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2.2.
SYNTACTIC GENERIC CONSTRUCTIONS
Clearly, every generative class has JEP since JEP means the
d
-amalgamation property over the empty set.
121
2.2.0.11. Denition.
respect to the class
any nite set
A ⊆ M
inM, i. e., there is a nite setBwith
such that
M |= X(C)
M |= Ψ(B)
and
Ψ(B) ⊆ X(C)
respect to the class
if each structure inPhas nite closures (with respect to
Clearly, an at most countable structure
with respect to
(D0; 6)
self-sucient sets
A structure
(D0; 6)
, or is
M ∈ P
has
nite closures
nitely generated over
(D0; 6)
is contained in some nite self-sucient set
and
(D0; 6)
A ⊆ B ⊆ M
Ψ(B) 6 X(C)
. A classPhas
, or is
nitely generated over
for any
and
Ψ(B) ∈ D
X(C) ∈ D0with
nite closures
(D0; 6)
(D0; 6)
M
has nite closures
S
Aifor some
i∈ω
.
Aiwith
if and only if
Ai6 A
i+1
,
i ∈ ω
M =
with
, if
with
Note that the nite closure property is dened moduloΣand
does not correlate with the cardinalities of algebraic closures. For
instance, ifΣcontains innitely many constant symbols then
acl(A)
is always innite whereas a nite setAcan or can not be extended
to a self-sucient set.
Besides, for the nite closures of setsAwe consider nite self-
sucient extensions
(D0; 6)
M
Ψ(B)
only andBcan be both a universe of a substructure of
or not. Moreover, it is permitted that corresponding diagrams
can have only nite, nite and innite, or only innite mod-
B
in a given structure
M
with respect to
els.
Thus, for instance, Example 2.2.0.5.3 with the generative class
(D0; ⊆)
has identical nite closures whereas each diagram in
D
has only innite models with innitely many distinct constants.
0
,
).
0
2.2.0.12. Denition.
is
(D0; 6)
denoted by
6
The
assumed to have some permissions and bans. Interpreting that organization by
a structure, these rules with permissions and bans are represented by diagrams
-generic6, or a
glim(D0; 6)
nature of each organization is under some rules. These rules are
At most countable structure
generic limit for the class
M ∈ K(D0)
(D0; 6)
, if it satises the following conditions:
and
122
(a)
M
has nite closures with respect to
(b) if
A ⊆ M
Φ(A) 6 Ψ(B)
and
M |= Ψ(B0)
Chapter
is a nite set,
2. GENERIC CONSTRUCTIONS
Φ(A), Ψ(B) ∈ D0,
, then there exists a set
.
D
B06 M
;
0
M |= Φ(A)
such that
and
A ⊆ B
0
Clearly, admitting uncountable
(D0; 6)
-generic structures they
can be non-isomorphic. Indeed, for instance, all innite structures
in the empty language are generic (satisfy (a) and (b)) for a given
generative class although these structures are non-isomorphic for
distinct cardinalities. But, as the following theorem shows, they
are isomorphic for an at most countable cases.
Similarly to the construction of a
given any generative class
by-step construction of a
(D0; 6)
(D0; 6)
(K0; 6)
-generic structure,
, we can embark on a step-
-generic structure
M
usingd-
amalgamations and local realizabilities. Moreover, the structure
M
is unique up to isomorphism.
2.2.0.13. Theorem
ture).
For any generative class
(on existence and uniqueness of generic struc-
(D0; 6)
, there exists
(D0; 6)
-generic
structure, unique up to isomorphism.
Proof.
construct a required structureM. We enumerate the set
all pairs
for some setsAandB,
At rst, using the generative class
(Φ(X), Ψ(X, Y )),X ∩ Y = ∅
A ⊆ B
, and in this case we assume that
, such that
(D0; 6)
, we
X
of
Φ(A) 6 Ψ(B)
any such pair is enumerated innitely often. Since the set of all
considered types
pairs is also countable:
(Φ0(X0), Ψ0(X0, Y0)) = (Φ0(∅), Φ0(∅))
X
, we construct, by the induction, some consistent set
of propositions in a language
of constant symbols that do not occur in
diagrams in
Φ(X)
is countable, then the set of enumerated
X = {(Φn(Xn), Ψn(Xn, Yn)) | n ∈ ω}
. Using the enumeration of
Σ ∪ˆK
, whereˆK
Σ
D
. In this case, for each stepi, the set
0
S
S =
S
i∈ω
is a countable set
and appear in the
Siwill equal
,
i
describing
(D0; 6)
with given rules.
the rules by formulas, and are dened by some generative class
. A
(D0; 6)
-generic structure is a realization of required organization
2.2.
SYNTACTIC GENERIC CONSTRUCTIONS
123
some set
Axiom (iii), we will have
S0 Φ0(∅)
Assume that for a stepi, the set
and
does not contain sets
S
i+1
Φi(A1), . . . , Φi(Am)
nd diagrams
•
•
and for some
embeddable in
•
j > 1
strongly embeddable in
Now we set
Since for any step of construction, the set
set
Θmcontain quantier free subdiagrams and the
form a upward directed set, the set
Σ
united with the set of all constant symbols inˆK
S
, has a quantier free diagram being a subdiagram ofS.
For
c2⇔ (c1≈ c2) ∈ S
elements
contains exactly one element in
Now we dene the structure
Xi(Ci) ∈ D0and
, where
Φ0(∅)
Xi(Ci) 6 X
Ci⊆ C
. In the initial step
i+1
is the diagram that exists by Axiom (v).
Si= Xi(Ci)
Ci⊂ˆK
Xi(Ci)
. Consider the pair
Φi(A)
. If such sets
such that
Φi(A)
(Φi(Xi), Ψi(Xi, Yi))
exist, we enumerate all of them:
. Now, using thed-amalgamation property, we
Θ1, . . . , Θm∈ D0satisfying the following conditions:
all new constant symbols in
the diagram
Xi(Ci)
B1with
is identically strongly embeddable in
Φi(A1) ≤ Ψi(A1, B1),Ψi(A1, B1)
Θ1, identically over
the diagram
, and for some
Θ
is identically strongly embeddable in
j−1
Bjwith
Θ1, . . . , Θmbelong toˆK
Φi(A1)
Φi(Aj) 6 Ψi(Aj, Bj),Ψi(Aj, Bj)
Θj, identically over
S
Θm.
i+1
S
S =
Siis consistent. Note also that since the diagrams
i∈ω
K0of all constant symbols in
K0, we dene an equivalence relation
. Since for any diagram
a1, a2∈ A
we have
¬(a1≈ a2) ∈ Φ(A)
K0.
M
with universe set of∼-classes
)
i+1(Ci+1
. In particular, by
i = 0
is already constructed
. If
Φi(A) 6 Xi(Ci)
is strongly
;
Φi(Aj)
.
Siis a diagram, the
Θm, by amalgams,
, that appear in
∼
such that
Φ(A) ∈ D0and distinct
, then each∼-class
, we set
Xi(Ci)
, we set
;
Θ
Θj,
is
c1∼
and with predicate and functional symbols interpreted by the fol-
lowing rules:
•ifP ∈ Σ
is an-ary predicate symbol then
1
˜c
M |= P(˜c1, . . . , ˜cn) ⇔ P(c1, . . . , cn) ∈ S;
•iff ∈ Σ
is an-ary functional symbol then
M |= (f(˜c1, . . . , ˜cn) ≈ ˜c) ⇔ (f (c1, . . . , cn) ≈ c) ∈ S.
124
Chapter
2. GENERIC CONSTRUCTIONS
A standard checking shows the correctness of the denition for
satisfaction. It should be mentioned that, by the local realizability
property, for any term
constant symbols
in
K0such that
(t(c1, . . . , cl) ≈ c) ∈ S
We are going to prove that for anyi, the structure
all formulas deducible from
Let
ϕ ϕ(˜c1, . . . , ˜cn)
quence of
[Xi(Ci)]
t(x1, . . . , xl)
c1, . . . , clin
K0, there exists a constant symbol
[Xi(Ci)]
be an arbitrary formula that is a conse-
C
i
. Consider a formula
˜
C
i
of the languageΣand for any
.
M
C
i
, where˜Ci= {˜ci| ci∈ Ci}
˜
C
i
ψ ψ(˜c1, . . . , ˜cn)
satises
that is
equivalent toϕand is in the prenex normal form. We shall show
that
M |= ψ
inψ. Ifψis quantier free then
bols in
Σ ∪ K0are interpreted as described in
, using the induction on the number of quantiers
M |= ψ
, because language sym-
S ⊇ Xi(Ci)
. Now
consider two possible induction steps.
Ifψequals
Xi(Ci) ` ∀x χ(x)
for any constant symbol
Ifψequals
∀x χ(x)
implies
∃x χ(x)
then
M |= χ(a)
S ` ∀x χ(x)
c ∈ K0.
then
M |= χ(b)
for any
a ∈ M
and hence we get
for some
b ∈ M
, since
S ` χ(c)
, because,
by the local realizability property and by the construction ofS,
Xi(Ci) ` ∃x χ(x)
implies
S ` χ(c)
for some constant symbol
c ∈
K0.
Now we shall show that
any nite set of elements in
C
M |= [Xi(Ci)]
i
, it suces to see that the set˜Ciis self-sucient
˜
C
i
inM. Indeed, by the construction, any diagram
subdiagram of
X
j+1(Cj+1
)
, and hence, by Axioms (iv), (viii), and
by transitivity of the relation6, the diagram
subdiagram of any diagram
∆(˜D)
.
Since, by the construction ofS, any diagram
where
Ψn(B0)
are satised and
Φn(A) 6 Ψn(B)
, is extended to some self-sucient diagram
, we get˜A ⊆˜B0,˜B06 M
Thus, all properties of
M = glim(D0; 6)
(D0; 6)
M
has nite closures. Since
M
is contained in some set˜Ciwith
Xj(Cj)
Xi(˜Ci)
∆(˜D) ∈ D0, where˜Ci⊆˜D
Φn(A) ∈ S
, and
M |= Ψn(˜B0)
.
-genericity for the structure
.
is a strong
is a strong
and
M |=
M
c
.
,
2.2.
SYNTACTIC GENERIC CONSTRUCTIONS
125
Now we argue to show that
We consider another countable structure
member that
Ai6 A
i+1
Φi(Ai) ∈ D0,
are copies
and
N =
i ∈ ω
. Now it is easy to construct step-by-step a chain of sequen-
M =
,
i ∈ ω
M |= Φi(Ai)
Φi(Bi)
S
Bifor some self-sucient sets
i∈ω
S
Aifor some self-sucient sets
i∈ω
, where each set
. SinceNis also
of the diagrams
tially embedded nite partial isomorphisms
witnessing that the diagrams
other. As
M
ontoNand preserving all denable relations and operations of
M
. Thus, the structures
fi⊆ f
i+1
,
i ∈ ω
, there is a bijection
M
By the denition, constructing a
we permit extensions of nite sets
that
Φ(A) 6 Ψ(B),M |= Φ(A)
M
is unique up to isomorphism.
N = glim(D0; 6)
Aiis a universe of a diagram
Φi(Ai)
Φi(Ai)
and
(D0; 6)
such that
Biwith
fi:
Φi(Bi)
-generic, there
N |= Φi(Bi)
Bi6 B
Ai→ Bi,
are copies to each
S
f =
i∈ω
andNare isomorphic.
(D0; 6)
A ⊆ M
-generic structure
by copies of setsBsuch
¤
, and forbid other extensions.
. Re-
Aiwith
i+1
i ∈ ω
fimapping
M
Note that [268, Theorem 2.1] implies that Theorem 2.2.0.13
does not hold for some generative classes
(D0; 6)
with uncountable
languages and corresponding to semantic amalgamation classes. It
means that for some uncountable languages there are amalgama-
tion classes
generic structures. In such a case the class
to an amalgamation class
The class
tures of a structure
and should have new isomorphism types with respect to
class
K
respect toM). Clearly,
question is: is it possible to nd the least
minimal one, with
arises for generative classes
(K0; ⊆)
0
K
can be dened taking all nitely generated substruc-
0
0
can be considered as a \closure" of the class
0
without Frasse limits, i. e., without
(K0; ⊆)
0
(K
; ⊆)
with
0
M
which collects copies of all structures in
0
K
can vary depending on
0
(K
0
; ⊆)
0
K
0
(K
; ⊆)
-generic structure? The same question
0
(D0; 6)
with uncountable languages,
can be extended
-generic structure.
0
⊃ K0, or at least a
0
K
M
(K0; ⊆)
K
. The
0
(with
0
and the
K
modulo classes generating isomorphic structures.
Considering these questions we note that the only obstacle for
the existence of a
(D0; 6)
-generic structure, actually noticed by
,
,
-
0
126
Chapter
2. GENERIC CONSTRUCTIONS
K. Zh. Kudaibergenov [268], is an imbalance between cardinalities
of denable sets forced by diagrams in
D
with their amalgams
0
and cardinalities of denable links forced by formulas in these dia-
grams. By Theorem 2.2.0.13 this imbalance can start with genera-
tive classes
(D0; 6)
having at least uncountably many copies which
are not linked by substitutions. In particular, it can be forced by
generative classes with uncountable vocabularies. Thus, if the least
(minimal) \closure" exists, then it can be achieved adding the least
(minimal) class of diagrams which removes that imbalance.
Consider the following modication ofd-amalgamation prop-
erty for a class
(
vi0) for any diagrams
injections
[Φ(A)]
A
g0(A)
(B \ A) ∩ (C \ A) = ∅
then there are a diagram
and
g1:
C → D
and
f0◦ f1= g0◦ g1.
f0:
A → B
6 X(C)
(D0; 6)
and
such that
for which
:
Φ(A), Ψ(B), X(C ) ∈ D0, if there exist
g0:
A → C
with
[Φ(A)]
A
f0(A)
6 Ψ(B)
B \A = B \f0(A),C \ A = C \g0(A)
and
[Ψ(B)]
f0(A)
A
∪ [X(C)]
Θ(D) ∈ D0and injections
[Ψ(B)]
B
6 Θ(D),[X(C)]
f1(B)
g0(A)
is consistent,
A
f1:
C
g1(C)
and
B → D
6 Θ(D)
,
Note that replacing thed-amalgamation property in the de-
nition of generative class by (
embeddings for the construction of
2.2.0.14. Denition.
structure
ifTis
A structure
M
(D0; 6)
is said to be
-generic for some generative class
Mis(D0; 6)-universal
A theory
vi0) it suces to use only identical
(D0; 6)
(D0; 6)
Th(M)
-generic
-generic structure.
of a
(D0; 6)
. A theoryTis
(D0; 6)
.
if each diagram in
strongly embeddable inM.
A structure
setsAandBin
and
M |= Ψ(B)
of a bijective mappingfrealizing a substitution
Ψ(B)
, insists on there being an automorphism ofM, containingf.
Clearly, any
and
(D0; 6)
2.2.0.15. Denition.
quantier-free diagrams is said to be
Mis(D0; 6)-homogeneous
M
with
Φ(A), Ψ(B) ∈ D0such that
, the equality
(D0; 6)
[Φ(A)]
-generic structure is
-homogeneous.
A generative class
if for any self-sucient
A
= Ψ(B)
B
and the existence
[Φ(A)]
(D0; 6)
(D0; 6)
quantier free
M |= Φ(A)
A
, equal to
B
-universal
consisting of
.
-generic
generic
D
is
0
2.2.
SYNTACTIC GENERIC CONSTRUCTIONS
If
(D0; 6)
is a quantier free generative class and
6
127
is de-
ned by the relation \to be a substructure", then the generic limit
glim(D0; 6)
is called the
Frasse limit
and denoted by
Flim(D0; 6)
.
The following theorem shows that constructing any
generic structure can be reduced to constructing some
(K0; 6)
(D0; 6)
generic one.
2.2.0.16. Theorem.
there exists a quantier-free class
(D0; 60)
class
type
-generic.
Proof.
(D0; 60)
The required class is the quantier-free generative
, where
tpqf(A), A ∈ K0},Φ(A) 60Ψ(B) ⇔ A 6 B.¤
For any
(K0; 6)
-generic structure
(D0; 60)
D0 {Φ(A) | Φ(X)
M
such that
M
is
is the quantier-free
Theorem 2.2.0.16 admits the following converse:
2.2.0.17. Remark.
of diagrams
Φ(A)
versesA, allows us to construct a class
Any generative class
(D0; 6)
, which consists
, corresponding to nite structuresAwith uni-
K
, which consists of all
0
nite structures isomorphic to structuresAsatisfying quantier-
free subdiagrams
the class
K
0
A 60B ⇔ Φ(A) 6 Ψ(B)
A |= Φ(A)qfand
The class
so there is a
Φ(A)qfof diagrams
we specify a relation
for some
B |= Ψ(B)qf.
(K0; 60)
(K0; 60)
satises the conditions 2.1.0.1 (1){(4),
-generic structure.
Φ(A) ∈ D0. Having dened
0
6
with the following condition:
Φ(A), Ψ(B) ∈ D0such that
However, in general case, that structure as well as the genera-
tive class
be non-isomorphic to the
if there are two (inconsistent) diagrams
(K0; 60)
may not exist (cf. Example 2.2.0.5), or existing
(D0; 6)
-generic structure. For instance,
Φ0(A), Φ00(A) ∈ D0with
distinct numbers of someΨ-extensions and such that quantier free
restrictions of
Φ(A)
for a structure
will contain sets
the
(K0; 60)
for
(A0, A00)
Φ0(A)
and
Φ00(A)
A ∈ K0, then the
A0and
A00satisfying
are equal to a common diagram
(D0; 6)
Φ0and
-generic structure
Φ00respectively while
-generic structure will not contain a corresponding pair
.
Illustrating that dierence we take a vocabulary with one bi-
-
-
,
128
Chapter
2. GENERIC CONSTRUCTIONS
nary relationR,
types (
a ≈ a
formula) while
The
(D0; 6)
from the class
corresponding diagrams
Φ0({a})
) but
Φ00({a})
and
Φ0({a})
Φ00({a})
assertsahas unique successor (a
may agree on quantier free
asserts it has at least two.
-generic structure is uniquely reconstructed
(K0; 60)
if for every structure
A ∈ K0and for
Φ(A) ∈ D0, the diagrams will bear the
Σ
0
2
information, written by a set of sentences, describing the number
of all possible extensionsBof the structureAwith
A 60B
, and
on interrelations of elements of those extensions.
For this aim (cf. [100, Denitions 1.8, 1.10]) it suces to
add for each
Ψ0(A, B \ A) ⊂finΨ(B)
mulas
Σ
χ∗(a)= ¬∃ZX0(A, Z)
-diagrams
inconsistent with some strong extension of
is witnessed by
Φ(A) ∈ D0all formulas
,7Φ(A) 6 Ψ(B)
, where
X(C) ⊃ Φ(A)
a)
.
χ∗(
(in some generative classes) which are
The extensions
∃Y Ψ0(A, Y )
, where
, as well as some for-
X0(A, C \ A) ⊂finX(C)
Φ(A)inD0and this fact
Φ0(A)
, obtained from
, for
Φ(A)
adding described formulas, and steps for the construction of the
(D0; 6)
guarantee that the generic structure
D
-generic structure
0
of diagrams
0
Φ0(A)
If the generative class
M
(as in the proof of Theorem 2.2.0.13)
N
with respect to the class
is isomorphic toM.
(D0; 6)
is quantier free, it not nec-
essary to extend diagrams as above: the numbers and links for
Ψ(B)
-copies over
diagrams
X(C)
structure all these diagrams extend
considering the diagrams
copies is bounded or not, and the quantier free formulas in
describe links between
We get a similar eect if
tion of the algebraic closure ofAand rather than adding to
it is requirements on
Φ(A)
, where
, where
Ψ(B) 6 X(C)
X(C)
Ψ(B)
Φ(A) 6 Ψ(B)
, are controlled by
, since constructing a generic
Φ(A)
-copies uniformly. Indeed,
we can see if the number of
-copies.
Φ(A) ∈ D0contains explicit descrip-
D
. Having nite bounds on the number
0
Ψ(B)
X(C)
D
0
of algebraic extensions we get a generalization of the Hrushovski
collapse [102, 225, 351].
-
-
,
7
Here
and elsewhere the notation
is a nite subset ofΨ.
Ψ0(A, B \ A) ⊂finΨ(B)
means that
Ψ
0
2.3.
SELF-SUFFICIENT CLASSES
2.3. Self-sucient classes
129
2.3.0.1. Denition.
smooth
, or
(ix) if
coherent
Φ, Ψ, X ∈ D0,
A generative class
if the following
Φ 6 Ψ
, and
(D0; 6)isself-sucient
coherence
X ⊆ Ψ
axiom holds:
, then
Φ ∩ X 6 X
.
The following example motivated by the original Hrushovski
examples [221, 225] and their modications [93, 94, 111, 191, 206,
269, 471] illustrates the notion of self-sucient generative class
from the semantic viewpoint.
2.3.0.2. Example
[96]. Let
A
be a nite structure of a nite
relational languageΣ. We assume that for any symbolRinΣif
a)
a)
then
all coordinates in
a.e(A)
(whereAis the universe ofA) such that
for
someRinΣand some (every) ordering
A |= R(
under all permutations of
A0= {a1, . . . , an}ofA
A |= R(
particular, ifAis a symmetric graph then
a
are
denotes
distinct, andRis closed
the number of subsets
e(A)
is the number of
aofA0.
In
edges inA.
Fix a
α, β ∈ R+. Ifαandβare rational,
imension function
prerank function
. We will write
δ
β,α
(u, v)
with the form
δ
(u, v)
β,α
δ
(A)
β,α
for
βu − αv
is called the
δ
(|A|, e(A))
β,α
where
pred-
and
replaceAbyAifAis a substructure of a xed structureN.
Below we will usually work with functions
δ
(u, v) = u − αv
1,α
with
nu − mv
For anyα,
. If
.
0 < α ≤ 1
α = m/n
, we set
is rational it is similar to work
δα(u, v)
of the form
,
Kα {A | |A| < ω
and for any
Clearly, the empty structure∅is in
If
A
is a nite substructure of a structure
dα(N , A)
the notation
and
for
dα(N , A)ifN
function
the value
B ⊆finN
B ⊆finN
inf{δα(B) | A ⊆ B ⊆finN}
means thatBis a nite substructure ofN,
means thatBis a nite subset ofN). We write
is xed. The function
, and ifαis rational then
A0⊆ A, 0 ≤ δ
K
since
α
dα(·)
dα(·)
is the
(A0)} .
1,α
δ
(∅) = 0
1,α
N
.
we denote by
(here and elsewhere
dα(A)
is called the
rank
dimension function
.
130
Chapter
2. GENERIC CONSTRUCTIONS
For structures
each
X ⊆ A,dα(A, X ) = dα(B, X )
with
A ⊂ Y ⊆ B
It follows from [96] that for any positive
A, B ∈ Kα, where
.
A ⊆ B
, i. e.,
, dene
A 6 B
δα(Y) ≥ δα(A)
α ≤ 1,hKα; 6i
if for
for any
Y
is
a generative class producing by Theorem 2.2.0.16 a self-sucient
generative class, where the amalgamation property is guaranteed
by
free amalgams
A = B ∩ C
(with
R(B ∪ C) = R(B) ∪ R(C)
A, B ∈ Kαand
δα(A ∩ C) ≤ δα(C)
of structuresBandCover their intersections
A 6 B
and
A 6 C
), i. e., by unions
B ∪ C
for allRinΣ. Axiom (ix) holds since if
A ⊆ B
then
A 6 B
if and only if for each
.
with
C ⊆ B
Note that, in particular, the arguments above hold for the lan-
guagesΣconsisting of binary relation symbols, i. e., for undirected
graphs. In such a case,
e(A)
is just the number of edges in the
structureA.
In Chapter 5 we will dene syntactic modications of the gen-
erative classes
hKα; 6i
allowing to obtain stable theories with all
possible basic characteristics for distributions of countable models,
described in Theorem 1.1.4.1.
2.3.0.3. Assumption.
2.5 we denote by
a
(D0; 6)
subclass of
-generic
K(D0)
K = Mod(T)
(D0; 6)
structure, byTthe theory
consisting of all models of the theoryT, i. e.,
.
¤
Below in this Section and Sections 2.4,
a self-sucient generative class, by
M)
Th(
,
and byKa
M
,
In order to illustrate the condition
Mod(T ) ⊆ K(D0)
two classes of examples.
Clearly, any quantier-free generative class
(D0; 6)
guage, which is closed under restrictions of diagrams
to any subset ofA, satises
Mod(T ) ⊆ K(D0)
also holds for any self-sucient class
having the followingd-covering property
(x) each type
some type
[ΨΦ(B)]
Φ(X)
B
X∪Y
Mod(T ) ⊆ K(D0)
of the theory
, where
ΨΦ(B) ∈ D0.
. The condition
:
T
is deduced from
, we present
of nite lan-
Φ(A) ∈ D
(D0; 6)
0
,
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