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Dimension of Orders

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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
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*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/
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1
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3
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rP wemeanano
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d
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. An e
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n
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f an o
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ss
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i
da l
i
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ES,
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i
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a
i
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o
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f
o
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d
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rd
e
f
i
n
e
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h
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e
t5
. Then
Us
Es
P,
u{
(x"xs
'
)! υ X
s,
EAs'ands
くs
'
(
Q
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}
i
sano
r
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i
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e
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t Us
ES A" whichi
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a
i
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h
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SES}a
c
c
o
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o
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np
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c
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r
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o
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h
e sum c
o
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n
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i
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s with Us巳 sPs and i
ss
a
i
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e
when Q i
Theo
r
d
i
n
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i
t
enumbero
fsummands
c
a
r
d
i
n
a
lsumanddenoted I
.
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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
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,
g(S))EPsforv
e
r
y
i
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e
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i
n
e
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e
tF whichi
ss
a
i
dt
h
ec
a
r
d
i幻a
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odudo
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h
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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
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nt
h
i
sc
a
s
et
h
ep
r
o
d
u
c
t 18 d
e
n
o
t
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a
l
lmappingso
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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
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l
e
m
e
n
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h
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i
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r
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e
f
i
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a
i
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h
eo
r
d
i
n
a
lp
r
o
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c
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h
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a
c
c
o
r
d
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n
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h
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e
l
l
o
r
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e
r W andd
e
n
o
t
e
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{
u{
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o
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h
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i
sa
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e
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et~ b
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n
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t
a
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h
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o
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n
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e
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i
s
t
sa
1
1 element L i
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r
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re
x
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e
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r
i
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r
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i
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a
re
x
t
e
n
s
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o
n
t
a
i
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u
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e
r
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i
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re
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n
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i
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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
.
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