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2.6.
FINITE CLOSURE PROPERTY IN FUSIONS
151
M. Ziegler [117, 118]; A. Hasson, M. Hils [200]; K. Holland [217,
218]; M. Ziegler [478]) are dened by non-negative linear predimen-
sion functions
δifor classes
(Di; 6i),i = 0, 1, 2
, with non-negative
linear predimension functions
δ(A) = δ1(A) + δ2(A) − δ0(A),
of fusions, where
number of tuples, connected by predicates onA,
general, these fusions do not have closures of form
since closed, relative to
ues
δ1(A)
and
rized number of weights of connections with respect to
δ2(A)
tion of
can exceed the number of elements that used in calcula-
δ(A)
). Moreover, iterative numbers
sup{nA} = ∞
Theories of
δi(A) = |A| − αi|Ri(A)|,αi∈ R+,
Cl1and
δ2(A)
) can be not closed relative to
Cl2, sets (with non-decreasing val-
.
Herwig graphs
[206] and theories of digraphs, that
Ri(A)
i = 0, 1, 2
is the
. In
hCl1, Cl2i
Cl3(summa-
δ1(A)
and
nAcan be unbounded:
will be described in Chapter 5, can be also considered as Hrushovski
fusions. Moreover, a countable graph structure, supplied by weights
of edges or arcs, allows to interpret resulted theoriesTas fusions
of countable set of theories
ClT6= hClki
model ofT, and
of
Tk,
k ∈ ω
, where
k∈ω
Clkare self-sucient closures in generic models
.
Inessential combinations
identical closures
Cl1and
Tkof languages
ClTis a self-sucient closure in a generic
M3of structures
Cl2generate identical closures
amples of fusions for generative classes with
(2)
{I
},k ∈ ω
k
M1and
, with
M2with
Cl3. Ex-
Cl3= hCl1, Cl2i
are
represented in Chapter 4.
Below, in this Section, we shall consider closure operations
generated by
Cl1and
Cl2. Fix a fusion of generative classes
Cl3,
,
(D3; 63) (D1; 61) F
(D0;60)
(D2; 62).
The following proposition represents an obvious (by compact-
ness) characterization for the preservation of the nite closure prop-
erty under transition from the classes
class
(D3; 63)
2.6.0.5. Proposition.
.
A generative class
the nite closure property if and only if, in
ture, there exists a sequence
An,
(D1; 61)
n ∈ ω
and
(D3; 63)
(D3; 63)
(D2; 62)
does not have
-generic struc-
to the
, of nite sets having the
152
Chapter
2. GENERIC CONSTRUCTIONS
same cardinality, such that closures
leastniterations with respect to
Cl3(An)
Cl1and
can be obtained by at
Cl2, and a description
of unbounded number of iterations for these sets is consistent with
(D3; 63)
-generic theory.
For the illustration, we represent an example of fusion
(D3; 63)
of generative classes, for which
Cl3= hCl1, Cl2i
and the
nite closure property is not satised.
2.6.0.6. Example.
languages
{Q
(2)
i
},i = 1, 2
Let
(Di; 6i)
be generative classes of graph
, with diagrams describing pairwise dis-
joint edges such that any vertex is either isolated or belongs to
unique edge, being not a loop. Moreover, we add the following
requirements:
(1) the number of edges and the number of isolated vertices are
unbounded;
(2) each nite graph with given number of isolated vertices and
given number of edges without common vertices is represented by
a diagram in
(3) if a vertexabelongs to a setA, where
description
[a, b]
, then
(4) 4)
if
(a, b) ∈ Qiand
Note that each self-sucient closure of nite set
a
(Di; 6i)
D
;
i
Φ(A) ∈ Diand the
Φ(A)
contains an information thatabelongs to an edge
b ∈ A
;
A 6iB
if and only if
b ∈ B
then
A ⊆ B
and, for any vertex
b ∈ A,i = 1,2
a ∈ A
.
A
in
-generic structure is obtained by adding to each endpoint
of edge, belonging toA, an opposite endpoint of the edge.
Now we dene a fusion of the generative classes
and
(D2; 62)
, allowing for each vertex to be either isolated, or
(D1; 61)
belong to unique edge, or belong to two edges of distinct colors
(
Q1and
formation about
Q2), such that descriptions of diagrams contain only in-
nite
chains, such that every nite length is rep-
resented.
Self-sucient sets of
(D3; 63)
-generic structure are nite
sets, closed under additions of opposite endpoints of edges. At
the same time, a presence of unbounded chains means that there
exists a countable model of
(D3; 63)
-generic theory, having an in-
nite chain. There are no elements in such chain belonging to
self-sucient sets, since such sets should be nite.
¤
,
2.6.
FINITE CLOSURE PROPERTY IN FUSIONS
153
For practical creation of operation
sures
Cl1and
Cl2and with preservation of the nite closure prop-
erty, it is appropriate to use the
or MI
-principal
minimization can be
pending on cardinalities of
, by which iterative numbers
majorized
|A|:nA≤ f (|A|)
by estimatesfof numbers
Cl3with non-identical clo-
minimization iterative principal
nAare minimal. This
nAde-
. if there exists a
majorizing estimate for number of iterationsffor all setsA, in-
cluded in self-sucient diagrams
is preserved for self-sucient amalgams in the class
estimate will be valid for all models of
Having a majorizing estimate for generative class
Φ(A) ∈ D3, and this estimate
D
, then this
3
(D3; 63)
-generic theory.
(D3; 63)
, the -
nite closure property for this class will be satised. Thus we have
the following:
2.6.0.7. Proposition.
neric class
(Di; 6i)
(D1; 61)F
have the nite closure property,
a majorizing estimate for iterative numbers for the class
then the class
(D3; 63)
Let a class
(D0;60)
(D2; 62),Cl3= hCl1, Cl2i
(D3; 63)
i = 1, 2
coincide with a ge-
, and there exists
has the nite closure property too.
. If classes
(D3; 63)
Now we consider a sucient condition for the existence of a
minimal majorizing estimate (
nA≡ 1
) of fusion
,
,
(D3; 63) (D1; 61) F
where closures
Cl1and
2.6.0.8. Denition.
generic structure
M3, there is (not necessary formula-denable)
(D0;60)
Cl2are, possibly both, non-identical.
Suppose that on the universe of
(D2; 62),
(D3; 63)
an equivalent relationE, satisfying the following conditions for
any nite set
(1)
Cl1(A) =
(2)
Cl2(C) = C
Having these conditions, we say that
stepped special closure system with minimality condition
an ESSM
A ⊆ M3:
-system
S
Cl1(A ∩ E(a))
a∈A
for any setCwith
.
;
Cl2(A) ⊆ C ⊆
a∈Cl2(A)
(Cl1, Cl2)
S
E(a)
is anE-
, or
-
.
154
If
(Cl1, Cl2)
is an ESSM-system, then there exists the mini-
Chapter
2. GENERIC CONSTRUCTIONS
mal majorizing estimate for iterative numbers of self-sucient class
(D3; 63)
generic theory. Then the set
the operation
implies thatBis
. Indeed, letAbe a nite set in a model of
B Cl1(Cl2(A))isCl1-closed, since
Cl1is transitive, and the inclusion
B ⊆
a∈Cl2(A)
Cl2-closed.
(D3; 63)
S
E(a)
The following generalization of concept of ESSM-system guar-
antees the existence of the majorizing estimate for iterative num-
bers in the fusion
(D3; 63)
.
-
2.6.0.9. Denition.
generic structure
Suppose that on the universe of
(D3; 63)
M3there is (not necessary formula-denable) an
equivalent relationE, satisfying the following conditions for any
nite set
D
:
3
(1)
(2) if
Cl2(C) = C
(3) there exists a nite number
dened by some formula in
Having these conditions, we say that
special closure system
If
A ⊆ M3, where
S
Cl1(A) =
a∈A
C ⊆
a∈Cl2(A)
;
(Cl1, Cl2)
is an ESS-system, then there exists a mini-
M3|= Φ(A)
Cl1(A ∩ E(a))
S
E(a)
Φ(A)
, or ESS
;
and
such that
-system
for some diagram
S
Cl2(C) ⊆
a∈Cl2(A)
mAofE-classes
Cl3(A) ⊆
(Cl1, Cl2)
.
Φ(A) ∈
E(a)
, then
E1, . . . , E
m
A
S
m
A
Ei.
i=1
is anE-stepped
mal majorizing estimate for iterative numbers of self-sucient class
(D3; 63)
generic theory. Then the iterative number is bounded by
since each iteration denes a subset of
. Indeed, letAbe a nite set in a model of
m
A
S
Ei, and since stabiliza-
i=1
(D3; 63)
mA+ 1
tion of number ofE-classes, containing the result of two sequential
iterations, the conditions (1) and (2) imply that the resulted set is
simultaneously
Cl1- and
Cl2-closed.
Thus we have the following:
-
,
-
,
2.7.
ON GENERATING ELEMENTS IN GENERIC ALGEBRAS
155
2.6.0.10. Theorem.
(Cl1, Cl2)
be an
ESS
the nite closure property. Then the class
Let
(D3; 63)
(D1; 61) F
(D0;60)
be a class
(D2; 62),
-system, and the classes
(Di; 6i),i = 1,2
(D3; 63)
has the nite
, have
closure property too.
We denote the class
(D3; 63)
, being in Theorem 2.6.0.10,
by
ESS
(D0;60)
(D2; 62).
, be generative classes satis-
Let
(Di; 6i),(D
(D1; 61) F
0
0
; 6
),i = 1, . . . , n
i
i
fying the following conditions:
0
(1)
(D
(2)
(D
n − 1
.
The generative class
0
; 6
) = (D1; 61)
1
1
0
; 6
i+1
0
i+1
) = (D
(D
0
i
;
; 6
0
n
0
ESS
)F
i
(D
0
; 6
)
is denoted by
n
0
0
;6
i
i
)∩(Di;6i)
(Di; 6i),i = 1,. . . ,
ESS)n
(F
(Di; 6i).
i=1
Theorem 2.6.0.10 implies, that the nite closure property is
preserved under nite iterations of creation of generative classes
on a base of ESS-systems, i. e., by transition from classes
i = 1,. . . , n
, to the class
(F
ESS)n
(Di; 6i).
i=1
(Di; 6i)
,
2.6.0.11. Corollary.
Any class of form
(F
ESS)n
(Di; 6i)
i=1
has the
nite closure property.
2.7. On generating elements
in generic algebras
In this Section, we consider questions of existence of various
generating sets in countable homogeneous algebras that have built
by syntactic generic constructions. It demonstrates some applica-
tions of syntactic generic constructions producing desirable prop-
erties of generic algebras.
Throughout this Section we suppose that
countable functional language with a set
C0of constants and con-
sider at most countable algebras of languageΣ. By
Sigma
T (Σ)
is at most
we denote
156
Chapter
2. GENERIC CONSTRUCTIONS
the set of terms of the languageΣ, and by
stant terms of the language
Σ ∪ A
for a constant setAdisjoint
T (Σ, A)
the set of con-
fromΣ.
We shall use the standard terminology of the universal algebra
in the book [42] by A. G. Pinus.
2.7.1. Bases of generators
2.7.1.1. Denition.
set
X ⊆ M
x ∈ X
X \ {x}
. Recall thatXis a
does not belong to the subalgebra ofM, generated by
.
Consider a generative class
Since
Φ(A)
contains some quantier free diagram overA, it has
Let
M
be an algebra, generated by some
basis of generators
(D0; 6)
and a diagram
if any element
Φ(A) ∈ D0.
a complete information of connections of values of terms
any
(t1(
x),
t1(
a) ≈ t2(a)) ∈ Φ(A)
diagram
A
generated byAand such that
Φ(A)
free formulas in
C06= ∅
x) ∈ T(Σ)
t2(
Φ(A)
(where
Φ(A)
and
for any
or
¬(t1(a) ≈ t2(a)) ∈ Φ(A)
A ∪ C06= ∅
. Having constant symbols inΣ(i. e., for
) we obtain a subalgebra
a ∈ A,l(a)
= l(
.
x)
Thus any
) allows to construct an algebra
A
A
of
C
0
satises all quantier
Φ(A)
A
generated by
Φ(A)
t(
,
a)
:
either
C0.
for
By the denition, any algebra
A
has a nite basis of gener-
Φ(A)
ators.
2.7.1.2. Denition.
Ψ(B),A
X
is an
A
, and
Φ(A)
be a subalgebra of
Φ(A)
A
-basisofA
Ψ(B)
Xisindependent
belong to the subalgebra of
is a
basisofΦ(A)ifX
Ψ(B)
, where
extensibleifX
Ψ(B)
, where
Φ(A) 6 Ψ(B)
can be extended to a basis of
Φ(A) 6 Ψ(B)
Dene external bases of
A
ifXgenerates an algebra containing
Φ(A)
,Xbe a set in
Ψ(B)
, i .e., any element
is a
A
A
, generated by
Ψ(B)
-basis of
Ψ(B)
. A basis
X
A
Φ(A)
of
Ψ(B)
.
A
. Let
Φ(A)
A
x ∈ X
X \ {x}
Ψ(B)
. A set
. The set
does not
for some diagram
Φ(A)iseverywhere
for any diagram
The following assertion produces a syntactical characterization
of existence of generator basis for
(D0; 6)
-generic algebras.
Φ(A) 6
X
2.7.
ON GENERATING ELEMENTS IN GENERIC ALGEBRAS
157
2.7.1.3. Theorem.
(D0; 6)
-generic structure. The following conditions are equivalent:
(1) M
(2)
has a basis of generators;
for any diagrams
there exists an everywhere extensible basisXof
can be extended to an everywhere extensible basisYof
Proof.
Φ(A)
Ψ(B0)
As
A
and
be diagrams such that
A
Φ(A0)
⊆ A
Φ(A0)
X0⊆ Y0,
A
X(C)
and
(1) ⇒ (2)
Ψ(B)
and
A
Ψ(B0)
X0is a basis of
A
X(C0)
Y0is a subset of some
Y
of
Ψ(B)
, corresponding to
Let
(D0; 6)
Φ(A)
be a generative class and
and
Ψ(B)
, where
. LetZbe a basis of generators forM,
be diagrams, where
Φ(A) 6 Ψ(B),Φ(A0)
Φ(A0) 6 Ψ(B0)
are nitely generated subalgebras of
Ψ(B0)
, there are nite subsets
X0and
Φ(A0),Y0is a basis of
are isomorphic for any diagram
A
X(C0)
⊆ M
X0and
, the bases
Y0respectively, are required
M
Φ(A) 6 Ψ(B)
Φ(A)
and
such that
Ψ(B)
.
M |= Ψ(B0)
M
Y0ofZsuch that
Ψ(B0)
. As algebras
X(C) ∈ D0and
X
of
Φ(A)
be a
X
and
and
and
everywhere extensible bases.
(2) ⇒ (1)
extensible basis
extensible basisYof
and
M
can be considered as a countable chain of diagrams
where
Φn(An) 6 Φ
everywhere extensible bases
n ∈ ω
. Thus
Obviously, if a
then
M
a basis stabilizes on some nite step, i. e., some algebra
. As each diagram
X
that can be extended again to an everywhere
Ψ(B)
for each diagram
Φ(A) ∈ D0has an everywhere
Ψ(B),Φ(A) 6 Ψ(B)
Φn(An)
Z
n+1(An+1
S
),n ∈ ω
Xnof
Xnis a basis ofM.
n∈ω
(D0; 6)
-generic structure
, we can step-by-step construct
Φn(An)
, where
Xn⊆ X
n+1
¤
M
is nitely generated
has a basis of generators. So the process of construction of
A
Φ(A)
has
a basisXthat coincides with a basis ofM. It means that for any
diagram
with
Ψ(B)
, where
A
. Thus we have the following
Φ(A)
Φ(A) 6 Ψ(B)
, an algebra
A
Ψ(B)
coincides
,
.
,
,
,
2.7.1.4. Theorem.
(D0; 6)
Ψ(B)
is a term
-generic structure. The following conditions are equivalent:
(1) M
(2)
has a nite basis of generators;
there is a diagram
, where
Φ(A) 6 Ψ(B)
a),a ∈ A
t2(
Let
(D0; 6)
be a generative class and
Φ(A) ∈ D0such that for any diagram
, and for any term
,
such that
(t1(
b) ≈ t2(a)) ∈ Ψ(B)
b),b ∈ B
t1(
M
be a
,
there
.
158
Chapter
2. GENERIC CONSTRUCTIONS
2.7.2. Free amalgams and free algebras
2.7.2.1. Denition.
generating a nontrivial variety,
in some generative class
I,Φ(A) 6 Ψ(B),Φ(A) 6 X(C),Φ(A) = Ψ(B) ∩ X(C)
amalgamofΨ(B)
X(C)
, is a diagram
Ψ(B) ∪ X(C)
and
a),a ∈ B ∪ C
t2(
and formulas
LetIbe a set of identities of languageΣ,
Φ(A),Ψ(B)
(D0; 6)
such that each diagram satises
, and
X(C)
be diagrams
. TheI-free
and
X(C)
over
Φ(A)
, denoted by
Ψ(B) ∗
Θ(B ∪ C) ∈ D0, satisfyingIand containing
a) ≈ t2(a))
¬(t1(
,
of distinct terms
t1(
x),
for
t2(
any values
x) ∈ T(L)
t1(
suc
I
Φ(A)
a)
that:
1)Idoes not imply
a)ort2(a)
2)
t1(
do
not belong to
If
I = ∅,I
denoted by
-free amalgams
Ψ(B) ∗
do
T (Σ, C)
X(C)
Φ(A)
x) ≈ t2(x))
(t1(
not belong to
.
Ψ(B) ∗
.
,
T (Σ, B)
I
X(C)
Φ(A)
, and
t1(
are called
a)ort2(a)
free
and
It is obvious for languages of unary functions that
A
where
free class, then
A
Ψ(B)∩AX(C)
Ψ(B) ∗
SinceI-free amalgams
Ψ(B)∗
= A
I
Φ(A)
Φ(A)
I
Φ(A)
=
X(C)
A
. If additionally
Ψ(B)
∪ A
X(C)
(D0; 6)
X(C) = Ψ(B) ∪ X(C)
Ψ(B) ∗
I
Φ(A)
X(C)
satisfyIwe observe
,
is a quantier
.
that forming generic algebras from diagrams satisfyingIand using
I
-free amalgams one get algebras in the variety dened byI.
If any diagram
tions of formulas
x)
suc
t2(
A
Φ(A)
I
. Thus the
Moreover, if
then
h that
is a free algebra (generated byA) of variety generated by
(D0; 6)
D
M
is countably and not nitely generated.
Now we consider free amalgams and observe that
A
A
X(C)
Ψ(B)∗
= A
Φ(A)
Φ(A)
X(C)
Φ(A)inD0satisesIand contains only nega-
a) ≈ t2(a))
(t1(
x) ≈ t2(x))
(t1(
-generic structure
is quantier free and closed under free amalgams
0
for
any distinct terms
is
not deduced fromI, then any
M
is also anI-free algebra.
t1(
A
for algebras
A
Ψ(B)
and
A
being subalgebras of
X(C)
.
x)
Ψ(B)
and
∩
h
2.8.
ON VARIETIES OF GENERATIVE CLASSES
159
A basisXof a diagram
Φ(A)isinternalifX ⊆ A
.
The following proposition asserts that the class of internal bases
is closed under free amalgams.
2.7.2.2. Proposition.
be an internal basis of
of
Φ(A) = Ψ(B) ∩ X(C),Ψ(B) ∗
Ψ(B)
Ψ(B) ∗
by the denition,
that
and
Φ(A)
Proof.
Y ∪Z
X(C)
X(C)
over
.
SinceYgenerates
Y ∪Z
is independent. Consider an arbitrary element
without loss of generality
for
b ∈ Z
If
some
b ∈ (Y ∪ Z) \
,
then
a ∈ A
independency ofZ. Soais not a value of terms
b ∈ Z
Y \
{a}
¬(a ≈ t(
or
b)) ∈ Ψ(B) ∗
.
Now by the denition of free amalgam, we have
elementahas been chosen arbitrarily in
independent.
¤
Let
Y
be an internal basis of
X(C),X Y ∩ Z
Φ(A)
Φ(A)
generates
a ∈ Y
{a}
X(C)
X(C)
Φ(A)
. Then
then, as
. Thus
A
Y ∪ Z
andZgenerates
Ψ(B)
A
Ψ(B)∗
. If
(a ≈ t(
a ∈ X
for
any
be an internal basis
X(C)
be a free amalgam of
is an internal basis of
A
. We argue to show
X(C)
Φ(A)
b)) ∈ Ψ(B) ∗
Y
is independent,
that contradicts the
b)
t(
b ∈ (Y ∪ Z) \
Y ∪ Z
, the set
Ψ(B),Z
X(C)
a ∈ Y ∪Z
Φ(A)
,
where
{a}
Y ∪ Z
Now, by Proposition 2.7.2.2, we immediately obtain the follow-
ing
, then,
X(C)
b 6∈ Y
b ∈
. As the
is
,
.
2.7.2.3. Corollary.
subclass of
that
D
D
, consisting of diagrams with internal bases, such
0
is a closure of
0
Let
(D0; 6)
D
under free amalgams. Then
1
be a generative class,
D
be a
1
(D0; 6)
generic structures have bases of generators.
2.8. On varieties of generative classes
We consider relations on generative classes that correspond to
basic closure operations in varieties of algebras (closures under sub-
algebras, under homomorphic images, and under Cartesian prod-
ucts), and prove characterizations of the relations on classes of
algebras in terms of relations on classes of generative classes.
-
160
Chapter
2. GENERIC CONSTRUCTIONS
As generative classes correspond to at most countable homo-
geneous structures, we consider closure operations for varieties re-
stricted on the class of at most countable homogeneous algebras
and call these restrictions of varieties by
bras
. By a
variety of generative classes
varieties of generic alge-
we call a class of generative
classes of a xed functional languageΣ, corresponding to algebras,
forming a variety of generic algebras.
Using Theorem 2.4.0.7 we obtain
2.8.0.1. Theorem.
Let
M
and
M0be algebras of a variety of
generic algebras of languageΣ. The following conditions are equiv-
alent:
(1)
the algebra
(2)
there are generative classes
guageΣsuch that
and
D0E D
0
0
2.8.0.2. Denition.
diagram
Φ(A)ifA0is a quotient-set ofAand an algebra
is a quotient-algebra of algebra
ary symbol
f ∈ Σ
and for any tuple
a)
a),
A
b
elong to a common class in
. . . , tn0(
Φ0(A0)
and
ti1(
f(t10(
class in
Thus equalities
correspond to equalities
classes, and
Let
The class
Φ0(A0)
(D0; 6)
(D
M
Mis(D0; 6)
.
, for any terms
a
a))
.
may contain some additional equalities.
and
0
; 60)
0
is isomorphic to a subalgebra of
-generic,
A diagram
A
of
elements ofA,
and
f(t11(a),
a) ≈ t2(a)) ∈ Φ(A)
(t1(
(t1(
0
(D
; 60)
0
is a
quotient-class
. . . , tn1(
a0) ≈ t2(a0)) ∈ Φ0(A0)
be generative classes of languageΣ.
(D0; 6)
Φ0(A0)
. It means that for anyn-
Φ(A)
x),i =
tij(
l(a)= l(
of
M0is
is a
A
Φ0(A0)
a))
on
(D0; 6)
and
quotient-diagram
,
b
elong to a common
M0;
0
(D
; 60)
of lan-
-generic,
(D
0
0
; 60)
0
of
A
Φ0(A0)
1, . . . , n,j = 0, 1
x)
,
if values
i = 1, . . . , n
a)
ti0(
, then
tuples of elements
on
tuples of
if the following
conditions hold:
1) for any diagram
Φ0(A0)
belongs to
2) any diagram
diagram
Φ(A) ∈ D0;
3) for any diagrams
Ψ(B)
is equivalent to
D
Φ(A) ∈ D0, some its quotient-diagram
0
;
0
Φ0(A0) ∈ D
0
is a quotient-diagram of some
0
Φ(A), Ψ(B) ∈ D0, the condition
Φ0(A0) 60Ψ0(B0)
.
Φ(A) 6
,
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