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(M\I)·(M\I)⊆M\I
IMI
K
(K, +,·, . . .)N
KMK|K|+
I=\
U∈N
U(M)
IOMK⊆ O ( M N
K b ∈K
{b+U:U∈ N }
X κ κ X
{φα(x;yα) : α < κ} {bi,j :i < κ, j < ω}
η:κ→ω aηX
φi(aη, bij )⇐⇒ j=η(i)i, j
X κ
κ X
dp-rk(X)
dp-rk(a/S) tp(a/S)
x=x∞
x=x
X A dp-rk(X) dp-rk(a/A)a∈X
dp-rk(X)>0X
dp-rk(a/A)=0 ⇐⇒ a∈acl(A)
n < ω dp-rk(a/A)≥n I1, . . . , In
A IiAa
X Y X −∞Y
X Y A (a, b)∈X×Ydp-rk(a, b/A) = 2
c=a−b
2 = dp-rk(ab/A) = dp-rk(bc/A)≤dp-rk(b/cA) + dp-rk(c/A)≤1+1
b /∈acl(Ac)c /∈acl(A)b∈Y∩(X−c)Ac
Y∩(X−c)c∈X−∞Y A X −∞Y
KM
N={X−∞X:X⊆M1K}
NM1N
K
N N X
Y K K Z
Z−∞Z?
⊆(X−∞X)∩(Y−∞Y)
X Y Z =X∩Y
Y Y −∞Y
Y X X −∞Y
MD
∃∞D K D0
D D−∞D0c∈K X := D∩(D0+c)
(X−c)∩X⊆(D0+c−c)∩D=D0∩D=∅,
c /∈X−∞X X −∞X K
KM∈M1K
∈X−∞X K X
K K X
X∩(X+)
KMIKK
IKK
0∈IK
a∈K a ·IK⊆IK
IK
IK6⊇ K
IK∩K={0}
X−∞X K X
K X X + 0
a= 0 a·IK=IK
a·(X−∞X) = (a·X)−∞(a·X)
IK∩K×K×∅
∅
K X ⊆MK t M
X K t X(K) + t X
t K X X
KK0Mt, t0∈Mtp(t0/K0)
tp(t/K)
•t K t0K0
•X K K t X K0t0
t K t0≡Kt t0K
K0K0X X ∩(X+t0)
tp(K0/Kt0)K∃∞
X K K X0X0∩(X0+t)
t0K
K X K t X K t0
X K0t0a∈X(K0)a+t0∈Xtp(a/Kt0)
K a0∈X(K)a0+t0∈X
X K t0
KM K Y K K
Y
K0=KK1K2 · · ·
0, 1, 2, . . .
•tp(i/Ki) tp(/K)
•Ki+1 i
iKiKiY
w∈ {0,1}<ω
Yw=
y∈Y:^
i<|w|
y+i∈w(i)Y
∈0/∈ ∈1∈Y001 y∈Y
y+0/∈Y
y+1/∈Y
y+2∈Y
Y∅=Y
YwYw0Yw1
Ywn=|w|YwKnYw(Kn)
a∈Yw(Kn)⊆Y(Kn)a+n/∈Y Y Knn
Yw(Kn)Yw0
nKnYwKnYw∩(Yw−n)
Yw∩(Y−n) = Yw1Yw1
Y∅
YwS⊆ω aS∈K
as+i∈Y i ∈S
as+i/∈Y i ∈ω\S
x+y∈Y
KMX K ∈MK
x∈K
x∈X⇐⇒ x+∈X
S⊆K x x ∈X x + /∈X
S K Y
Y(K)⊆S X ∩Y K K
S
S x ∈K
x∈X=⇒x+∈X
X
G G00
G G00
G H (M,+) G=G00
H=H00 G⊆H H ⊆G
G∩H G H
h(ai, bi)ii<ω+ωG×H ai
G∩H biG∩H
a0, a1, . . . bω, bω+1, . . .
c:= a0+bωc−ai∈H i = 0
bω+i−c∈G i = 0
KMK IK
OKMOK⊇K
I00
KK K I00
K
IKI00
K
IK⊆I00
K−I00
K=I00
K.
IK=I00
Ka∈M×
IK⊆a·IKa·IK⊆IK
IKa·IK
a−1·IK⊆IKa·IK⊆IKa∈M×.
Oa∈Ma·IK⊆IKO
•IK
•
•K K ·IK⊆IK
•a a−1a∈M
OMOK v :M→Γ
IKOM
Ξ Γ
x∈IK⇐⇒ v(x)>Ξ
c∈K K ⊆ O v(c)≥0
Ξ>0IK⊆ O IKO
γ > Ξγ≥2·γ0γ0>Ξ
S⊆ O S·IKIK·IK
I2
KK
IK⊆I2
K−I2
K=I2
K
IK=I2
K=IK·IKγ > Ξγ1, γ2>Ξ
γ=γ1+γ2γ0= min(γ1, γ2)
IK
0≤γ1, γ2<Ξγ1+γ2>Ξγ= max(γ1, γ2)
0≤γ < Ξ<2·γ.
2·γ≥2·γ0γ0>Ξγ0≤γ
γ < Ξ< γ0.
IKOΞ
0+x0< v(x)<Ξx∈ O \ IK
x·(O \ IK)⊆ O \ IKx·(M\ O)x
IK
x·(M\IK) = x·(M\ O)∪x·(O \ IK)⊆(M\IK)∪(O \ IK) = M\IK
x·(M\IK)⊆(M\IK)x−1·IK⊆IKx−1∈ O
x
K
K
{X−X:X⊆K K }
U
{a·U:a∈K×}
{a·U+b:a∈K×, b ∈K}
S·IK
Ss∈Ss·IK
Y1, Y2, Y3, . . . ⊆M
Xi=Y2i4Y2i+1 0∈Xi
YiXi⊆M×
{Xi}k k XiX2i∩X2i+1
0/∈X1∩X2log2k
X1, X2, . . .
V a1, a2, . . . V ai∈
Xj⇐⇒ i=j
0/∈Xi∩XjV V ∩Xi∩Xj
(i, j)i6=j
0∈XiV∩XiaiV∩Xi
V∩Xiai/∈V∩Xjj6=i
bi·U ai,j i
bi·U ai0,j i0< i
ai,j bi0·U i0≤i
ai,j ∈Xj0⇐⇒ j=j0
ai,j ∈bi0·U⇐⇒ i≥i0
ai,j ∈Xj0⇐⇒ j=j0
ai,j ∈bi0·U\bi0+1 ·U⇐⇒ i=i0
Xj0bi0·U\bi0+1 ·U
M
p X Mp∈X
x∈X p(x)p
X p ∈X
X Y MX4Y
p∈X4Y p
p∈X⇐⇒ p∈Y p
n
Mnn
MnU
{a·U:a∈M×}
KM K
a∈K S ⊆K
a∈acl(S) =⇒a∈acl(S).
U K ·U K
a φ(M;)S φ(x;y)
acl (∅) tp(/K) acl (∅)S ψ(x)
|=φ(b;)⇐⇒ |=ψ(b)b∈K
a ψ(K)a∈acl(S)
OKK IK
I−1
KOKOK∨
K X IKOK
IK⊆X⊆ OK.
X⊇IKX K
U a ∈M×a·U⊆X
X U K a ∈K×
U⊆a−1·X⊆a−1· OK=OK,
K N
K
·U⊆· OK⊆IK· OK=IK⊆N,
(a1, . . . , an)S
ai/∈acl(a1, . . . , ai−1, ai+1, . . . , an, S)
i
S U X ⊆Mn
S
X n
X S
XMn
M1
Mnn=⇒
=⇒X n ~a = (a1, . . . , an)
X n S aiS
a1∈acl(a2a3. . . S)
dp-rk(~a/S)≤dp-rk(a1/a2a3. . . S ) + dp-rk(a2a3. . . /S)≤0+(n−1)
~a
=⇒n n = 1
n > 1 (a1, . . . , an)∈X
S S~a
a1, . . . , anS
a1acl(Sa2· · · an)
(a0
1, a2, . . . , an)∈X
a0
1a1
(a1+·U)× {(a2, . . . , an)} ⊆ X.
a2, . . . , ana1S
N⊆Mn−1
(a1+·U)×N⊆X.
X=⇒
XMnX
n
X n
X n
X
X⊆Mnm≤ndp-rk(X)≥m
π:Mn→Mmπ(X)