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3.15.
POSTC
Note
that
are atomic.
-MONOIDS
POSTC
-monoids with ordinals
231
sup{si(u) | u ∈ U}
3.15.0.3. Theorem.
oryTwith a type
ν(p)
such that
M
and the operations of
T
is small.
Proof
follows the same scheme as the proof of Theorem 3.6.0.2
and, for the structure
For any
p(x) ∈ S1(T )
= M
ν(p)
. If the alphabet is at most countable
M
do not force continuum many types then
hP(U) \{∅}; ·i
POSTC
-monoid
M
there is a the-
and a regular labelling function
, it is identical to this proof
1,2
word for word. Since the proof of Theorem 3.6.0.2 is voluminous
we only point out the distinctive features leading to the proof of
this theorem.
1. A binary predicate
{∅}
. This predicate links only elements of the same colors if
and denes a
Qu-ordered coloring
2. For any elements
Quis dened for each element
Colifu ∈ U−∪ U0;
u, v ∈ U ∪ {∅}
the following condition is
u ∈ U ∪
u ≥ 0
Q∅= ∅
.
u E0v ⇔ Qu⊆ Qv.
3. For any elements
u, v ∈ U ∪ {∅}
the following conditions
hold:
u1∨ u2= v ⇔ Q
u1∧ u2= v ⇔ Q
u1∧ ¬u2= v ⇔ Q
u
u
1
1
u
1
∪ Q
∩ Q
\ Q
u
2
u
2
u
2
= Qv,
= Qv,
= Qv,
,
u1◦ u2= v ⇔ Q
In particular, the predicates
if
u1∧ u2= ∅.¤
3.15.0.4. Remark.
Since labels
mulas admit complements in
Q
ρ
◦ Q
u
1
and
u
1
u ∈ ρ
only for principal typesp(and
ν(p)
= Qv.
u
2
Q
are disjoint if and only
u
2
for semi-isolating for-
ν(p)
these complements are dened relative to the isolating formula of
p
), unlikeI-groupoids, if a
set
U≥0, it admits a representation in a transitive theoryTwith
a (unique) type
such that
M
form a Boolean algebra.
M
p(x) ∈ S (T )
= M
ν(p)
POSTC
-monoid
M
is constructed by a
and a regular labelling function
ν(p)
if and only if the set-theoretic operations in
232
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.16. Partial
POSTC
-monoid on a set of real-
izations for a family of1-types of a com-
plete theory
In this section, the results above for a structure of a type, as
well as for isolating formulas, are generalized for a structure on the
set of all realizations for a family of types.
3.16.0.1. Denition.
S1(T )
. We denote by
ν(p, q):SICF(p, q)/SICE(p, q) → U, p, q ∈ R,
As in Proposition 3.13.0.1, the partial (for
SI
on the set
R × P(U ) × R
(p1, u1, p2), . . . , (pk, uk, p
ρ
ν(pk,p
k+1
∪ {∅}
)
, to the set of triples
v ∈ SI(p1, u1, p2, u2, . . . , pk, uk, p
LetRbe a nonempty family of types in
ν(R)
a regular family of labelling functions
[
ρ
ν(R)
p,q∈R
ρ
ν(p,q)
.
|R| > 1
) function
, which maps each tuple of triples
)
k+1
, where
u1∈ ρ
(p1, v, p
ν(p1,p2)
k+1
k+1
∪ {∅}, . . . , uk∈
)
, where
),
is associative:
SI(SI(p1, u1, p2, u2, p3), u3, p4) =
= SI(p1, u1, p2, u2, p3, u3, p4) =
= SI(p1, u1, SI(p2, u2, p3, u3, p4))
for
u1∈ ρ
ν(p1,p2)
∪ {∅},u2∈ ρ
ν(p2,p3)
∪ {∅},u3∈ ρ
ν(p3,p4)
∪ {∅}
Consider the structure
M
hR × P( U ) × R; ·, E, ∨, ∧, (· ∧ ¬ ·), ◦i
ν(R)
with the partial operations·and◦such that
(p1, X1, p2) · (p2, X2, p3) =
=[{(p1, u1, p2) · (p2, u2, p3) | u1∈ X1, u2∈ X2},
(p1, u1, p2) · (p2, u2, p3) = {(p1, v, p3) | v ∈ SI(p1, u1, p2, u2, p3)},
(3.10)
.
3.16.
PARTIAL
POSTC
-MONOID
(p1, X1, p2) ◦ (p2, X2, p3) =
=[{(p1, u1, p2) ◦ (p2, u2, p3) | u1∈ X1, u2∈ X2},
(p1, u1, p2) ◦ (p2, u2, p3) = {(p1, u ◦ v, p3)},
233
u1∈ ρ
ν(p1,p2)
∪ {∅}, u2∈ ρ
ν(p2,p3)
∪ {∅},
as well as the relationEof preorder, being induced by the par-
tial order, of the same name, on the set of labels and the partial
operations
∨, ∧, (· ∧ ¬ ·)
such that
(p, X, q) ∨ (p, Y, q) =[{(p, u, q) ∨ (p, v, q) | u ∈ X, v ∈ Y },
(p, u, q) ∨ (p, v, q) = {(p,u ∨ v, q)},
(p, X, q) ∧ (p, Y, q) =[{(p, u, q) ∧ (p, v, q) | u ∈ X, v ∈ Y },
(p, u, q) ∧ (p, v, q) = {(p,u ∧ v, q)},
(p, X, q) ∧ ¬(p, Y, q) =[{(p, u, q) ∧ ¬(p, v, q) | u ∈ X, v ∈ Y },
(p, u, q) ∧ ¬(p, v, q) = {(p,u ∧ ¬v, q)},
u, v ∈ ρ
The
POSTC
-monoids
M
ν(p)
,
this structure. The structure
monoids
M
ν(p)
,
p ∈ R
, relative to the family
functions and it is denoted by
L
the join
POSTC
M
p∈R
ν(p)
-monoids
is
M
free
, it is represented as the disjoint union of
and denoted by
ν(p)
p ∈ R
∪ {∅}.
ν(p,q)
, are naturally embeddable into
M
L
p∈R
ν(R)
M
ν(p)
is called a
. If
ρ
ν(p,q)
join of
ν(R)
= ∅
of labelling
for all
F
M
.
p∈R
ν(p)
POSTC
p 6= q
-
By (3.10) we have
3.16.0.2. Proposition.
any nonempty family
lar family
SI(p1, ·, p2, ·, p3. . . , pn, ·, p
ν(R)
of labelling functions, eachn-ary partial operation
L
a term of the structure
p∈R
For any complete theoryT, for
R ⊂ S(T )
)
n+1
M
ν(p)
of1-types, and for any regu-
on the set
with xed types
P(U)
is interpretable by
p1, . . . , p
n+1
∈ R
.
234
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Denote by
SI
the restriction of
ν(R)
M
to the partial oper-
ν(R)
ation·.
Using Proposition 3.10.0.4 we obtain the following analogue of
Proposition 3.13.0.4.
3.16.0.3. Proposition.
nonempty family
ν(R)
of labelling functions, the restriction of the structure
to the set
U≤0(
R ⊂ S(T )of1
respectively
For any complete theoryT, for any
-types, and for any regular family
U≥0,
U≥0∪ U0)
is closed under the
partial operation·.
By Proposition 3.16.0.3 the structure
tures
with
≤0
SI
ν(R)
,
SI
u ≤ 0,u ≥ 0
Here, for any triple
attributed to
SI
≥0
ν(R)
≥0
and
ν(R)
, and
(p, u, q)inSI
.
≥0,neu
SI
ν(R)
u ∈ U≥0∪ U0respectively,
≥0
ν(R)
, generated by triples
, the triple
SI
has substruc-
ν(R)
(q, u−1, p)
p, q ∈ R
Replacing for the denition in Section 3.14 the function
to the family
deterministic,α-deterministic, almostα-deterministic
deterministic
ν(R)
of functions we obtain the notions of
structures
SI
ν(R)
, and
¹ U0.
Below we formulate a series of assertions that immediately
transformed from the class of structures
tures
SI
ν(R)
.
SI
to the class of struc-
ν(p)
SI
ν(R)
(p, u, q)
is also
ν(p)
(α, n)
(α, ω)
.
-
-
3.16.0.4. Proposition
is(almost)α
structure
-deterministic andβis a positive ordinal then the
(SI
ν(R)
¹ U0) ¹ β
3.16.0.5. Proposition.
α, β
, where
α, β > 0,β ∈ ω + 1
(Monotony).
is also(almost)α
For a structure
, the following conditions are
If a structure
equivalent:
(1)
the structure
(2)
for any types
ρ
ν(q,r),α,β
then
, the inequality
deg(u1◦ u2) < β
3.16.0.6. Corollary.
SI
ν(R)
is
(α, β)
p, q, r ∈ R
si(u1◦ u2) ≤ α
.
For a structure
-deterministic;
and labels
holds, and if
SI
ν(R)
nalα, the following conditions are equivalent:
SI
ν(R)
-deterministic.
SI
u1∈ ρ
ν(R)
ν(p,q),α,β
and ordinals
,
si(u1◦ u2) = α
and a positive ordi-
¹ U
u2∈
0
3.16.
ρ
ν(p,q),α
PARTIAL
(1)
the structure
(2)
si(u1◦ u2) ≤ α
,
u2∈ ρ
POSTC
ν(q,r),α
-MONOID
SI
is almostα-deterministic;
ν(R)
for any types
.
p, q, r ∈ R
and labels
235
u1∈
3.16.0.7. Corollary.
SI
ν(R)
is almost
si(R)
3.16.0.8. Proposition.
ministic then
SI
3.16.0.9. Proposition.Ifsi(R)
SI
ν(R)
is
si(R)
-deterministic if and only if the value
If
si(R)
-deterministic.
If a structure
SI
ν(R),α
is an ordinal then the structure
SI
ν(R)
¹ α
ν(R)
is also
(α, β)
is an ordinal then the structure
dened or equals1.
3.16.0.10. Proposition.
istic if and only if
SI
3.16.0.11. Denition.
1
-types of a theoryT,
tions, andαbe an ordinal,
locallyα-deterministic
is a natural number
n ≥ 2
ν(R),1,2
A structure
is(almost)1
P
is(almost)determin-
ν(R)
-deterministic.
LetRbe a nonempty family of complete
ν(R)
be a regular family of labelling func-
α > 0
. The structure
if for any nonempty nite set
such that the structure
deterministic.
Repeating the proof of Proposition 3.14.0.9 we obtain
is
(α, β)
-deter-
-deterministic.
deg(R)
SI
ν(R)
is called
R0⊆ R
SI
ν(R0)
is
(α, n)
is not
there
-
3.16.0.12. Proposition.
plete1-types of a theoryT,
functions,
(1)
(2)
(3)
(4)
for any
si(R) < ω
the structure
the set
the set
the set
p, q ∈ R
ρ
ν(p,q)
ρ
ν(p,q),1
ρ
ν(p,q),1,2
.
. The following conditions are equivalent:
SI
is nite for any
Let
R
be a nonempty family of com-
ν(R)
be a regular family of labelling
is locally1-deterministic;
ν(R)
p, q ∈ R
is nite for any
(
consisting of all atoms
p, q ∈ R
Lemma 1.3.1.3 and Proposition 3.16.0.12 imply
3.16.0.13. Corollary.IfR
types of a theoryT,
ν(R)
is a nonempty family of complete1-
is a regular family of labelling functions,
;
;
u ∈ ρ
ν(p,q)
)
is nite
236
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
and all
the structure
3.16.0.14. Corollary.IfT
ν(S1(∅))
SI
ordinalα, we denote by
of
Considering this equality and the equality
(p1, p2)
-types, where
SI
is locally1-deterministic.
ν(R)
p1, p2∈ R
is an almostω-categorical theory and
is a regular family of labelling functions then the structure
ν(S1(∅))
For a nonempty family
is locally1-deterministic.
R
of1-types in
SI
(in a model
R,α
SIRto the set of formulas ofsi-ranks
SI
{(a, b) | tp(a), tp(b) ∈ R
R,α
by a formula
Clearly,
IR= SI
θ
tp(a)
for any nonempty family
R,1
,u,
tp(b)
(x, y),
, are
MofT
≤ α
(p1, p2)
S(T)
:
-principal then
and a positive
) the restriction
andasemi-isolates
with asi-rank
≤ α}.
R
SI
ν(R),1,2
= P
b
of1-types.
, the
ν(R)
following proposition generalizes Proposition 3.14.0.13 as well as
Propositions 3.4.0.5 and 3.8.0.5.
3.16.0.15. Proposition.
1
-types of a theoryT,
LetRbe a nonempty family of complete
ν(R)
be a regular family of labelling func-
tions, andαbe a positive ordinal. The following conditions are
equivalent:
(1)
the relation
any model
(2)
the structure
M |= T)is transitive;
SI
(
on a set of realizations of types
R,α
SI
ν(R),α
is almostα-deterministic.
p ∈ R
in
Proof
word for word.
is identical to the proof of Proposition 3.14.0.13 almost
¤
Propositions 3.10.0.4 and 3.16.0.15 imply the following asser-
tions.
3.16.0.16. Corollary.
1
-types of a theoryT,
LetRbe a nonempty family of complete
ν(R)
be a regular family of labelling func-
tions, andαbe a positive ordinal. The following conditions are
equivalent:
(1)
ρ
ν(R),α
the relation
(2)
the structure
⊆ U≤0.
SI
, in any model
R,α
SI
ν(R),α
M |= T
, is a partial order;
is almostα-deterministic and
3.17.
POSTCR-STR
UCTURES
237
The partial order
The non-identical partial order
if
|ρ
ν(p),α
| > 1
for some type
of pairwise distinct types inRsuch that
or
|ρ
ν(p
n+1
| ≥ 1,n ∈ ω
,pn)
3.16.0.17. Corollary.
1
-types of a theoryT,
SI
is identical if and only if
R,α
SI
has innite chains if and only
R,α
p ∈ R
or there is a sequence
|ρ
ν(pn,p
ρ
| ≥ 1,n ∈ ω
),α
n+1
ν(R),α
pn,
.
LetRbe a nonempty family of complete
ν(R)
be a regular family of labelling func-
= {0}
n ∈ ω
tions, andαbe a positive ordinal. The following conditions are
equivalent:
(1)
the relation
any model
(2)
the structure
ρ
ν(R),α
⊆ U≥0.
M |= T
SI
on a set of realizations of types
R,α
is an equivalence relation;
SI
ν(R),α
is almostα-deterministic and
p ∈ R
in
The results above substantiate that the diagram in Figure 3.2
admits the transformation replacing the typepby a nonempty
family
3.17.
R ⊆ S1(∅)
.
POSTCR-structures
.
,
,
3.17.0.1. Denition.
be an alphabet consisting of a set
U+of
positive elements
LetRbe a nonempty set,
−
U = U
˙
∪ {0}˙∪ U
U−of
, a set
U0of
neutral elements
Ifpandqare elements inR, we write
for any element
u ∈ U+;
all pairs
U≤0 U−∪ {0},U≥0 U+∪ {0}
(p, q),p, q ∈ R
µ(p, q) ⊆ U
• 0 ∈ µ(p, q)
• µ(p, p) ∩ µ(q, q) = {0}
• µ(p, q) ∩ µ(p0, q0) = ∅ifp 6= q
u ∈ U−,
such that
if and only if
u > 0
and
, we consider a
p = q
;
for
p 6= q
and
(p, u, q) > 0
;
S
•
p,q∈R
µ(p, q) = U
.
+
0
˙
∪ U
negative elements
u < 0
and
for any element
. For the set
regular
family
(p, q) 6= (p0, q0)
, a set
, and zero0.
(p, u, q) < 0
R2of
µ(R)
of sets
;
238
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Below we write
operation·on the set
(p, u, q) · (q, v, r)
µ(p)
instead of
R × P(U) × R
instead of
µ(p, p)
, and considering a partial
we shall write, as above,
(p, {u}, q) · (q, {v}, r)
.
A structure
M = hR × P(U ) × R; ·, E, ∨, ∧, (· ∧ ¬ ·), ◦i
with a regular family
µ(R)
of sets is said to be a
POSTCR-structure
if the following conditions hold:
•
the partial operation·of the structure
values
(p, X, q)·(p0, Y, q0)
only for
p0= q,X ⊆ µ(p, q),Y ⊆ µ(p0, q0)
hR × P(U) × R; ·i
has
and it is generated by the function·for elements inU: for any sets
X, Y ∈ P(U),∅ 6= X ⊆ µ(p, q),∅ 6= Y ⊆ µ(q, r)
, the following
equality is satised:
(p, X, q) · (q, Y, r) =[{(p, x, q) · (q, y, r) | x ∈ X, y ∈ Y },
and if some of
•
each restriction
morphic to a
atoms
u ∈ µ(p)inM
X, Y
POSTC
is empty then
M
µ(p)
of
(p, X, q) · (q, Y, r) = ∅
M
to
{p} × P(µ(p)) × {p}
-monoid with the universe
are calledp-atoms
µ(p)
;
;
is iso-
P(µ(p)),p ∈ R
,
;
•
each restriction
M
µ(p,q)
,
p 6= q
, ofMto
{p}× P(µ(p, q))× {q }
has empty partial operations·and◦; the restriction of
the relationEis a preordered set
with the least element
(p, ∅, q)
is induced by the partial order
(forming a upper semilattice if
if
X, Y ∈ P(µ(p, q))
label
v ∈ Y
•
if
v E
u ∈ X
there is a label
a label
p,q
there is a label
u ∈ µ(p, q)
u
implies
µ(p, q)∩(U−∪U+)
lay under each label in
then
X E
p,q
v ∈ Y
u ∈ X
with
, where
v = u
may be
for any label
(p, q)
µ(p, q) ∩ U0, moreover, if only labels
µ(p, q) ∩ U≥0lay under a label
greatest labels among labelsv; only labels in
each label in
µ(p, q) ∩ U0;
h{p} × P(µ(p, q)) × {q}; E
, the preorder
0
E
on the set
p,q
µ(p, q) 6= ∅
Y
if and only if
with
u E
p,q
p 6= q
, is said to be a
-atoms; some labels in
E
of this structure
p,q
µ(p, q)
) by the following rule:
X = ∅
u E
v
v ∈ µ(p, q)
v
and for any label
p,q
;
; only labels in
µ(p, q)∩U
u ∈ µ(p, q) ∩ U0then there are no
µ(p, q) ∩ U0lay over
M
µ(p,q)
to
p,q
of labels
, or for any
(p, q)
-atom
≥0
v ∈
i
3.17.
POSTCR-STR
•
the operations
{∅}
in the structure
UCTURES
∨, ∧, (· ∧ ¬ ·)
M
relative complements on
u, v ∈ µ(p, q) ∪ {∅}
,
are dened on each set
and form a distributive lattice with
µ(p,q)
µ(p, q) ∪ {∅}
, moreover, for any elements
239
µ(p, q) ∪
u E
•
the relationEon the set
minimal elements
the union
preorders
labels in these structures: if
if and only if
there is a label
a label
u ∈ X
•
the partial operations
R×(U ∪{∅})×R
operations on the sets
{∅}inM
•
the partial operation◦is dened on the set
v ⇔ u ∧ v = u ⇔ u ∨ v = v ⇔ u ∧ ¬v = ∅;
p,q
R × P(U) × R
(p, ∅, q),p, q ∈ R
E
of preorders
U
E
in the structures
p,q
p = p0,
v ∈ Y
with
u EUv
E
in the structures
p
X, Y ∈ P(U )
q = q0, and
with
u EUv
;
∨, ∧, (· ∧ ¬ ·)
; this preorder is induced by
M
M
µ(p,q)
,
p, q ∈ R,p 6= q
then
X = ∅
or for any label
and for any label
are dened on the set
in the structureMbeing unions of corresponding
µ(p,q)
,
p 6= q
µ(p)∪ {∅}inM
;
and on the sets
µ(p)
is a preorder with
,
µ(p)
p ∈ R
, and of
, on sets of
(p, X, q)E (p0, Y, q0)
u ∈ X
v ∈ Y
there is
µ(p, q)∪
R×(U ∪{∅})×R
in the structureMbeing obtained from the union of corresponding
operations in the structures
sion: if
v ∈ µ(p, r)
is the
a
u1∈ µ(p, q)
such that
E
-greatest label in the set
p,r
composition
and
(p, u1, q) ◦ (q, u2, r) = (p, v, r)
of elements
M
µ(p)
u2∈ µ(q, r)
u1and
u2and it is denoted by
,
p ∈ R
, by the following exten-
then there is a unique element
; this element
(p, u1, q) · (q, u2, r)
, it is called
u1◦ u2;
v
(p, u1, q) ◦ (q, ∅, r) = (p, ∅, q) · (q, u2, r) =
= (p, ∅, q) · (q, ∅, r) = (p, ∅, r);
•
the partial operations
∨, ∧, ◦
on the set
R × P(U ) × R
are
induced by the corresponding partial operations on the set
R×(U ∪ {∅})×R
{∨, ∧, ◦}
then the value
dened and coincides with the set
Y }
, in which all values are dened; the partial operation
the set
R × P (U) × R
: if
(p, X, q), (p0, Y, q0) ∈ R× P(U) ×R
(p, X, q) τ (p0, Y, q0)
is not dened or it is
{(p, u, q) τ (p0, v, q0) | u ∈ X, v ∈
(· ∧ ¬ ·)
is also induced by the corresponding partial
and
τ ∈
on
240
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
operation on the set
R×P(U)×R
p = p0,
q = q0,
u ∈ X, v ∈ Y }
•
each of the sets
operations
∨
; if
u ∈ U−and
•
repeating the denition in Section 3.12, each label
then the value
X, Y ⊆ µ(p, q)
;
∨, ∧, (· ∧ ¬ ·)
obtains inductively the
degree of semi-isolation
R × (U ∪ {∅}) × R
(p, X, q)∧¬(p0, Y, q0)
and it is equal to
U−∪ {∅}
; the set
v ∈ U≥0then
(u ∨ v) ∈ U0;
rank of semi-isolation
deg(u),si(∅) = 0,deg(∅) = 1
: if
(p, X, q), (p0, Y, q0) ∈
is dened only for
{(p, u, q)∧¬(p, v, q) |
and
U≥0∪ {∅}
is closed under
U0is closed under the operation
u ∈ U
si(u) ≥ 1
and the
, as well as
the following attributes are dened: the equivalence relations
restrictions
0
M
,
M
α
ofsi-ranks
si
-degree
•
the restriction
M
is an
•ifu ∈ µ(p, q)
(r, v0, p) · (p, u, q)
and
v0∈ (r,p)
Xαand
0
for restrictions
α,β
≤ α
, and for labels ofsi-rankαto the set of labels of
< β
;
IR-structure;
;
X
of sets
α,β
M
hR × (P(U ) \ {∅}) × R; ·i
and
u < 0
X ∈ {U,U ∪ {∅}}
0
of the structureMto the set of labels
then the set
, and restrictions
of the structure
1,2
(p, u, q) · (q, v, r)
consist of negative elements for any
v ∈ µ(q, r)
∼α,
and
•ifu ∈ µ(p, q),v ∈ µ(q, r),u > 0
(p, u, q) · (q, v, r)
consists of elements in
, and
U≥0;
•ifu ∈ µ(p,q)∩ (U≥0∪U0),v ∈ µ(q, r) ∩(U≥0∪U0)
or
v ∈ U0, then
•
for any element
u−1of
and
then
(q, v2, r)
inverse
(q, 0, q) ∈ (q, u0, p) · (p, u, q)
u−1⊆ v−1;
•
if an element
, where
By the denition, each
POSTCR-substructures
the sets
U≤0and
(p, u, q) · (q, v, r) ⊆ U0;
u ∈ µ(p,q)
elements
u0> 0
with
such that
, moreover, if
(p, u, r)
, where
v1◦v2∈ U+, then
POSTCR-structure
≤0
M
and
U≥0respectively.
u > 0
there is a nonempty set
(p, 0, p) ∈ (p, u, q)·(q, u0, p)
u > 0
, belongs to a set
(r, u−1, p) ⊆ (r, v
≥0
M
being restrictions of
v > 0
0
u E
p,q
−1
2
, then the set
, and
u ∈ U
v
and
v ∈ U
(p, v1, q) ·
, q)·(q, v
M
−1
, p)
1
contains
M
to
0
+
.
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