1TheSCiel1C
巳 Report日 oftheKa.l1ρZ乳 waUnl.
V{;，rf.
江ty VoL ll，:No. 1，pp.1‑‑3，iYlarch，1953.
宅A Note on Mr. Komm's 'l'heore
皿 S
By Toshio HIRAGUCHI(Receiv
edJune，27，1952)

告1. Mr¥Horace Komm (1)hasproved that(1) dz・mPn'CEn)= nproz!Z'dedn手拭 οbyshowingthat (2) dimPn.'(E"，)手11(11三派。)and (の i.
lP is a demmzerab!e partial!yordereds
et0.
1dimeJtsioJl 11，theree:xi・tsasubut1
V.
I;， o.
lEnsucJztlzatP~Pn/ (111;
ふEnmea
l1St
h巴 se
tofa
ll sequ
巴nc
esofr
ea
1.numbers {xd，k.
=1， 2，…

Here

1er
'e刀
7.
Z三三五b
怜
=0刀
f 叫 児巴仇
釘閃i
ωはLn
吋
川仁
clPi
託
fa
肌胤
n
町
c
ビc
l 0凶 y 託fXk.与乙，y"ん forall ka
吋れ宇ム forS01l1.
ei. 1nthesequelweuseぬtheω仁onv巴 山na山bbrω
己v
i
勾乱L川l
吐t
1
仙0
11"pos総附
巴伐t
円 fo1' P出ぷr
ti
M
勾
品al
ly0αr
c
白l
巴re
白
吋
ピdse
t出an
吋clt
hen悦ot品刷
ti∞on D CP)f
机ord
仙l
泊i
imP. Nowle
t.
a
lp
r
ocluc
tof pose低u
sde
finea cardin

Let Xs be a posetassociatec
lwith each elerl1entsofa setS，x={XsIs::=S} asetof
ιX8 selected one ata time frorneachset Xs 仙 clX the. Thensetofa
1
1 such x
X={(，x: y)l(xs，ys)
ξX sfora
11sξS}
1is乱 paτtialorcler)on thesetX whereX'
8i
sthepartialorc
1erassociateclwith each posetX8 ・ Bythecarclinal productJls X8 wemea
l1thesetX together wuh thepartialor
市 r

X. ¥Vhen"'¥
:‑
8ar
ea
1
1 isomorphic w
ithaposet Y，I1sX" isthecarclina
lpower yswhich可rillalsobe denoted by Y wherem isthenumber ofelementso
fS. Thusi
fthechainof
1
出1by R thet]
日出 propositionsof Mr.K01
11111will be for
111ulate1c asreal numbersi
sdenotec. (1勺 DCR"
つ= m，m三ぬ。 (2り ρCRm)手1ll，1ll三;;'~(1 (3勺 EZleryposdP witli ?lfollolVsCP)め手代。 altdD CP)= m isisomOlアhicwi・tlta subposet 0.
1R恥.

Thc purpose ofthisnot巴 isto prove thatth巴 propositions(1持)， (2''<‑) and (3促) hold inmoregeneralforms. That iseacardi1Za!proぬ ct0.
1
posetsXs，sξS仰 d 抗 s={.M;(8)Theorem 1. Let刀sXs=X bIt(s)εT，，} a 1Jli・nima!，'ealz・zer2) 0.
1X

3 •Tlu1Z D CX)
三五1l CT) 7Cル re T=US T

8 •(Ageneralizationof (2勺.)Theorem2.ζfX" isa chai12ρl' everysξSI D CI1)= llCSJ; (AgenemlizationsXsof (1勺.)Theorem3. EveryposdP τvzγ!
tD CP)= m is isomol
アんlCτv
itlza sztbposet 0.
1somecardilla!product 0.
1m CItαiガ s'whou di・lI
lensiollislJ.
ム(A generalizationof(3り.)Theorenl 3 is aspe
cialc
a.
seof

satoutwhichis d-reducib!e~) to a poset 波~hiclz i
sisomol
アliic
Theorem4.ζfXsi
with21'8 .
101' eZ1erysξS，then D CI
1sX8)=nCT) wlzere 2'=U8T~.
骨〉見Jlne2.
nslhenu
mberofelementsofasetwr
ittenint
hebr
ackets.

)，3)，4) Astot
heter
minologiesrefertot
heauthor'spreviou
spaper[2J.
1)，22T.HIRAGUCHI

32. 1.Proo/0/thetheorem1. I
tmaybeassumedt
hatS andT.arewell‑orderedsetswithout108sofgenerality. Letto(s)bethe白rs
te1ementof T.・ Thenμ.，y.)ξMt(8)}L~ (.
)= {(x，x)jxEX}U{(x，y)jx.キ:
y8

U{(x，y)jX8=y.，(x.
σ.ya)ξMto(a)forthe白rstσsucht
hatxσキ:ya}4) ois alinearordernt
hesetX. Thatthereflexivity and antisymmetry ho1d isev
ident.

Toshowt
hatthetransitivityho
1ds1et(x，y) ξL~ ωand (y，z)ξL!(S). Then there arefollowingfour.
cases:(1) x.中 y.，(x8，y.) 巳 M t(.) ;ysキ zs，(y.，Z8)

ξ Mt
(.
)・

(2) x.宇 ys，(x.，ys)

E Mtω ;

ys=Zs，(xa，ya)

ξ Mto(a)fort
he firstσsucht
hatya宇 Za・(3) Xs=ys，(xσ，yσ〉 巴 Mto(σ)forthe五rstσsucht
hatxσ宇yσ;s，(vs，ZS)
ys宇 Z(4) x.=ys，(xa，ya)

巴 Mt(s)ε Mto(σ)forthe白rstσsuchthatxa勺 σ;，Z〆〉 ξ Mt0(a/)forthefirst〆 sucht
hatYaly.=z.. (Yalキザ・Whenone ofthefir
stt
hreec
ase
soccursi
ti
sev
identthat(x.，ZS)ξMtω. Hence(x，z)ξL~ ωWhen t
he1
astc
aseoccur
sx.=Z8' I
fσ豆〆， thenσisthefirstsu
自x sucht
hatxσ=ドZσandforthis(Xσ，Zσ〉ξMto(σ・〉Simi
1arlyif〆く σ，thenσIisthe 白rst Se:伍x suchthatx〆 宇 Zql，andforthis(Xql，Zσ1)ξMto的 ・Hencei
neither caseωρξL~ (S). Thusbtaine
dbyassoc
Iating L~ ωwith X i
sa1inearthet
ransi
tiv
ityisverified. Thec
hainL!(8) oextensionoftheposetX

1nfact，if(X，v)ξX and x宇 v，theneither x.宇y.，(X.，y.)EMtωor x.=ys，(Xq，yq)ξMtoω forthefirstσsuchthatXa勾 σhence(X，y)EL~ω.
Moreover thesystem

m= {L~(') jsES，t(s)ET.}
ofalllinearextensionsL~ ωis arealizeroftheposetX

To show thisletx cpy inXorsomesξS or(Theneither(1)X8P
Y. inXsf2)x.宇 Ys.(X，. Y.)ξX.forsomesandY.I宇 Xal，(ys'，Xs')ξ X〆 forsome〆ゎ. If(1)， then (x.，ys) ξMtωfor somet(s)ξTsal1d (y8'x.) ξMt〆(s)forsome t'(s) ET. hence(x，y) εL~ (s) and (;ザ〉

εL!ぺ.) If(2)，thel 1 仇，y8)ξMtωand (y.，x.
，)EMt(s'); hence(x，y)弘子 s)and(v，x)ξL:〈sつ. Thus m isarealizerofXThereforeDCX)= n CT8 ) .

2. Proo/0/t
he theorem 2. By t
he theorem 1 we have theinequalityD Cllsx.
)~nCS).

Tohavethei
l1verseinequalityitsuffice
stoshowt
hatIIs.X. contains

A 仰 teonM シ.Komm'sTheorems

3

asubpose
tofdimensionn CSJ. Butev
idently28 isasubpose
tofIIs x.・'1'0 provethat
DC2'~J = nCSJ let2 be {O，I}， a. the elemento
f 2s whose s‑componentis I and otherheelemento
f 28 whoses‑componenti
s0andothe
rcomponents
componentsare0andβ. tareI. '1'henthesubposetof28 composedo
fal
la. and allん is isomorphic w
itht
he
pose
t composed ofallelementso
f S and their complernents in S and ordered bytherelationofset‑inclusion. Ithasbeenknown C3Jthatthe dimensiono
fthelatterposetisnCS} HencewehaveDC28J= n(S}3. Proofofthetheorem3. LetaminimalrealizerofP bem= {L.ls
ξS，} nCs)=nz.
'
1'henP Is isomorphic w
ithasubpose
tofIIsL.・ In factP is isomorphic w
ith thesubposetP*ofIIsL. composedofallelementssuchthatallthecomponentsareequalto叫anelemexEP.

4. Proofoft
hetheorem4.
D(llsX.J<n(TJisevidentbythetheorem1. 011 theotherhand
BJ=DC2TJ=n(T}D(llsx.J?PClls2T
REFERENCES
1. H.Kornrn，，01tth
edimmsi
01toff>artiallyor
deredse.
's"，Arn
6ricanJournalofMathernatics，vol.7
1く1948)，pp.5
07‑
520.

.Hiraguch!，"0
抑 揚edimC
1tsi
・
0'
1tofp
a
1'fiallyordC
iョdsels"，ScienceRe
por
tsoft
heKanazawaUniv
ersity，2. Tv
ol.1，No.2(1951)，pp.7
7‑
94.
3. B.DushnikandE. W. Miller，P山 lial
か，orderetIsels"ArnericanJournalofMathernatic
町 vol.63(1941)，

pp.660‑610.