on the cardinality of the family of precomplete classes in pe

Mathematical Notes, vol. 78, no. 6, 2005, pp. 801–806.Translated from Matematicheskie Zametki, vol. 78, no. 6, 2005, pp. 864–869.

Original Russian Text Copyright c©2005 by S. S. Marchenkov.

On the Cardinality of the Familyof Precomplete Classes in PE

S. S. Marchenkov

Received March 18, 2003

Abstract—Let E be an infinite set of cardinality m, and let PE be the set of all functionsdefined on E . We prove that the cardinality of the family of all classes precomplete in PE is

equal to 22m

. If CR is the set of all continuous functions of real variables, then the cardinality

of the family of all classes precomplete in CR is equal to 22ℵ0.

Key words: precomplete classes, functional system, criterial family, superposition, ultrafilter.

Let E be a set containing at least two elements, and let PE be the set of all functions definedon E , i.e., the set of all functions f(x1 , . . . , xn) of the form

f : En → E, n = 1, 2, . . . .

The set PE is regarded as a functional system with the operation of superposition (see [1]).Superposition generates the closure operator [ · ] on PE ; if Q ⊆ PE , then [Q] denotes the set ofall functions that can be obtained as arbitrary superpositions of functions from Q . Sets of theform [Q] are ususally called closed classes. The set of all closed classes from PE is denoted byCl(PE) .

A set Q ⊆ PE is said to be complete in PE if [Q] = PE . A closed class Q is said to beprecomplete in PE if Q �= PE and the set Q ∪ {f} is complete in PE for any function f fromPE \ Q .

One of the main problems in the study of functional systems PE is that of completeness: todetermine, given an arbitrary system of functions from PE , whether it is complete in PE . Thefunctional approach to the solution of the completeness problem consists in the construction ofwhat is called a criterial family. The collection T of closed classes from Cl(PE) is called a criterialfamily for PE if an arbitrary system S of functions from PE is complete in PE if and only if Sdoes not belong entirely to any of the classes of the family T .

It is readily seen that the family Cl(PE) \ {PE} is always a criterial family for PE . Also,it is easy to understand that any criterial family must contain all precomplete classes. From thestandpoint of the functional approach, the optimal situation is the one in which the criterial systemconsists only of precomplete classes. In this case, the completeness criterion in PE amounts to theverification of the smallest possible number of properties (a property is the fact that the systemof functions is not contained in a precomplete class). There are criteria of this form for finitesets E (see [2–4]). For |E| = 2, a criterial family consists of five precomplete classes [2, 3] and for|E| = 3, it consists of 18 classes [5, 6]. As |E| grows, the number of classes precomplete in PE

grows approximately as the double exponential of |E| [7].For infinite sets E , only the case of a countable E was investigated. Gavrilov [8] proved that

in this case the cardinality of the family of all classes precomplete in PE equals 22ℵ0 (however,

0001-4346/2005/7856-0801 c©2005 Springer Science+Business Media, Inc. 801

802 S. S. MARCHENKOV

so far it remains unknown whether a criterial family for a countable set E will consist only ofprecomplete classes). In [9], the author proposed a method that allows one to prove statementsof the following form: if E is a countable set, Q is a closed class of an infinite cardinality m ,Q ⊆ PE , and Q satisfies certain additional conditions, then the cardinality of the family of classesprecomplete in Q is equal to 2m .

In [10, 11], it is proved that in certain classes of functions that are close to continuous ones, thefamilies of precomplete classes are of cardinality 22ℵ0 .

In this paper, we establish a general fact: for infinite sets E of cardinality m , the cardinalityof the family of all classes precomplete in PE is equal to 22m

. On the basis of the same ideas,we prove that the cardinality of the family of all classes precomplete in the class of all continuousfunctions of real variables is 22ℵ0 .

Recall that a filter over a nonempty set E is any nonempty collection F of subsets of the set Esatisfying the following conditions:

(1) if A1 , A2 ∈ F , then A1 ∩ A2 ∈ F ;(2) if A1 ∈ F and A1 ⊆ A2 ⊆ E , then A2 ∈ F ;(3) ∅ /∈ F .A filter F over E is called an ultrafilter over E if F is not contained in any other filter over E .

It is known (see, e.g., [12]), that each filter over E can be extended to an ultrafilter over E . Inaddition, if F is an ultrafilter over E , then for any subset A ⊆ E the ultrafilter F contains one(and only one) of the sets A , E \ A .

Lemma 1. The cardinality of the family of classes precomplete in PE is not less than that of thefamily of ultrafilters over E .

Proof. For any function f(x1 , . . . , xn) from PE , we set

∆(f) = {x : f(x, . . . , x) = x}.

To each ultrafilter U over E we assign a set of functions

QU = {f : ∆(f) ∈ U}.

Obviously, the sets QU1 , QU2 assigned to different ultrafilters U1 , U2 are different as well. Further,since not all of the subsets of E belong to the ultrafilter U , we have QU �= PE . Let us show thatthe set QU is closed under superposition.

Let f0 , f1 , . . . , fm ∈ QU , and let f = f0(f1 , . . . , fm) (the distribution of the variables in thefunctions f1 , . . . , fm is of no importance). Then

∆(f) ⊇ ∆(f0) ∩ ∆(f1) ∩ · · · ∩ ∆(fm).

Since we have∆(f0), ∆(f1), . . . , ∆(fm) ∈ U ,

using properties (1) and (2) of ultrafilters, we obtain ∆(f) ∈ U . From the condition E ∈ U , itfollows that the set QU contains the selector functions

eni (x1 , . . . , xi , . . . , xn) = xi.

Hence the set QU is closed under superposition.Let us prove that the class QU is precomplete in PE . Take an arbitrary function f(x1 , . . . , xn)

from PE and an arbitrary function g from PE \ QU . Then ∆(g) /∈ U and, therefore, by the

MATHEMATICAL NOTES Vol. 78 No. 6 2005

ON THE CARDINALITY OF THE FAMILY OF PRECOMPLETE CLASSES 803

property of ultrafilters, we shall have (E \ ∆(g)) ∈ U . Let h be a function from QU satisfyingthe condition ∆(h) = E \ ∆(g) . Set

d(x, y, z) ={

x if x = y,

z otherwise

(in a universal algebra the function d is called the dual discriminator [13]). The function

f1(y, z , x1 , . . . , xn) = d(y, z , f(x1 , . . . , xn))

belongs to the class QU . This follows from the relations

E ∈ U , ∆(d) = E, ∆(f1) = E.

Hence the functionf1(g(x1 , . . . , x1), h(x1 , . . . , x1), x1 , . . . , xn)

belongs to the set [QU ∪{f}] . It is readily seen that it coincides with the function f(x1 , . . . , xn) ,because for any x1 from E ,

g(x1 , . . . , x1) �= h(x1 , . . . , x1).

This completes the proof of the lemma. �For any set A ⊆ E , we denote by χA(x) the characteristic function of the set A:

χA(x) ={

1 if x ∈ A,

0 if x /∈ A.

If A ⊆ E , then we denote by A the set E \ A .

Lemma 2. There exists an injective operator Φ such that for any distinct subsets A1 , . . . , Ak ofthe set E , any nonempty collection of sets chosen from

Φ(A1), . . . , Φ(Ak), Φ(A1), . . . , Φ(Ak)

that does not contain pairs of the form Φ(Ai), Φ(Ai) has a nonempty intersection.

Proof. Suppose that A ⊆ E and χA is the characteristic function of the set A . Let us define thecharacteristic function χB of the set B = Φ(A) .

Consider the set E′ that consists of all the elements of the form

((e0 , . . . , en−1), (a0 , . . . , a2n−1)), (1)

where n ≥ 1 , e0 , . . . , en−1 are distinct elements of E , a0 , . . . , a2n−1 ∈ {0, 1} , and the sets(e0 , . . . , en−1) , (a0 , . . . , a2n−1) are assumed to be ordered. Let ϕ be a one-to-one map of Eonto E′ .

Take an arbitrary element x from E . Let ϕ(x) be of the form (1). Set χB(x) = ai , where

i =n−1∑j=0

χA(ej) · 2j .

We shall prove that this definition implies the injectivity of the operator Φ. Let A1 , A2 bedistinct subsets of the set E and, for instance, e0 ∈ A1 , e0 /∈ A2 . Let us take an x such thatϕ(x) = (e0 , (0, 1)) . If

B1 = Φ(A1), B2 = Φ(A2),



then, by the above definition, we have χB1(x) = 1 and χB2(x) = 0.If A ⊆ E and σ ∈ {0, 1} , then let

Φσ(A) ={

Φ(A) if σ = 0,

Φ(A) if σ = 1.

We shall prove that for any subsets A1 , . . . , Ak of the set E and for any σ1 , . . . , σk from {0, 1} ,the set

Φσ1(A1) ∩ · · · ∩ Φσk(Ak) (2)

is nonempty. To this end, we choose elements e0 , . . . , en−1 in E such that all the sets

A1 ∩ {e0 , . . . , en−1}, . . . , Ak ∩ {e0 , . . . , en−1} (3)

turn out to be different. For any j , 1 ≤ j ≤ k , we set

ij =n−1∑l=0

χAj (el) · 2l.

Since the sets (3) are distinct, all the numbers i1 , . . . , ik are distinct and, obviously, do not exceed2n − 1 . Let (a0 , . . . , a2n−1) be a binary set such that

aij = σj for 1 ≤ j ≤ k and ai = 0 for i /∈ {i1 , . . . , ik}.

Now, if x is an element from E such that ϕ(x) coincides with (1) (for the numbers e0 , . . . , en−1

and a0 , . . . , a2n−1 defined above), then, by the definition of the operator Φ, the element x willbelong to the set (2). This completes the proof. �Theorem 1. Let E be an infinite set of cardinality m . Then the cardinality of the family of allclasses precomplete in PE is equal to 22m

.

Proof. Denote by 2E the Boolean of the set E . To each subset D ⊆ 2E we assign the filter FD

over E generated by the sets Φ(A) for A ∈ D and Φ(A) for A /∈ D . Lemma 2 ensures that theempty set does not belong to the filter FD , and hence FD does not coincide with the Boolean 2E .For each filter FD , we choose an ultrafilter UD over E that contains FD . For distinct setsD1 , D2 , the filters FD1 , FD2 contain complementary sets. Therefore, the corresponding ultrafiltersUD1 , UD2 are distinct. Hence the cardinality of the family of ultrafilters over E is at least 22m

.By Lemma 1, we see that the cardinality of the family of all classes precomplete in PE is also atleast 22m

. On the other hand, it obviously does not exceed 22m

. This proves the theorem. �Let us make a remark to the theorem we have proved. It is rather easy to extend it to other

closed classes Q ⊂ PE . To this end, we must require that the following conditions be satisfied:first, the set

∆(Q) = {∆(f) : f ∈ Q}

must coincide with the Boolean 2E . Secondly, the class Q must must be closed with respect tothe application of the operator Φ. Finally, the class Q must contain the function d .

Let R be the set of real numbers, and let Z be the set of integers. Denote by CR the set of allfunctions continuous on R . We shall regard the set CR as a functional system with the operationof superposition.

Lemma 3. The cardinality of the family of classes precomplete in CR is not less than that of thefamily of ultrafilters over Z .


ON THE CARDINALITY OF THE FAMILY OF PRECOMPLETE CLASSES 805

Proof. For any function f(x1 , . . . , xn) from CR , we set

∆(f) = {x : x ∈ Z, f(x, . . . , x) = x}.

To each ultrafilter U over Z , we assign the set of functions

QU = {f : ∆(f) ∈ U}.

Obviously, the sets QU1 , QU2 assigned to different ultrafilters U1 , U2 over Z will be different. Inthe same way as in Lemma 1, we can make sure that the sets QU are closed under superposition.

Let us prove that the class QU is precomplete in CR . Take an arbitrary function f(x1 , . . . , xn)from C and an arbitrary function g from CR \ QU (obviously, we can assume that g is a unaryfunction). Then ∆(g) /∈ U and hence, by a property of ultrafilters, we shall have (Z \ ∆(g)) ∈ U .Let h(x) be a function from QU satisfying the condition ∆(h) = Z \ ∆(g) . Define a function

df (y1 , . . . , yn , z1 , . . . , zn , x1 , . . . , xn)

in the class QU by the following requirements: ∆(df ) = Z and

df (g(x1), . . . , g(xn), h(x1), . . . , h(xn), x1 , . . . , xn) = f(x1 , . . . , xn).

Let us verify that these requirements can be satisfied. Take m ∈ Z . If m /∈ ∆(g) , then let εm bea positive real number such that

|m − g(x)| ≥ 12|m − g(m)| for |m − x| ≤ εm.

Similarly, if m /∈ ∆(h) , then we denote by εm a positive real number such that

|m − h(x)| ≥ 12|m − h(m)| for |m − x| ≤ εm.

We can assume that the inequality εm < 1/2 holds for any integer m .One of the ways to define the function df in the class CR is as follows. Let m /∈ ∆(g) . On the

set{m}n × R

n × {m}n , (4)

we put df equal to m . Further, on the set[m − 1

2g(m), m +

12g(m)

]n

× Rn × [m − εm , m + εm]n , (5)

we define df in such a way that it continuously changes from its value m in the “center” (4) to thevalue f(x1 , . . . , xn) on the boundary of the set (5) (a point (y1 , . . . , yn , z1 , . . . , zn , x1 , . . . , xn)lies on this boundary if either one of the coordinates y1 , . . . , yn coincides with m± g(m)/2 or oneof the coordinates x1 , . . . , xn coincides with m ± εm/2).

The values of m from Z for which m /∈ ∆(h) are treated similarly. In all the other cases, weset

df (y1 , . . . , yn , z1 , . . . , zn , x1 , . . . , xn) = f(x1 , . . . , xn).

The proof is complete. �Theorem 2. The cardinality of the family of all the classes precomplete in CR is 22ℵ0 .

Proof. The proof is an almost word-for-word repetition of the proof of Theorem 1. Instead ofLemma 1, we use Lemma 3, and for the set E in Lemma 2 we take the countable set Z . Asa result, we derive that the cardinality of the family of all classes precomplete in CR is not lessthan 22ℵ0 . On the other hand, the cardinality of the class CR is equal to 2ℵ0 . Therefore, thecardinality of the family of all classes precomplete in CR is at most 22ℵ0 . �



ACKNOWLEDGMENTS

This research was supported by the Russian Foundation for Basic Research under grant no. 03-01-00783.

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M. V. Lomonosov Moscow State University


on the cardinality of the family of precomplete classes in pe

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