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3.18.
ALGEBRAS FOR ISOLATED TYPES
241
3.17.0.2. Denition.
if for any label
v Epu
(p, q)
, and for any label
-atom
u ∈ µ(p),p ∈ R
v ∈ U
such that
A
POSTCR-structure
M
, there is ap-atom
u ∈ µ(p, q),p, q ∈ R,p 6= q
v E
p,q
u
.
is called
v ∈ U
such that
, there is a
atomic
Combining the proof of Theorems 3.6.0.2 and 3.9.0.2 as well as
the proof of Theorem 3.15.0.3, we obtain the following theorem.
3.17.0.3. Theorem.
theoryTwith a family
ν(R)
of labelling functions such that
and the familyRare at most countable, and the operations of
For any
POSTCR-structure
R ⊂ S(T )of1
M
there is a
-types and a regular family
M
ν(R)
= M
. If the alphabet
M
do not force continuum many types, thenTis small.
Note that, using the operation
be transformed for an arbitrary family of types in
·eq, the constructions above can
S(T)
.
3.18. Algebras of distributions of binary semi-
isolating formulas for families of isolated
types and for countably categorical the-
ories
In this Section, we apply a general approach for distributions of
binary isolating and semi-isolating formulas to families of isolated
types and to the class ofω-categorical theories.
3.18.0.1. Proposition.
R ⊆ S1(∅)
of isolated types, and a regular family
functions for semi-isolating formulas, the
consists of positive labels and zero, and each labeluhas a
u
such
R = {p}
then the monoid
labels)is generated by a Boolean algebra, for which
For any theoryT, a nonempty family
ν(R)
of labelling
POSTCR-structure
u = ∅
SI
and
ν(p)
u ∨ u
= hM
is
, ·i(for compositions of
ν(R)
u ∨
M
ν(R)
comple-
u
c
orre-
sponds to isolating formulas ofp.
S
Proof.
tion 3.10.0.2. By the denition, for any pair
The inclusion
p,q∈R
ρ
ν(p,q)
⊆ U≥0is proved in Proposi-
(p, q) ∈ R2of types,
242
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
each label
q(y)
, is a maximal element (among labels in
label
mains to note that for any isolated type
vq, corresponding to an isolating formula
u ∈ ρ
, the label
ν(p,q)
¬u ∧ vqis the complement
p(x)
ρ
). Then for any
ν(p,q)
, labels in
ϕq(y)
ρ
ν(p)
of type
u
.
form a
It re-
Boolean algebra with the least element∅and the greatest element
vpsuch that for any label
By
Ryll-Nardzewski theorem and Proposition 3.18.0.1, we get
3.18.0.2. Corollary.
family
R ⊆ S1(∅)
, and a regular family
u ∈ ρ
For anyω-categorical theoryT, a nonempty
for semi-isolating formulas, the
u = ∅
,
u ∧
ν(p)
ν(R)
POSTCR-structure
and
u ∨ u = vp.
¤
of labelling functions
M
ν(R)
is nite,
consists of positive labels and zero, and each labeluhas a comple-
u
ment
3.18.0.3.
.
Theorem.
For any
POSTCR-structureM, in which
each label is positive or zero and has a complement, there is a theory
T
, a nonempty family
family
that
ν(R)
M
ν(R)
Proof.
of labelling functions for semi-isolating formulas such
= M
.
Consider the construction for the proof of Theorem
R ⊆ S1(∅)
of isolated types, and a regular
3.17.0.3. We dene a family of isolated1-types bijective with the
setRby disjoint unary predicates
that if
|Colp| = 1
ρ
consists of one (nonempty) label, i. e.,
ν(p)
, and if
|ρ
ν(p)
| > 1
then
ements. Besides, we assume that each formula
Colp,
p ∈ R
. Here we assume
ρ
ν(p)
= {0}
, then
Colpcontains innitely many el-
Colp(x)
isolates a
type marked byp. Further scheme is based on a generic construc-
tion coordinated with operations in the
POSTCR-structureMand
with aE-ordering (isomorphic to the ordering of labels inM) of
formulas
realizations ofq,
cates
formulas
and any realizationaofp, we can dene formulas
θ
u,q
p,
θ
p,u,q
θ
(x, y)
p,u,q
, witnessing that realizations ofpsemi-isolate
u ∈ ρ
ν(p,q)
,
p, q ∈ R
, and forming with predi-
Colpthe language of theory under construction. Since the
Colq(y),q ∈ R
(a,
y)
so that these formulas complement each other in
(a, y)∧θ
(a,
u,q
p,
, are isolated, then for any label
y) `
and
θ
p,u,q
(a, y)∨θ
(a,
u,q
p,
u ∈ ρ
θ
(a, y)
p,u,q
Colq(y)
y) ≡ Colq(y)
ν(p,q)
. The
and
generic construction allows to get a required theory with quantier
elimination.
¤
:
3.19.
ALGEBRAS FOR STRONGLY MINIMAL THEORIES
243
Having a nite
POSTCR-structure
M
the schema above, gen-
erating a theory with quantier elimination, produce a required
theory beingω-categorical:
3.18.0.4. Corollary.
For any nite
POSTCR-structureM, in
which each label is positive or zero and has a complement, there is
anω-categorical theoryT, a nonempty family
regular family
las such that
3.18.0.5. Remark.
ν(R)
M
of labelling functions for semi-isolating formu-
= M
ν(R)
.
Note that if
u1, . . . , unare all labels linking
R ⊆ S1(∅)
, and a
realizations of1-typespandqby principal arcs, then for any label
u = u
where
∨ .. . ∨ u
i
1
the label
i
k
{i1, . . . , ik}, {j1, . . . jl}
u = u
is a partition of the set
∨ .
. . ∨ u
j
1
is its complement,
j
l
{1, . . . , n}
Thus Corollaries 3.18.0.2 and 3.18.0.4 can be reformulated omit-
ting phrases about complements.
Besides, since in any nite
reduced to labels of isolating formulas, the structure
POSTCR-structureMall labels are
M
is uniquely
determined by its subalgebra of distributions for isolating formulas.
3.19. Forcing of innity and algebras of distri-
.
butions of binary semi-isolating formu-
las for strongly minimal theories
In this Section, we describe properties of algebras of distribu-
tions of binary semi-isolating formulas for strongly minimal theo-
ries [89]. Recall that strongly minimal theories on binary level are
investigated in [210, 240, 241].
LetTbe a theory,
be a regular family of labelling functions for semi-isolating formulas
forming a setUof labels. Denote by
set of labelsu, each of which, being in
(co)nite set of labels in
By the denition all almost deterministic labels belong to
R ⊆ S1(∅)
ρ
.
ν(p,q)
be a nonempty family, and
Ufin(respectively
S
ρ
,E-dominates a
p,q∈R
ν(p,q)
U
cofin
ν(R)
) the
Ufin.
244
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.19.0.1. Denition.
innite set of solutions for the formula
We say that a label
θ
(a, y),|= p(a)
p,u,q
u ∈ ρ
ν(p,q)
forces
an
, if for any
theoryTwith a familyRof1-types, containingpandqand having
a
POSTCR-structure, including all labels that areE-dominated by
u
, the formula
By the denition a label
tions if and only if for any
the set of solutions of
|= p(a)
, there exists new element
and links between
θ
p,u,q
(a, y)
a
n+1
has innitely many solutions.
θ
u ∈ ρ
n ∈ ω
(a, y)
p,u,q
and
(in an arbitrary structure), where
a1, . . . , anare dened by some labels.
forces an innite set of solu-
ν(p,q)
and somenelements
a
, for which
n+1
a1, . . . , anin
|= θ
p,u,q
(a, a
n+1
Clearly, almost deterministic labels do not force innite sets
of solutions and any labelu,E-dominating innitely many la-
bels
vi, forces an innite set of solutions. Moreover, each label
u ∧ ¬v
∧ . . . ∧ ¬v
i
1
also forces an innite set of solutions. An-
i
n
other examples with labels, which force innite sets of solutions,
are series of theories of powerful digraphs, in particular, the theory
Th(hQ, <i)
, for which any nonzero label (dening the strict order
property) corresponds to formulas having only innitely many so-
lutions. An innite set of solutions can be forced by labels in
U
fin
for formulas in stable theories. Such an example is produced by
any label corresponding to a special element of an innite group
for an everywhere nitely dened polygonometry [54].
)
3.19.0.2. Denition
oryTis called
strongly minimal
guage obtained by adding parameters of
to the language ofT, either
(J. T. Baldwin, A. H. Lachlan [89]). A the-
a)
if for any formula
a
a)
ϕ(x,
,
or
¬ϕ(x,
(in
some model
a)
ϕ(x,
has
nitely many
of
lan-
M |= T
solutions.
An example of strongly minimal theory with the forcing of in-
nite set is represented by structure
hM; si
with
successor func-
tions(having exactly one preimage for any element, and do not
having cycles). Since
bel
u ∈ U
E
-dominates innitely many labelsvcorresponding to formulas
for the semi-isolating formula
cofin
(y ≈ sn(x)),n ∈ Z
for the formulas
Th(hM; si)
has unique1-type, there is a la-
(x ≈ x)
. This label
, and thus,uforces an innite set of solutions
θu(a, y),a ∈ M
.
)
3.19.
ALGEBRAS FOR STRONGLY MINIMAL THEORIES
245
3.19.0.3. Theorem.
ily
R S1(∅)of1
For any strongly minimal theoryT, the fam-
-types, and a regular family
ν(R)
of labelling
functions for semi-isolating formulas, the following conditions hold:
(a)
the
POSTCR-structure
Ufin∪ U
(b)
;
cofin
there is unique type
tions; in particular, any set
q 6= r0, all labels
u ∈ ρ
ν(p,q)
M
r0∈ R
ρ
ν(p,q)
consists of labels belonging to
ν(R)
having innitely many realiza-
is nite, where
p, q ∈ R
and
are almost deterministic and belong to
Ufin;
(c)ifR
is nite, i. e., all types in
R
are principal, then all
nonzero labels are positive and all labelsu, including zero, have
complements
,
and for any pair of labels
u ∈ ρ
u,
ν(p,r0)
,
exactly
u
one of them is almost deterministic and, in particular, belongs to
Ufin, and the other label marks a formula
many solutions, where
(d)ifR
is innite, i. e.,
|= p(a)
, and belongs to
r0is unique nonprincipal1-type, then
all nonzero labels, linking realizations of
in
R \ {r0}
realizations of types in
u
belongs to
does not have complements and
ρ
ν(r0)
(e)
, are positive, and labels, linking realizations of
ρ
ν(p,r0)
R\ {r0}
thenuis positive or zero, almost deterministic,
, are negative; in this case, if a label
p = r0, moreover,
is nite, and
only labels in
U
cofin
ρ
ν(p,r0)
= ∅ifρ
ν(r0)
with the principal type
θ
p,·,r
(a, y)
0
U
with innitely
;
cofin
r0or realizations of types
r0with
Ufin= U
is innite;
r0can force
cofin
innitely many solutions.
if
Proof.
formula
isolation for a type
solutions. For the nite set
the formula
only nitely many labels in
label
u ∈ ρ
ρ
except for a nitely many
ν(p,q)
plement
of labels
Ufin∪ U
By the denition of strongly minimal theory, each
ϕ(a, y)
, where
ϕ(x, y)
ν(p,q)
with
for the formula
|= p(a)
q(y)
, (in particular, witnessing the semi-
) has a nite or a conite set
ϕ(a, M)
ϕ(a, y) ` q(y)
ρ
ν(p,q)
, the label
and
. If
ϕ(a, M)
ϕ(x, y) E
|= p(a)
-dominates all labels in
ϕ(a, M)
u ∈ ρ
ν(p,q)
, marking
canE-dominate
is conite, then the
u1, . . . , uk, anduhas the com-
u
,
which is obtained fromuby disjunctive attachment
ui. Thus the condition (a) holds: all labels belong to
.
cofin
SinceTis strongly minimal we also have that there is unique
type
r0∈ R
, principal or nonprincipal, with innitely many re-
of
246
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
alizations: if there are nitely many1-types it is implied by the
property that models are innite and there are no two principal for-
mulas with innitely many realizations, and if there are innitely
many1-types, then the nonprincipal type
r0(x)
, having innitely
many realizations by Compactness, is isolated by the set of all for-
mulas
¬ϕ(x)
, where
ϕ(x)
are principal formulas and none of these
formulas can not have innitely many solutions.
Since the type
then any set
labels
u ∈ ρ
ν(p,q)
r0with innitely many realizations is unique,
ρ
is nite, where
ν(p,q)
p, q ∈ R
and
q 6= r0. Here all
are almost deterministic and belong to
Ufin. Thus,
we have the condition (b).
If
r0is isolated then all1-types are isolated and by Proposition
3.10.0.2 all nonzero labels are positive. Since each isolating formula
ϕ(x)
has a label, all labels, including zero, have complements. In
this case for any pair of labels
u,
u ∈ ρ
labels is almost deterministic and, in particular, belongs to
and the other label marks a formula
solutions, where
|= p(a)
, and belongs to
θ
p,·,r
,
ν(p,r0)
0
U
exactly one of these
(a, y)
with innitely many
. Hence, the condition
cofin
Ufin,
(c) holds.
If
r0is nonisolated, then all nonzero labels, linking realizations
of
r0are positive, since having a non-positive nonzero labelu,
linking realizations of
and as
the formula
r0is nonisolated there are innitely many solutions for
θu(x, a)
r0we have the non-symmetric relation
, where
|= r0(a)
. This contradicts the strong
SI
r
minimality of theoryT. By Proposition 3.10.0.2, nonzero labels
linking realizations of types in
linking realizations of
r0with realizations of types in
R \ {r0}
, are positive, and labels,
R \ {r0}
are negative. In this case, since for nonprincipal type there are
only relative complements, if a labelubelongs to
u
is positive or zero, almost deterministic and does not have a
complement. Moreover,
p = r0since realizations of principal types
cannot semi-isolate realizations of nonprincipal type
ρ
is nite, then any label in
ν(r0)
i. e.,
Ufin= U
deterministic and
cofin
, and if
U
cofin
ρ
= ∅
condition (e) is implied by previous items.
Ufinbelongs to
is innite, then all labels are almost
ν(r0)
U
cofin
. Thus, the condition (d) holds. The
¤
ρ
ν(p,r0)
, then
r0. If the set
and vice versa,
0
,
3.19.
ALGEBRAS FOR STRONGLY MINIMAL THEORIES
If
M
is a
POSTCR-structure and there is a theoryTwith a
family
for semi-isolating formulas such that
M
R = S1(∅)
and a regular family
is representedbyT
ν(R)
M
ν(R)
and also say thatTrepresents
of labelling functions
= M
, then we say that
the
POSTCR-
247
structureM. If all types ofRare realized in a modelNofT, then
we say that
Note that the syntactic representability of
M
(by a theory) is equivalent to the semantic representability of
M
(by a model).
Notice also that there is a representationTfor the
M
is represented
byN.
POSTCR-structure
POSTCR-
structureMsuch that a labeluis almost deterministic if and only
ifudoes not force an innite set of solutions.
3.19.0.4. Theorem.
Let
M
be a
POSTCR-structure satisfying
the following conditions:
(a)
M
consists of labels belonging to
(b)
there is an element
nite, where
p, q ∈ R
and
r0∈ R
q 6= r0, all labels
deterministic(in some representation
to
Ufin;
(c)ifR
u
, including zero, have complements
u ∈ ρ
u,
in particular, belongs to
θ
(a, y)(forN)
p,·,r
0
and belongs to
(d)ifR
ments of
in
R\ {r0}
is nite then all nonzero labels are positive and all labels
,
µ(p,r0)
exactly one of them is almost deterministic and,
Ufin, and the other label marks a formula
with innitely many solutions, where
U
;
cofin
is innite then all nonzero labels, linking
R \ {r0}
, are positive, and labels, linking
, are negative; in this case, if a labelubelongs to
Ufin∪ U
such that any set
u ∈ ρ
cofin
;
µ(p,q)
ρ
µ(p,q)
is -
are almost
mathcalNofM)and belong
u
,
and for any pair of labels
|= p(a)
r0or ele-
r0with elements
ρ
µ(p,r0)
thenuis positive or zero, almost deterministic, does not have com-
plements and
U
= ∅ifρ
cofin
(e)
only labels in
Then there is a strongly minimal theory
POSTCR-structure
p = r0, moreover,
is innite;
µ(r0)
ρ
M
µ(p,r0)
and with
and having unique1-type
Ufin= U
|R| < ω
if
cofin
ρ
µ(r0)
can force innity.
T
representing the
r0with innitely
is nite, and
many realizations.
,
,
248
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Proof.
3.9.0.2 and 3.17.0.3. We identify
Consider the construction for the proof of Theorems
R
with the set of1-types for
the required theoryT. Now we add to the types describing links
between elements with respect to binary relations
Qu,
u ∈ U
, an
information for the cardinality of sets of solutions for formulas
θ
(a, y)
p,u,q
D
of diagrams guarantees that the generic theory of the language
0
, where
{Qu| u ∈ U }
structureM.
|= p(a)
. The generic construction for the class
is strongly minimal and represents the
¤
POSTCR-
3.20. Absorbing structures
Recall that we have dened deterministic and almost determin-
istic structures, i. e., structures for which sets
labelsuandv. Now we dene opposite cases.
3.20.0.1. Denition.
ω \ {0}
µ(pn, p
, if whenever
)
, respectively,
n+1
An
IR-structurePisn-absorbing
u1, . . . , unare nonzero labels in
p1, . . . , p
n+1
∈ R
hold:
•
if some
equal to the set
•
if all
tains the set
uiis negative then
µ−(p1, p
)
of all negative labels in
n+1
uiare positive then
µ+(p1, p
)
of all positive labels in
n+1
P (p1, u1, p2, u2, . . . , un, p
P (p1, u1, p2, u2, . . . , un, p
u · v
are nite for all
, for
n ∈
µ(p1, p2), . . . ,
, the following conditions
)
n+1
µ(p1, p
µ(p1, p
n+1
n+1
n+1
)
)
(i. e.,
)
;
con-
is
P (p1, u1, p2, u2, . . . , un, p
or
P (p1, u1, p2, u2, . . . , un, p
An
IR-structure
if whenever
µ(pn, p
n+1
u1, . . . , unare nonzero labels in
)
, respectively,
tions hold:
•
if some
uiis negative then
P (p1, u1, p2, u2, . . . , un, p
is nite;
) = µ+(p1, p
n+1
P
is
almostn-absorbing
p1, . . . , p
) = µ+(p1, p
n+1
n+1
∈ R
n+1
) \ µ−(p1, p
n+1
)
n+1
) ∪ {0}
, for
).
n ∈ ω \ {0}
µ(p1, p2), . . . ,
, the following condi-
)
n+1
,
3.20.
ABSORBING STRUCTURES
•
if all
uiare positive then
249
P (p1, u1, p2, u2, . . . , un, p
is nite.
A
POSTCR-structure
whenever
respectively,
•
u1, . . . , unare nonzero labels in
p1, . . . , p
if some
equal to the set
•
if all
uiare positive then
tains the set
•
if the labels
belongs to
(µ+)0(p1, p
A
POSTCR-structure
{0}
, if whenever
µ(pn, p
n+1
µ+(p1, p
U0then
)
n+1
)
, respectively,
hold:
•
if some
uiis negative then
SI(p1, u1, p2, u2, . . . , un, p
is nite;
•
if all
uiare positive then
) \ µ+(p1, p
n+1
M
isn-absorbing
µ(p1, p2), . . . , µ(pn, p
∈ R
n+1
uiis negative then
µ−(p1, p
, the following conditions hold:
SI(p1, u1, p2, u2, . . . , un, p
)
of all negative labels in
n+1
SI(p1, u1, p2, u2, . . . , un, p
)
of all positive labels in
n+1
uiare positive or belong to
SI(p1, u1, p2, u2, . . . , un, p
of all labels of
U+∪ U0laying in
M
is
almostn-absorbing
u1, . . . , unare nonzero labels in
p1, . . . , p
n+1
∈ R
, the following conditions
) \ µ−(p1, p
n+1
)
n+1
, for
n ∈ ω \ {0}
µ(p1, p
µ(p1, p
U0and some
)
contains the set
n+1
µ(p1, p
, for
µ(p1, p2), . . . ,
)
n+1
n+1
n+1
)
n+1
)
.
n+1
n ∈ ω \
n+1
)
;
, if
n+1
)
)
con-
)
,
is
;
u
i
SI(p1, u1, p2, u2, . . . , un, p
is nite;
•
to
U0then
if the labels
uiare positive or belong to
SI(p1, u1, p2, u2, . . . , un, p
is nite.
The following properties are obvious:
1. Anyn-absorbing (
n
-absorbing.
IR- or
POSTCR-) structure
) \ µ+(p1, p
n+1
) \ (µ+)0(p1, p
n+1
)
n+1
U0and some
n+1
M
uibelongs
)
is almost
250
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
2. A (
IR- or
POSTCR-) structureMis both almost determin-
istic and almostn-absorbing if and only if, inM, all sets
µ+(p, q),(µ+)0(p, q)
3. A (
IR- or
only if every set
are nite.
POSTCR-) structure
Mis1
µ−(p, q),µ+(p, q),(µ+)0(p, q)
-absorbing if and
contains at most one
nonzero element.
3.20.0.2. Proposition.
structure
M
is(almost)n
For all
n ∈ ω \ {0}
-absorbing then
, if an associative
M
is(almost)(n + 1)
absorbing.
Proof.
u1. . . unu
is dened. If some
n
-absorbability,
in
µ(p1, p
a fortiori contains (almost) all labels in
uiu
i+1
i
. It means that
for oddn. Since
is an (almost) unique label in
a unique (respectively, nitely many) label in
u1. . . u
labels in
Take nonzero labels
SI(p1, u1, p2, u2, . . . , un, p
n+1
uiu
u1. . . u
)
of corresponding sign and by
n+2
= {0}
for allithen all
u1. . . un= {0}
M
= {u1}
n+1
µ+(p1, p
n+2
u1, . . . , un, u
contains a nonzero labelv, then since, by
i+1
i−1vui+2
. . . u
contains (almost) all labels
n+1
µ(p1, p
uiare positive and
for evennand
is (almost)n-absorbing,0or
µ+(p1, p
n+1
, for which
n+1
, u
n+1
n+1
vE uiu
n+2
,
u1. . . unu
i+1
)
of same sign. If
u
= u
i+1
u1. . . un= {u1}
u1, respectively,
)
. Therefore, there is
µ+(p1, p
or
u1. . . u
n+1
= {0}
containing (almost) all
).¤
, p
µ−(p, q)
)
n+2
−1
for each
i
n+2
)
,
-
n+1
and
Now we denote by
spectively) the class of associativen-absorbing
absorbing
n
-absorbing
SIR-structures, almostn-absorbing
SIR-structures). By proposition 3.18.0.1, we have in-
AbI
R,n
(
AbSI
R,n
,
AAbI
R,n
,
AAbSI
IR-structures (n-
IR-structures, almost
R,n
, re-
clusions
AAbI
R,n
AbI
AbI
⊆ AAbI
R,n
R,n
⊆ AAbI
⊆ AbI
, AAbSI
R,n+1
R,n
R,n+1
, AbSI
, AbSI
⊆ AAbSI
R,n
⊆ AAbSI
R,n
⊆ AbSI
R,n
R,n
R,n+1
, n ∈ ω \ {0}.
R,n+1
,
,
Using constructions for the proof of Theorems 3.6.0.2, 3.9.0.2,
3.15.0.3, and 3.17.0.3 we obtain that all these inclusions are strict.
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