(J Inductive definitions

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1967)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1967)

W. Wa liszew sk i (Łódź)

Inductive definitions

The purpose of this paper is to give a proof of a theorem on defi­ nitions by induction with respect to a general binary relation. We prove that the only condition which is essential for the validity of such a defi­ nition is a minimum condition.

The set of all functions of domain A and of values in В will be denoted by B A; we shall often write /: A В instead of f e B A.

Let R be an arbitrary binary relation. Denote

Dr = {oo\ Vy<x, y}eR}, CR = {x\Vy<x, y>eRv<y, x>eR}, Ry = {x\(x, y}eR} for yeCB,

and

mR{N) = {x\ xcNaN ^ R ' x = 0 } for N c CR.

R' is the relation defined by equality R' = {<x, y}\ <x, у } е В л х Ф y ] .

We shall say that the relation R is of type (M) iff mR(N) Ф 0 for every nonempty subset N of CR.

We shall say that R is of type (ID) iff for every set Z , for every func­ tion p: mR(CR) -> Z and for every function

(*) h:

(J

x*CR- mR(CR)

there exists a unique 99: CR -> Z, such that

<p(x) = h(cp\R'x) for all xeCR—mR{CR) and (p\ mR{CR) = p.

It is clear that if R is of type (M), then for every P <= CR we have

p = CR whenever for every xeCR from R 'x <= P it follows that xeP.

Let us consider an arbitrary set Z, an arbitrary function p : mR(CR) ->Z and an arbitrary function h satisfying the condition (*). Let R ‘ {p, h , Z) denote the set of all functions у such that Dn cz CR, у : Dn -a- Z , Rx a Dn for x eDn, and

20 W. W a l i s z e w s k i

Lemma 1. Let R be of type (M) and let G <= R ’{p, h, Z). Then

U

P roof. Let y, ipeR'{p, h, Z) . There exist sets A a CR and В <= CR such that y: A - ^ Z , у: В -> Z , Rx с= A for xeA, Rx с В for xeB,

y{u) = p {'ll) for U e А mR(CR), ip {'ll) = p {u) for U € В гл mR{CR), y{u)

= h{y \R'u) for ue A ~ m R{CR) and ip{u) = h{ip\R'u) for u e B — mR{CR). Put

N — {u\ Ue А гл В л у { и ) Ф ip{u)}.

Let us assume that N Ф 0. Then mR{N) Ф0 and there exists an u0

in mR{N). Therefore u0eA гл В and y{u0) Ф ip{u0). Let u e R 'u 0. Thus

u e A гл В and y{u) = ip{u). Hence it follows that R 'u0 cz A r\ В and

y \R 'u0 = ip\R'u0. If u0emR{CR), then y(u0) = p{u0) = ip{u0). This, how­

ever is impossible. Thus, u04mR{CR). Consequently, v {u0) = h { y \ R' u 0) = h{ip\R'u0) = ip{u0),

which is also impossible. In other words, if у and ip belong to R * {p ,h , Z), then y{u) = ip{u) for u e B n o By,. Hence it immediately follows that U G is a function. Putting у = U G we obtain Dv = B n. Thus R x a Dv

rjeO

for x e B y . Let x e B Y. There exists an yeG such that x e B n. If xem R{CR), then y{x) = y{x) =p { x ) . If oc^mR{CR), then R 'x c B n c: B y. Thus,

y \R 'x = y \R 'x . Consequently,

y{x) = y{x) = h{y\R’x) = h{y\R'x).

Therefore у eR ‘ {p, h, Z).

Lemma 2. I f R is of type (M), then the union of all B n for y e R ' { p , h , Z) is equal to CR.

Proof. Let P denote the union of all sets B n where y e R ’ { p, h, Z) . Since p \ { x } e R ’ {p, h, Z) for xem R{CR), we have mR{CR) с= P. Let

xeCR — mR{CR) and R 'x <= P. Let F denote the set of all functions y e R ‘{ p, h, Z) for which R ' x гл B v Ф 0 and let у denote the union of all

functions belonging to F. It follows from Lemma 1 that y e R ’ {p ,h , Z). Since R 'x Ф0 , there exists an x0 in R 'x. Thus, x0eP. According to the

definition of P, there exists y0 belonging to R ' { p , h , Z ) and such that

x0eBVo. Therefore x0eR' x гл В щ. Consequently, F Ф 0 . Hence it follows

that yeF. Let t eR' x. Thus, teP and there exists an yeR' {p, h , Z ) such that t eВ n. Hence it follows that R ' x гл B n ф 0. Thus yeF, and therefore

t e B M. In other words, R ' x c= B^. Let

h{y\ R' x) for и = x,

Inductive definitions 21

Then B e = w {x}. Since jueB' (p, h, Z ), we obtain В' и c= for u e B e. We notice that 0(w) — p(u) for ueB^. Indeed, let и е В ц. If и — x, then

В 'и c Bp and p(u) = h(p\B' x), because p e B ' {p ,h , Z). From the defi­

nition of В we obtain B{u) = p(u). Let u e B e ^ mR(CB). Thus, и Ф x. Then u e B fi. Therefore B(u) = p(u) — p(u). Now, let us consider an arbitrary u e B e—mR{CR). If и = x, then

B(u) = h(p\ B' x) = Ji(6\B'x) = h (6\B'u).

If и Ф x, then u e B ^ — {x}, В 'и <= B^, and

6(u) = p(u) — h{p\B' u) — h(6\B'u).

Thus, 6eB'{p, h, Z) , x e B Q, and xeP. We have shown that x e P if xeCR

and B 'x c= P. Thus, we obtain P = CR.

Lemma 3. I f В is of type (M), then В is of type (ID).

Proof. Let us assume that В is of type (M). Put cp — B ’ (p , h, Z).

By Lemma 1, <peB’( p , h, Z). In turn, by Lemma 2, В v = CR. If ipeB’(p,

h , Z ), then 9o(u) = гр(и) for u e B v; if B v — CR, then cp = yj. Thus, В

is of type (ID).

Lemma 4. Let В satisfies the following condition: for every function

h which maps the union of all sets (0,1}йж, where x belongs to CR—mR(CR),

into the set {0,1}, and for any functions y, y> which map CR into {0,1}

and satisfy <p(u) = y>(u) for и of mR{CR), <p(x) = h(<p | B' x) and y){x)

= h(ip\B'x) for x of CR—mR{CR), we have cp =■ %p. Then В is of type (M).

Proof. Assume that В is not of type (M). Let A denote the union of all sets N such that I с Од and mR{N) is empty; let В = CR—A. It follows immediately that А Ф 0 . We shall prove that mR(A) is empty, the sets mR{CR) and B 'x are contained into В if xeB.

Indeed, we first suppose that mR{A) Ф 0. Then there exists an

x emR(A). Thus x e A and therefore there exists an N a CR such that x e N and mR(N) = 0 . From the definition of mR{N) we obtain that N гл B ' x Ф 0 . Since N cz A, A ^ B ' x Ф 0. Therefore x4mR{A)\ and we

get a contradiction. Thus, mR (A) = 0 . Suppose now that there exists an x e mR(CR) гл A. Then there exists an I с Ой such that x e N and

mR(N) = 0. Consequently, B ' x ф 0 , and x4mR(CR). This is again impos­

sible. Let now xeB . Suppose that B ' x —В Ф 0. Then A B ' x Ф 0 . If

there existed a ue mR(A ^ {ж}), и would belong to 1 и {x } and (A w M ) О В ' и would be empty. Hence it would follow that и ф х and ueA. Since mR{A) is empty, u4mR{A). Then А г ^ В ' и ф 0 - , this, however,

is impossible. Thus mR(A w {ж}) = 0 and A w {x} c CRt and it follows that A w {x} is contained into A. Therefore xeA.

arbi-22 W. W a l i s z e w s k i

trary function which maps the union of {0 ,1 }Rx for x of CR—mR(CR) ' into {0, 1}. From the definitions of functions y0 and <Pi follows that 9oQ(u) = <рг(и) = 0 for uem R(CR). Let X€CR—mR(CR). Then Ji((p0\R'x) = 0

= <p0(x). If xeB, then R ' x c B. Thus, (cp1\R' x)(u) = (px{u) = 0 if u e R 'x .

Consequently, h(p1\R'x) = 0 = 9ox(x). If xeA^ there exists a и in A ^ R 'x.

Therefore h((p1\R'x) = <px(u) = 1 = (px(x). In other words, щ{х) =

= h{(pi\R'x) for x of CR—mR(CR) , i = 0,1. Lemmas 3 and 4 yield the following:

Theorem. A binary relation R is of type (M) if and only if R is of type (ID).

From Lemma 3 we immediately obtain the theorem on definitions by transfinite induction (see, e.g. [1]). The weak asymmetry of the rela­ tion R follows from the condition (M), but it need not be reflexive or connective.

R eferences

[1] K. K u r a to w s k i, A. M o sto w sk i, Teoria mnogości, Warszawa 1966.