Combinations related to classes of finite and countably categorical structures and their theories

0.21 MB·

by  Sergey V. Sudoplatov

Combinations related to classes
7
1 of finite and countably categorical structures
0
2 and their theories∗

n
a
J
Sergey V. Sudoplatov†
3

]
O
L
h. Abstract
t
a We consider and characterize classes of finite and countably cate-
m
gorical structures and their theories preserved under E-operators and
[ P-operators. We describe e-spectra and families of finite cardinalities

2 for structures belonging to closures with respect to E-operators and
v
P-operators.
5
0 Key words: finite structure, countably categorical structure, el-
2 ementary theory, E-operator, P-operator, e-spectrum.
0
0
. We continue to study structural properties of E-combinations and P-
1
0 combinations of structures and their theories [1, 2, 3, 4, 5] applying the
7 generalcontexttotheclassesoffinitelycategoricalandω-categoricaltheories.
1
Approximations of structures by finite ones as well as correspondent ap-
:
v
proximations of theories were studied in a series of papers, e.g. [6, 7, 8]. We
i
X
consider these approximations in the context of structural combinations.
r
a We consider and describe e-spectra and families of finite cardinalities for
structuresbelongingtoclosureswithrespecttoE-operatorsandP-operators.
∗Mathematics Subject Classification: 03C30, 03C15,03C50, 54A05.
The research is partially supported by the Grants Council (under RF President)
for State Aid of Leading Scientific Schools (Grant NSh-6848.2016.1) and by Committee
of Science in Education and Science Ministry of the Republic of Kazakhstan (Grant No.
0830/GF4).
†[email protected]
1
1 Preliminaries
Throughout the paper we use the following terminology in [1, 2] as well as
in [9, 10].
Let P = (P ) , be a family of nonempty unary predicates, (A ) be
i i∈I i i∈I
a family of structures such that P is the universe of A , i ∈ I, and the
i i
symbols P are disjoint with languages for the structures A , j ∈ I. The
i j
structure A ⇋ S A expanded by the predicates P is the P-union of the
P i i
i∈I
structuresA ,andtheoperatormapping(A ) toA istheP-operator. The
i i i∈I P
structure A is called the P-combination of the structures A and denoted
P i
by Comb (A ) if A = (A ↾ A ) ↾ Σ(A ), i ∈ I. Structures A′, which
P i i∈I i P i i
are elementary equivalent to Comb (A ) , will be also considered as P-
P i i∈I
combinations.
Clearly, all structures A′ ≡ Comb (A ) are represented as unions of
P i i∈I
their restrictions A′ = (A′ ↾ P ) ↾ Σ(A ) if and only if the set p (x) =
i i i ∞
{¬P (x) | i ∈ I} is inconsistent. If A′ 6= Comb (A′) , we write A′ =
i P i i∈I
Comb (A′) , where A′ = A′ ↾ T P , maybe applying Morleyzation.
P i i∈I∪{∞} ∞ i
i∈I
Moreover, we write Comb (A ) for Comb (A ) with the empty
P i i∈I∪{∞} P i i∈I
structure A .
∞
NotethatifallpredicatesP aredisjoint,astructureA isaP-combination
i P
and a disjoint union of structures A . In this case the P-combination A
i P
is called disjoint. Clearly, for any disjoint P-combination A , Th(A ) =
P P
Th(A′ ), where A′ is obtained from A replacing A by pairwise disjoint
P P P i
A′ ≡ A , i ∈ I. Thus, in this case, similar to structures the P-operator
i i
works for the theories T = Th(A ) producing the theory T = Th(A ), be-
i i P P
ing P-combination of T , which is denoted by Comb (T ) . In general, for
i P i i∈I
non-disjoint case, the theory T will be also called a P-combination of the
P
theories T , but in such a case we will keep in mind that this P-combination
i
is constructed with respect (and depending) to the structure A , or, equiv-
P
alently, with respect to any/some A′ ≡ A .
P
Foranequivalence relationE replacing disjoint predicates P by E-classes
i
we get the structure A being the E-union of the structures A . In this
E i
case the operator mapping (A ) to A is the E-operator. The structure
i i∈I E
A is also called the E-combination of the structures A and denoted by
E i
Comb (A ) ; hereA = (A ↾ A ) ↾ Σ(A ), i ∈ I. Similarabove, structures
E i i∈I i E i i
A′, which are elementary equivalent to A , are denoted by Comb (A′) ,
E E j j∈J
where A′ are restrictions of A′ to its E-classes. The E-operator works for
j
2
the theories T = Th(A ) producing the theory T = Th(A ), being E-
i i E E
combinationofT , whichisdenotedbyComb (T ) orbyComb (T ), where
i E i i∈I E
T = {T | i ∈ I}.
i
Clearly, A′ ≡ A realizing p (x) is not elementary embeddable into A
P ∞ P
and can not be represented as a disjoint P-combination of A′ ≡ A , i ∈ I.
i i
At the same time, there are E-combinations such that all A′ ≡ A can be
E
representedasE-combinationsofsomeA′ ≡ A . Wecallthisrepresentability
j i
of A′ to be the E-representability.
If there is A′ ≡ A which is not E-representable, we have the E′-
E
representability replacing E by E′ such that E′ is obtained from E adding
equivalence classes with models for all theories T, where T is a theory of a
restriction B of a structure A′ ≡ A to some E-class and B is not elementary
E
equivalent to the structures Ai. The resulting structure AE′ (with the E′-
representability) is a e-completion, or a e-saturation, of A . The structure
E
AE′ itself is called e-complete, or e-saturated, or e-universal, or e-largest.
For a structure A the number of new structures with respect to the
E
structures A , i. e., of the structures B which are pairwise elementary non-
i
equivalent and elementary non-equivalent to the structures A , is called the
i
e-spectrum of A and denoted by e-Sp(A ). The value sup{e-Sp(A′)) |
E E
A′ ≡ A } is called the e-spectrum of the theory Th(A ) and denoted by
E E
e-Sp(Th(A )).
E
If A does not have E-classes A , which can be removed, with all E-
E i
classes A ≡ A , preserving the theory Th(A ), then A is called e-prime,
j i E E
or e-minimal.
For a structure A′ ≡ A we denote by TH(A′) the set of all theories
E
Th(A ) of E-classes A in A′.
i i
By the definition, an e-minimal structure A′ consists of E-classes with a
minimal set TH(A′). If TH(A′) is the least for models of Th(A′) then A′ is
called e-least.
Definition [2]. Let T be the class of all complete elementary theories
of relational languages. For a set T ⊂ T we denote by Cl (T) the set of
E
all theories Th(A), where A is a structure of some E-class in A′ ≡ A ,
E
A = Comb (A ) , Th(A ) ∈ T . As usual, if T = Cl (T) then T is said
E E i i∈I i E
to be E-closed.
The operator Cl of E-closure can be naturally extended to the classes
E
T ⊂ T as follows: Cl (T ) is the union of all Cl (T ) for subsets T ⊆ T .
E E 0 0
For a set T ⊂ T of theories in a language Σ and for a sentence ϕ with
3
Σ(ϕ) ⊆ Σ we denote by T the set {T ∈ T | ϕ ∈ T}.
ϕ
Proposition 1.1 [2]. If T ⊂ T is an infinite set and T ∈ T \ T then
T ∈ Cl (T ) (i.e., T is an accumulation point for T with respect to E-closure
E
Cl ) if and only if for any formula ϕ ∈ T the set T is infinite.
E ϕ
Theorem 1.2 [2]. If T′ is a generating set for a E-closed set T then the
0 0
following conditions are equivalent:
(1) T′ is the least generating set for T ;
0 0
(2) T′ is a minimal generating set for T ;
0 0
(3) any theory in T ′ is isolated by some set (T′) , i.e., for any T ∈ T′
0 0 ϕ 0
there is ϕ ∈ T such that (T ′) = {T};
0 ϕ
(4) any theory in T ′ is isolated by some set (T ) , i.e., for any T ∈ T′
0 0 ϕ 0
there is ϕ ∈ T such that (T ) = {T}.
0 ϕ
Definition [2]. For a set T ⊂ T we denote by Cl (T) the set of all
P
theories Th(A) such that Th(A) ∈ T or A is a structure of type p (x) in
∞
A′ ≡ A , where A = Comb (A ) and Th(A ) ∈ T are pairwise distinct.
P P P i i∈I i
As above, if T = Cl (T) then T is said to be P-closed.
P
Using above only disjoint P-combinations A we get the closure Cld(T )
P P
being a subset of Cl (T ).
P
The closure operator Cld,r is obtained from Cld permitting repetitions of
P P
structures for predicates P .
i
Replacing E-classes by unary predicates P (not necessary disjoint) being
i
universes for structures A and restricting models of Th(A ) to the set of
i P
realizations of p (x) we get the e-spectrum e-Sp(Th(A )), i. e., the number
∞ P
of pairwise elementary non-equivalent restrictions of M |= Th(A ) to p (x)
P ∞
such that these restrictions are not elementary equivalent to the structures
A .
i
Definition [9, 10]. A n-dimensional cube, or a n-cube (where n ∈ ω) is
a graph isomorphic to the graph Q with the universe {0,1}n and such that
n
any two vertices (δ ,...,δ ) and (δ′,...,δ′) are adjacent if and only if these
1 n 1 n
vertices differ exactly in one coordinate. The described graph Q is called
n
the canonical representative for the class of n-cubes.
Let λ be an infinite cardinal. A λ-dimensional cube, or a λ-cube, is a
graph isomorphic to a graph Γ = hX;Ri satisfying the following conditions:
(1) the universe X ⊆ {0,1}λ is generated from an arbitrary function f ∈
X by the operator hfi attaching, to the set {f}, all results of substitutions
for any finite tuples (f(i ),...,f(i )) by tuples (1−f(i ),...,1−f(i ));
1 m 1 m
4
(2) the relation R consists of edges connecting functions differing exactly
in one coordinate (the (i-th) coordinate of function g ∈ {0,1}λ is the value
g(i) correspondent to the argument i < λ).
The described graph Q ⇋ Q with the universe hfi is a canonical repre-
f
sentative for the class of λ-cubes.
Note that the canonical representative of the class of n-cubes (as well as
the canonical representatives of the class of λ-cubes) are generated by any
its function: {0,1}n = hfi, where f ∈ {0,1}n. Therefore the universes of
canonical representatives Q of n-cubes like λ-cubes, is denoted by hfi.
f
2 Closed classes of finitely categorical
and ω-categorical theories
Remind that a countable complete theory T is ω-categorical if T has exactly
one countable model up to isomorphisms, i.e. I(T,ω) = 1. A countable
theory T is n-categorical, for natural n ≥ 1, if T has exactly one n-element
model up to isomorphisms, i.e. I(T,n) = 1. A countable theory T is finitely
categorical if T is n-categorical for some n ∈ ω \{0}.
The classes of all finitely and ω-categorical theories will be denoted by
T and T , respectively.
fin ω,1
Let T be a set (class) of theories in T ∪T , T be a theory in T . By
fin ω,1
Ryll-Nardzewski theorem, Sn(T) is finite for any n. Then, for any n, classes
⌈ϕ(x¯)⌉ = {ϕ′(x¯) | ϕ′(x¯) ≡ ϕ(x¯)} of T-formulas with n free variables and
⌈ϕ(x¯)⌉ ≤ ⌈ψ(x¯)⌉ ⇔ ϕ(x¯) ⊢ ψ(x¯) form a finite Boolean algebra B (T) with
n
2mn elements, where m is the number of n-types of T.
n
The algebra B (T) can be interpreted as a m -cube C (T), whose ver-
n n mn
tices form the universe B (T) of B (T), edges [a,b] link vertices a and b such
n n
that a 6-covers b or b 6-covers a, and each vertex a is marked by some
u ⇋ ⌈ϕ(x¯)⌉, where a 6 b ⇔ u ≤ u . The label 0 is used for the vertex
a a b
corresponding to ⌈¬x¯ ≈ x¯⌉ and 1 — for the vertex corresponding to ⌈x¯ ≈ x¯⌉.
Obviously, the sets ⌈ϕ(x¯)⌉ and the relation ≤ depend on the theory T
but we omit T if the theory is fixed or it is clear by the context.
Clearly, algebras B (T ) and B (T ), for T ,T ∈ T, may be not coordi-
n 1 n 2 1 2
nated: it is possible ⌈ϕ(x¯)⌉ < ⌈ψ(x¯)⌉ for T whereas ⌈ψ(x¯)⌉ < ⌈ϕ(x¯)⌉ for
1
T . If ⌈ϕ(x¯)⌉ < ⌈ψ(x¯)⌉ for T and ⌈ϕ(x¯)⌉ < ⌈ψ(x¯)⌉ for T , we say that T
2 1 2 2
witnesses that ⌈ϕ(x¯)⌉ < ⌈ψ(x¯)⌉ for T (and vice versa).
1
5
At the same time, if a countable theory T does not belong to T ∪T
0 fin ω,1
then for some n ≥ 1, B (T ) is infinite and therefore there is a formula
n 0
ϕ(x¯), for instance (x¯ ≈ x¯), such that for the label u = ⌈ϕ(x¯)⌉ there is an
infinite decreasing chain (u ) of labels: u < u < u, witnessed by
k k∈ω k+1 k
some formulas ϕ (x¯). In such a case, if T ∈ Cl (T), then by Proposition
k 0 E
1.1 for any finite sequence (u ,...,u ,u) there are infinitely many theories
l 0
in T witnessing that u < ... < u < u. In particular, cardinalities m for
l 0 n
BooleanalgebrasB (T)andforcubesC (T)areunboundedforT: distances
n mn
ρ (0,u) are unbounded for the cubes C (T), i.e., sup{ρ (0,u) | T ∈
n,T mn n,T
T} = ∞. It is equivalent to take (x¯ ≈ x¯) for ϕ(x¯) and to get sup{ρ (0,1) |
n,T
T ∈ T } = ∞.
Thus we get the following
Theorem 2.1. Let T be a class of theories in T ∪T . The following
fin ω,1
conditions are equivalent:
(1) Cl (T) 6⊆ T ∪T ;
E fin ω,1
(2) for some natural n ≥ 1, Boolean algebras B (T), T ∈ T, have un-
n
bounded cardinalities and, moreover, there is an infinite decreasing chain
(u ) of labels for some formulas ϕ (x¯) such that any finite sequence
k k∈ω k
(u ,...,u ) with u < ... < u is witnessed by infinitely many theories in T ;
l 0 l 0
(3) the same as in (2) with u = 1.
0
Corollary 2.2. A class T ⊆ T ∪ T does not generate, using the
fin ω,1
E-operator, theories, which are neither finitely categorical and ω-categorical,
if and only if for the Boolean algebras B (T), T ∈ T, there are no infinite
n
decreasing chains (u ) of labels for some formulas ϕ (x¯) such that any
k k∈ω k
finite sequence (u ,...,u ) with u < ... < u is witnessed by infinitely many
l 0 l 0
theories in T.
Remark 2.3. Corollary 2.2 together with Proposition 1.1 allow to deter-
mine E-closed classes of finitely categorical and ω-categorical theories. Here,
since finite sets of theories are E-closed, it suffices to consider infinite sets.
ConsideringasetT oftheorieswithdisjointlanguages, fortheE-closeness
it suffices to add theories of the empty language describing cardinalities, in
ω +1, of universes if these cardinalities meet infinitely many times in T .
In such a case we obtain relative closures [4] and have the following as-
sertions.
Proposition 2.4. A class T of theories of pairwise disjoint languages is
E-closed if and only if the following conditions hold:
6
(i) for any n ∈ ω\{0} whenever T contains infinitely many theories with
n-element models then T contains the theory T0 of the empty language and
n
with n-element models;
(ii) if T contains theories with unbounded finite cardinalities of models,
or infinitely many theories with infinite models, then T contains the theory
T0 of the empty language and with infinite models.
∞
Corollary 2.5. A class T ⊂ T ∪T of theories of pairwise disjoint
fin ω,1
languages is E-closed if and only if the conditions (i) and (ii) hold.
Corollary 2.6. A class T ⊂ T of theories of pairwise disjoint lan-
fin
guages is E-closed if and only if the condition (i) holds and there is N ∈ ω
such T does not have n-categorical theories for n > N.
Corollary 2.7. A class T ⊂ T of theories of pairwise disjoint lan-
ω,1
guages is E-closed if and only if the condition (ii) holds.
Corollary 2.8. For any class T ⊂ T ∪ T of theories of pairwise
fin ω,1
disjoint languages, Cl (T) is contained in the class ⊂ T ∪T , moreover,
E fin ω,1
Cl (T ) ⊆ T ∪{T0 | λ ∈ (ω \{0})∪{∞}}.
E λ
Remark 2.9. Using relative closures [4] the assertions 2.4–2.8 also hold
if languages are disjoint modulo a common sublanguage Σ such that all
0
restrictions of n-categorical theories in T ∩ T to Σ have isomorphic (fi-
fin 0
nite) models M and all restrictions of theories in T ∩T have isomorphic
n ω,1
countable models M . In such a case, the theories T0 should be replaced by
ω n
Th(M ) and T0 — by Th(M ).
n ∞ ω
It is also permitted to have finitely many possibilities for each M and
n
for M .
ω
The following example shows that (even with pairwise disjoint languages)
ω-categorical theories T with unbounded ρ (0,1) do not force theories out-
n,T
side the class of ω-categorical theories.
Example 2.10. Let T be a theory of infinitely many disjoint n-cubes
n
(2)
withagraphrelationR , R 6= R form 6= n. Fortheset T = {T | n ∈ ω}
n m n n
we have Cl (T ) = T ∪ {T0}. All theories in Cl (T ) are ω-categorical
E ∞ E
whereas ρ (0,1) = n+2 that witnessed by formulas describing distances
2,Tn
d(x,y) ∈ ω ∪{∞} between elements.
7
Similarly, taking for each n ∈ ω exactly one n-cube with a graph relation
(2)
R , we get a set T of theories such that Cl (T) ⊂ T ∪T .
n E fin ω,1
Remark 2.11. Assertions 2.1 – 2.5 and 2.7 – 2.9 hold for the operators
Cld and Cld,r replacing E-closures by P-closures. As non-isolated types
P P
always produce infinite structures, Corollary 2.6 holds only for Cld with
P
finite sets T of theories.
3 On approximations of theories with (in)finite
models
Definition [6]. An infinite structure M is pseudofinite if every sentence true
in M has a finite model.
Definition (cf. [11]). A consistent formula ϕ forces the infinity if ϕ does
not have finite models.
By the definition, an infinite structure M is pseudofinite if and only if
M does not satisfy formulas forcing the infinity.
We denote the class T \T by T .
fin inf
Proposition 3.1. A theory T ∈ T belongs to some E-closure of theo-
inf
ries in T if and only if T does not have formulas forcing the infinity.
fin
Proof. If a formula ϕ forces the infinity then T ⊂ T for any T ⊆ T .
ϕ inf
Thus, having such a formula ϕ ∈ T, T can not be approximated by theories
in T and so T does not belong to E-closures of families T ⊆ T .
fin fin
Conversely, if any formula ϕ ∈ T does not force the infinity then, since
T ∈/ T , (T ) is infinite using unbounded finite cardinalities and we can
fin fin ϕ
choose infinitely many theories in (T ) , for each ϕ ∈ T, forming a set
fin ϕ
T ⊂ T such that T ∈ Cl (T ). ✷
0 fin E 0
Note that, in view of Proposition 1.1, Proposition 3.1 is a reformulation
of Lemma 1 in [7].
Corollary 3.2. If a theory T ∈ T belongs to some E-closure of theories
inf
in T then T is not finitely axiomatizable.
fin
Proof. If T is finitely axiomatizable by some formula ϕ then |T | ≤ 1
ϕ
for any T ⊆ T and ϕ forces the infinity. Thus, in view of Proposition 3.1,
8
T can not be approximated by theories in T , i. e., T does not belong to
fin
E-closures of families T ⊂ T . ✷
0 fin
In fact, in view of Theorem 1.2, the arguments for Corollary 3.3 show
that Cl (T ), for a family T of finitely axiomatizable theories, has the least
E
generating set T and does not contain new finitely axiomatizable theories.
Note that Proposition 3.1 admits a reformulation for Cld repeating the
P
proof. At the same time theories in T can not be approximated by theories
fin
in T with respect to Cl (in view of Proposition 1.1) whereas each theory
inf E
in T can be approximated by theories in T with respect to Cld:
fin inf P
Proposition 3.3. For any theory T ∈ T there is a family T ⊂ T
fin 0 inf
such that T belongs to the Σ(T)-restriction of Cld(T ).
P 0
Proof. It suffices to form T by infinitely many theories of structures A ,
0 i
i ∈ I, with infinitely many copies of models M |= T forming E -classes for
i
equivalence relations E , where E is either equality or complete for j 6= i.
i j
Considering disjoint unary predicates P for A we get the nonprincipal 1-
i i
type p (x) isolated by the set {¬P (x) | i ∈ I} which can be realized by the
∞ i
set M with the structure M witnessing that T belongs to the restriction of
Cld(T ) removing new relations E . ✷
P 0 i
Remark 3.4. We have a similar effect removing all relations E in the
j
structures A and obtaining isomorphic structures A′: by compactness the
i i
P-combination of structures A′ (where disjoint A′ form unary predicates
i i
P ) has the theory with a model, whose p -restriction forms a structure
i ∞
isomorphic to M. In this case we have Cld,r({Th(A′)}). ✷
P i
Remark 3.5. As in the proof of Proposition 3.1 theories in T can be
0
chosen consistent modulo cardinalities of their models we can add that e-
Sp(T) = 1 for the E-combination T of the theories in T .
0
As the same time e-Sp(T′) is infinite for the P-combination T′ of A in
i
the proof of Proposition 3.3, since p (x) has infinitely many possibilities for
∞
finite cardinalities of sets of realizations for p (x). ✷
∞
4 e-spectra for finitely categorical
and ω-categorical theories
We refine the notions of e-spectra e-Sp(A ) and e-Sp(T) for the theories
E
T = Th(A ) restricting the class of possible theories to a given class T in
E
9
the following way.
For a structure A the number of new structures with respect to the
E
structures A , i. e., of the structures B with Th(B) ∈ T , which are pairwise
i
elementary non-equivalent and elementary non-equivalent to the structures
A , is called the (e,T)-spectrum of A and denoted by (e,T)-Sp(A ). The
i E E
value sup{(e,T)-Sp(A′)) | A′ ≡ A } is called the (e,T)-spectrum of the
E
theory Th(A ) and denoted by (e,T)-Sp(Th(A )).
E E
The following properties are obvious.
1. (Monotony) If T ⊆ T then (e,T )-Sp(Th(A )) ≤ (e,T )-Sp(Th(A ))
1 2 1 E 2 E
for any structure A .
E
2. (Additivity) If the class T of all complete elementary theories of rela-
tional languages is the disjoint union of subclasses T and T then for any
1 2
theory T = Th(A ),
E
e-Sp(T) = (e,T )-Sp(T)+(e,T )-Sp(T).
1 2
We divide a class T of theories into two disjoint subclasses T fin and T inf
having finite and infinite non-empty language relations, respectively. More
precisely, for functions f: ω → λ , where λ are cardinalities, we divide T
f f
intosubclasses T f oftheoriesT suchthatT hasf(n)n-arypredicatesymbols
for each n ∈ ω.
For the function f we denote by Supp(f) its support, i.e., the set {n ∈
ω | f(n) > 0}.
Clearly, the language of a theory T ∈ Tf is finite if and only if ρ ⊂ ω
f
and Supp(f) is finite.
Illustrating (e,T )-spectra for the class T of all cubic theories and taking
the class Tfin ⊂ T of all theories of finite cubes we note that for an E-
0
combination T of theories T in T fin, (e,T )-Sp(T) is positive if and only if
i 0
thereareinfinitely many T . Insuch acase, (e,T )-Sp(T) = 1 andnew theory,
i
which does not belong to T fin, is the theory of ω-cube.
0
The class T is represented as disjoint union of subclasses T of the-
fin fin,n
ories having n-element models, n ∈ ω \{0}. For N ∈ ω, the class S T
fin,n
n≤N
is denoted by T .
fin,≤N
Proposition 4.1. For any T ⊂ T, Cl (T )\T 6= ∅ if and only if for
E fin
any natural N, T 6⊂ T .
fin,≤N
10