl}u{(的 + l ' Y i )1i E l } . U{X ThenD[WJ=2. P r o o f . LetL iandL 'i bel i n e a ro r d e r sont h es e t s{x什 l ' Y i }and{九 y ; }r e s p e c t i v e l y s p e c i f i e dby and L.={( 約十 1 ,X i + 1 , )( Y i ' Y i ) '( お +l, Y i ) } L ' i = { ( X i ' X ; ),( Y i ' Yム ( X i ' Y i ) } I ) be t h e 面l e a ro r d e rd e f i n e d on 1 i nt h en a t u r a lf a s h i o nandQ'( 1 )t h e and l e t Q( i n v e r s eo r d e ro fQ( 1 ) . ThenLニ L : .Q(I)LiandL'=L : .Q'(I)L a r e l i n e a r e x t e n s i o n s o f W i , L ' }i sar e a l i z e ro fW. S i n c eD[WJ二 三 2i se v i d e n t ,wehaveD[WJ=2. and{L 7 . 2 . LetX={xsISES}a ndY={Ys1SES} b e twod i s j o i n ts e tand PS t h eo r d e ron p e c i f i e dby XU Y s Ps ={( X ,.xs)I SES}U {(y ,.Ys)I SES} u{白 川s ' , )IS , S'ESands チs ' } . Psi si s o m o r p h i ct ot h eo r d e ront h es e tcomposedo fa l le l e m e n t so fS andt h e i rcom p l i m e n t si nS d e f i n e d by t h er e l a t i o no fs e ti n c l u s i o n . Asi sa l r e a d yknown [ 4 Jt h e d i m e n s i o no ft h el a t t e ri s1S1 . H enceD[P s Jニ [S1 . T h e r e f o r ei f1S1i sa t r a n s f i n i t e ,PS i sd r e d u c i b l e . But i f 1S[i s an i n t e g e ri ti sd i r r e d u c i b l ei n c a r d i n a l number g e n e r a l ,i .e .wehavet h ef o l l o w i n gt h e o r e m . 回 7 . 3 . THEOREM. Theo r d e rPnd e f i n e dont h es e t A {X1, X2, . . ., Xn; Y 1, Y2, . ' ,, Y n } b ys p e c i f y i n gt h a t P ={(Xi, X i )i ニ 1 , 2, . . . . n } u { ( Y ; ' Y i ) . i = 1, 2, . . . , n} u{ ( x ; ' Y j )i , j=l, 2, . . . ,n ;i チj} i sd i r r e d u c i b l e1 り r o v i d e dt h a tn 主3 . 勿 ニ 旬 1 1 P r o o f . Wes h a l lb e g i nw i t hp r o v i n gt h ef o l l o w i n glemma. 1 3 01 't h eDかn e n s i o nofOrders LEMMA. lfanypairGゲ c o n s e c u t i v e ( P " Je l e m e n t si sd e l e t e dfromA",t h edimension diminishesb y2J りr o v i e dt h a t nミ4 . Ones e e se a s i l yt h a twhateverpa 仕切: y ) maybe d e l e t e dt h es u b o r d e rPバA,, -x-y) i si s o m o r p h i ct ot h es u b o r d e rPバAnXn-Yn 1). Thereforet ob eprovedi st h a t → D[PCA"x "-Yn--l)J=n 仏 s i n c eD[P.J=nby7 . 2 . Themathmaticali n d u c t i o ni su s e df o rt h ep r o o f . I nthefirst η p l a c eweh a v ε D[YIJニ 4andD[P4CA 生-x 生 y g ) ]ニ 2, s i n c eP4(A 壬X'l-yg)ニ W(Xl, X : !, X3 , Yl , y~ , y ,1') i sas u b o r d e ro ft h eo r d e rW d e f i n e di n7 . 1 . Hencet h elemmai st u r ef o r 1 ' 1 = 1 .1 nt h enextp l a c eassumet h a tt h elemmai st r u ef o rn=k-l,. ie . D[P ,,l(Ak l andshowt h a tD[PkCAk-xん Y"-l)]=k-2h o l d s . Fort h eb r e v i t y -xト l-yト 2)J=k-3, p u t P' =Pk-l(Ak-l-Xk-l-Yk-2)=Pk(Ak-Xk--Yk-Xk-l-Yk2 ), P"= Pk (A ,, x"-Yk-1), p =P ,,( Ak-Xk-Yk-1xト 2 Y k 2 ) . P 'andP beingisomorphic,wehave 附 川 k-3=D[P'Jニ D[P' 勺 , ( 1 ) andbyt h e theorem4 . 4wehavekニ D[P"J主D [ P ' ' ' J+2, hence ( 2 ) k-2 三 二D [P"J. S i n c eP i so b t a i n e dbyd e l t i n gX k-2andY k : !fromP whichareincomparableminimal (P") andmaximal(P") e l e m e n t s,wehavebyt h etheorem4 . 3 リ 附 D[P"J三 D[P 川]←ト I ( 3 ) From ( 1 ) パ2 )and( 3 ) weo b t a ! I n D[P"J=k-2. 1 no r d e rt oprovet h etheoremi ts u伍 c e st oshowt h a tt h ee q u a l i t yD[P バA,, -Yn-l)] o l d s ,s i n c e PバAn-x) i si s o m o r p h i co ri n v e r s e l yi s o m o r p h i ct oPバA -Y 1 ) =n-1 h f o ranyxEA・ Whenn=3 ,D[P3 ]=3andD[Pg(Ag-Y2)]=2 s i n c e Pg(Ag-Y2)=W 叫 旬 η C X 1, X2, Xg, Yl, Y 3 ),a s u b o r d e ro fW i n7 . 1 . Hencet h etheoremi st r u ef o r nニ 3 . When n乙4,wehavebyt h etheorem4 . 2andt h elemma, n-1 三D[PnCAn--Yn-l)]三D[PnCA -Xn-Y -l)]+l=n~.l , 旬 hence " , D[P CA Yn-1)]=n-1 . " , 叫 Wes h a l lc l o s et h i ss e c t i o nwithp r o v i n gt h ef o l l o w i n gtheorem. 7 . 4 . THEOREM. L e tA b eas e t whose c a r d i n a l i t yd o e sn o te x c e e dδ. Thenfor 乙 lnotherwords in order t od e f i n e anorder e v e r yorderP d e f i n e donA,D[PJ三 whose dimension i sg r e a t e rt h αn 2,a s e t whose c a r d i n a l i t yi sg r e a t e rt h a nδ i s n e c e s s a r y . Proo f . Wheni A :=2i ti st r i v i al . L et3 手 !A 三 ふ Everyo r d e r whichi sdecompos- ,s i n c ei ti sd r e d u c i b l e by the a b l et oa c a r d i n a lsummaybel e f to u to fc o n s i d e r a t i o n theorem G .2 . C l a s s i f y i n ga l lo r d e r s under c o n s i d e r a t i o n by t h ec o m b i n a t i o no f the b t a i nt h ef ol Iowingt a b l e : numbero fmaximale l e m e n t sandt h a to fminimalo n e sフ weo T .H!RAGUT! 1 4 [ ; ι ι h i J J J Byi n t e r c h a n g i n gt h enumbero fmaximale l e m e n t sandt h a to fminimalonesweo b t a i n o t h e rc l a s s e sthant h o s el i s t e di nt h et a b l ewhichmaybel e f to u to fc o n s i d e r a t i o non a c c o u n to ft h ed u a l i t y . Every o r d e rb e l o n g i n gt ot h ec l a s s e so t h e rthanIV会 V5and Vh may be a l s ol e f to u to fc o n s i d e r a t i o n ,s i n c ei thast h eg r e a t e s telementwhichi s d-removablebyt h ec o r o l l a r y6 . 3 . The domain o f each o r d e rb e l o n g i n gt ot h ec l a s s sdecomposablet oa u n i o no ftwod i s j o i n tl i n e a rs u b s e L Therefore t h ed i m e n s i o n IV4 i i sa tmost2 t h ec o r o l l a r y3 . 4 The domain o f each o r d e rb e l o n g i n gt ot h ec l a s s 陶 V5 c o n t a i n sal i n e a rs e to ft h r e ee l e m e n t s . Theremainingtwoelementsa r ee i t h e r f they arecomparable,thenthedomaino ft h eo r d e ri s comparableo ri n c o m p a r a b l e . I au n i o no ftwod i s j o i n tl i n e a rs u b s e 胎, h encet h ed i m e n s i o ni sa tmost2by3 . 4 . I fthey a r eincomparable,thenoneo fthemi smaximalandt h eo t h e rminima. l T herefore t h e h etheorem4 . 3,a tmost2 s i n c et h ed i m e n s i o no ft h eo r d e ro b t a i n e d d i m e n s i o ni s,byt byd e l t i n gthemi sL Every o r d e rb e l o n g i n gt ot h ec l a s sVhi si s o m o r p h i ct ooneo f f o u ro r d e r sr e p r e s e n t e dbyt h ef o l l o w i n gd i a g r a m s : 肉、 [ も り h¥ 日 ¥ Thef i r s tt h r e eo fthema r ed r e d u c i b l ebyt h et h e o r r n5 . 2 . The l a s ti si s o m o r p h i ct o X2, Y l, v2 )o ft h eo r d e rW i n7 . 1,hence t h ed i m e n s i o ni s2 . Thus t h es u b o r d e rW(Xl, t h etheoremi sprovedc o m p l e t e l y . 8 . TheLeastUppersound of theDimensions of the 'd ers d i f i n e d on a f i x e dS e t . 01 se!n Weknowt h a tf o rt h ed i m e n s i o n so ft h eo r d e r sd e f i n e d onaf i x e ds e tA, Ai i 1 u p p e rboundb u tn o tt h el e a s tone 2 ) andt h a tf o r every c a r d i n a l number n t h e r e e f i n e dona s e to fc a r d i n a l i t y2n (7.2),whichshows e x i s t s an o r d e ro fd i m e n s i o n nd e to fc a r d i n a l i t y2ni ss u i 五c i e nt . t h a ti no r d e rt od e f i n e an o r d e ro fd i m e n s i o n na s Herea r i s et h ef o l l o w i n gtwoq u e s t i o n sc l o s e l yconnected with each o t h e r : “What i s t h el e a s tupperboundf o rt h ed i m e n s i o n so ft h eo r d e r sd e f i n e donaf i x e ds e tA?" and “I sa s e to fc a r d i n a l i t y2nn e c e s s a r yi no r d e rt od e f i n eano r d e ro fd i m e n s i o nn?ぺ To answert ot h e s eq u e s t i o n si st h es u b j e c to ft h i ss e c t i o n . i t eO r ( たr S . Wes h a l lb e g i nwithp r o v i n gtwolemmasont h e五n 1 5 Ont h enirnension0 /Orders 8 . 1 LEMMA ャ 1 . Fore v e r yo r d e 1 'P dりinedonas e tA whosec a r d i n a l i t yi sa tmost 7,t h e r ee x i s t sa tl e a sapair0 /twoconsective(P) elementswhoseranki sa tm o s t1 . f the rank o f every p a i ro fc o n s e c u t i v e P r o o f . Letu sprovet h ec o n t r a p o s i t i o n : I elementsi sa t1.e a s t2,thenIA i 与8 . Denole t h es e t {a1α (: b ), =P} b y A(b) f o rbEA : c =A. Nowl e t( a 1 : b 1 )b eap a i ro fc o n s 巴c u t i v e andt h es e t{ b :α (: b ) : = P }byB(α )f o ra y p o t h e s i s,a tl e a s t2,t h e r ee x i s t st h r e eelementse I t h e r e l e m e n t s . S i n c ei t sranki s,byh 1 ) α2 ;EA(b1)-a1,b3EB(α 1 }-bl and b4EB(al)-bl-b3 s u c ht h a t α2併 b3(P) and a2 ;併以 P) o r 2 ) b2EB(M)-bl,a3三 A( ! J l )-al and a,/己 A(bl)-a1-a3 such t h a tα 1 1併 2 ;( P) and d〆1~b2(P) ー Wemayassume,w i t h o u t1 0 8 8o fg e n e r a l i t y,t h a tt h eC a 8 e1 )o c c u1's S i n c eα (2 : b] ) , ( α 1: h J )and ( a 1: b 2 )a1'e a tl e a s to f rank 2,t h e r e must e x I s t three element8 圃 b 2EB(αョ)b ls u c ht h a ta ]併 2(P人 a3EA(b3)-a1 s u c ht h a ta3併 l(P), a4EA(b4)-al s u c ht h a ta 4 ψbl(P). 2, 8, 4 )a ¥ r ep a i i w i s ed i s t i n c t and山 手 的 f o r i=2, . ' 3 , 4anda2 ; チa if o r E v i d e n t l yb (i=1, i , 3 ' .4 . Hencewhena 8チα必 的 anc 1b j( i, j=1, ' c ! 8, 4)a r ep a i ' 1 w I s ec 1 i s t i n c t . Whenα日=α4 i= =A(b8)-alan1 cforεverya 4 ' =A ( b ' l )-a l,n e c e s s a r i l y A(ba)-al=A(b , ' / ' ) f o reveryaa, a l . Hencei f( a a, b ; 2 )己 P an c 1t b e r ee x i s t s an elementc8uch t h a ta aく : cく : b 2 C P ),then 2, . ' 3 ) , b/j=1, 2JJ, 4) andca rep a i r w i s ec 1 i s t i n ct . Anc 1i fe i t h e ' 1( a , p: b 2 )εP or ai(i=1, ? ( P ),t h e n,s i n c et h eranko fα (3: b4 , )i sa tl e a s t2 ,t h e re x i s t sa tl e a s t anelement a3併; l ( P ) . S i n c e b ; u c ht b a tb , ポ b f o r i= , l 2 J ] , 4 , a バ i =1, 2, 8) an c 1 b0三 B(ag)-bg--b4 s α i 5世 . c b i Cj=1, 2よ 4 , δarep a i r w i s ec 1 i s t i n ct . ThusA mustc o n t a i na tl e a s t3 1c 1 i s t i n c te 1 e m e n t s . 8 . 2 . LEMMA 2 . L e tP b e an order d e f i e d on a s e tA s a t i s f y i n gt h efollowing c o n d i t i o n s : 1 " Everyl i n e a rs u b s e tofA i sC 0 1 J ψosedofa tmost3 e le m e n t s . x i s i sa tl e a s tαl i n e a rs u b s e tofA com ρosedof3 e l e m e n t s . 2 Theree 0 30 Noρ αirofconsecutiveelementsi sofrankO . 40 Everyminimale l e m e n li sC 0 1 1 2 j り αr a b l ewithe v e r ymaximal e l e m e n t . Thent h e r ee x i s t sa tl e a s t apairoftwoorderd i s j o i n tl i n e a rs u b s e tofA. 1t h es e to fa l lminimal P r o o f . Denote t b es e to fa l l maximal elements by B anc 1 i t i o n2 l e t {a , a l, bl }beal i n e a rs u b s e tcomposedo fthree elementsbyM. Byt h econc 0 elements whe1'eαE M ,btEBan 1 c( α : α 1 ),( α I:bl)EP. Since( α 1: b l )i snoto frank0, t h e r ee x i s t two elements a2EA(bl)-a1 an1 cb aεB(al)-blcB s u c ht h a ta 2 , . gb 3CP). h e r ee x i s t sanelement b ; ? E B ( a 2 )-bl s u c h that Then s i n c e( α 2 : b l )i sn o to frank0,t b 2併ω ( P ) . Evi c 1 e n t l yb l, b 2andb:~ a r ep a i r w i s ec 1 i s t i n c t,andA(ba)-a]チo s i n c e( a 1 :b s ) u c ht h a te i t h e ra a併l(P) i sn o to frapkO . Hence i ft h e r ee x i s t s an a3巴 A(ba)-al s a ,p : b 3 )i so r1 c e 1 ' c 1 i s j o i n tw i t he i t h e r( a 立: b l )0 ' 1α (2 : b 2 ) . 1 ncase o r a8φ2(P),then ( T .HIRAGUTI 1 6 ( a 3 : b 1 ),(a3:b2)EPf o ral Ja 3EA(b3) a 1,t a k eanya 3andl e ti tb ef i x e d . The rank o f( a 1: b 3 )n o tb e i n g0 ,there must beab4εB(α1)-b1-b2-bSsuchthat b4併a3(P). E v i d e n t l yb 4 チb i f o ri=1, 2, 3anda 3牢M byt h e . c o n d i t i o n4 T h e r e f o r e( a 1 : b )EPf o r i a l li=1, 2, 3byt h ec o n d i t i o n1 Theranko f( a 1 :む)n o tb e i n g0,A(b 心 α 1チO . Hence 心 α1s u c ht h a ta4 φバP) for a value ofi=1,2,3,then i ft h e r ee x i s t s an a4EA(b 2, 3 . 1 nc a s e( a 4 : b )EPf o r ( a 4 : b 4 )i so r d e rd i s j o i n tw i t hα (4 : bi )f o rav a l u eo fi=1, i e v e r ya4EA(b 心 -a1andf o re v e r yv a l u eo fi=1, 2, 3,t a k e an a 4and l e ti tb ef i x e d . ,theremustbea boEB(α1)-b1-b2-b3-b4suchthat Theranko f( a 1 : b 4 )n o tb e i n g0 h卯4(P). Evidentlyboチ f o re v e r yi 三4anda4牢M bythecondition40 • Therefore ム ( a 4 :ム)EPf o ri 三3bythecondition1 Theranko fα (l : b i ' ; )n o tb e i n g0 ,A(bo)-aデ O . Hencei ft h e r ee x i s t sanaoEA(b ) a 1 s u c h t h a t a . . 1 . b バ P )f o r a v a l u e o f i 三 4 , ( a o : b ) o o i so r d e r d i s j o i n tw i t h(向 : b )f o rav a l u eo fi=1, 2, 3, 4 .1 nc a s e(向 : b βfore v e r yi~4 , i a p p l yt h e same r e a s n i n g andc o n t i n u et h esamep r o c e d u r ea sa b o v e . Wew i l lo b t a i n twos e q u e n c e s b2"" , b , . . .( bi EB(a1)-b1-b2・ . .-bi 1 ), b 1, i anda 1, a2"" , a ;, . . .s u c ht h a tadb;(P)f o re v e r yiandα (i 1: bk)巴 P f o raf i x e diand f o re v e r yξ~i -1. Buts i n c eA i saf i n i t es e tandb ' sa r ea l ld i s t i n c tt h es e q u e n c e { bi} mustbef i n i t e . Lett h el a s ttermb eb " . Thent h e r emustb ean仇 EA(b ρα1 s u c h 三n--1,andhenceα ( : b ρiso r d e r d i s j o i n tw i t h( a ,,l : b ) t h a ta " c p b ; ( P )f o rav a l u eo fi i f o ra v a l u eo fi 手n-1. I nf a c t,assumet h a t( a ", b βε P f o ra l li 三n-1andfora l l 仇 εA (b ρ-a1・ Then s i n c e( a 1 :b , , )i sn o to frank0 ,t h e r emustbea九 十1 εB(α1)U bi・ Thisc o n t r a d i c t st h ed e f i n i t i o no fb " . ← 0 • 0 • 0 • 旬 包 倫 8 . 3 . THEOREM. L e tA b eas e tw h o s ec a r d i n a l i t yi sg r e a t e rt h a n3 andP any [AI/2Jω' h e r e1 [A1 / 2 J meanst h ei n t e g r a lρa r tof o r d e rd e f i n e donA.Then D[PJ三1 i t eand1 A1i t s e l finc a s ei ti st r a n s f i n i t e . I A I / 2inc αs eI A Ii s舟 z P r o o f . When1 AIi st r a n s f i n i t ei ti se v i d e n t by t h e theorem 3 . 2 . When 1 AIi s f i n i t ewes h a l lp r o v ei tbyt h em a t h e m a t i c a li n d u c t i o na c c o r d i n gt ot h ec a r d i n a l i t yo f 5 . Let1 A1=6o r7 ;t h e n A . Byt h etheorem6 . 5t h ep r o p o s i t i o ni st r u ef o r1 A1 = 4, t h e r ee x i s t s ,by the lemma 1 ,a p a i roftwo c o n s e c u t i v ee l e m e n t sα (: b )t h eranko f . Henceb yt h etheorem4 . 7wehave w h i c hi s0O r1 ヰ3=[IAI/2J D[PJ三D[P(A-a一 b)J+1 c o n s i d e r i n gt h a t1 A-a-b二 4o r5 . Thust h ep r e p o s i t i o ni st r u ef o r1 AI 孟7 . Now l e t I A I 孟8andassumet h a tt h ep r o p o s i t i o ni st u r ef o rt h es e t s whose c a r d i n a l i t ya r el e s s fP i sd r e d u c i b l e ,thentheree x i s t sane l e m e n tas u c ht h a tD[PJ= t h a nt h a to fA. I D[P(A-a)J. C o n s i d e r i n gt h a tt h ep r o p o s i t i o ni st r u ef o rP(A-a) byt h ea s s u m p t i o n wehave [A-a1 /2 J手 [IAI2J. D[PJ=D[P(A-a)J三1 L e tP bed i r r e d u c i b l e . I fthere e x i s t sa p a i ro fi n c o m p a r a b l em i n i m a landm a r x i m a l ,then we have,by the theorem 4 . 3 and t h ea s s u m p t i o no ft h e e l e m e n t s a and b i n d u c t i o n, Ont h eDimensionofOrders 1 7 D[PJ三D[P(A-a-b)J+1 手CIA-a-bI/2J+1=[1A1 / 2 J . ft h e r ee x i s t s Lete v e r ymaximale J e m e n ti sc o m p a r a b J ew i t he v e r yminimale l e m e n t . I l i n e a rs u b s e tC composedo f4e l e m e n t s ,t h e nbyt h et h e o r eqJ. 4 . 4andt h ea s s u m p t i o n o ft h ei n d u c t i o nwehave D[PJ手D[P(A-C)J+2 三1 [A- C1/2J+2=[ 1A1 / 2 J . Lete v e r yl i n e a rs u b s e to fA becomposedo fa tmostt h r e ee l e m e n t s .I ft h e r e re x i s t s nol i n e a rs u b s e to ft h r e ee l e m e n t s ,then by the theorem5.2,P i sd r e d u c i b l es i n c e i nt h i sc a s et h e r ei snoe J e m e n to t h e rt h a n maximal o rminimale l e m e n t sa n devery maximal e l e m e n ti sc o m p a r a b l e with e v e r y minimal e l e m e n t . Hence i ts u 伍c e st o c o n s i d e rt h ec a s e where a tl e a s ta l i n e a rs u b s e to ft h r e ee l e m e n t se x i s t s . Nowi f α :b )o frank0,t h e nb yt h etheorem 4 . 7 t h e r ee x i s t sap a i ro fc o n s e c u t i v ee l e m e n t s( : ve weha 五D[P(A-a-b)J+1 三[IA-a-bI/2J+1=[IA1 2 J . D[PJ三 L e te v e r yp a i ro fc o n s e c u t i v ee l e m e n t si sn o to frankO . Thent h e r ee x i s t s ,by t h e lemma2 ,twoorder-disjointl i n e a rs u b s e t sB andC . Hencewehave ,bythetheorem 4 . 5, D[PJ手D[P(A-B-C)J+2 三1 [A-B-C1 / 2 J+ 三2 [ ( 1AI-4)/2J+2=[ iA1 / 2 J . Thust h ep r o p o s i t i o ni se s t a b l i s h e dc o m p l e t e l y . Thef o l l o w i n gp r o p o s i t i o ni se q u i v a l e n tt ot h el a s ttheorem. e tP b eano r d e rd e f i n e d ona s e tA. lfD[PJ三3,t h e n2D[PJ 8 . 4 . THEOREM. L 詞 A. 1 lnotherwords,inordert od e f i n e an o r d e r of dimensionnas e to fc a r d i ρrovidedn<3. n a l i t y2ni sn e c e s s a r y, The example 7 . 2 shows t h a tf o re v e r yc a r d i n a l numbern (血i t eo rt r a n s f i n i t e ), t h e r ee x i s t sano r d e ro fd i m e n s i o nn d e f i n e d on a s e to fc a r d i n a l i t y2 n . B u tt h i s w i l lbeg e n e r a l i z e da sf o l l o w s . 8 . 5 . THEOREM. Fore v e r yc a r d i n a lnumbern三2,t h e r ee x i s t sano r d e rofdimen s i o n[ n / 2 Jd i f i n e donas e tofc a r d i n a l i t yn . 剖 P r o o f . I ts u伍c e st oc o n s i d e rt h ec a s ewheren i s an odd i n t e g e r . LetP' be an o r d e ro fd i m e n s i o n(n-1) 2d e f i n e donas e tA'o fc a r d i n a l i t yn-1. Thent h eorder , f o rexample,s p e c i f i e dby P=P'U { ( b , b ) }U { ( x, b )I xEA};b$A i soneo fr e q u i r e do r d e r s . Asanimmediater e s u l to ft h etheorems8 . 3and8 . 5wehavet h efoUowingtheorem whichi st h ea n . swert ot h ef i r s tq u e s t i o nmentioneda tt h eb e g i n i n go ft h i ss e c t i o n . 8 . 6 . THEOREM. Among t h ed i m e n o s i o n s of t h eo r d e r sd e f i n e donaf i x e ds e t A, [ I A ! / 2 Ji st h eg r e a t e s t,p r o v i d e dI A I 三4 . , T .HIRAGUTI 1 8 9 . ThedimensionoftheProductofO r d e r s . 9 .1 . THEOREM. L e tf l w ( s )PSb et h ep r o d u c toft h eo r d e r sPSaccordingt oaωe l l sd i f i n e donas e tAsfore a c hSES. 1fD[P σ ] o r d e rW d i f i n e donas e tS,wherePSi 孟D[PsJfora l lSES,t h e nD [ f l w C s ) P s J=D[PaJ. P r o o f . Let佼s={Ltω [ t ( s )εTs }beaminimalr e a l i z e ro fPs .S i n s eD[Pσ ]乙D[PsJ t h e r ee x i s t s a mapping C f !s o f s t ao n t o 民. Let { C f !s[SES} be a system o fs u c h mappings ,wherec f ! σist h ei d e n t i c a lmapping ,andf o rs チ17, , ! f C maybetakena r b i t r a r i l y . ThenMtρ=flw加 料 ( L t c 川 i sal i n 回 re x t e n s i o no ff lwωPsand貸 ={ M t c σ )I t ( σ)ETa } i sa r e a l i z e ro ff l w cρP, ・I nf a c t ,l e tfく g(flw∞Ps ) ,then t h e r ee x i s t sane l e m e n t o ra l lSES'(W)andf( めくg ( s ' )(Ps , ) . S i n c eψ, , (L t ( a ) )i s S'ES s u c ht h a tf(s)=g(s)f al i n e a re x t e n s i o no fPs ' ,wehave( f ( s ' ) , g ( s ' ) )巴ヂピ ( L t cρ)f o re v e r yt (σ〉εTa ・ HenceWe have(f , g)EL'tC σ )f o re v e r yt ( σ〉εTσ・ Nowl e tf ' 世 i g ( f l w ( s ) P ふ Thent h e r ee x i s t ss 'ES s u c ht h a tf ( s ' )ゆ g(s')(P, , )and α) ( f ( s )ニ g ( s )f o re v e r ys くど (W). Hencet h e r ee x i s tt ( s ' ),t ' ( s ' )εT S Is u c ht h a t (f ( s ' ) , g ( s ' ) )εL t ( S I )and ( g ( s ' ) , f ( s ' ) ) E L ! ' ( s ' )・ L e tL山 1)ニヂs , (Lt ndL t ! ( , / ) =ヂ ピ( L t l ( σ ) ) . Then(f ( s ' ) , g ( s ' ) ) EC f ! s , (L ( σ ) )and( g ( s ' ) , f ( s ' ) )ε 】) a σ l t ヂ バ ムω ),whichimply,togetherwith(α),( f ,g)EM f )EMtfcσ ) ) . t ω and(g, Thuss t r e a l i z e r[fw ( s ) s D . HenceD[flw的 p s J三D[Pσ Jands i n c ei ti se v i d e n tt h a tt h ei n v e r s e J=D[Pσ J . i n e q u a l i t yh o l d swehaveD[flwωP, 9 . 2 . THEOREM. L e t{AslsES}b eas y s t e mofs e t s, Psano r d e rd e f i n e donA" s t = s }α minimalr e a l i z e rofP,αndP=flsP~ t h ec a r d i n a lρr o d u c tof t h e {Lt ω1t(s)εTs , IS ES}. Then D[PJ三 五1UsT I u n d e r t h e c o n d i t i o n t h a t T , ' s a r e ρ a i r w i s e s y s t e m {P s .e.D[PJ豆L :sD[P, J d i s j o i n t,i P r o o f .C o n s i d e ranyw e l l o r d e rWandWsd e f i n e donS andoneachTsr e s p e c t i v e l y o ( s )t h eleast(W lemento fTs andl e tt ) e s ・ Then L, tくs)={( f , f )IfEF}U{ ( f , g)I f , gεF andf ( s )く g ( s )( L t C s ) ) } U{( f , g)lf , gEF ,f(s)=g(s) andf(σ〉くg( σ) ( L t o Cσ)) fort h eleast(W) e l e m e r i tσ s u c ht h a tf( σ〉 チ g( σ)}, Fb e i n gt h es e to fa l l mappingsf o fS i n t o UsA ,suchthatf(S)EAs( s e e1 .5 ),i sa l i n e a ro r d e ronF andal i n e a re x t e n s i o no fP andt h esystem 総 ={L , t C ' )I sESandt(s)ET, }i sar e a l i z e ro fP. HencewehaveD[PJ手IUsT , I . I nt h econc Iu s i o no ft h el a s ttheoremt h ee q u a l i t yd o e sn o talwaysh o l d . 9 . 3 . T証 EOREM. 1fPsi sal i n e a rorderfore a c h-SES ,thenD[flsP.J= I S [, ρrovided o n t a i n sa tl e a s tt ωoe l e m e n t s . t h a tt h edomainofe a c hP,c I no r d e rt oprovet h i swes h a l lp r o v et h ef o l l o w i n gl e m n : : t a . 9 . 4 . LEMMA. 1f Q b eal i n e a ro r d e rd e f i n e d on a s e t of two e l e m e n t s,t h e n D[QsJ=[S[ . Ont l z eDimension0 1Orders 1 9 P r o o l . LetQ bet h eo r d e rde 白l e dont h es e t{ a , b }s ot h a tα (, b)EQandl e tf ,and g s bet h ef u n c t i o n ss p e c i f i e dby I s ( s )=b ,I s ( s ' )=αlore v e r ys 'チs ; g s ( s )=a ,g s ( s ' )=blore v e r yダチs r e s p e c t i v e l y . PutF={f slsES}andG={gsISES}. Thent h es u b o r d e rQS(FuG)o f QS onFuG i se q u a lt ot h eo r d e r { C f s , f s )[SES}U { ( g " g s )1SES}U {(f s , g . )1S , S 'ESands チs ' } whosed i m e n s i o ni s ,by7 . 2,[S. HencewehaveD[QS]=!Si. Ont h eo t h e rh a n d we havebyt h etheorem9 . 2 ,theinversei n e q u a l i t y . o n t a i n sas u b o r d e rwhichi si s o m o r p h i ct ot h巴 o r d e r Nows i n c et h el i n e a ro r d e rP c Q,t h eo r d e rP c o n t a i n sas u b o r d e rwhichi si s o m o r p h i ct ot h eo r d e rQ S . Hencewe sP ] 孟I S !,andt h ei n v e r s ei n e q u a l i t ybyt h etheorem9 . 2 . havet h ei n e q u a l i t yD[ll s S Mr.H .kommh a sp r o v e dt h a tD[P' ,,( E n ) J=nf o rn三 派 。 [ 5 J . P u t t i n gPs=Rwhere Ri st h el i n e a ro r d e rd e f i n e dont h es e to fa l lr e a lnumbersi nt h en a t u r a lf a s h i o n ,we ,by9 . 3 ,D[RsJ=IS[. Andwhen[S[=n 三辻し RS=P' ,,(E , ふ ThustheMr.Komm's have theoremi sas p e c i a lc a s eo ft h etheorem8 .3 . Thetheorem9 . 3w i l lb eg e n e r a l i z e da sf o l l o w s : 1 1P i s an o r d e rw h i c l zc o n t a i n sas u b o r d e ri s o 明 o r ρhict ot h e s o r d e sQT ,ω' l z e r eTsi sas e tw h o s ec a r d i n a l i t yi se , 抑 a lt ot h edimensionofPS and Q i st h e same o r d e ra si nt h el a s tlemma,t h e nD[llsPsJ= 1UsTslundert l z ec o n d i t i o n t l z a tT;sa r ep a i r w i s ed i s j o i n t . 9 . 5 . THEOREM. S For e v e r yo r d e rP 0 /dimensionm ( j i n i t eo rt r a n s j i n i t e ),t h e r e e x i s t sac a r d i n a lρr o d u c t0 1m linearorderswhichcontainsasuborderisomorPhic t o 9 . 6 . THEOREM. P . P r o o l . Letaminimalr e a l i z e ro fP b e{ L s SES}whereIS[=m ,andI xthec o n s t a n t mappings u c ht h a tI x ( s )=xf o re a c hxEA,A b e i n gt h edomaino fP. Thent h es u b o r d e ro ft h ep r o d u c tn s L s on{ I xIxEA}i si s o m o r p h i ct oP. Problemss t i l lo p e n . ,but can not prove nor dis Thea u t h o rc o n j e c t u r e st h ef o l l o w i n gtwop r o p o s i t i o n s p r o v e . 帽 1 . L e tP b e an o r d e rd e f i n e d on a s e t A and aamaximal(P)e l e m e n t0 1A. 1 1t l z e r ee x i s t so n eandon か oneelemenibs u c l zt h a t( b :α)EP,t h e nai sd r e m o v a b l e . I tmaybeprovede a s i Iyt h a ti f,m oreover,e i t h e rno element o t h e rt h a nbe x c e e d s (P) ao rt h es u b o r d e rP(A-a)i sd i r r e d u c i b l e ,thenai sd r e m o v a b l e . 2 . l ti sn o tJ う o s s i b l et od e f i n ead i r r e d u c i b l eo r d e r onas e tw l z o s ec a r d i n a l i t yi s anoddi n t e g e r . T .HIRAGUTI 20 References 1 . T .Hiraguti, “ O河 t h ed i m e n s i o no fρ ' a r t i a l l yo r d e r e ds e t s " Science Reportso ft h eKanazawa U n i v e r s i t y,v ol .1 ,No.2(1951),p p .7 7 9 4 . “A n o t eo nMr.Komm'sTheorems",i b .,vol .I I,No. 1( 1 9 5 3 ),p p .1 3 . 2 . T .H i r a g u t i, “ Surl ' e x t e n s i o nd el ' o r d r e仰 r t i a l ",Fundamenta Mathematica,vol .1 6( 1 9 3 0 ), 3 . E .S z p i l r a j n, p p .3 8 6 3 8 9 . 4 . B .DushnikandE .W.M i l l e r, “P a r t i a l l yo r d e r e ds e t s ",Am e r i c a nJ o u r n a lo f Mathematics, v o l .6 3( 1 9 4 1 ),p p .6 0 0 6 1 0 . 5 . H.Komm , “ O蹄 t h ed i m e n s i o no ft a r t i a l l yo r d e r e ds e t s ",Am e r i c a n J o u r n a lo fMathematics, v o l,7 1( 1 9 4 8 ),p p .5 0 7 5 2 0 .