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3.20.
ABSORBING STRUCTURES
251
Indeed, for instance, for
I
-structure
{p}
innite sets
M
with an innite set
X1, . . . , Xn⊆ U−such that
ω,k = 1, . . . , n − 1,Xn= U−, and for all labels
u1. . . u
are innite subsets of
k+1
The latter means that
u1. . . un, but can not be covered by sets
k < n
.
AbI
R,n
⊂ AbI
R,n+1
U−of negative labels and
Xk⊂ X
Xk,
k = 1, . . . , n,u1. . . u
U−is covered by sets
wu
k+1
, we can dene an
k+1|Xk+1
u1, . . . , u
vu
n+1
, where
\ Xk| ≥
n+1
n+1
, where
w ∈ u1. . . uk,
∈ U−,
= Xn.
v ∈
The following theorem shows that assuming the non-validity of
the strict order property (i. e., with NSOP), we can not construct
a special, in sense of Section 3.7, monoid
terministic, with bounded cardinalities for products
P
being almost de-
ν(p)
u1. . . um, or
almost absorbing. Hence, these monoids can not be too small or
too large with respect to their operations.
3.20.0.3. Theorem.
a special monoid
bounding cardinalities of sets
somen, then
Proof.
P
is almostn-absorbing. By the denition, there are elements
ν(p)
P
ν(p)
Take a negative PIP-element
v1, . . . , vm∈ U−such that the label
E
-dominates all negative labels for
θw(b, M)
and
a ∈ θw(a, M) \ θw(b, M)
andbof typepin a model
P
generates the strict order property.
ν(p)
Now we assume that
IfTis a small theory with a typep, and
P
is almost deterministic, with a constant
ν(p)
u1. . . um, or almostn-absorbing for
generates the strict order property.
u0in
P
. Suppose that
ν(p)
w = u
ν(p)
n
∨ 0 ∨ v1∨ . . . ∨ v
0
. We get
for every realizations
M |= T
P
ν(p)
, where
M |= θ
u
0
does not generate the strict order
θw(a, M) ⊆
(a, b)
. Thus,
C
m
property. Using the arguments above, we have that negative labels
m
u
are pairwise distinct,
0
many (negative) labels
p
, such that
0
u
= 0
. But since
0
induction,
the cardinalities
ϕm(a, y) ` θ
u0is a PIP-element, we have
k
0
E u
|u
m
0
m
0
u
for
|
m ∈ ω \ {0}
. So there are innitely
vmfor isolating formulas
m
(a, y) ∧ ¬θ
u
0
1 ≤ k ≤ m
. Whence,
are unbounded.
¤
u
m−1
0
ϕm(a, y),a |=
(a, y),m ∈ ω \ {0}
u0∈ u
v1, . . . , vm∈ u
2
and, by
0
m
0
and
Note that the arguments for Theorem 3.20.0.3 hold for monoids
SI
with special labels
ν(p)
u0being disjunctions of negative labels
for isolating formulas.
a
,
252
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
In the following chapter, we construct a graph with a small
theory having NSOP and a typephaving a special monoid
generated by PIP-special element
P
is negative and has a form
ν(p)
u0, i. e., each nonzero label in
∧ ¬u
m−1
. Thus, the result of
0
m
u
0
P
ν(p)
Theorem 3.20.0.3 can not be improved.
3.21. Structures of distributions of isolating
formulas as derivative structures:
for acyclic graphs
We have considered the operatorsPand
oryT, for a nonempty familyRof types in
family
ν(R)
of labelling functions, structures
SI
dening, for a the-
S1(T )
, and for a regular
P
ν(R)
and
SI
ν(R)
re-
spectively being partial groupoids of (semi-)isolating formulas on
a set of realizations of types inR. The structures
are derivative structures relative to models
MofT
P
ν(R)
.
and
SI
ν(R)
Natural problems arise on possibilities of reconstruction, spec-
tra, and structures corresponding to given derivative structures
P
ν(R)
and
SI
ν(R)
.
Below we consider a partial solution of these problems for de-
terministic structures
graphs
hM; Qi
which do not have cycles with respect to
We also prove that the structures
P
corresponding to
ν(R)
P
acyclic
for acyclic graphs are
ν(R)
graphs, i. e.,
Q ∪ Q−1.
almost deterministic.
3.21.0.1. Denition.
We say thatThas
paths with marked intermediate vertices
ϕ(x1, . . . , xn)ofT
mulas
d
ϕ0,δ1,ϕ1,...,δk,ϕ
formulas having one free variable such that
if and only if there is a shortest
lengthk) in
l = 0, . . . , k
T
is based by the set of formulas
hM; Q ∪ Q−1i
. By the denition this quantier reduction means that
Let
Γ = hM; Qi
be a graph,
T = Th(Γ)
quantier reduction relative shortest undirected
if every formula
is equivalent to a Boolean combination of for-
(xi, xj)
k
, where
(a, b)
such that
d
ϕ0,δ1,ϕ1,...,δk,ϕ
δ1, . . . , δk∈ {−1, 1}
|= d
ϕ0,δ1,ϕ1,...,δk,ϕ
-path
|= Q
S = (c0, . . . , ck)
δ
l
(c
, cl)
(x, y)
and
.
l−1
k
and
ϕlare
k
|= ϕl(cl)
(a, b)
(of
.
,
3.21.
ACYCLIC GRAPHS
253
It is easy to see that all theories of acyclic (with respect to
Q ∪ Q−1) graphs and many similar theories, in particular, theories
of acyclic graphs expanded by unary predicates, have this quantier
reduction.
If all formulas
d
ϕ0,δ1,ϕ1,...,δk,ϕ
k
ϕl(x)
are
(x ≈ x)
and write simply
dkif
, we write
Q−1= Q
d
δ1,...,δ
instead of
k
.
Clearly that the diameter of a connected componentCinΓis
nite if and only if for
m
a formula
W
dk(x, b)
k=0
Q ∪ Q−1the relation
for any
b ∈ C
).
x ∈ C
is denable (by
3.21.1. Deterministic structures for acyclic graphs
Below we shall consider theories
T = Th(Γ)
expanded by some unary predicates.
For a theoryTand a nonempty family
sider indexes in formulas
Having isolating formulas
|= p(a)
the structures
, and
θ
p,(ϕ0,δ1,ϕ1,...,δk,ϕk),q
P
ν(R)
dk(x, y)
θ
and
d
p,(ϕ0,δ1,ϕ1,...,δk,ϕk),q
(a, y) ` q(y),
are dened by the following inclusions
(p, (ϕ0, δ1, ϕ1, . . . , δm, ϕm), q) · (q, (ϕm, δ
⊆ {(p, (ϕ0, δ1, ϕ1, . . . , δm, ϕm, δ
(p, (χ0, δ
00
, χ1, . . . , δ
1
0
, ψ
1
00
|m−n|
where
(χ0, δ
if
m ≥ n,ϕ
00
, χ1, . . . , δ
1
m−i
= ψ
(χ0, δ
00
|m−n|
m+i
, χ
|m−n|
, and
00
1
δ
m−i
, χ1, . . . , δ
) = (ϕ0, δ1, ϕ1, . . . , δ
= −δ
00
|m−n|
R ⊂ S1(∅)
ϕ0,δ1,ϕ1,...,δk,ϕ
(a, y)
0
1
m+1
, χ
, χ
, ψ
0
m+i
, . . . , δ
m+1
, . . . , δ
), r)},
|m−n|
,
i = 1,. . . , n
) =
|m−n|
of acyclic graphs
, we con-
(x, y)
0
, ψ
n
m+n
as labels.
p, q ∈ R
(∗)
), r) ⊆
m+n
), r),
, ϕ
|m−n|
:
),
k
, where
0
, ψ
n
|m−n|
;
,
if
m < n,ϕ
= (ψ2m, δ
= ψ
m−i
0
2m+1
m+i
, ψ
2m+1
, and
δ
m−i
. . . , δ
= −δ
0
2m+|m−n|
0
,
m+i
, ψ
2m+|m−n|
i = 1,. . . , m
),
.
254
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Replacing labels
paths, we have
there are realizations
|= θ
In particular, for
realizations ofp, we have
for nonprincipalp, either
degree0in
only edges in
p(M)
p(M)
innite paths). IfQis non-symmetric we preserve1for
p(a)
, and add the label
ϕ0, δ1, ϕ1, . . . , δk, ϕkby lengthskof shortest
(p, |m ± n|, r) ∈ ((p, m, q) · (q, n, r))
p,m,q
a, b, c
(a, b) ∧ θ
of types
q,n,r
R = {p}
p, q, r
(b, c) ∧ θ
respectively such that
p,|m±n|,r
with only principal edges linking
m · n = {m + n, |m − n|}
), or
U = {0}
U = {0, 1}
(i. e., all vertices of
with addition modulo2(there are
without common vertices), or
1−1to the setUfor
Q(y, a)
if and only if
(a, c).
. In this case,
U = ω
. Ifpis principal
p(M)
have
(there are
Q(a, y),|=
we add the label2if there are at least two connected components on
p(M)
is denable by a formula
p(x)
p(M)
least three connected components the product
2
loops,
degree either0or1, and ifpis isolated then
(the relation \to belong to a common connected component"
, the formula
ϕ(x, y)
¬ϕ(a, y) ∧ ψ(y)
and for an isolating formula
isolates the set of elements in
that are not linked withaby paths). In this case, having at
2 · 2
contains0and
, and
P
is not deterministic.
ν(p)
Hence, for undirected acyclic graphs on
P
is deterministic if and only if all vertices of
ν(p)
p(M)
p(M)
and without
contains at most
p(M)
ψ(x) `
have
two connected components for degree0, and it contains the unique
edge for degree1.
More generally, if all principal arcs on a familyRof1-types
are dened by some formulas
vertices of shortest(a, b)-path,a∈p(M),b∈q(M),|=
p, q ∈ R
, may be not satisfy types inR) then
if and only if for any principalm-arcinR(M)
and
q ∈ R
principaln-arc in
either
parameters
(i. e., pairs
(m + n)
-arcs or
(a, b) ∈ p(M) × q(M )
R(M)
linking typesqand
|m − n|
ϕ0, δ1, ϕ1, . . . , δk, ϕkof these arcs are uniquely dened.
dk(x, y)
-arcs in
(where some intermediate
dk(
a, b),
P
R(M)
is deterministic
ν(R)
linking types
with
|= dm(a, b)
r ∈ R
, there are only
p ∈ R
) and
linkingpandr, and
Thus, we get the following
3.21.1.1. Proposition.
For any theoryTof an acyclic graph with
unary predicates, for any nonempty family
lated types, and for a regular family
ν(R)
R ⊆ S1(T )
of noniso-
of labelling functions,
3.21.
ACYCLIC GRAPHS
255
the structure
palm-arc in
n
-arc in
arcs or
P
R(M)
R(M)
|m − n|
is deterministic if and only if for any princi-
ν(R)
linking types
linking typesqand
-arcs in
R(M)
p ∈ R
r ∈ R
and
q ∈ R
and principal
, there are either
linkingpandr, and parameters
(m + n)
ϕ0, δ1, ϕ1, . . . , δk, ϕkof these arcs are uniquely dened.
3.21.1.2. Corollary.IfT
acyclic graph with a(unique)1
function
ν(p)
then
P
ν(p)
has degree either0or1, and ifpis isolated then
is a transitive theory of an undirected
-type
p(x)
and a regular labelling
is deterministic if and only if each vertex
p(M)
contains at
most two connected components for degree0, and it contains the
unique edge for degree1.
Considering a transitive theory of directed acyclic graph, we
immediately get unbounded distances and non-denability of con-
nected components. Hence, every isolating formula
form
d
δ1,...,δ
(x, y)
k
. For the (unique)1-type
p(x)
θu(x, y)
has a
we repeat the ar-
guments for undirected graphs and get that the determinacy of the
algebra
asymmetricQ-imageb, i. e., with
asymmetricQ-preimagec, i. e., with
more than one
Therefore,
groupZor to the free product
P
means that each realizationaof
ν(p)
symmetricQ-imaged, i. e., with
P
is generated by a group isomorphic either to the
ν(p)
Z ∗ Z2.
p(x)
has the unique
|= Q(a, b) ∧ ¬Q(b, a)
|= Q(c, a) ∧ ¬Q(a, c)
|= Q(a, d)∧Q(d, a)
, the unique
, and not
Thus we get
-
.
3.21.1.3. Corollary.
acyclic graph with a(unique)1
function
ν(p)
then
ated by eitherZor
If
T
is a transitive theory of a directed
-type
p(x)
and a regular labelling
P
is deterministic if and only if it is gener-
ν(p)
Z ∗ Z2.
As an illustration conrming that the transitivity of theory is
essential in Corollaries 3.21.1.2 and 3.21.1.3, we consider
3.21.1.4. Example.
graph without loops, with the transitive theory
Let
Γ = hM; Qi
be an acyclic undirected
Th(Γ)
, and such
that each vertex has degree2. ExpandΓby disjoint unary predi-
cates
Col1,
(1)
Col2satisfying the following conditions:
Col1∪ Col2= M
;
256
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
(2) ifais a vertex in
Col1and the other is in
(3) ifais a vertex in
The formulas
p1(x)
and
p2(x)
Col1(x)
respectively. Taking a vertex
naturalkwe have a unique vertex
Col1then one adjacent vertex belongs to
Col2;
Col2then adjacent vertices belong to
and
Col2(x)
isolate some complete types
Col1.
a ∈ Col1and a
b ∈ Col1with
|= dk(a, b)
. Hence
for any labelsmandnof isolating formulas witnessing distances
m
andnrespectively,
½
{m + n},
{|m − n|},
Thus
P
m · n =
is deterministic. At the same time,
ν(p1)
ministic since any two verticesaandbof
connected components have the distance3k, for some
two-element sets
The algebra
with generating elements
2
in
Col1(x)
{c ||= d3k(a, c)}
P
is generated by noncommutative group
ν(p1)
g1,
g2(corresponding to distances1and
) and dening relations
isomorphic to the free product
3.21.1.5. Example.
Take the graph
3.21.1.4 and expandΓby disjoint unary coloring predicates
Col1,
Col2such that each vertexaof colorihas neighbors of colors
j
andk,
only if
i = 0, 1, 2
the groupZ, where1and
{i, j, k} = {0, 1, 2}
|= d3k(a, b)
for somek.
Again the formulas
. Each algebra
, and an elementbhas the coloriif and
Coli(x)
P
ν(pi)
−1
correspond to formulas
if
m + n
if
m + n
is odd,
is even.
P
ν(p2)
is not deter-
Col2lying in a common
k ∈ ω
, and
are subsets of
2
g
= e,g
1
Z2∗Z2of two copies of group
Γ = hM ; Qi
Col2.
2
= e
2
, i. e.,
Z2.
in Example
Col0,
isolate some complete types
pi(x)
is deterministic and generated by
G
G
is
¤
¡
∃z1, z
Q(x, z1) ∧ Q(z1, z2) ∧ Q(z2, y)∧
2
,
∧Coli(x) ∧ Col
∃z1, z
∧Coli(x) ∧ Col
(i+1)(mod 3)
(z1) ∧ Col
(i+2)(mod 3)
¡
Q(x, z1) ∧ Q(z1, z2) ∧ Q(z2, y)∧
2
(i−1)(mod 3)
(z1) ∧ Col
(i−2)(mod 3)
(z2) ∧ Coli(y)¢,
(z2) ∧ Coli(y)¢. ¤
3.21.
ACYCLIC GRAPHS
257
Example 1.2.3.5 of free directed pseudoplane produces the de-
terministic algebra
P
for the acyclic directed graph with colored
ν(p)
vertices each of which has innitely many images and innitely
many preimages. Here the colors
tion and the set
p(x)
. The algebra
{¬Coln(x) | n ∈ ω}
P
is generated by
ν(p)
Colnare ordered by graph rela-
implies the nonisolated type
hω∗; +i
.
Modifying Examples 3.21.1.4, 3.21.1.5 and the example with
the monoid
mediate vertices we can construct the algebra
free product of unboundedly many copies ofZ,
By inclusions
exhaust all possibilities for algebras
hω∗; +i
by several disjoint unary predicates for inter-
P
generated by
ν(p)
Z2, and
(∗)
and the acyclicity of graphs, these free products
P
corresponding to acyclic
ν(p)
hω∗; +i
graphs (directed or undirected) with unary predicates:
.
3.21.1.6. Theorem.IfT
with some unary predicates, a1-type
bra
P
, then
ν(p)
hω
∗
; +i
k
∗
k∈K
and copies
P
is generated by a free product
ν(p)
for some copies
∗
hω
; +i
of monoid
k
is a theory of an acyclic graph
p(x)
, and a deterministic alge-
∗
i∈IZi∗∗j∈JZ2,j
Ziof groupZ, copies
hω∗; +i
. If there are
Z
2,j
hω
of group
∗
; +i
k
hM; Qi
Z2,
then the
typepis not isolated.
Proof.
for isolating formulas
Q−1)
-pathsS,
p(x)
then
Take a realizationaof type
|= p(b)
c = a
ϕ(a, y)
, such that ifaisolates
or
c = b
corresponding to shortest
. If
p(x)
p(x)
and a set
is isolated,
c ∈ S
hM; Qi
U0of labels
(a, b)-(Q ∪
andcrealizes
has nite
diameter for the set of realizations ofp, and several (in fact two)
connected components with elements realizing
label
u0of order2corresponding to isolating formulas
link these connected components. By the denition the set
generates the algebra
P
. At the same time elements of
ν(p)
independent and each of them generates either a copy ofZ, or
or
hω∗; +i
free product of these groups generates
of
hM; Qi
u ∈ U0generates
since
hM; Qi
is acyclic and
and determinacy of
hω∗; +i
thenuwitnesses that the relation
P
P
ν(p)
P
. Finally we note that if a label
ν(p)
semi-isolation is non-symmetric and thereforepis not isolated.
p(x)
we add to
ψ(a, y)
is deterministic. The
ν(p)
again by the acyclicity
U0a
that
U
U0are
Z2,
SIpof
¤
∗
0
258
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.21.2. On almost determinacy
3.21.2.1. Proposition.
For any theory
T
of an acyclic graph
with bounded diameter and with unary predicates, for a nonempty
familyRof types in
functions, the structure
Proof.
By the inclusions
S1(T )
P
and a regular family
is almost deterministic.
ν(R)
(∗)
, there are nitely many possi-
ν(R)
of labelling
bilities for products of given labelsuandvin a theory of acyclic
graphs, where labels symbolize some shortest paths. In fact, there
are at most two possibilities for
(p, u, q) · (q, v, r)
: with a sum or a
dierence of distances.
Let a labelusymbolize that some elements
with
|= θ
tinct connected components
pothesis,
isolating formula
for
(p, w, r) ∈ (p, u, q) · (q, v, r)
with
|= θ
nected components
or
Ca= Cdwith nitely many possibilities for distances between
(a, b)
p,u,q
and principal formula
Caand
θ
p,u,q
Cbrespectively. By the hy-
Cahas bounded diameter orbis separated from
ϕ(y)
. Taking a product
(p, u, q) · (q, v, r)
and an isolating formula
(a, d)
p,w,r
, either againaanddbelong to distinct con-
Caand
Cd(in particular,
a |= p
(a, y)
, belong to dis-
w = u
for
and
b |= q
Caby an
we get,
θ
(a, y)
p,w,r
p = q = r
andd(in fact, for nite diameters there are at most two possibili-
ties with respect to typer: with a sum or a dierence of distances
froma).
Now we take the labeluand the elementsa,b, and consider
a product
(p, u, q)
(r, v, p) · (p, u, q)
and an element
, a labelwwith
e |= r
satisfying
(r, w, q) ∈ (r,v, p) ·
|= θ
(e, a) ∧ θ
r,v,p
r,w,q
(e, b)
Ifvwitnesses thateandalay in the same connected component:
Ce= Ca, thenwwitnesses that
two possibilities: either
Ce= Cbwith nitely many variants for
distances betweeneandb(since
Ca6= Cb. Forwandethere are
Cehas bounded diameter), or
Ce6= Cbandwwitnesses this fact.
In any case, we have nitely many (in fact, at most three)
possibilities forw. Thus,
P
is almost deterministic.
ν(R)
¤
,
)
a
.
3.21.
ACYCLIC GRAPHS
The following example illustrates that the condition of bounded
diameter is essential for Proposition 3.21.2.1.
259
3.21.2.2. Example.
3.21.1.4 and add new vertexbhaving degree0. Any vertexain
isolates the vertexbby a formula
b
isolates vertices inMby a formula
Take the graph
ϕ(x)
ψ(x)
Γ = hM ; Qi
in Example
M
describing degree0. In turn,
describing degree2. But
vertices inMhave unbounded distances from the vertexa. Hence
for labelsuandvof formulas
product
P
p,q
u · v
contains innitely many labels. Thus, the structure
, wherepandqare1-types, isolated by formulas
respectively, is not almost deterministic.
ϕ(x)
and
ψ(x)
¤
respectively, their
ϕ(x)
and
ψ(x)
The reason above for the absence of almost determinacy is in
fact unique, andsi-ranks of products
formulas
ϕ(a, y)
are bounded by value2.
u·v
for labelsu,vof isolating
Note that for any theory of graph having quantier reduction
till the formulas
and for any regular family
ture
P
is almost deterministic. Moreover,
ν(R)
{(p, i, r) | 0 ≤ i ≤ m + n}
dk(x, y)
, for any nonempty familyRof1-types,
ν(R)
of labelling functions, the struc-
(p, m, q) · (q, n, r) ⊆
, where
m, n, i
witness the shortest paths
of corresponding lengths.
Similarly to acyclic graphs, if
lengthkthen
m · n = {m + n(mod k), m − n(mod k)}
p(M)
consists of disjoint cycles of
. The results
above on deterministic and almost deterministic structures stay
true for theories of disjoint cycles with unary predicates (having
unbounded cardinalities for products of labels if there are innitely
n
manynwith pairs
all elements
tp(bn)
) as well as for acyclic graphs, where relationsQare replaced
an, bn∈ C
(C
by disjoint unions of some relations
n
, C
)
1
2
n,1∪Cn,2
of cycles of lengthn, where for every
satisfy same isolated type:
tp(an) =
S
Qi,
i ∈ I
, such that
Q =
i∈I
Qi.
They are also true for disjoint unions of cycles and acyclic graphs.
n
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