One abstract characterization of intervals of cardinal numbers

A C T A U N I V E H S I 'Г A T I S L O D Z I E N S I S FOLIA MAT HEMATIC-A 9, 1997

A rc hil K ip ia n i

O N E A B S T R A C T C H A R A C T E R I Z A T I O N O F I N T E R V A L S

O F C A R D I N A L N U M B E R S

YYV g iv e , in t hé la n g u a g e o f f u n c t io n s a n d a u t o m o r p h is m « o f b i-nary r ela t io n s , a c h a r a c t er iz a t io n o f t h e in te rv al o f car din al n um b er s [u>o,*2u'u] an d o f t h e in t e rv al o f ca rd in al n u m b e r s ( к , A], w her e к a n d A are u n c o u n t a b l e bet.hs.

In this paper we shall use the standard set theoretical and graph theoretical term inology. The* cardinality of a set A is denoted by |A |. By P ( X ) we den ote the fam ily of all subsets ol the set A . By ш = cu0

we denote th e first infinite ordinal num ber. Sim ultaneously, üj0 denotes the cardinality o f the ordinal num ber w. We shall identify any ordinal number С with th e set. o f all tho se ordinal num bers which are strictly sm aller than ( . and we shall identify som e functions with their graphs.

For any ordinal num ber £, the £ -th cardinal num ber from the hier-archy o f bet.hs is denoted by </(.-. Nam ely, we put by recursion

and, for a lim it ordinal num ber C,

«< = U K : Ć < C b

If and Л are arbitrary cardinal num bers, then by (л-., A] we denote the set,

{//, : Ц is a cardinal number and к < ц < A}.

In works [I], [2] and [5] som e com binatorial properties of isom or-phism s and autom oror-phism s of binary relations (such as graphs, parti-tions, partial orders and trees) are considered.

In this paper we will use th e corresponding com binatorial properties of trees for finding a characterization of various intervals of cardinal num bers.

By an oriented tree we shall m ean a tree with a root. In addition, we shall assum e, in our further considerations, that a tree is oriented from the root. W e shall identify a, tree T with the corresponding set of ordered pairs and by V ( T ) we shall denote the set o f all vertices of T .

W e announced in [3] the follow ing theorem .

T h e o r e m 1. Le/, E be a .set su ch that, |/ i | > 3. T h e n th e follo wing

tw o co n d itio n s a re equiva len t: 1 ) u * < \ E \ < 2 “'°.

2) th e re e xis ts a fu n c tio n f : E —> E su ch that

(a ) there e xist s .r„ 6 E su ch th a t | / —1 ( ;Co)I — | E\ and, for e ve ry

•V € E \ {.r0}, we h a v e | / - l (:r)j < 2,

( b ) th e g ro u p o f all a u to m o rp h is m s o f th e s tru c t u re ( E , f ) is

trivial.

Proof. Suppose that un < \E\ < 2^°. Let .V be a s qqubset of P(ui)

of the cardinality |Z?|. We dehne the following tree Y:

{((.S', V ),(>', И .0)) : S G X к n G .S'}

u {(0,

T h e function

V' -1 U { ( 0 ,0 ) } : V ( Y ) —» V'(V')

has the required properties and. since |V'(V’)| = |/? |, the proof of im -plication ( I ) =»> (2) is done.

Suppose now thal, condition (2) holds. Lot ./' : E -+ E

he a. function which has the properties (a) and (h) and suppose that *Yi < | £ | < 2U'"

does not hold. There are t wo possibilities: |£ |< u .- o or |C | > 2“'°.

Since |/ i | > :L the possibility \ E\ < u>0 co ntradicts the conjunction o f the properties (a ) and (b).

Suppose now that \ E\ > 2W0 holds and .rn G E is such an elem ent that

i r V o l l =

\E\-Let

z = . \ Г/ V . l ) g -.-} .

It is d ea r that the sot Z has the cardinality |/?|. For every elem ent

z G Z , lot T- bo a tree given by the formula:

T s = { ( * J ( x ) ) : .r G E fc ( 3 k G w \ { 0 } ) ( / * ( * ) = z ) } .

It is clear tha t | l ’(T’-)| < u.’,, for every c G Z. Hence, there are at m ost 2U,° m any non-isom orphic trees in the fam ily { T: : z G Z } . T hus, there are two distinct elem ents and ~2 in the set, Z such th at the

trees 7'-, and Г-2 a ie isom orphic. From this fact we deduce tha t the elem ents Ci and are sim ilar in the graph

{ ( x , f ( x ) ) : x G E }

and thus there ex ists a nontrivial autom orphism of the structure ( E, / ) . H ence, T heorem 1 is proved.

In the characterization of intervals of the form we shall apply the oriented t гее T from [4], which was used there for the solution o f one of U hun’s problem s and one problem about uniform ization.

Let R u ( E) denote relation (2) from T heorem 1 and, for any nonzero ordinal num ber £, let R ^ ( E ) denote the following property o f th e set

E:

( i) (V C )( C < £ = ► ( £ ) ) ,

( ii) ( 3 ./•)(./• : E -+ E к (3.r„ 6 E ) ( \ f - ' ( . v 0)\ = \E\ к (Vy G E \ I/ “ 1 ( //) I G и,- V (3Ç < O d W - ' Ш ) ) к the structure

( E , f ) has no nontrivial autom orphism ).

T h e o r e m 2. Let \E\ > 3 a n d let, £ be a n on zero ordinal n u m b er.

T h en

\E\ G (fl(;,rt£+i] R-z(E).

Proof. W e shall prove the theorem by transfinite induction on £.

At. the beginning we shall prove that

\E\ G & R i { E ) .

Let us remark tha t the inductive step is very sim ilar to the proof o f th is equivalence.

Suppose tha t \ E\ G (« i,« * ]. T h en, by Theorem 1, we get - iR o ( E ) . W e shall show now that condition (ii) holds for the case £ = 1.

Let u>c, be the initial ordinal num ber o f cardinality « ,. For any

71 £ üJ \ {()}. we define:

A.+i = {(o?^‘ + . . . + w6' , ^ 1 + ...•+■ w6, + <^’,+1 ) :

Let 7' be an oriented tree defined by ( lie form ula

T = \ J { A n : » 6 w \ {()}}. T he tree T has th e following properties:

1) (71-1 ) U {(().())} is the graph of a function,

2) every vertex o f the tree T which is not the root (i.e. which is not

equal to 0) has less than ч.\ successors and the root has precisely «i successors.

3 ) T is a rigid tree (i.e. the oriented tree /' has a trivial group of auto m orphism s).

T he proof of properties 1) and 2) is easy. T he property 3) o f the tree T is proved in [4].

Let ( Mj ) j çi be a on e-to -on e enum eration o f all subsets o f шс,. For every i £ / . we deline t he following trees:

7 « . = {((£ , '/,) .( » /. Л /,)): ( * ,,,) € 7 ’} U {((£ . Л/,-). (£. Л /,,0 )) : £ 6 Л/,}.

It is easy to see th a t, for every i £ / , the tree 7д/, is rigid. M oreover, if i , j E I and i ф j , then the sets V{'I'm, ) <«id V ( Tm,) are disjoint and

th e trees 7л/, and IM, are non-isom orphic.

Suppose now that .1 is a subset o f / o f cardinality |/ i | £ ( «1,112]. It

is easy to show tha t the tree

N = и {7 д /, : ; e ./} U { ( 0 ,( 0 , Л/,)) : i 6 J )

has properties analogical to 1 ). 2) and 3 ). where in the property 2) the cardinal num ber «1 is replaced by th e cardinal num ber \E\.

T he function

has the required properties and, sin ce the set V ( 8 ) o f all vertices o f the tree S has cardinality \ E\ , we get the required function

/ : E -> E. H ence, the im plication

\E\ £ ( « ь «a] =* R- t (E)

is proved.

T he proof ol t he o ppo site im plication is analogical to the proof of the im plication

fi o ( E) => \E \ £ [do,r/j]

from Theorem I, but it is necessary to use Theorem 1 in this proof. C o r o lla r y . For a n y tw o n on zero ordin al n u m b e r s £ a n d ?/, th e follow ing eq u iva le nc e holds:

\E\ e («*,«„] (3C)(£ < с < V к R ( ( E ) ) .

Re f e r e n c e s

[1]. A .B .K h a r a z is h v ili , E l e m e n t s o f C o m b i n a t o r i a l T h e o r y o f Inf i ni te Se t s , Izd. T b il . (Jos . U n iv . . T b il is i , 1 i) 8 1, (in R us s ia n) .

[2]. A .K ip ia n i. S o nn r t n n b in a l o i i a l p r o b l e m s c o n n e c t e d wit h p r o d u c t - i s o m o r - p l i i s m s o f b i na r y r e la ti ons , A c t a U niv e rsit ät is C a r o lin a e , M a t h e m a t ic a et, P h y s ic a 2 0 no. 2 ( 1 9 8 8 ) .

[3]. A .K i p i a n i, On one u n if o r m s ub s e t in u>„ x u>a , B u lle t in o f t h e A c a d e m y o f S c ie n c e s o f t h e G e o r gia n S S R 1 3 5 no. 2 ( 1 9 8 9 ) , (in R us s ia n) .

[4]. A .K i p i a n i, U ni f o r m s e t s a n d i s o m o r p h i s m s o f tree s, M a t h e m a t ic a l I n s t it u t e , U n iv e r s it y o f W r o cla w , P re prin t no . 107 ( 1 9 8 9 ) .

[5]. S .U I a m , A C o l le c t i o n o f M a t h e m a t i c a l P r o bl e m s , In t e rs c ie n c e P u blis he r s, N e w Yo rk, 1900.

A r c h i l K i p i a i i i

O A B S T R A K C Y J N E J C H A R A K T E R Y Z A C J I P R Z E D Z I A Ł Ó W L I C Z B K A R D Y N A L N Y C H

W pracy przedstaw iona zosta ła algebraiczna charakteryzacja, w języku funkcji i autom orlizm ów relacji binarnych, przedziałów liczb

kardinalnych.

I n s t it u t e o f A p p lie d M a t h e m a t ic s U n iv e r s it y o f T b ilis i U n ive r s it y S ir . 2, 3 й 0 0 4 3 T bil is i 43 , G e o r g ia