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3.8.
PARTIAL GROUPOID OF BINARY FORMULAS
201
In view of Proposition 3.8.0.3, the structure
tures
u ≥ 0
the triple
≤0
P
ν(R)
and
respectively,
(q, u−1, p)
3.8.0.4. Denition.
ministic
triples
if the set
(p, u, q)
The deterministic structure
0
P
(q, v, r)}
= hR × U × R; ¯i
ν(R)
for
≥0
P
, generated by triples
ν(R)
p, q ∈ R
. Here, for any triple
is also attributed to
A structure
(p, u, q) · (q, v, r)
and
(q, v, r)inP
, where
p, q, r ∈ R,u, v ∈ U
P
ν(R)
is a singleton (nite) for any
with
ν(R)
P
is generated by the structure
ν(R)
(p, u, q) · (q, v, r) = {(p, u, q) ¯
.
u ∈ ρ
P
ν(R)
(p, u, q)
≥0
P
.
ν(R)
is called (
ν(p,q)
has substruc-
with
u ≤ 0
(p, u, q)inP
almost)deter-
and
v ∈ ρ
Adapting the proof of Proposition 3.4.0.1 to a familyRof1-
types we obtain
3.8.0.5. Proposition.
nonempty family
p ∈ P
, and for any regular family
R ⊂ S(T )of1
For any complete theoryT, for any
-types having models
ν(R)
of labelling functions, the
Mpfor each
following conditions are equivalent:
(1)
the relation
(2)
the structure
IRis transitive for any model
P
is almost deterministic.
ν(R)
M |= T
and
≥0
,
ν(R)
.
ν(q,r)
;
Note that the absence of principal edges linking distinct realiza-
tions of types inRis equivalent to the antisymmetry of the relation
IR. Since
the family
3.8.0.6. Corollary.
family
for any regular family
IRis reexive (by the formula
ν(R)
and Propositions 3.1.0.6, 3.8.0.5 imply
For any complete theoryT, for any nonempty
R ⊂ S(T )of1
-types having models
ν(R)
of labelling functions, the following
(x ≈ y)
), the denition of
Mpfor each
p ∈ P
, and
conditions are equivalent:
(1)
the relation
of types ofRin any model
(2)
the structure
The partial order
The non-identical partial order
if
|ρ
ν(p)
| > 1
for some
pairwise distinct types inRsuch that
|ρ
ν(p
n+1
,pn)
| ≥ 1,n ∈ ω
IRis a partial order on the set of realizations
M |= T
P
is almost deterministic and
ν(R)
IRis identical if and only if
;
ρ
⊆ U≤0.
ν(R)
ρ
ν(R)
= {0}
IRhas innite chains if and only
p ∈ R
or there is a sequence
|ρ
ν(pn,p
| ≥ 1,n ∈ ω
)
n+1
pn,
n ∈ ω
, of
, or
.
.
202
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Lemma 3.1.0.3 and Proposition 3.8.0.5 imply
3.8.0.7. Corollary.
family
R ⊂ S(T )of1
for any regular family
For any complete theoryT, for any nonempty
-types having models
ν(R)
of labelling functions, the following
Mpfor each
p ∈ P
, and
conditions are equivalent:
(1) IRis an equivalence relation on the set of realizations of
types ofRin any model
(2)
the structure
An element
u ∈ U
to the regular family
izationaof a type inRand for some type
θ
tp(a),u,q
(a, y)
is consistent and has a unique solution (has nitely
M |= T
P
ν(R)
is called (
ν(R)
;
is almost deterministic and
almost)deterministic
ρ
⊆ U≥0.
ν(R)
with respect
of labelling functions if for some real-
q ∈ R
, the formula
many solutions).
Repeating the proof of Proposition 3.4.0.9 we have
3.8.0.8. Proposition.
P
ν(R),d
(
respectively
For any structure
P
ν(R),ad
)
to the set of(almost)determinis-
P
its restriction
ν(R)
tic elements is closed under the partial operation of the structure
P
.
ν(R)
Using the proof of Proposition 3.4.0.11 the following proposition
holds.
3.8.0.9. Proposition.
Mpand
if and only if
Mqexist then an element
(q, u−1, p) · (p, u, q) = {(q, 0, q)}
3.8.0.10. Proposition.
then the structure
ρ
,
p ∈ R
ν(p)
set
Proof
ρ
ν(p)
, consists of non-negative elements.
is identical to the proof of Proposition 3.4.0.13 for each
.
¤
P
0
ν(R)
3.8.0.11. Corollary.
S1(T )
structure, and
p ∈ R
, there are models
0
P
is a join of groups, then all elements in
ν(R)
, are deterministic.
If for the types
u ≥ 0inρ
p, q ∈ S1(T )
is deterministic
ν(p,q)
the models
.
If the structure
P
is deterministic
ν(R)
is a join of groups if and only if each set
If
R
is a nonempty family of1-types in
Mpfor
p ∈ R,P
is a deterministic
ν(R)
P
0
ν(p)
,
3.9.
IR-STR
UCTURES
203
Proof.
of non-negative elements, the determinacy of the structure
and Proposition 3.8.0.9 imply that each element in
is deterministic.
Since, by Proposition 3.8.0.10, the sets
¤
P
ρ
0
ν(p)
ν(p)
consist
P
,
p ∈ R
ν(R)
Repeating the proof of Proposition 3.4.0.15 we obtain
3.8.0.12. Proposition.IfR
S1(T )
, there exists models
family of labelling functions, then for the structure
≥0
ρ
ν(R),d
which the elements
istic substructure
of all non-negative deterministic elementsuin
u−1are also deterministic, forms the determin-
≥0
G
ν(R),d
is a nonempty family of1-types in
Mpfor
p ∈ R
, and
ν(R)
P
of
P
ν(R)
such that
(G
≥0
)0is a join of
ν(R),d
is a regular
the set
ν(R)
ρ
, for
ν(R)
groups.
The results above substantiate the transformation of the di-
agram in Figure 3.1 replacing the typepby a nonempty family
R ⊆ S1(∅)
3.9.
3.9.0.1. Denition.
.
IR-structures
LetRbe a nonempty set,
,
U = U
−
be an alphabet consisting of a set
U+of
R
(p, u, q) > 0
p, q ∈ R
positive elements
, we write
, we consider a
u < 0
for any
and a zero0. Ifpandqare elements in
and
(p, u, q) < 0
u ∈ U+. For the set
regular
such that
• 0 ∈ µ(p, q)
• µ(p, p) ∩ µ(q, q) = {0}
• µ(p, q) ∩ µ(p0, q0) = ∅
if and only if
for
p = q
for
p 6= q
S
•
p,q∈R
µ(p, q) = U
.
˙
∪ {0}˙∪ U
U−of
for any
family
;
p 6= q
;
and
+
negative elements
u ∈ U−,
u > 0
R2of all pairs
µ(R)
of sets
µ(p, q) ⊆ U
(p, q) 6= (p0, q0)
;
, of a set
and
(p, q)
,
204
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Below we write
operation·on the set
above,
(p, u, q) · (q, v, r)
A left semi-associative structure
with a regular family
partial operation·of
p0= q,∅ 6= X ⊆ µ(p, q),∅ 6= Y ⊆ µ(p0, q0)
by the partial function·for elements inUwhere
forms a nonempty set of triples
and
y ∈ µ(q, r)
∅ 6= Y ⊆ µ(q, r)
µ(p)
instead of
R × (P(U ) \ {∅}) × R
instead of
µ(R)
P
: for any sets
,
µ(p, p)
, and considering a partial
we shall write, as
(p, {u}, q) · (q, {v}, r)
P = hR× (P(U) \{∅})×R; ·i
of sets is called an
has values
(p, X, q) · (p0, Y, q0)
IR-structure
, and is generated
(p, x, q) · (q, y, r)
(p, z, r),z ∈ µ(p, r)
X, Y ∈ P(U ) \ {∅},∅ 6= X ⊆ µ(p, q)
(p, X, q) · (q, Y, r) =[{(p, x, q) · (q, y, r) | x ∈ X, y ∈ Y },
as well as the following conditions hold:
•
each restriction
{p}
is isomorphic to anI-groupoid with the universe
p ∈ R
;
•ifu ∈ µ(p, q)
(r, v0, p) · (p, u, q)
and
v0∈ (r,p)
;
P
and
ofPto the set
µ(p)
u < 0
then the sets
{p} × (P(µ(p)) \ {∅}) ×
(p, u, q · (q, v, r)
consist of negative elements for any
.
if the
only for
, if
x ∈ µ(p, q)
P(µ(p))\{∅}
and
v ∈ µ(q, r)
,
,
•ifu ∈ µ(p, q),v ∈ µ(q, r),u > 0
(p, u, q) · (q, v, r)
•
for any element
inverse
element
(p, u, q) · (q, u−1, p)
•
if an element
(q, v2, r)
(q, v
then the element
−1
, p)
;
1
•
for any elements
consists of non-negative elements;
u ∈ µ(p, q)
u−1∈ µ(q, p),u−1> 0
and
(q, 0, q) ∈ (q, u−1, p) · (p, u, q)
(p, u, r)
is positive and belongs to the set
(r, u−1, p)
(p, u1, q), (q, u2, r), (r, u3, t)
with
, and
u > 0
v > 0
there is the unique
, such that
belongs to the set
, then the set
(p, 0, p) ∈
;
(p, v1, q)·
−1
(r, v
2
the following in-
clusion holds:
((p, u1, q) · (q, u2, r)) · (r, u3, t) ⊇ (p, u1, q) · ((q, u2, r) · (r, u3, t)),
, q) ·
3.9.
IR-STR
UCTURES
and the strict inclusion
((p, u1, q) · (q, u2, r)) · (r, u3, t) ⊃ p, u1, q) · ((q, u2, r) · (r, u3, t))
205
may be satised only for
•
the structurePcontains the
u1< 0
and
|(q, u2, r) · (r, u3, t)| ≥ ω
deterministic
substructure
;
P
≥0
d
being the restriction to the set
≥0
U
= {u ∈ U≥0| (q, u−1, p) · (p, u, q) = {(q, 0, q)}
d
for some
every set
v ∈ U
By the denition, any
P
µ(p)
ofPto the sets
3.9.0.2. Theorem.
T
with a family
(p, u, q) · (q, v, r)
≥0
∩ µ(q, r)
d
,
p ∈ R
, and
is a singleton for
.
IR-structurePcontainsI-subgroupoids
IR-substructures
U≤0and
U≥0respectively.
For any
R ⊂ S(T )of1
labelling functions such that
p, q ∈ R};
≥0
∩ µ(p, q)
d
≥0
being restrictions
and
P
≤0
and
u ∈ U
P
IR-structurePthere exists a theory
-types and a regular family
P
ν(R)
= P
. If the alphabet and the
ν(R)
of
familyRare at most countable, and the operation ofPdoes not
force continuum many types, thenTis small.
Proof
follows the schema for the proof of Theorem 3.6.0.2 ex-
tended by links between types. In view of bulkiness of this proof
we only point out the distinctive features leading to the proof of
this theorem.
1. For each symbol
which intersects all predicates
of realizations of complete1-type
{Rp(x)} ∪ {¬Coln(x) | n ∈ ω}
corresponding to theI-groupoid
predicates
Rpare disjoint.
2. For the elements
elementsain
Col(a) = Col(b)
Rpwith elementsbin
, and if
p ∈ R
we introduce a unary predicate
u ∈ µ(p, q)
u < 0
then
Rp,
Coln,
n ∈ ω
, and forms, on the set
p0(x)
, being isolated by the set
, a structure of isolating formulas
P
. Moreover, we suppose that
µ(p)
the predicates
Rq. Moreover, if
Col(a) ≤ Col(b)
Qulink only
u > 0
then
and the coloring
ColisQu-ordered.
,
206
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3. The relation
that its classes are ordered by the relation
Note that using the operation
be transformed for an arbitrary family of types in
Q≥0=
Quis an equivalence relation such
u≥0
Q<0=
S
Qu.
u<0
·eq, the constructions above can
S(T)
.
¤
3.10. Notions, notations, and properties
S
3.10.0.1. Denition.
by
SICF(p, q)
{ϕ(a, y)}
Let
mulas in
SICE(p, q)
the set of all
is consistent for some (all)awith
SICF(p, q)
sets of solutions for
Clearly,
SICF(p, q)
either a set of
SICE(p, q)
. Notice that each
(p, ϕ, q)
or simultaneously a set of
arcs linking realizations ofpandqby any (some) formulaϕinE.
Thus the quotient
union of sets
SICFE(p, q)
edges,
SICFA(p, q)
SICF(p, q)/SICE(p, q)
SICFE(p, q),SICFA(p, q)
consists of
sets of irreversible arcs, and
classes corresponding to sets containing edges and irreversible arcs.
The sets
SICFM(p, p)
SICFA(p)
SICF(p, p),SICE(p, p),SICFE(p, p),SICFA(p, p)
are denoted by
, and
SICFM(p)
LetTbe a complete theory,
alphabet of cardinality
u−∈ U−,
positive elements
and zero0. As usual, we write
any
u ∈ U+. The set
denoted by
Let
U≥0. Elements ofUare called
ν(p, q):SICF(p, q)/SICE(p, q) → U
functions,p(x), q(y) ∈ S(∅)
For types
(p → q)
be the set of all pairs
p(x), q(y) ∈ S(∅)
-formulas
ϕ(x, y)
|= p(a)
.
(ϕ(x, y), ψ(x, y))
, we denote
such that
of for-
such that for any (some) realizationaofpthe
ϕ(a, y)
and
ψ(a, y)
coincide.
is an equivalence relation on the set
SICE(p, q)
-edges, or a set of irreversible
(p, ϕ, q)
-edges and of irreversible
-class
E
corresponds to
(p, ϕ, q)
(p, ϕ, q)
-arcs,
is represented as a disjoint
SICE(p, q)
consists of
, and
-classes corresponding to sets of
SICE(p, q)
SICFM(p, q)
SICFM(p, q)
, where
-classes corresponding to
consists of
SICE(p, q)
, and
SICF(p),SICE(p),SICFE(p)
respectively.
≥ |S(T )|
u+∈ U+,
U−∪ {0}
U = U
−
, consisting of
neutral elements
u < 0
for any
is denoted by
˙
∪ {0}˙∪ U
u ∈ U−and
U≤0and
labels
be an injective
+
˙
∪ U0be an
negative elements
u0∈ U0,
u > 0
U+∪ {0}
.
labelling
, for which negative elements
for
is
-
-
,
3.10.
NOTIONS, NOTATIONS, AND PROPERTIES
207
correspond to classes in
ements and0correspond to classes in
such that0is dened only for
by the formula
(x ≈ y)
SICFM(p, q)/SICE(p, q),ν(p) ν(p, p)
that
ρ
ν(p)
∩ ρ
ν(q)
= {0}
ρfthe image of the functionf) and
and
(p, q) 6= (p0, q0)
. Labelling functions with the properties above
as well families of these functions are said to be
SICFA(p, q)/SICE(p, q)
, positive el-
SICFE(p, q)/SICE(p, q)
p = q
and is represented
, and neutral elements code classes in
. We additionally assume
for
p 6= q
(where, as usual, we denote by
ρ
ν(p,q)
∩ ρ
ν(p0,q0)
= ∅ifp 6= q
regular
. Below
we shall consider only regular labelling functions and their regular
families.
The labels, corresponding to isolating formulas, are said to be
isolating
whereas each label in
the denition, each isolating label belongs to
S
p,q∈S1(∅)
ρ
ν(p,q)
is
semi-isolating
−
˙
U
∪ {0}˙∪ U+, i. e.,
. By
it is not neutral.
We denote by
u ∈ ρ
θ
(x, y)
p,u,q
. If the typepis xed and
ν(p,q)
is denoted by
Note that if
that for realizationsaandbofpandqrespectively the pairs
and
(b, a)
belong to
θ
q,v,p
(y, x)
witnesses that
is principal and
θ
(a, b),|= p(a)
p,u,q
U≥0corresponds uniquely tousuch that
formula with
reciprocally inverse
In general case, each label
labels in
label
is a
[b, a]
U≥0, denoted also by
u ∈ U≥0, for which a formula
(p, ϕ, q)
is a
-edge, the set
(q, θ
Neutral labels correspond, for instance, the formulas
θ
(x, y)
p,v,q
, where
For types
θ
(x, y)
p,u,q
θu(x, y)
θ
p,u,q
(x, y)
SI
and
, then the formula
{p,q}
[a, b]
θ
(a, y)
p,u,q
, then the labeluis
|= θ
q,v,p
(b, a)
, and vice versa. The labelsuandvare
and are denoted by
u ∈ U≥0has a (nonempty) set of
u−1includes all labels
, p)
q,v,p
p1, p2, . . . , p
-edge.
u < 0
and
k+1
formulas in
SICF(p, q)
p = q
then the formula
with a label
.
θ
q,v,p
(x, y)
are formulas witnessing
(a, b)
is a
(p, ϕ, q)
ϕ(x, y) θ
-edge. If the edge
p,u,q
(x, y)∧
is an isolating formula such that
invertible
θ
v−1and
and the label
(b, y)
q,v,p
is an isolating
u−1respectively.
inverse
u−1. Note that independently on a
θ
p,u,q
(x, y)
witnesses that
v ∈ U≥0such that
θ
(x, y)∨
p,u,q
v ≥ 0
.
∈ S1(∅)
and sets
X1, X2, . . . , Xk⊆ U
[a, b]
|=
v ∈
[a, b]
of labels we denote by
SI(p1, X1, p2, X2, . . . , pk, Xk, p
k+1
)
208
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
the set of all labels
satisfying, for realizationsaof
uk∈ Xk∩ ρ
ν(pk,p
θ
p1,u,p
u ∈ U
corresponding to formulas
p1and some
, the following condition:
)
k+1
(a, y) ` θ
k+1
p1,u1,p2,u2,...,pk,uk,p
u1∈ X1∩ ρ
(a, y),
k+1
where
θ
p1,u1,p2,u2,...,pk,uk,p
∃x2, x3, . . . xk(θ
. . . ∧ θ
p
k−1,uk−1,pk
p1,u1,p
(x
(x, x2) ∧ θ
2
, xk) ∧ θ
k−1
(x, y)
k+1
pk,uk,p
p2,u2,p
(x2, x3) ∧ .. .
3
(xk, y)).
k+1
In view of transitivity of semi-isolation, each formula
θ
p1,u1,p2,u2,...,pk,uk,p
has a label in
Thus the Boolean
ρ
ν(p1,p
.
)
k+1
P(U)ofU
of distributions of binary semi-isolating formulas
(x, y)
k+1
is the universe of an
withk-ary oper-
ations
SI(p1, ·, p2, ·, . . . , pk, ·, p
where
to any family
of binary
p1, . . . , p
isolating
∈ S1(∅)
k+1
R ⊆ S1(∅)
.4This algebra has a natural restriction
as well as to the algebras of distributions
formulas. Besides, if
then the restriction of the universe ofAto the set
restrictions for values of operations to the set
partial, algebra
A ¹ U0.
),
k+1
U0is a subalphabet of
P(U0)
U0forms, possibly
θ
p1,u,p
k+1
ν(p1,p2)
algebra
and the
(x, y)
, . . . ,
A
U
Note that if some set
if it is empty then
SI(p1, X1, p2, X2, . . . , pk, Xk, p
and if each
Xihas common elements with
SI(p1, X1, p2, X2, . . . , pk, Xk, p
4
Later
(Section 3.16) it will be shown that it is sucient to consider only
SI(p1, ·, p2, ·, p3)
.
Xiis disjoint from
ρ
ν(pi,p
k+1
ρ
ν(pi,p
k+1
, in particular,
)
i+1
) = ∅,
then
)
i+1
) 6= ∅.
3.10.
NOTIONS, NOTATIONS, AND PROPERTIES
209
Note also that if
= SI(p1, X1∩ρ
ν(p1,p2)
Xi6⊆ ρ
ν(pi,p
for someithen
)
i+1
SI(p1, X1, p2, X2, . . . , pk, Xk, p
, p2, X2∩ρ
ν(p2,p3)
, . . . , pk, Xk∩ρ
k+1
) =
ν(pk,p
In view of the previous equality, below considering values
SI(p1, X1, p2, X2, . . . , pk, Xk, p
we shall assume that
If each set
use
uiinstead of
Xiis a singleton consisting of an element
Xiin
SI(p1, u1, p2, u2, . . . , pk, uk, p
Xi⊆ ρ
ν(pi,p
,
i = 1,. . . , k
)
i+1
SI(p1, X1, p2, X2, . . . , pk, Xk, p
k+1
k+1
)
.
k+1
).
By the denition the following equality holds:
SI(p1, X1, p2, X2, . . . , pk, Xk, p
= ∪{SI(p1, u1, p2, u2, . . . , pk, uk, p
Hence the specication of
duced to the specications of
also that
Clearly, if
SI(p, X, q) = X
ui= 0
SI(p1, X1, p2, X2, . . . , pk, Xk, p
SI(p1, u1, p2, u2, . . . , pk, uk, p
for any
then
pi= p
) | u1∈ X1, . . . , uk∈ Xk}.
k+1
X ⊆ ρ
i+1
ν(p,q)
for nonempty sets
k+1
.
) =
, p
k+1
)
k+1
uithen we
)
and write
)
is re-
k+1
)
. Note
k+1
).
SI(p1, u1, p2, u2, . . . , pi, 0, p
and the following conditions hold:
SI(p1, 0, p1) = {0},
SI(p1, u1, p2, u2, . . . , pi, 0, p
i+1
= SI(p1, u1, p2, u2, . . . , pi, u
If all types
and
SIp(u1, u2, . . . , uk)
piequal to a typepthen we write
as well as
du1, u2, . . . , uke
, . . . , pk, uk, p
, u
i+1
i+1
i+1
, p
i+2
, p
, . . . , pk, uk, p
i+2
, . . . , pk, uk, p
k+1
SIp(X1, X2, . . . , Xk)
dX1, X2, . . . , Xkepand
p
)
k+1
k+1
) =
).
210
instead of
and
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
SI(p1, X1, p2, X2, . . . , pk, Xk, p
SI(p1, u1, p2, u2, . . . , pk, uk, p
k+1
k+1
)
)
respectively. We omit the index
case, we write
3.10.0.2. Proposition.
then
ρ
ν(p,q)
(2)Ifp, q ∈ S1(T ),p
type then
Proof.
∪ ρ
ρ
ν(p,q)
θ
u1,u2,...,u
ν(q,p)
= ∅
(1) If
k
⊆ U≥0.
and
ρ
ν(p,q)
(x, y)
(1)Ifp, q ∈ S1(T )
is a principal type andqis a nonprincipal
ρ
ν(q,p)
contains a label
·pif the typepis xed. In this
instead of
θ
p,u1,p,u2,...,p,uk,p
are principal types
⊆ U−.
u /∈ U≥0then there are
realizationsaandbofpandqrespectively such that
and
(b, a) /∈ SI
ϕ(x)
, this formula witnesses that
implies that
(2) Let
and
(a, b) ∈ SI
ρ
ϕ(x)
∃x(ϕ(x) ∧ θu(x, y))
ρ
ν(p,q)
= ∅
. By the same reason,
3.10.0.3. Corollary.Ifp(x)
3.10.0.4. Proposition.
. But since
{p,q}
p(x)
(b, a) ∈ SI
⊆ U≥0. Similarly we obtain
ν(p,q)
be a principal formula of
that witnessed by a formula
{p,q}
isolates
q(y)
. Sinceqis not isolated we obtain
ρ
ν(q,p)
is a principal type then
Let
p1, p2, . . . , p
contains a principal formula
. The contradiction
{p,q}
ρ
ν(q,p)
p(x)
. If
θu(x, y)
⊆ U−.
¤
be types in
k+1
The following assertions hold.
(1)Ifui∈ ρ
ν(pi,p
,
i = 1,. . . , k
)
i+1
, and some
uiis negative then
(x, y)
.
(a, b) ∈ SI
{p,q}
⊆ U≥0.
|= p(a),|= q(b)
, the formula
ρ
⊆ U≥0.
ν(p)
S1(∅)
,
.
(2)Ifui∈ ρ
(3)Ifui∈ ρ
belongs to
SI(p1, u1, p2, u2, . . . , pk, uk, p
ν(pi,p
i+1
∩ U≥0,
)
i = 1,. . . , k
SI(p1, u1, p2, u2, . . . , pk, uk, p
ν(pi,p
∩ (U≥0∪ U0),i = 1, . . . , k
)
i+1
U0then
SI(p1, u1, p2, u2, . . . , pk, uk, p
) ⊆ U−.
k+1
, then
) ⊆ U≥0.
k+1
k+1
) ⊆ U0.
, and some
u
i
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