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3.6.I-GR
·
on elements in
nonempty set
OUPOIDS
•
the set
•
the operation·of the groupoidPis generated by the function
{0}
is the unit of the groupoidP;
U
such that every elements
(u · v) ⊆ U
: for any sets
u, v ∈ U
X, Y ∈ P(U) \ {∅}
following equality holds:
X · Y =[{x · y | x ∈ X, y ∈ Y };
191
dene a
the
•ifu < 0
for any
•ifu > 0
then the sets
v ∈ U
;
and
v > 0
u·v
and
then the set
elements;
•
for any
that
0 ∈ (u · u−1) ∩ (u−1· u)
•
if a positive elementubelongs to a set
−1
to
v
· v
2
•
for any elements
−1
1
u > 0
;
there is a unique
;
u1, u2, u3∈ U
(u1· u2) · u3⊇ u1· (u2· u3),
and the strict inclusion
(u1· u2) · u3⊃ u1· (u2· u3)
may be satised only for
•
the groupoidPcontains the
u1< 0
and
(being a monoid) with the universe
v·u
consist of negative elements
u · v
consists of non-negative
inverse
element
v1·v2then
u−1> 0
u−1belongs
the following inclusion holds:
|u2· u3| ≥ ω
deterministic
≥0
P(U
) \ {∅}
d
;
subgroupoid
, where
such
≥0
P
d
≥0
U
= {u ∈ U≥0| u−1· u = {0}};
d
any set
u · v
is a singleton for
u, v ∈ U
≥0
d
.
By the denition eachI-groupoidPcontainsI-subgroupoids
≤0
P
and
{0}) \ {∅}
≥0
P
with the universes
respectively. The structure
P(U−∪ {0}) \ {∅}
≥0
P
is a monoid.
and
P(U+∪
192
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.6.0.2. Theorem.
a type
P
p(x) ∈ S1(T )
= P
ν(p)
. If the alphabet is at most countable and the operation
For anyI-groupoidPthere is a theoryTwith
and a regular labelling function
ν(p)
such that
ofPdoes not force continuum many types thenTis small.
Proof.
We x anI-monoid
P = hP(U ) \ {∅}; ·i
. The con-
struction of a required theory will be fullled in accordance with
a construction of a generic structure
n ∈ ω} ∪ {Q
an ordered coloring
mula
Qu(x, y)
p(x)
(isolated by the set
(2)
| u ∈ U }
u
, where
with pairwise disjoint predicates
Col:M → ω ∪ {∞}
u < 0
, and with a unique nonprincipal1-type
{¬Coln(x) | n ∈ ω}
out loss of generality we assume that
M
of language
Σ = {Col
with respect to each for-
of formulas). With-
|U| ≤ ω
(for
|U| > ω
(1)
n
Qu, with
, the
construction diers by cardinalities of diagrams describing links
for elements of nite sets and by cardinalities of sets of diagrams
forming generic structures).
Consider a generative class
diagrams
Φ(A)
over nite setsAsuch that each
maximal consistent set of quantier-free formulas
united
with a set of formulas
Quv(x, y) = ∃z(Qu(x, z) ∧ Qv(z, y)),u, v ∈ U
(D0; 6)
δ
Q
uv
consisting of all possible
Φ(A)
contains a
a),a ∈ A
ϕ(
(a, b),a, b ∈ A,δ ∈ {0, 1}
, and
Φ(A)
includes
formulas with parameters inA, without free variables, and describ-
ing the following properties:
|
,
,
(1) for any
of some elements by the relation
(2) the relation
(3) if
all
Qu-preimages ofahave the colors
(4) if
and
Col(b) = Col(a)
(5) if
(6) for any
a graph
n > 0
hB; Qui
;
(7) if
u ∈ U
Q0on the setAis identical;
a ∈ A
then all
u > 0,a ∈ A
v ∈ (u1·u2)
u 6= 0
with a cycle if and only if
u ∈ U
≥0
then each element
d
, any element inAis an image and a preimage
Qu;
Qu-images ofahave colors
≤ Col(a)
, and
Qu(a, b) ∈ Φ(A)
then
;
Q
≥ Col(a)
−1
u
;
and
Qv(a, b) ∈ Φ(A)
some diagram
then
Q
(a, b) ∈ Φ(A)
u1u
2
Ψ(B) ⊇ Φ(A)inD0denes
a ∈ A
0 ∈ u · . . . · u
has a unique
|
{z }
n
times
the following inductive condition describes the least set
of non-negative elements
u ∈ U
for which the sets of
and
(b, a) ∈ Φ(A)
for
some
Qu-image;
≥0
U
ad
⊇ U
≥0
d
Qu-images
;
3.6.I-GR
OUPOIDS
193
and of
of nitely many elements belonging to
u−1· u
each elementahas nitely many
nitely many elements belonging to
nitely many
Qu-images and of
element
of sets of
elements that are not
Qu-preimages ofaare nite: if
(u · u−1) ∪ (u−1· u)
≥0
U
then
ad
consists of nitely many elements belonging to
Qu-images; if
≥0
U
then each elementahas
ad
Qu-preimages; for other elementsuthe numbers of
(8) if
Qu-preimages for elements
u1, u2∈ U
a ∈ A
the set of
and the set
Q
-images ofais represented as a union
u1u
2
Qv-images for all elements
u1· u2is (in)nite then for any
v ∈ u1· u2(and some set of
a ∈ A
Qu-images ofaon any of the relations
(9) for any element
v ∈ ((u1· u2) · u3) \ (u1· (u2· u3))
description forming Example 3.3.0.1.
If
Φ(A), Ψ(B)
pose, by the denition, that
(i. e.,
Φ(A) 6 Ψ(B)
descriptions (for numbers andQ-links) of its
where
u−1· u
For the checking that
it suces to observe that for any diagrams
D
with
0
Φ(A) 6 Ψ(B),Φ(A) 6 X(C)
is a diagram
X(C) 6 Θ(B ∪ C)
For the diagram
are diagrams in
Φ(A)
) if
Φ(A)
, with each elementainA, contains all
D
and
0
is a
Φ(A) ⊆ Ψ(B)
strong subdiagramofΨ(B)
Qu-images in
consists of nitely many labels belonging to
(D0; 6)
is a self-sucient generative class,
Φ(A), Ψ(B), X(C) ∈
, and
Θ(B ∪ C) ∈ D0such that
A = B ∩ C
Ψ(B) 6 Θ(B ∪ C)
.
Θ(B ∪ C)
we choose the set
extended by the following formulas for elements
c ∈ C \ A
:
consists
u, u−1∈ U
≥0
; if
ad
≥0
U
then
ad
u · u−1consists of
is unbounded;
Qu);
there is a
, we sup-
Ψ(B)
≥0
U
.
ad
there
and
Ψ(B) ∪ X(C)
b ∈ B \ A
and
,
(a)
θ
(b, c)
u,v
some
a ∈ A
(b)
for all
¬θ
u,v
a ∈ A
;
(c) some formulas
X(C)
for some
(d) formulas
, where
(b, c)
, where
;
a ∈ A,v0∈ u · v
¬θ
Qu(b, a) ∈ Ψ(B)
¬Qu(b, a) ∈ Ψ(B)
0
θ
(b, c)
v
0
(b, c),v0∈ U
v
, where
and
Qv(a, c) ∈ X(C)
or
¬Qv(a, c) ∈ X(C)
Qu(b, a) ∈ Ψ(B)
, and the set
u · v
and
is nite;
, if the previous items do not
for
Qv(a, c) ∈
imply a converse.
We claim that if the operation ofPdoes not force continuum
many types then, applying the generic construction, one obtains
194
a
(D0; 6 )
T = Th(M)
function
P
. By Proposition 1.2.3.3, each formula
non-symmetry of the relation
0
, links realizations ofponly with realizations of the same type
and, being a principal formula of the structure on the set
realizations ofp, has the inverse principal formula
p(M)
the saturation of
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
-generic saturated structure
, the type
p(x) ∈ S(T )
ν(p):PF(p)/PE(p) → U
SIp, and each formula
.
Now we argue to show that
M
is implied by Theorem 2.5.1.2 in view of the
M
with the generic theory
, and the regular labelling
satisfying the condition
Qu(x, y),u < 0
Qu(x, y),u >
Q
M
is saturated. If
P
, witnesses
p(M)
−1
(x, y)
u
≥0
U
is nite
ad
ν(p)
=
of
on
uniformd-amalgamation property that holds by the formula den-
ability of self-sucient closure of any nite set.
Using the proof of the same theorem, we shall observe thatMis
saturated for
Qu,
u ∈ U:Qm,
Let
M0be anω-saturated model of
[Φ(A)]
Φ(A0)
A
be diagrams in
0
A
. If
the construction of
extendingAand satisfying
isomorphismf:
isomorphismg:
Now, let
≥0
|U
| = ω
ad
m ∈ ω
. For this aim we enumerate all predicates
.
Th(M),Φ(A)
D
Ψ(B0) ∈ D0,
M
A → A0between
such that
0
Φ(A0) 6 Ψ(B0)
implies that there exists a set
M |= Ψ(B)
M
M |= Φ(A)
, and
M0|= Ψ(B0)
. It means that for a partial
and
M0there exists a partial
B → B0between these structures extendingf.
Ψ(B) ∈ D0,
Φ(A) 6 Ψ(B),M |= Ψ(B)
and
Φ(A0) =
and
M0|=
then
B ⊂ M
, andXand
Y
be disjoint sets of variables, which are in bijective correspondence
with setsAand
respectively),
B \A
. Assume that the formula
n ∈ ω
, describes the following:
ϕn(X)(ψn(X, Y )
(i) nite colors of elements ofA(ofB);
(ii) negations of colors not exceedingnfor elements ofA(ofB)
that are innite in color;
(iii) the existence, colors of arcs, the existence and colors of
some arcs of pathes of length2(including all possibilities for colors
≤ n
of intermediate arcs) connecting elements ofA(ofB), and
the colors
for which
u ∈ U
m ≤ n
of arcs outgoing from vertices
a ∈ A(a ∈ B
∃yQm(a, y) ∈ Φ(A)(∃yQm(a, y) ∈ Ψ(B)),Qm= Qu,
≥0
;
ad
,
)
3.6.I-GR
OUPOIDS
195
(iv) the non-existence of arcs of colors
length2(including all possibilities for colors
≤ n
and of pathes of
≤ n
of intermediate
arcs) connecting elements ofA(ofB), if these elements are not
linked by the pathes, as well as the absence of colors
arcs outgoing from vertices
Φ(A)(¬∃yQm(a, y) ∈ Ψ(B)),Qm= Qu,
a ∈ A(a ∈ B
) for which
u ∈ U
≥0
ad
.
m ≤ n
for
¬∃yQm(a, y) ∈
By the construction ofM,
M |= ∀X (ϕn(X) → ∃Y ψn(X, Y )).
Hence
M0|= ∀X (ϕn(X) → ∃Y ψn(X, Y )).
This implies that the set
cally realizable in
ω
-saturated. Therefore there exist a set
M0; hence, it is realizable in
and a partial isomorphismg:
{ψn(A0, Y ) | n ∈ ω}
of formulas is lo-
M0since
B0⊂ M0containing
B → B0extending the partial iso-
M0is
A0,
morphismf.
The possibility for extending any partial isomorphismsf:
A →
A0and the known back-and-forth method show that the structure
M
with distinguished constants for the elements in
A ⊂ M
is iso-
morphic to a countable elementary substructure of the structure
M0with distinguished constants for the elements in
nite setsAand
serving a type
conclude that
and
Th(M)
Note that for the
A0connected by a partial isomorphism and pre-
Φ(X)
M
are chosen arbitrarily, and
M0is saturated, we
realizes any type over a nite set,
is small.
(D0; 6 )
-generic structureM, the possibil-
A0. Since the
M
is saturated,
ity for extending any nite partial isomorphisms preserving types
Φ(X)inD0implies that if
then there is an automorphism of
isomorphism betweenAandB. Consequently,
In particular, for any realizationaofpand for any
formula
Qu(a, y)
is isolating and these formulas exhaust the list
of all pairwise non-equivalent isolating formulas
ϕ(a, y) ` p(y)
.
Similar arguments are valid for the general case.
A, B ⊂ M,M |= Φ(A)
M
extending the initial partial
tpM(A) = tpM(B)
and
ϕ(a, y)
¤
M |= Φ(B)
u ∈ U
, the
for which
.
196
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
3.6.0.3. Remark.
If anI-groupoidPis constructed by a set
≥0
U
then by the construction above (restricting the construction to a
set of realizations of the type innite in color) there is a transitive
theoryTwith a (unique) type
function
ν(p)
such that
P
p(x) ∈ S(T )
= P
ν(p)
and a regular labelling
.
3.7. Groupoids of binary isolating formulas on
sets of realizations of types of special the-
ories
In this section, we present a specicity of groupoids
typespof
special
theories that are described in the following Chap-
ters and used for the classications of countable models of Ehren-
feucht theories, theories with nite Rudin{Keisler preorders, small
theories,ω-stable theories with respect to numbers of limit mod-
els over types, as well as for the investigations of graph links for
limit models over types that obtained by quotients of numerical
sequences.
Below we dene the property of powerfulness for a directed
graph
1
Γ = hX ; Qi
in terms of the groupoid
-typepof the theory
T = Th(Γ)
P
ν(p)
assuming that the theory is
small.
At rst we note that
so
P
is a monoid.
ν(p)
Since the formula
n
W
θ
(x, y)
, the acyclicity ofΓmeans that
∈ {u1, . . . , un}
i
k
i=1
u
i
1
u
i
, . . . , u
is equivalent to that any sets
U−= ∅
Q(x, y)
in view of Corollary 3.1.0.5 and
is equivalent to some disjunction
. The condition
−1
−1
u
u
i
i
1
2
. . . u
0 /∈ u
u
i
i
1
2
acl({a}) ∩ 4(a) = {a}
−1
do not contain almost
i
k
deterministic elements. The pairwise intersection property means
that for any
tains an element
v ∈ U
. In this case we say that the element
intersection property
ui,
i = 1, . . . , n
uj. In particular, if
, or it is a
, and any
n = 1
PIP
-element
v ∈ U
then
, the set
u1∈ u1v
u1induces the pairwise
.
The characterizations above imply the following
P
ν(p)
for
for the unique
. . . u
for any
i
k
uiv
con-
for any
3.7.
GROUPOIDS OF BINARY FORMULAS
197
3.7.0.1. Proposition.
is a theory of a powerful graph
A small theoryTof the language
Γ = hX; Qi
if and only ifThas
the unique1-typepwith a regular labelling function
for some elements
satised:
(1) ` Q(x, y) ↔
(2) 0 /∈ u
(3)
an element
i
for any
uj.
1
3.7.0.2. Denition.
U−6= ∅
and for any elements
0, . . . , un< 0,v ≥ 0
is an element
A special monoid
u ∈ ρ
ν(p)
is a
PIP
u1, . . . , un∈ ρ
n
W
θ
i=1
u
. . . u
i
ui,
i
2
k
i = 1,. . . , n
A monoid
, and for any element
v0≥ 0
such that
P
-element, i. e.,
(x, y)
u
i
for any
is called
ν(p)
the following conditions are
ν(p)
;
u
, . . . , u
i
1
, and any
P
ν(p)
∈ {u1, . . . , un}
i
k
v ∈ U
, the set
is called
u1, u2, . . . , un, v ∈ ρ
u0∈ u1u2. . . unv
u0∈ v0u1u2. . . un.
PIP
-special
u ∈ uv
for any
v ∈ ρ
ν(p)
uiv
specialifρ
, where
ν(p)
if each negative
.
ν(p)
(2)
{Q
such that
;
contains
∩
ν(p)
u1<
, there
Having a special monoid (for a special small theoryT) the pro-
cess of construction of a limit model over a typepis reduced to a
sequence of
realizations ofp: for any limit model
tary chain
|= θ
(a
u
n
type of
As shown in the following Chapter, if a
θ
-extensions,
u
n
(M(an))
, an)
n+1
M
is dened by the sequence
n∈ω
is satised,
un< 0,n ∈ ω
,
|= p(an)
n ∈ ω
, of prime models over
M
overpthere is an elemen-
, such that its union forms
M
. In this case the isomorphism
(un)
n∈ω
.
PIP
-special monoid
and
exists then, by adding of multiplace predicates, each prime model
over a tuple of realizations ofpis transformed to a model isomor-
phic to
Mp. Thus, the typepis connected with the unique, up to
isomorphism, prime model over a realization ofpand with some
(nite, countable, or continuum) number of limit models overp,
which is dened by some quotient for the set of sequences
un∈ U−∩ ρ
ned by some identications
U−∩ ρ
any
with
ν(p)
v ∈ U≥0∩ρ
u0∈ v0u
,
n ∈ ω
ν(p)
such that if
and
ν(p)
0
0
. . . u
2
0
n
u
1
. The action of these quotients is de-
(w ≈ w0)
w = u1. . . umand
u0∈ u1. . . umv
of words in the alphabet
w0= u
, there exists
0
. . . u
1
v0∈ U≥0∩ρ
.
(un)
0
then for
n
n∈ω
ν(p)
}
,
198
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
To conclude this section, we describe some connections of
monoids with the strict order property.
3.7.0.3. Denition.
the model
Mp,
P
having a cardinalityλ. We say thatXis (formula)
a realizationaofpthe set of solutions of
W
θu(a, y)inMpis
u∈X
In this case we say that the formula
LetTbe a theory with a typephaving
an
I
ν(p)
-groupoid, and
ν(p)
X
a subset of
denable
L
-formula
λ+,ω
L
-denable in
ω,ω
Mpby a formula
ψ(x, y)
witnesses
ofX.
We say that a groupoid
if for some denable set
and for some realizationsaandbofpsatisfying
label
v ∈ ρ
, the inclusion
ν(p)
3.7.0.4. Proposition.IfT
the groupoid
element
u < 0
P
has a denable set
ν(p)
with
P
ν(p)
X ⊆ ρ
ν(p)
ϕ(a, Mp) ⊂ ϕ(b,Mp)
is a small theory with a typep, and
u · X ⊆ X
generates the strict order property
, for a witnessing formula
|= θv(b, a)
holds.
, then
X ⊆ ρ
P
ν(p)
generates the strict
ν(p)
order property.
Proof.
witnessing formula
and, for any realizationsaandbofpwith
ϕ(a, Mp) ⊆ ϕ(b, Mp)
ϕ(b, Mp)
by
u < 0
strict order property.
Take a denable set
ϕ(x, y)
. At the same time,
, and if
. Thus,
b ∈ ϕ(a, Mp)
ϕ(a, Mp) ⊂ ϕ(b, Mp)
¤
. Since
Y = X ∪ {0}
u · X ⊆ X
and consider a
then
M |= θu(b, a)
0 ∈ Y
thenaisolatesbthat is impossible
and
P
ν(p)
I
ν(p)
ρ
ν(p)
if for
ϕ(a, y)
ψ(a, y)
denability
ϕ(x, y)
with a
containing an
u · Y ⊆ Y
, we have
implies
b ∈
generates the
-
.
,
3.7.0.5. Corollary.
LetTbe a small theory with a typep, and for
some nonempty nite set
n
n
such that
the groupoid
Proof.
and
Y
element
n+1
X
P
ν(p)
Clearly, the nite setXis denable and the sets
n
S
Xiare also denable. Since
i=1
u ∈ X
we have
3.7.0.4 the groupoid
S
⊆
i=1
generates the strict order property.
P
ν(p)
X ⊆ U−∩ρ
Xi, where
u·Y ⊆ Y
X1= X,X
. Since
there be a natural number
ν(p)
i+1
= Xi· X
n+1
X
u < 0
⊆ Y
then for any
then by Proposition
generates the strict order property.
. Then
X
¤
i
3.8.
PARTIAL GROUPOID OF BINARY FORMULAS
199
3.7.0.6. Corollary.
U−∩ρ
is a nonempty nite set then the groupoid
ν(p)
IfTis a small theory with a typepand
P
generates
ν(p)
the strict order property.
Proof.
denable. SinceXcontains all negative labels in
tion 3.1.0.6, we have
Proposition 3.7.0.4, the groupoid
property.
Consider the set
u·X ⊆ X
¤
X = U−∩ ρ
for any
u < 0inρ
P
ν(p)
. AsXis nite it is
ν(p)
ρ
, by Proposi-
ν(p)
. Therefore, by
ν(p)
generates the strict order
3.8. Partial groupoid of binary isolating for-
mulas on a set of realizations for a family
of1-types of a complete theory
In this section, the results above for a structure of a type are
generalized for a structure on a set of realizations for a family of
types.
3.8.0.1. Denition.
S1(T )
. We denote by
LetRbe a nonempty family of types in
ν(R)
a regular family of labelling functions
ν(p, q):PF(p, q)/PE(p, q) → U, p, q ∈ R,
[
ρ
ν(R)
p,q∈R
ρ
ν(p,q)
.
Similarly to Proposition 3.3.0.1, we obtain that having atomic
models
functionP, being partial for
R
, which maps each tuple of triples
where
(p1, v, p
associative
Mpfor all types
u1∈ ρ
)
, where
k+1
ν(p1,p2)
, . . . , uk∈ ρ
v ∈ P (p1, u1, p2, u2, . . . , pk, uk, p
:
p ∈ R
|R| > 1
(for instance, ifTis small), the
, on the set
R×(P(U )\{∅})×
(p1, u1, p2), . . . , (pk, uk, p
ν(pk,p
, to the set of triples
)
k+1
)
, is
k+1
left semi-
P (P (p1, u1, p2, u2, p3), u3, p4) =
= P(p1, u1, p2, u2, p3, u3, p4) ⊇
⊇ P(p1, u1, P (p2, u2, p3, u3, p4))
for
u1∈ ρ
ν(p1,p2)
,
u2∈ ρ
ν(p2,p3)
,
u3∈ ρ
ν(p3,p4)
.
k+1
(3.8)
)
,
200
Chapter
3. ALGEBRAS OF DISTRIBUTIONS FOR FORMULAS
Having the models
ture
P
·
such that
hR × (P(U ) \ {∅}) × R; ·i
ν(R)
Mpwe consider the semi-associative struc-
with the partial operation
(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 ∈ P(p1, u1, p2, u2, p3)},
u1∈ ρ
The groupoids
P
ν(p)
structure. The structure
p ∈ R
denoted by
is
groupoids
, relative to the family
free
, it is isomorphically represented as the disjoint union of the
By
(3.8)
L
P
ν(p)
p∈R
P
and denoted by
ν(p)
, we have
3.8.0.2. Proposition.
nonempty family
p ∈ P
, and for any regular family
n
-ary partial operation
R ⊂ S(T )of1
ν(p1,p2)
,
p ∈ R
P
. If
ρ
ν(p,q)
For any complete theoryT, for any
, u2∈ ρ
ν(p2,p3)
.
, are naturally embeddable into this
ν(R)
ν(R)
= ∅
is called a
of labelling functions and it is
for all
join of groupoids
L
p 6= q
the join
p∈R
F
P
.
ν(p)
p∈R
-types having models
ν(R)
of labelling functions, each
Mpfor each
P
P
ν(p)
ν(p)
,
on the set
L
P
ν(p)
p∈R
P (p1, ·, p2, ·, p3. . . , pn, ·, p
P(U) \ {∅}
with xed types
is interpretable by a term of the structure
p1, . . . , p
n+1
∈ R
)
n+1
.
By Proposition 3.1.0.6, we obtain the following analogue of
Proposition 3.3.0.6.
3.8.0.3. Proposition.
nonempty family
ν(R)
of labelling functions, the restriction of the structure
R ⊂ S(T )of1
For any complete theoryT, for any
-types, and for any regular family
P
ν(R)
to
the set of negative(respectively non-positive, non-negative)labels
is closed under the partial operation·.
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