Generalization of Abian’s fixed point theorem

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I : PRACE MATEMATYCZNE X I I I (1969) ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : COMMENTATIONES MATHEMATICAE X I I I (1969)

A.

and

(Warszawa)

Generalization of Abian’s fixed point theorem

Eecently Abian [1] stated the following theorem:

Let E be a finite set and f — a mapping from E into E. Then f has a fixed point i f and only i f E is not the union of three mutually disjoint sets E x, E 2, E

such that

(1) E t n f { E t) = 0 f o r i = 1 , 2 , 3 .

We show that the assumption of finiteness of the set E is superfluous.

The “only if” part of the theorem is trivial. I t remains to prove that if / has no fixed point, then E is the union of three mutually disjoint sets Ei (i = 1 , 2 , 3 ) such that (1 ) holds.

P roof. Let a0, a x, a 2, . . . , % , ... (| < a) be a sequence (with distinct terms) of all elements of E (we employ here the well-ordering principle).

We define f°(x) = x, f k{x) = f [ f k~l {x)) f°r & = 1 , 2 , 3 , ... Consider the sequences /°(%),/1(%),/2(%), ... (| < a ). Every sequence has at least two elements and is either (a) ultimately periodic or (b) consists of mu­

tually distinct terms. Evidently, if E is finite, only (a) is possible.

Let be the least non-negative integer for which there exist a non­

negative integer m and an ordinal number r\ < I such th at/*£(%) = f m{ari).

If such a number does not exist and (a) holds, we define Jcs as the least integer for which there exists an integer \ such that 0 < Ц < Jcs and f* { a s) = f k4 aę). However, Ją may be not defined for some | < a.

Now we construct sets Ei (i = 1 , 2 , 3 ) .

Elements f

c(a0) we put into E x, f

c+

(a0) — into E

unless the sequence

( 2 ) f ^ o ) , f 4 a 0) , P ( a 0) , . . .

is ultimately periodic and JcQ — l

is odd. In the last case we put f ° { a 0)

into E

and handle other elements f m{a0) for m < Jc

in the same way

as in the first case. Elements /*0+р(а0) f°r P — 0 , 1 , 2 , . . . occur among

the elements /m(a0) (w < Jc0).

64 A. M ąkow ski and К. W iśn ie w sk i

Suppose that the sequences f { a n) i f («О, ...(? ?< £) are split into the sets Е г, E 2, E 3. If for some m and ) j< ^ we have / кЦщ)

= f m(a^)eEi, then we put (1 into E i+j (we consider indices mod2 ), otherwise we proceed as in the case of sequence (2).

From the method of construction of sets Ei it follows that E = Е г ^ Е

~> E z, Ei (i = 1 , 2 , 3 ) are mutually disjoint and (1) is satisfied.

Let us denote by (I) the implication proved above. We notice that the axiom of choice was used in the proof of (I).

Now we show that the use of this axiom in the proof of (I) is essen­

tial. In the sequel we use the notation of Mostowski [2] and denote by ZF' the axiomatic system of set theory which differs from ZF (the Zer- melo-Fraenkel system) by containing the constant 0 for the void set and the axiom stating the existence of an infinite set of individuals.

The axiom of extensionality in ZF' is restricted only to sets. We denote by [n] the axiom of choice for families of sets of power n < со.

Proposition (I) is independent of ZF' w {[n]: n > 1}.

Indeed, if (I) were a theorem of the above system, then by the com­

pactness theorem there would exist a finite set M of integers > 1 such that ZF' w [M] h (I). We show that it is impossible. Let p Ф 3 be a prime number greater than every number in Ш. We use the model from [2]

taking as О the cyclic subgroup (of the order p) of the group Sp . Now we sketch the construction of the model. Let E k = {(&—l ) p + l , .. ., Jcp}

00

for к = 1 , 2 , ... and K

= N — { J N k. Let K s (£ denotes here an ordinal

*=i

number > 0 ) be the subset of P {\ JK ) consisted of all elements x for which there exists a positive integer q — q(x) such that if (peGm and

<px = ... = <pq = 1, then (p (x) = x. As individuals we assume the ele­

ments of K

and as sets the elements of any K s with | > 0. Evidently, for an arbitrary subgroup H of the group Gm and for an arbitrary sequence

Г

K x, K r of proper subgroups of H we have £ [ I I : K i] > p, hence i=l

(cf. [2 ], proof of Theorem 3) the proposition [p —1 ]! is true in the model, therefore the proposition [M] is also true, because M <= { 2 , 3 , . . . , p —1).

If (ilf i 2, . .. , ip_l ,p )eG , we put

oo

/ = +

(k — l)p + i2y,

kp},

<kp, (к-1)р + {г)}.

I t is evident that the function / maps E = K

onto itself, that it

has no fixed point and is a set of the model. Suppose that there exist

in the model the disjoint sets E u E 2, E z such that E t ^ E

^ E

= E

and (1) holds. For every к > 1 the sets Е г ^ Nk , E

Nk , E

^ Nk are

A bian 's fix e d p o in t theorem 65

disjoint, different from Nk and E x r\ Nk ^ E

Nk ^ E z Nk — Nk.

3

Then A = U {Ei r\ Жк: JceN} is a set of the model. Elements of this i=l

set have at most p —1 elements. Because we showed that in the model the proposition [p—1]! is true, there exists a choice function for the family A. This implies the existence of a choice function for the family

{Жк: JceNj, which is not true (cf. [2], proof of Theorem 3).

R eferences

[1] A. A b ia n , Afixed point theorem, Nieuw Arch. Wisk. (3), 16 (1968), pp. 184-185.

[2] A. M o stow sk i, Axiom of choice for finite sets,Fund. Math. 33 (1946), pp. 137-168.

INSTITUTE OF MATHEMATICS, UNIVERSITY OF WARSAW

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