Generating schemes for full mappings

MATHEMATICS

GENERATING SCHEMES FOR FULL MAPPINGS BY

B. VAN ROOTSELAAR (Communicated by Prof. A.

HEYTING

at the meeting of June 25, 1955)

Every full mapping of a finitary spread (for the notion of spread see [l]) onto a second spread induces i~ a natural way a mapping of a subspecies of the underlying spreaddirection of the first spread onto the spreaddirection of the second spread (cf. [2], th. l). The thus induced mappings of the spreaddirections may be characterised by several formal requirements and thus be defined independently. This gives a generating scheme for full mappings. However, a predet~rmined mapping of a spreaddirection may generate a full mapping of the corresponding spread without being of the above-mentioned kind. It is the purpose of this note to determine necessary and sufficient conditions for such a mapping of a spreaddirection in order to generate a full mapping of the corresponding spread. The notation of [l] and [2] will be used. Let A and B denote finitary spreaddirections and let 'lfJ be a unique mapping of a subspecies A 0 of A onto B. If an element IX E P(A) contains a sequence {IX,.} of nodes with indefinitely increasing n, such that IX,i E A 0 and {'lfJIX,.}={.B.} is an element of P(B), we say that 'lfJ induces an image of IX and ,8={,8.} is called a 'ljJinduced image of IX : ,B=tpiX. A unique mapping 'lfJ of a subspecies A 0 of A onto B is said to induce a full mapping I of P(A) if for every IX E P(A) the species of 'ljJ-induced images is identical with IIX. Next we introduce the notion of 'ljJ-descendant ('lfJ-ascendant): If a and a' are nodes of A 0 such that a' is either a descendant or an ascendant of a, then a' is called 'ljJdescendant ('tjJ-ascendant) of a if "Pa' E B is a descendant (ascendant) of "Pa E B. Finally we introduce the notion of inessential subspecies by the following definitions: A node a E A is called inessential if there exist no IX E P(A) passing through a which contain infinitely many 'ljJ-descendants of a. A subspecies of A is called inessential if it consists of inessential nodes. After these preliminaries we state the following Theorem l : If a unique mapping 'lfJ of a subspecies A 0 of a finitary spreaddirection A onto a finitary spreaddirection B induces a full mapping I of P(A) onto P(B), then a. A 0 contains a removable subspecies A 0r such that every a E A 0 either belongs to A 0• or is a descendant and 'ljJ-descendant of some element -1

in A Or, for every b E B the species 'lfJ (b)

n

-1

A or is thin, for every n 'lfJ ( B,) n A or

G47 contains a finite removable crude bar S., of A, 1pS.,=B., and the species 00

A or - U S., is inessential. b. for all n every 8E S,. has a 1p-ascendant in s.,_l and a v-·-descendant in s..+l. c. for every n there exists an N .. such that for all b and b' =1= b in B., the -I 00 -I 00 species 1p (nb) n (uS.,) n A 0• and 1p (nb,) n (uS.. ) n A 0' have no nodes of order m ;?; N., in common. Proof. a. Since the mapping f induced by "Pis supposed to be a full mapping, there exists according to BROUWER ([1], p. 15) a bar A,..,l for P(A) of maximal order n 1 (cf. [2], p. 558) such that for every
Thus for every
that such a first node of 1p (B1) will appear after a finite number of choices. From this follows (by the fantheorem, [1], p. 15), that we may suppose A,.,,l to be such that for every
of "P (B1) is contained in rA,.,)A 0 ). (rA,.,)A 0 ): free stump carried by A,..,l, cf. [1], p. 9). The species S1 of these determining nodes which are contained in T A (n,) will be called the first essential crude bar. It is clear that S 1 is a finite removable subspecies of A. Moreover we have 1pS1 =B1 • The same argument shows the existence of S., for any n, having the required properties. One should notice that S., is in general not a bar; -I

for every bE B.,, however, "P (b) n s.. is a finite thin species. By AOr will be indicated the subspecies of A 0 which is obtained from A 0 by removing from A 0 for every a E s.. and any n those nodes which are descendants of a for which 1p(a')=1p(a). Now consider A 0 r -Sv then from the definition of A 0r and S 1 it follows that no node of this species can determine a node of B having at the same time infinitely many 1p-descendants. In fact let a be such a node of A or- S 1 and let 1pa = b E B 1 , then a neither can be an .

-I

ascendant of some element of 'If (b) n S 1 nor a descendant of such an element. Let
-I

(in "P (B1 ) n S 1 ) for which 1pa0 =F 1pa, but because this element a 0 has infinitely many descendants in
any k A 0r - uS, cannot contain an element a having infinitely many i=I

00

'If-descendants and determining a node of Bk. Thus the species A Or- u si is inessential. b. From the fact that f is a full mapping, i.e. the existence of the bars A,..,l and the definition of S, follows immediately, that every a E S,

648

has at least one '!jJ-descendant in si+1 and (exactly) one 'ljJ-ascendant in si-1• c. The full mapping f induced by 'IJ' is unique, which may be formulated as follows: if {3=/IX E P(B) and {3'=/IX' E P(B) are distinct, then IX and IX' are distinct (cf. [1], p. 6). If /1X={3 then IX contains an infinite subsequence of nodes 1Xmk E Sk, such that 'IJ'IXmk =(/1X),=f3k· Suppose b; E B;, then the elements IX of P(A) for which (/IX);=b} form a subspread Pj(A) of P(A). The elements of P(A) for which {fiX); =1= b} form a subspread PJ(A) of P(A). Now consider two elements IX E Pf(A) and liE Pj(A) then we have {fiX);=!= {fli); and because the mapping is unique it follows that a number nj(IX, ~)exists, such that IX and li are distinguished by their nodes of order n;(IX, &). Now consider the full function n}(IX, li) defined on Pj(A) x P;(A). By application of Brouwer's fantheorem ([1], p.15) one sees that nHIX, li) has a maximal value N; on Pj(A) xPj(A). If i runs through the finite row of indices of order j, there exists M; = max NJ . i

Then a number k(j) exists, such that the nodes of Ac,k{j) > are all of order k greater than M;. Evidently for every

IX

nlc(i)

~

the species of nodes

j; further u sic TA oo

U

i=l

IX;

(nk(J))

(A 0•) and thus

contains only '!jJ-descendants of

i>MJ

one element inS;, which argument holds for every j, and thus proves c. Next we prove the converse of theorem I: Theorem 2: A unique mapping 'IJ' of a subspecies A 0 of a finitary spreaddirection A onto a finitary spreaddirection B satisfying conditions a., b. and c. of th. 1 induces a full mapping fv· of P(A) onto P(B). Proof: From c. follows the existence of bars AcN,.> for every n, such that all elements IX E P(A) that pass through an element n E A eN > contain 'lj!00 " descendants in uS, corresponding to only one node in B, moreover for every n holds S, C -rA (from which follows that AcN >actually is a bar of (~) . " A). The bars AcN,.> are finite because the number of nodes in Bn is finite. Let IX be arbitrary in P(A), then suppose 1Xp, E A(N,)> let sq, be the 00

crude bar of u S, such that the least order which occurs in Sq, n IX is the least one greater than p 1 ; by b. and the definition of AcN,> there exists one 'ljJ-ascendant in Sq,-1 which is 'ljJ-ascendant of every node of sq, n IX, to the thus uniquely determined node in sq,-1 there exists by b. exactly one 'ljJ-ascendant in sq,-2, and so on, hence a sequence of nodes: O"q,(1X) E Sq, nIX, O"q,-1(1X), "., a1(1X) E 81 n ()(

is obtained, and a1 (1X) is uniquely determined. In the same way we proceed for AcN,> and arrive at a uniquely determined node a12 (1X) in 8 2 nIX which is a '!jJ-descendant of a1(1X), by definition of the AcN,.>· Thus for every n there is uniquely determined a node a 12 ..... (1X) E S, nIX such that for every n a 12... n,cn+I>(1X) is a '!jJ-descendant of a 12 ..... (1X). Thus to every IX E P(A)

649

a uniquely determined 1p-induced image is obtained by putting 00

Because A Or-usn is inessential no infinite subsequence in

(X

of successive

00

1p-descendants exists in A Or and outside uSn. If {xnJ is an infinite subsequence of ex consisting of successive 1pdescendants in A 0 - A or, i.e. determining a 1p-induced image of ex, then we must show {1f!cxn)=f'Px. By definition of A 0r, to every cxn; there exists an ascendant in A 0 r which is at the same time a 1p-ascendant. From this follows that to every cxn; there exists a node in A 0 rn ex having the same 00

image under 11'· The elements of this sequence then must belong to uSn, which species has in common with ex only one such sequence, namely the constructed sequence {a1 ...n(cx)}. From this follows that the species of 1p-induced images is a single element in P(B). The function f'P is consequently a full mapping of P(A) onto P(B). From theorems 1 and 2 it follows that conditions a., b. and c. are necessary and sufficient for a given function of a finitary spreaddirection onto a finitary spreaddirection to induce a full mapping of the corresponding spreads. In fact conditions a., b., and c. give a criterion for equivalence of a given mapping scheme with a natural mapping scheme as described in the proof of theorem 1 of [2], which is the mapping scheme of a full mapping. REFERENCES l. BROUWER, L. E .•J., Points and spaces. Canadian J. Math. 6, 1-17 (1954).

2.

B. VAN, On the mapping of spreads. Proc. Kon. Ned. Ak. v. Wet. (Amsterdam) 58, 557-563 (1955); lndagationes Math.17, 557-563 (1955).

RooTSELAAR,