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3.10.
NOTIONS, NOTATIONS, AND PROPERTIES
211
(4)Ifui∈ ρ
the set
set
X SI(p1, u1, p2, u2, . . . , pk, uk, p
X−1 ∪{v−1| v ∈ X}
ν(pi,p
SI(p
Proof.
(1){(3) follow by transitivity of semi-isolation.
(4) All elements inXare invertible by (2). Let
in
v−1⊆ X−1. Then for any
that
[b, a]
is a
(p
k+1
pi,
i = 1,. . . , k+1
i = 1,. . . , k
Since
[a
.
i+1
, such that
, ai]
i+1
, u
k+1
, θ
is an
∩ U≥0,
)
is contained in
−1
k
, pk, u
−1
k−1
(p1, θ
,v0,p
, p1)
1
p
k+1
a0= a,a
0
u
-edge for some
i
i = 1, . . . , k
, . . . , p2, u
, then all elements of
)
are invertible and the
k+1
−1
, p1).
1
v0be an element
, p
)
p1,v,p
k+1
k+1
-edge
-edge, there are realizations
= b,|= θ
k+1
u
0
i
∈ u
pi,ui,p
i+1
−1
,
i = 1, . . . , k
i
[a, b]
(ai, a
such
aiof
i+1
then
0
whence,
θ
p
k+1
v0∈ SI(p
,v0,p
k+1
1
(b, x) ` θ
−1
, u
, pk, u
k
p
k+1
−1
k−1
,u
0
,pk,u
k
k
, . . . , p2, u
...,p2,u
1
0
,p
1
−1
, p1).¤
1
(b, x),
1
Note that the inclusion
X−1⊆ SI(p
can be strict since labels in
to
X−1) labels for
SI(p
k+1
k+1
, u
, u
−1
k
u
−1
k
Indeed, by the denition,
formulas satisfy all edges for
whereas labels in
SI(p
k+1
, u
−1
k
, pk, u
−1
, pk, u
−1
i
, pk, u
, . . . , p2, u
k−1
may compose new (with respect
−1
, . . . , p2, u
k−1
−1
1
−1
1
, p1)
, p1)
.
X−1should consist of labels whose
SI(p1, u1, p2, u2, . . . , pk, uk, p
−1
, . . . , p2, u
k−1
−1
1
, p1)
may corre-
k+1
spond to formulas disjoint from these edges. Taking, for instance,
k = 2
the labelvfor
does not satisfy edges for
and a disjunction
θ
p3,v2,p2,v1,p
w ∈ SI(p3, u
θ
p3,v2,p2,v1,p
(x, y)
1
(x, y)∨θ
1
belongs to
SI(p1, u1, p2, u2, p3)
−1
2
, p2, u
−1
1
p3,w2,p2,w1,p
X−1and
, we have
, p1) \ X−1,
(x, y)
1
θ
p3,w2,p2,w1,p
, where
)
,
,
)
,
1
wherewis the label for
3.10.0.5. Corollary.
θ
p3,w2,p2,w1,p
(x, y)
1
.
Restrictions ofUto the sets
U≤0,
U≥0, and
U≥0∪ U0form subalgebras of the algebra of distributions of binary
semi-isolating formulas. The operation of inversion is coordinated
with the operations of the algebra.
212
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.11. Preordered algebras of distributions of
binary semi-isolating formulas
3.11.0.1. Denition.
For the setUof labels in the algebraAof
binary semi-isolating formulas of theoryT, we dene the following
relationE: if
for some types
(any) realizationaofp. If
u, v ∈ U
p, q ∈ S1(∅)
then
uEv
if and only if
and
u E v
θ
p,u,q
and
(a, y) ` θ
u 6= v
we write
u = v
p,v,q
, or
(a, y)
u, v ∈ ρ
for some
u C v
ν(p,q)
.
By the denition the relationEis reexive and transitive. It
is antisymmetric since distinct labels correspond to non-equivalent
formulas.
Below we consider some properties for the substructures
hρ
3.11.0.2. Proposition.
tially ordered set
realizationaofp, the formula
{−, +}
θ
θ
minimal by the denition. If the formula
then there is a formula
θ
the labels
v2C u
θ
Hence, for any solutionbof
is also an irreversible arc and soubelongs to
by edges, the same arguments show that
ply
θ
erty is preserved for any labelvwith
; Ei
ν(p,q)
(2)
An element
(3)
(monotony).
, implies
Proof.
(x, y)
p,u1,q
(x, y)
p,u2,q
(2) If
p,u,q
θ
(a, y) ∧ ϕ(a, y)
v1and
.
(3) If
p,v,q
(a, y)
v ∈ ρ
, where
u ∈ U+. If
(a, y)
p,u,q
contains both irreversible and reversible arcs. This prop-
of the partially ordered set
(1)
For any types
hρ
u ∈ Uδ, and if
(1) If
and
θ
p,u2,q
; Ei
ν(p,q)
u ∈ ρ
If
u, v ∈ ρ
ν(p,q)
u1, u2∈ ρ
(x, y)
forms a upper semilattice.
isE-minimal if and only if for a
θ
p,u,q
ν(p,q)
u ∈ U0then
ν(p,q)
the labelvfor the formula
hU; Ei
(a, y)
and
is the supremum for the labels
p,u,q
(a, y)
is an isolating formula then the labeluisE-
ϕ(a, y)
and
such that the semi-isolating formulas
θ
(a, y) ∧ ¬ϕ(a, y)
p,u,q
v2of these formulas, we have
∩ U−then for any solutionbof the formula
ν(p,q)
|= p(a)
, the pair
θ
(a, y)
p,u,q
(a, b)
, where
u ∈ U0then the set of pairs
u E v
.
p, q ∈ S1(∅)
, the par-
is isolating.
u E v
then
v ∈ Uδ,
δ ∈
v ∈ U0.
then for the formulas
θ
(x, y)∨
p,u1,q
u1and
θ
p,u,q
u2.
(
a, y)is not isolating
are consistent. For
v16= v2,
v1C u
, and
is an irreversible arc.
u E v
, the pair
(a, b)
U−. Replacing arcs
u E v
, whence
and
(a, b)
for the formula
v ∈ U0.
v ∈ U+im-
¤
3.11.
PREORDERED ALGEBRAS
213
3.11.0.3. Denition.
sion to a preorder on the set
put
X E YifX = ∅
and for any
A
is transformed to the preordered algebra
y ∈ Y
tonic property with respect to its restrictions to the sets
and
U0.
The partial orderEhas a natural exten-
P(U)
, or for any
there is
x ∈ X
: for any sets
x ∈ X
with
there is
x E y
X, Y ∈ P(U)
y ∈ Y
. Thus, the algebra
hA; Ei
with the mono-
with
U≤0,
Another natural expansion of the already preordered algebra
hA; Ei
and
θ
p,u2,q
¬θ
θ
p,u1,q
inU. We denote these labels by
u1∧ ¬u2respectively. The last label is also denoted by
The label
the
intersectionorconjunction;u1∧ ¬u2is the
u2in
is based on the properties mentioned that if
v ∈ ρ
(x, y)
(x, y)
p,u2,q
(a, y) ∧ ¬θ
then the formulas
ν(q,r)
as well as
(if the formulas
(a, y)
p,u2,q
θ
θ
(x, y) ∧ θ
p,u1,q
are consistent for
p,u1,q,v,r
p,u2,q
θ
p,u1,q
u1◦ v,u1∨ u2,
union
u1◦ v
or the
is the
composition
disjunction
of labels
of labels
u1and
u1.
Clearly,
u1E u1∨ u2,
u2E u1∨ u2,
u1∧ u2E u1,
u1, u2∈ ρ
(x, y)
(x, y)
(a, y) ∧ θ
and
and
p,u2,q
|= p(a)
θ
p,u1,q
θ
p,u1,q
(a, y)
) have labels
u1∧ u2, and
u1andv;
u2;
u1∨ u2is
u1∧ u2is their
relative complement
u1∧ u2E u2,
¬u2∧ u1.
u1∧ ¬u2E u1.
We set
½
(p, (u1◦ v), r )
{u1◦ v},
∅,
if
u1∈ ρ
if
u1/∈ ρ
ν(p,q)
ν(p,q)
and
or
v ∈ ρ
v /∈ ρ
ν(q,r)
ν(q,r)
we
x E y
U≥0,
ν(p,q)
(x, y)∨
(x, y) ∧
and
of
,
,
(p, (u1∨ u2), q)
(p, (u1∧ u2), q)
(p, (u1∧¬u2), q)
{u1∨ u2},
{u1},
{u2},
∅,
{u1∧ u2},
∅,
{u1∧ ¬u2},
∅,
if
u1∈ ρ
if
u1∈ ρ
if
u1/∈ ρ
if
u1/∈ ρ
if
u1∈ ρ
|= ∃y(θ
otherwise
if
u1∈ ρ
|= ∃y(θ
¬θ
p,u2,q
otherwise
ν(p,q)
ν(p,q)
ν(p,q)
ν(p,q)
ν(p,q)
p,u1,q
,
p,u1,q
(a, y)),
and
and
and
and
, u2∈ ρ
(a, y) ∧ θ
, u2∈ ρ
ν(p,q)
(a, y)∧
,
u2∈ ρ
u2/∈ ρ
u2∈ ρ
u2/∈ ρ
ν(p,q)
p,u2,q
ν(p,q)
ν(p,q)
ν(p,q)
ν(p,q)
ν(p,q)
and
(a, y)),
and
,
,
,
,
214
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
(p, (X1τ X2), q) ∪{(p,(u1τ u2), q) | u1∈ X1, u2∈ X2},
τ ∈ {◦, ∨, ∧},
(p, (X1∧ ¬X2), q) (p,(¬X2∧ X1), q)
∪{(p, (u1∧ ¬u2), q) | u1∈ X1, u2∈ X2}, X1, X2∈ P(U).
Labels
the labels
The preordered algebra
(p, (· τ ·), q),τ ∈ {∨, ∧, ◦}
called a
the composition
For any types
u1and
u1and
u2are
consistentifu1∧ u2∈ U
u2are called
hA; Ei
, and
. If
u1∧ u2= ∅
inconsistent
.
equipped with binary operations
(p, (· ∧ ¬ ·), q),p, q ∈ S1(∅)
, is
preordered algebra with relative set-theoretic operations and
or briey a POSTC
p, q ∈ S1(∅)
-algebra
the structure
.
hρ
∪ {∅}; ∨, ∧, ∅i
ν(p,q)
with operations∨and∧on labels, being extended by equalities
u∨∅ = u,u∧∅ = ∅
i. e., a
distributive lattice with zero∅and relative complements
such that for any
then
u ∧ u0= ∅
then
u = v
.
3.11.0.4. Denition.
labelifu
v E u
is aE-minimal element inU, i. e., for any label
then
v = u
, where
u, v ∈ ρ
and
u ∨ u0= v
.
u ∈ ρ
ν(p,q)
A label
if
ν(p,q)
u E v
∪{∅}
, is an Ershov algebra,
and
u0= ¬u ∧ v
, and if the label
u ∈ U
is an
is a label
u0does not exist
atom
or an
atomic
v ∈ U
[13]
, if
By Proposition 3.11.0.2, the set of atoms equals the set of iso-
lating labels and, thus, each atom
isolated formula
θ
p,u,q
(a, y)
, where
u ∈ ρ
|= p(a)
is represented by an
ν(p,q)
.
3.11.0.5. Denition.
LetRbe a nonempty family of types in
S1(∅),ARbe a restriction of POSTC-algebraAto the familyR.
The structure
label
u ∈ ρ
POSTC-algebraAis calledR-atomic
then theR-atomic POSTC-algebra is called
A
is
R
there is an atom
ν(p,q)
atomic
if for any types
v ∈ ρ
ν(p,q)
if
A
is atomic. If
R
p, q ∈ R
such that
atomic
.
and for any
v E u
. The
R = S1(∅)
Using the denition of atomic structure, ofR-atomic POSTC-
algebra, and of small theory we obtain the following assertions.
3.12.
RANKS AND DEGREES OF SEMI-ISOLATION
215
3.11.0.6. Proposition.
S1(∅)
and for any type
a realization ofp, then the
3.11.0.7. Corollary.
If
R
is a nonempty family of types in
p ∈ R
, there is an atomic model
POSTC
-algebraAisR-atomic.
IfTis a small theory then the
Mpover
POSTC
algebraAis atomic.
3.12. Ranks and degrees of semi-isolation
The following denition is a local variation of Morley rank [310].
3.12.0.1. Denition.
u ∈ U ∪ {∅}
isolation
(1)
si(p, u, q) = 0ifu /∈ ρ
(2)
si(p, u, q) ≥ 1ifu ∈ ρ
, we dene inductively the
:
(3) for a positive ordinalα,
{vi| i ∈ ω}
of pairwise inconsistent labels such that
si(p, vi, q) ≥ α,i ∈ ω
(4) for a limit ordinalα,
β ∈ α
.
As usual, we write
si(p, u, q) 6≥ β
for
α ∈ β;si(p, u, q) ∞ifsi(p, u, q) ≥ α
ordinalα.
If typespandqare xed, we write
and this value is said to be the
of the labeluor of the element
(p, q)
). For a formula
For triples
;
θ
p,u,q
(p, u, q)
;
ν(p,q)
;
ν(p,q)
si(p, u, q) ≥ α + 1
, where
rank
p, q ∈ S1(∅)
si(p, u, q)
if there is a set
si(p, u, q) ≥ αifsi(p, u, q) ≥ β
si(p, u, q) = α
if
si(p, u, q) ≥ α
si(u)
instead of
rank of semi-isolation
(x, y)
u = ∅
we set
(with respect to the pair
si(θ
(x, y)) si(u)
p,u,q
of semi-
viC u
si(p, u, q)
or thesi-rank
-
,
and
for any
and
for any
.
Clearly, if the theory is small then thesi-rank of each label is
a countable ordinal (having a labeluwith
continuum many complete types
r(x, y) ⊃ p(x) ∪ q(y)
si(p, u, q) ≥ ω1, we get
).
By the denition we have the following inequality for any for-
mula
θ
(x, y)
p,u,q
the Morley rank of the formula
and any realizationaofpgiving a low bound for
θ
si(θ
(x, y)) ≤ MR(θ
p,u,q
p,u,q
(a, y)
by thesi-rank:
(a, y)) + 1.
p,u,q
The inequality (3.9) implies
(3.9)
216
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.12.0.2. Remark.
si
-ranks of labels in
MR(x ≈ x) + 1
.
3.12.0.3. Denition.
{∅}}, si(p) si(p, p).
si(R) sup{si(p, q) | p, q ∈ R}
si(R) = 1
or thesi-rank
or thesi-rank
. The value
of pair
of the familyR.
Since there are
If a theoryThas a nite Morley rank then
S
p,q∈S1(T )
We set
ρ
si(p, q) sup{si(p, u, q) | u ∈ U ∪
are bounded by the value
ν(p,q)
For a nonempty familyRof1-types, we put
. A familyRis calledsi-minimal
si(p, q)
(p, q)
|T |
is said to be the
, and
si(R)
is the
rank of semi-isolation
rank of semi-isolation
formulas of a theoryTand the inequality
(3.9) holds we obtain
3.12.0.4. Proposition.
to∞or less than
MR(x ≈ x)
more than
α + 1
min{|T |+, (MR(x ≈ x)+1)+}
is equal to an ordinalαthen anysi-rank inTis not
.
Eachsi-rank in a theoryTis either equal
. If the Morley rank
The estimation forsi-ranks in Proposition 3.12.0.4 can be far
from exact. For instance,si-ranks inω-categorical theories are
nite while there are non-ω-stableω-categorical theories.
if
3.12.0.5. Proposition.
For any types
p, q ∈ S1(∅)
assertions are satised.
(1)Ifu, v ∈ ρ
(2)Ifu, v ∈ ρ
si(u ∧ v) ≤ min{si(u), si(v)}
the equality if and only if there is a label
and
si(v0) = si(u)orsi(v0) = si(v)
(3)
The equality
∪ {∅}
ν(p,q)
∪ {∅}
ν(p,q)
si(p, q) = 0
and
u E v
then
si(u ∨ v) = max{si(u), si(v)}
then
si(u) ≤ si(v)
. The last inequality is transformed to
v0such that
.
holds if and only if there is no
realization ofpsemi-isolating realizations ofq.
(4)
The equality
(p → q)
-formula witnessing that a realization ofpsemi-isolates a
realization ofqand each such a
alent to a disjunction of formulas
ϕi(a, y)
is isolating, where
Proof
is obvious.
si(p, q) = 1
|= p(a)
¤
holds if and only if there is a
(p → q)
ϕi(x, y)
-formula
such that each formula
.
, the following
.
v0E u,v0E v
ϕ(x, y)
is equiv-
and
,
3.12.
RANKS AND DEGREES OF SEMI-ISOLATION
217
3.12.0.6. Proposition.
For any nonempty family
R ⊆ S1(∅)
the
following assertions are satised.
(1) si(R) ≥ 1
(2)
The family
p, q ∈ R
each
of formulas
where
|= p(a)
Proof.
type
p ∈ S1(∅)
isolates itself, where
Proposition 3.12.0.5, (4).
3.12.0.7. Remark.
set of solutions for any formula
semi-isolating formula
of some isolating formulas
junction of isolating formulas
a nonprincipal type then the representation of
.
R
issi-minimal if and only if for any types
(p → q)
ϕi(x, y)
-formula
ϕ(x, y)
is equivalent to a disjunction
such that each formula
.
(1) is implied by the inequality
since the formula
|= p(a)
. (2) is an obvious corollary of (1) and
(a ≈ y)
witnesses thatasemi-
¤
Since for a strongly minimal theoryTthe
ψ(a, y)
ψi(a, y)
ϕ(a, y)
is represented as a nite disjunction
or as a negation of a nite dis-
ψi(a, y)
is nite or conite, any
. If
ψ(a, y) ` p(y)
ϕi(a, y)
si(p) ≥ 1
ψ(a, y)
is isolating,
for any
and
p(y)
is
is possible
only as a nite disjunction of isolating formulas. It means that
si(p) = 1
pairwise non-equivalent isolating formulas
ψ(a, y) ` p(y)
pairwise non-equivalent isolating formulas
. If
p(y)
is a principal type and there are nitely many
then
si(p) = 1
ψ(a, y)
, too. If there are innitely many these
ψ(a, y)
with
then
si(p) = 2.¤
|= p(a)
and
3.12.0.8. Denition.
labels in
ρ
ν(p,q)
such that
Letαbe a positive ordinal,
si(u1) = si(u2) = α
u2areα-almost identicor∼α-equivalent
si(u1÷ u2) < α
3.12.0.9. Proposition.
tion for any set of labels in
Proof.
checking transitivity we assume that
, where
u1÷ u2 (u1∧ ¬u2) ∨ (¬u1∧ u2)
The relation
ρ
ν(p,q)
Clearly the relation
having thesi-rankα.
∼αis reexive and symmetric. For
u1∼αu2and
(u1∧ ¬u2∧ u3) E (u1∧ ¬u2) E (u1÷ u2)
si(u1∧ ¬u2∧ u3) ≤ si(u1÷ u2) < α.
As
u1∧u3= (u1∧u2∧u3)∨(u1∧¬u2∧u3)
for
u1∼αu3, it is enough to prove that
si(u1∧u2∧u3) = α
. The labels
(denoted by
u1and
u1∼αu2) if
u2be
u1and
.
∼αis an equivalence rela-
u2∼αu3. Since
we have
and
si(u1∧¬u2∧u3) < α
. Suppose
,
218
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
on contrary that
si(u1∧ u2∧ u3) < α
. Then
si(u1∧ u2) = α
and
u1∧ u2= (u1∧ u2∧ u3) ∨ (u1∧ u2∧ ¬u3)
imply
(u2÷ u3)
obtained contradiction means that
3.12.0.10. Denition.
having thesi-rankα, there is a greatest number
si(u1∧ u2∧ ¬u3) = α
, and
si(u2÷ u3) < α
By the denition, for any label
. But
gives
(u1∧ u2∧ ¬u3) E (u2∧ ¬u3) E
si(u1∧ u2∧ ¬u3) < α
u1∼αu3.
¤
. The
u ∈ ρ
n ∈ ω \ {0}
of pairwise inconsistent (or, that equivalent, of pairwise non-
equivalent) labels
i = 1, . . . , n
. This numbernis called the
or thesi-degree
by
deg(u)
. We have
3.12.0.11. Proposition.
deg(u)
un∈ ρ
is equal to the number of pairwise inconsistent labels
having thesi-rankα, thesi-degree1, and such that
ν(p,q)
u1, . . . , unsuch that
uiE u
and
degree of semi-isolation
of the labeluand it is denoted by
si(∅) = 0
and put
(1)Ifu ∈ ρ
deg(∅) 1
and
ν(p,q)
si(ui) = α
deg(p, u, q)
.
si(u) = α
u1, . . . ,
u = u1∨ . . . ∨ un.
(2)Ifu, v ∈ ρ
deg(v)
.
(3)Ifu, v ∈ ρ
ν(p,q)
ν(p,q)
,
si(u) = si(v)
and
si(u) = si(v)
, and
then
u E v
then
deg(u) ≤
ν(p,q)
∼α-
,
or
then
deg(u ∨ v) ≤ deg(u) + deg(v).
The equality in this inequality holds if and only if
If
si(u ∧ v) = si(u)
then
deg(u ∨ v) = deg(u) + deg(v) − deg(u ∧ v).
(4)Ifu ∈ ρ
an atom, then
(5)
If for a label
si(u) = 1
is a label for an isolating formula, i. e.,uis
ν(p,q)
and
u ∈ ρ
deg(u) = 1
,
ν(p,q)
si(u) = 1
.
and
deg(u) = 1
is not neutral.
(6).Ifu ∈ ρ
ber of pairwise inconsistent labels
formulas such that
Proof
is obvious.
and
ν(p,q)
si(u) = 1
u = u1∨ . . . ∨ un.
¤
then
deg(u)
u1, . . . , un∈ ρ
is equal to the num-
si(u ∧ v) < si(u)
, then
for isolating
ν(p,q)
.
u
3.12.
RANKS AND DEGREES OF SEMI-ISOLATION
219
3.12.0.12. Denition.
si(u)
then the
the pair
(p, q)
degree of semi-isolation
is
sup{deg(u) | u ∈ ρ
deg(p) deg(p, p)
.
If there is a label
or thesi-degree
, si(p, q) = si(u)},
ν(p,q)
u ∈ ρ
If for a nonempty familyRof1-types there is a label
p, q ∈ R
thesi-degree
, with
Clearly, if
si(R) = si(u)
deg(R)ofR
is
sup{deg(u) | u ∈ ρ
deg(p, q)ordeg(R)
then the
ν(R)
degree of semi-isolation
, si(R) = si(u)}.
exist then these values are posi-
tive natural numbers or equalω.
3.12.0.13. Denition.
and a set
X ∈ {U,U ∪ {∅}}
For an ordinalα, a natural number
we put
X ¹ (α, n) {u ∈ X | si(u) ≤ α
and if
si(u) = α
then
deg(u) < n},
[
X ¹ (α, ω) X ¹ α
X ¹ (α, n).
n≥1
ν(p,q)
with
deg(p, q)
u ∈ ρ
si(p, q) =
of
ν(p,q)
or
n ≥ 1
,
,
Clearly, if
limit ordinal then
3.12.0.14. Denition.
{0}
, and for the algebraAof distributions of binary semi-isolating
α = β + 1
X ¹ (α, 1) =
then
X ¹ (α, 1) = X ¹ β
S
X ¹ β
β<α
For ordinals
.
α, β
, where
, and ifαis a
β ∈ (ω + 1) \
formulas of a theoryTas well as for expansions and restrictions
0
A
ofA, dened in the previous sections, we denote by
and
A0¹ (α, β)
as well as by
algebras to the set
are denoted by
called the
A ¹ α,A0¹ α,Aα, and
(α, β)
-restrictions
A
and
α,β
(U ∪ {∅}) ¹ (α, β)
and theα-restrictions
0
A
the restrictions of these
α,β
. If
β = ω
A
, these restrictions
0
. The restrictions are
α
A ¹ (α, β)
respectively.
220
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Since thesi-rank of each label is positive, nontrivial restrictions
(i. e., with nonempty sets of used labels) are only the restrictions of
algebras with
ation the inequality
α > 0
. If
si(S1(∅)) = α0then, taking into consider-
α > 0
, all essential (i. e., reecting links of sets
of labels of semi-isolating formulas with respect to theirsi-ranks)
restrictions of these algebras are formed only for
0 < α < α0.
In view of Proposition 3.12.0.11 we obtain
3.12.0.15. Proposition.
isolating
The algebra
formulas of theoryTcoincides with the algebra
A ¹ 1
consists of labels being disjunctions of labels of
The algebra of distributions of binary
A ¹ (1,2)
isolating formulas.
3.13. Monoid of distributions of binary semi-
isolating formulas on a set of realizations
of a type
Consider a complete theoryT, a type
labelling function
sets
SIp(u1, . . . , uk)
u1, . . . , uk∈ ρ
ν(p):SICF(p)/SICE(p) → U
of labels of binary semi-isolating formulas,
,
k ∈ ω
ν(p)
.
Below we show some basic properties for sets
du1, . . . , uke SIp(u1, . . . , uk).
3.13.0.1. Proposition.
ρ
, the following equalities hold:
ν(p)
(Associativity).
p(x) ∈ S(T)
, and a family of
For any
, a regular
u1, u2, u3∈
.
ddu1, u2e, u3e = du1, u2, u3e = du1, du2, u3ee.
Proof
of inclusions
ddu1, u2e, u3e ⊆ du1, u2, u3e
du1, du2, u3ee ⊆ du1, u2, u3e
is identical to the proof of Proposition 3.3.0.1.
The reverse inclusions are satised since, taking labels
for the formulas
θ
u1,u
(x, y)
2
and
θ
u2,u
(x, y)
3
, we obtain, for
and
v1and
|= p(a)
v
2
,
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