Projective Algebra I

Projective Algebra IAuthor(s): C. J. Everett and S. UlamSource: American Journal of Mathematics, Vol. 68, No. 1 (Jan., 1946), pp. 77-88Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2371742 .

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PROJECTIVE ALGEBRA IL*

C. J. EVERETT and S. ULAM.

1. Projective algebra of subsets of a direct product. Let X and Y be any two sets of points and [X; Y] the class of all pairs [x; y], x E X, y E Y. Fix [x0,; yj] arbitrarily in [X; Y]. Suppqse now that e is a boolean algebra of subsets A of [X; Y], including as the identity I the entire set [X; Y], and the null set 0. Define for every A E 0, the set Ax as the class of all pairs [x; yo] for which there exists a y E Y such that [x; y] E A. Define Ay similarly. For every A = [X,; yo] in 5, X1 C X, and B - [x0; Y1] e , Y, !C Y., define the " product " set A El B as the class of all pairs [x1; yi], x1 e X1, y e Y,. Note that if A or B is 0, A El B = 0. We say that e is a projective algebra of subsets of the product [X; Y] in case for all A of 0, AI and Ay also are in 0, and A O B is in e for A, B of the above type. Thus we demand that the boolean algebra e be closed under projection and Ol-product formation. In particular, the boolean algebra of all subsets of [X; Y] is such a projective algebra.

It may be remarked that the following is a modest beginning for a study of logic with quantifiers from a boolean point of view, since, for example, the set A,, is essentially (t.he yIo being a dumLmy) the class of all x for which there exists a y such that the proposition A (x, y) is true.

Our object is to discover a set of properties of the above model which, adopted as postulates for an abstract " projective algebra," will permit a representation theorem to the effect that every such abstract algebra is isomorphic, with preservation of union, intersection, complement, x- and y- projection, and E-product, to a projective algebra of subsets of some direct product [X; Y]. This is accomplished only for the atomic case in the present paper. The authors hope to study the general problem and related matters subsequently.

We have derived many of the more immediate properties of the abstractly defined projective algebra, .and have shown, in particular, that every such algebra is embeddable in a complete ordered projective algebra.

The essential properties of the above model which we use later as axioms are the following.

PRoPERTY 1. (A v B), = Ax!u Bx ; (A u B)y=Ay v By.

* Received February 16, 1945. ry ry



78 C. J. EVERETT. AND S. ULAM.

For if [x;yO3 E (AvB)), then for some y, [x;y] is in A or B; hence Ex; yoJ is in Ax or B,,. Similarly, A v B,, C (A , B),,. If either A or B is 0, Property 1 is trivial. (See Property 3.)

PROPERTY 2. Ixy = [xO; yo] = Ii,,, where [x0; yo] is an atom in 3, i. e., an element minimal over 0 in 8.

For I,, [;X; yo]. and I-,,y is the class of all [xo; y] such that for some x, [x; y] EI I; hence IxJ= [x; yo].

PROPERTY 3. Ax = 0 if and only if A. = 0, and similarly for y-projection.

If A = 0, then the set of all [x* yJ for which there exists an [x; y] in A is empty. If A , 0, there is at least one point [x; y] E A, and, hence, at least one Ex; yO] in A$.

PROPERTY 4. A-,,,= A,; Ay,= Ay.

If [x,; yol] EA,,, then [x1, yJ 4EAxx. If [xi; yo] EA,,,,,, then for some y, [x1; y] E A; hence y = yo, and [xi; y0] E A,. If A = 0, Property 4 is trivial.

PROPERTY 5. For 0 A- [ X,; yo] C I,,, 0 B -= [xo; Y1] C I,, the set A O B {[x1; y1]; x1 EX1, y1, EY1} has the property

(A OB) x =A, (A OlB)y B,

and if SEO, S,, = A, Sy = B, then SC A O B.

For (A O B),, is the set of all [x; yo] such that [x; y] E [X1; YJ] for some y; hence (A Li B),= [EX;yo] =-A. Suppose now Sx = [Xi;yo], Sy = [xo; Y1] and [x; y] ES. Then [x; yo] ESx , [X1; yo] and [xo; y] ESy - [xo; YJ]. Hence xEX1, yEY,, and [x; y] E [X1; YJ= A O B. Thus SCA LiB.

PROPERTY 6. I Po IP,, and po I y = Iy,, where po [xo; yo.

This is immediate from the L-definition.

PROPERTY 7. For A1= [X1; yj], A2 - [X2; yO] in 3, one has (A1 u A2)

=Iy= (A1 Li Iy) u (A2 El sy), and similarly for B* = [xo; YJ and I,,.

For (A1l A2) 0 IY = [X1lZ X2; yO] Li [xO; Y] [EX v X,; Y] and (A1 L Iy) u (A42 L IV) = [-X1; Y] u [XI; Y] = [Xlv XI; Y].

If A1 or A2 is 0, Property 7 is trivial.

2. Projective algebra. Let 0 be a boolean algebra with unit i, zero 0, i > 0, so that for all a E Q3, 0 - a _ i. 3 is said to be a projective algebra



PROJECTIVE ALGEBRA I. 79

if two mappings a -> ax and a -* a, of 0 into 8 are defined, satisfyinig the following postulates.

Pl. (a-V'b) x=ax V'bx; (a'Vb)ya yVby. P2. xy = =- ivx where po is an atom of 0, that ig, an element minimal

over 0 in 0. P3. ax ~O if and only if a 0, and ay =O if and only if a 0. P4. axx ax.; avy > av. P5. For 0 < a ? i.x, 0 < b i y, there exists an element a O b such that

(a O b)x a, (a El b) v b, with the property that t E , tx= a, ty= b imnplies t ? a Ol b.

P6. i,x O npo = i; npo F1 iy,= i,J. P7. 0 < a1, a2 ? iw imtplies. (a1 V a,) El iy = (a1 E iy) V (a2 E] iY); and

O < b1, b, _ i1y implies ix [1] (b1-V b2) = (ix E b1) V (ix El b2)

We prove a number of immediate consequences of these postulates.

C1. a ? b implies ax ? bx and av ? by. From a7 b = b one has (PI) (aV-b)x b= = ax'V'bx. C2. For all aE a?, _ :faE ? i q.

By Cl_ a i; hence ax? ix.. C3. iA /\ (a)-' < (a') x and similarly foor y-projection. For i a zV a', ix ao V (a') (Pl), i /A (ax)' =) (ax'V (a') x) A (a,.)'

O '\j ( (a/) x /\ (ax)-) --- (a') X. C4. (aAb),_aAbx; (aAb)v-a.Aby For, (aA b) a, b, hence (CI) (a/Ab) ax, bx. CG5. a > 0 implies axv po ,p ay.

Since a?; i 'axy : ixy po (Cl, P2). Since po is an atom, axy =po fcr a,= 0 is impossible by P3.

C6. (po ) x po = (po ), --- ix$ /\ i,v

For po = x and hence (p,) y =-- ixy po, (P2, P4). The final inequality follows from C2.

In connection with the El -produLct, it is convenient to make two definitions. DI. For a ix, b ;i, define a Z O, 0 b , and OZ 0O all to be 0. It is seen that for all of these the x- and y-projections are 0. D2. For all a E , de-ine the " closure " a* of a to be ax LZ ay. C7. a E b O if and only if a 0 or b O. For if a > 0, b > 0, by P5, (a 0 b)x= a > 0, hence a E b > O by P3.

80 C. J. EVERETT AND S. ULAM.

08. The following are equivalent: (a) a ? ix; (b) there is a t sutch that tG =- a; (c) a, =- a; and similarily for y-projections.

(a) implies (b), for if a = 0, ax = a by P3, whereas fot a > 0, (a O ijy) a by P5. Next, (b) implies (c), since tx = a yields tx - ax - tx a

(P4). Finally, (c) implies (a) by C2. C9. 0 < a, a2 ? iX, 0 < b, < b2 _ iV if and only if 0 < a,L b

_ a2 LI b2. If 0 < a1, E b1 a2 LJ b2, then (Cl, P5) 0 < aj <-a2, 0 < bi b2.

Conversely, assume the first inequalities and let u = (a1 LI b,) V7 (a2 El b2). Then ux=--a1 V a2= a, uy b1 V b =b2 (Pl, P5); hence it C a2 0 bz , by PS, and a, LI b, ? a, 0i b2.

C0O. The (*) operation has the following closure properties:

0* ~0; a*> a; a** a*; a>b implies a*?kb*; ij* j.

First, 0* 0E00- Ox O =0. Second, a*= a,, ay ? a by the

maximality property of El -product in P5, for a > 0. For a 0, this becomes trivial. Third, a** *(a. O ay) O (ax O av)v= axO ay a* for a >0. Fourth, a i, hence it -

C0l. a,, aC f ix, b, b2 <i imp ly (a /A a2) 0 (b, A b2) (a, L b1) A (a2 Ey b2).

If any one of the ai, bi is 0, Cll is trivial. Otherwise, a, A a2 < a1, a2,

b1A b2 _ b1, b2, hence (a1 A a2) L (b1 A b2)? a, b, a20 b2 by C9. Hence the left member is contained in the right. Let t denote the right member. We have tx ? a1 A a2, ty, ? b1 A b2 (C4, P5); hence t ? t*

tx O tx ' (a, A a-) El (b /A b2) by ClO, C9. (C12. acix,, b?ii implies aL]b= (a[ii) A (ix [l]b). In C11, let a1 = a, a2 = z, b1 i, b2 = b. C13. t ? a I b implies tx 7< a, ty c- b- This requires only C( and PS for t > 0. It is trivial for t = 0. C14. ((a1 1 b1) V (a., E b2))*t (a1 V a2) O (b, V b2) for a,, b, not 0. We require Pl, J)5. C01. a ix implies (a O i) A/ix a. For a?C ia,, a a* a I po < a L iy. Hence a ? (a O iy,) A i,x. Now

let d be this intersection. Then d ?ix dx == d a a Aix a. (C16. All a, b satisfying 0 < a f ix, 0 < b ?- i, are closed, i. e., a O po

-a, po El b b.

a* =a L po since ax a and a= ay= po. By C12, a O po (a l iy) A (iL E po) = (a Ehi) A ix by P6. Then a E po =a by C15.

Cl7. (po) po polpo (po rIiy) A (ix M po) is, Aiy.

This- follows from 06, C16, C12, P6. C18. If ty, po then t?5x i$. If tG = po then t iy. For t t t i , El po o i, by C10, C9, P6. C19. For a,, a2 -x, b iy, one has (a, V a2) 0b (a1 0 b)

V (a2 0 b). Similarly for b1, b2 ? iy, a ? i. By Cl 2, P7, (a, \V a,) Ol b ((a1 V a2) 0 iY) A (x O b) ( ((a1 O iy)

'V (a, E iy) A (ivJ] b) ((a1 0 iv) A (ix O b)) V ( (a2 ? iV) A (ix El b)) (a1 El b) V (a2 0 b).

020. For a1, a2 < ix b1, b2 <- i one has (a1 V a2) 0 (b1 V b2)

(a1 E b1) V (a1 0 b,) V (a., E b1) V (a2 0 b2). This requires two applications of Cl9. C721. (a O iy,)':= (a' A ix) O: iv; (ix O b)' ;ix O (b A iy)

For (a El iv) V ((a' A ix) O iy) = (a V (a' A i,) El iv y x= ixlR iy

i, by P7. and (a El iy) A ((a' A ix) r1 i,) = (a A (a' A ix)) ? iy -0 E1 iy = 0, ,by Cll. Since complementation is unique in a boolean algebra, C21 follows.

C22. (a O b)'= ((a' A ix) O iy) V (ix El (b' A iv)) This follows from C12 and C21. C23. For c =- c, cy, < iy, one has (c')x = ix.

If c= 0, this is trivial. If 0 < c =c rEl cy, then c'= ((x(c)' A ix) O iy) V (i$ x ((c.)' A iy)) by C22, and (c') x (((Cx)' A ix) O iv)x V i$x= in

by Pl, P5. We need to, know that (cy)' A iy, # 0 but this is so since cX, < iv. C24. a ? ix, po < iv implies (a')x = ix.

For by C16, a ?-a*, and ax a, a" = po = . C24 now follows frorn 023. (For a 0, C24 is trivial).

C25. If po < iy then ((ix)')2, = ix, and similarly for iv. Let a -= ix in 024. C26. po iv if and only if i- iS.. If Po= --i, then i-= [ iv El iy ix po-ix (C10, P6). If i ix, tben

i = 'txy = Po. C27. ((ix)')v p'o A iv.

If po = iv, by C26, (ix)')v = ((i)')-y Ov O =0 (iy)'t A iy --p"o A iv. If p,, < iy, one has frnor i$x ix El po (P6) that (ix)' = (i$ E po)'

i, El (p'o A iy) where p'o Aiy#0 (C21). Hence ((ix)')v= p'0 A iv by P5.

C28. The class of all x-projections is an ideal in 9 and the correspondence ax -* ax El iy is an isomorphism.

The set of all ax, a E O is exactly the set of all a ? ix, by C8, and hence is an ideal, since the union of any two al, a, ? iX is < ix, as is al A c for all

6

CE 3. Aloreover the correspondence above preserves union (P7), intersection (Cll), and complement (C21).

029. For c E , 0 < a ? c, implies (a E iX) A c > 0.

Suppose (a P iy) A c= 0. Then c ? (a EIl iy)Y (a' A ix) [1 y. If riow a' A ix = 0, a =i iand (a O iy) A c i A c =c 0. But then cx 0 and a = 0, which is a contradiction. If a' A ix > 0 then a < cx ? a' A i? a' whence a 0, also a contradiction.

C30. If p is an atom in -9, so are px and py.

Suppose 0 < a < px. By C29, 0 < (a El i,) A p < p. Hence (a O iy) A p = p and p < a [1 i, px ? a, a contradiction.

C31. If 0 < c, and 0 _ p A Cx =p

3. Completion of projective algebras. We shall prove the following general theorem:

THEOREM 1. If e8 is a projective algebra, then P is embeddable with preservation of (unrestricted) union, intersection, complement, x- and y- projections, and El-products, in a projective algebra C which is complete- ordered in the sense that every set of elements of d: has a 1. u. b. and a g. 1. b.

Let s be a projective aloebra with elements 0 ? a ? i. We recall that if U(A), L(A) are the sets of all upper and all lower bounds, respectively, of all elements of A C 3, then LU(A) is a closure operation.' Specifically, AC LU(A), LUL,U(A.) LU(A), (indeed one has LUL(A) =L(A) and UT U(A) =U(A)), and A CB implies LU(A) CLU(B). The class E of all LU (A), A C P, is a complete ordered lattice which embeds e under the correspondence a -* LU (a) = L (a,), ac , with preservation of unrestricted union and intersection. For any class of elements LU (A,) of C, set intersection n L U(Aa) is effective as g. 1. b. A LU(A,,), and closure of set union L U ( L U (A,,,) ) is effective as l. u. b. V LU (A.). Note that LU (i) = L (i) = c contains all LU(A) E (E and is thus the identity of (i. Also LU(0) =L (0) = (0) is the zero of W. It is emphasized that while the elements

of (E are sets, and the order is that of set inclusion, the boolean algebra C is not the ordinary one of subsets of a set, in particular, union is not set union, and the zero of (I is not the null set.

1 G. Birkhoff, " Lattice theory," American Mathematical Society Colloquium Publica- tions, vol. 25 (1940), p. 25; H. MacNeille, " Partially ordered sets," Transactions of the American Mathematical Society, vol. 42 (1937), pp. 416-460.


MaeNeille2 has shown that in E, LU(A) V L(alla',aEA) =LU(i), and L U (A ) A L(all a') = LU(O), and that the correspondence LU(A) -> L (all a') is a dual isomorphism of E onto all of itself. From this it follows- that (E is distributive and hence a boolean algebra, with L (all a') effective as complement of LU(A). The correspondence a ->LU(a) obviously preserves complements, since (LU (a) )' LU (a').

In this section we shall understand that if A C i , A,, is to mean the set of all a,, aEA. In C define (LU(A)) LLU(A,,). We note that projection is well defined, i.e., LU(A) =LU(B) implies LU(Ax) =--LU(Bx). It is sufficient to prove A,, C LU(B,). If B (O) we have AC LU(A) LU(O) ==L(O)==(O), hence A=(O), and A,,S(0,S)=(O)CLU(B,,). Now let B contain at least one b > O, and let u ? Bx. In particular, u ?lb b, > 0, hence u A/i,,? z b,, > O. Then (u A i,,) O iY) ?bL a by b*> b, all bEB. But U(A) - U(B), hence (u/ A i,,) O iy ? A, and u 'l_ u A ix, A,,. Thus AS C LU(B,,). Similarly B,, C LU(A,). Moreover the correspondence a-*LU(a) preserves projection, since LU(a,,) (LU(a))x.

Finally, in (E we define, for (0) #- LU(A) C LU( ix), (0) # LU(B) C LU(iy), the direct product LU(A) C] LU(B) as the (closed) set:

L(allu L v; u?ix, v iy, u El v all a Oi b, iELU(A), bELUT(B)).

The form of this definition is convenient in that it is clearly an element of E, since for any X, L (X) = LUL (X); it is cumbersome, however, and we prove the following equivalences.

LEMMA 1. For LU(A) C LU(i,,), LU(A) Lx,Ux,(A) where the sub-x means that the operator is restricted to the elements ? i,,, that is to the ele- mnents of the sub-boolean algebra of all x-projections.

Clearly U (A) D UL, (A) ; hence LU (A) C LU,, (A). But LUx (A) =L,xUx(A). For LUx$(A) D L,Ui,.U(A), and if I ? U,,(A), then ? i,x since

ix.? (A); (recall that LU(ix,) =L(ix,) DLU(A) DA). Hence LU(A) C L.,Ux (A). Now let- 1x ? Ux,(A), u> A; then ix,, u> u ix , A, hence 1,, ? u A ix, u.

LEMMA 2. LU(A) O LU(B) L(allu L v; uEUx (A), vE (U, (B))

For in the L -definition u ranges over all U0Lx Ux (A) - U, (A).

I,EMMA 3. LU(A) L LU(B) = (alll; I?a El b for some aELU(A), b ELU(B)).

For let I EL(u L1 v; uEUx (A), vEUy (B)). Then l ? u L1 v, Ix _ all it,

2 H. MacNeille, loc. cit.



y ?all v, ,E Lx Ux (A), lyELyUv(B); hence 1.= a, ly = b, and 1? (1)* =a O b. Conversely, let I la b, for a eLU(A) = Lx U (A), bELU(B)

LvUy(B). Since U,(A) UxLxUx(A), UvLyU,(B), I < a EI b ?allu O v. Thus IEL(all u O v).

It is now clear that a->L(U(a)) preserves E-product, forw LU(a) ElLU(b)==LU(arlb) L(aElb). For LU(a) ElLU(b) (1;1 <soyne alb,aEL,U(a),bELU(b)) (1;I< some a b,aa, b?b).

We proceed now to verify Pl-7 in &.

Pl. (LU(A) VLU(B)) (LU(A))x V (LU(B))a. If A (0) or B (0), Pi is trivial. Using the definitions, we reduce

P1 to LU((LU(A) LU(B)) x) ==LU(LU(A x) ULU(B)). Operating on both sides with U and recalling that ULU(A) = U(A), we have U((LUJ(A) uLU(B)))=- U(Ax)nU(Bx). Since ACLU(A), BCLU(B), Ax and Ba, are in (LU (A) LU (B) ) , and the left member is contained in the right. Now let u ?Aa, B*, and I ELU(A). Then u A ix ?A, Bx, and (u Ai,) Eiy i?ax, Eaj >a for all aEA. Since at least one a>O, uAiai > 0. Thus I ? (u A ix) E iy, and Ix < (u A ia,) < U. Hence the left member is in the right.

P2. (LU(i))x v (LU(i) )yx is a point in L; indeed both equal LU(po) (O,Po).

For (LU(i))xy = LU(iy) = LU(po) =- L(po) (0,pO), which is minimal over LU(O).

P3. (LU(A) )a, = LU(0) if, and only if, LU(A) = LU(0). If LU(A) =LU(0) clearly (L U(A)) x=LU((Ox) = LU(0)- Con-

versely if LTU(A*x) (0) IDAa, then ax a 0 and a =0 for all aEA. Then LU(A) =LU(O).

P4. (LU(A))xx, (LU(A))$. The proof is trivial. P5. For (0) 74 LU(A.) C LU(i,) and (0) # LU(B) C LU(iy), the

direct product has x-projection LU(A), y-projection LU(B), and is maximal in CY with these properties.

First, LU((L(u El v; u E v ? all a El b, a E LU(A), b E LU(B))x) =LU(A). Let ? <all u El v, uo E U(A) = ULU(A), hence u0 > all a, and

UO A ia > all a, hence uo/ A ix > 0 since LU(A) 7: (0). Then (uo A'ix) El iy ? allaEO b, and I1? (u(0tAix,) rEliy. Thus 1x<-uoAi,?<uo. Hence the left side is in the right. Now let 0< aoEA, 0< boEB; then ao El bo >j all u El v, aO - (aO El bo)a,, and A C (L(u El v) )a.

Second, suppose (LU(S))X,LU(A), (LU(S))v LU(B), and let sES. Then s8fE LU(A), svELU(B), and u El v s El sy$ s s. Thus SC L(u I v) LU(A) CI LU(B).

P6. LU (i,) C LU (po) =LU (ix). This follows since L -product is preserved under a -> LU (a). Before proving P7, we note the following:

LEMIMA 4. If S C LU(i3,), u1 ELU(S), u0, ? S D iy, then u1l Ci ?; u_ .

First, if i E S., then u, ?- ix L i, = i, and the conclusion is trivial. Sup- pose i, ? S. We prove u',0 ? (u1 EZ is)' = (u'1 A i,,) [1 iy. Since u'o <? (u'o). El (ut'o) y, it is sufficient to prove (uo) x < ud1 A ix, i. e., (ueo) x < u'l. But it is given that u'o -< (S' A i,) [1 iy; hence (u'o), <? S' A ix <- S'. For if any s' A ix = 0, s i, ES contrary to our present hypothesis. But it is giveni that ul ?-U (S); hence u'1 ? (U(S))' L (S). Thus u'1 ? (ueo) x.

P7. (LU(A) V LUT(B)) [1 LU(iv)

= (LU(A) [1 LU(iy)) V (LU(B) [1 LU(iy)).

If either A = (0) or B = (0), P8 is trivially satisfied. Moreover, in anly case, the right member is contained in the left by C9 which follows from Pl-6.

We remark that if X and Y are any two subsets of 3, then LU(X u Y) LU(allx V y; xEX,yEY).

Now note the equalities: (LU(A) V LU(B)) rC1 LU(iv) = LU(LU(A)

L U(B) ) nI L(iy) LZU(all a V a E LZU(A), b E LZU(B)) C1 L(iy) = (glll?someu 'lI v; uELU(a Vb),vEL(i))v (alllIsome u ] iy;

u ELU(all a V b)).

We remark that U(all I < some a El iv,) = U(all a l isy), and then note the equalities: LU(A) El LU(iv) V (LU(B) E LU(i,)) = LU((all 1

_ some a rJ iy) u (all I _ some b r i)) L(U(I ? some a El iy)

n U (I some b l isy) = LU( (alla al iy) u (all b rJ iv) = LU(all a [] ij,

V iv O L) LU(all (ca V, b) Elq) We must therefore prove,that (all I ? some u El iy; u E LU(all a V b))

C L U(all (a V/ b) El y). Let I ? some u1 El iy, u1 E LU(all a V b), and

u0, ? all (a V b) rJ iy. Now use Lemma 4 with S = (all a V b). It follows that u1 0 iv -< u0. Since I ?< u1 Z ity _? u0, the theorem is proved.

4. Representation theory. We shall prove a representation theorem for the case where Q3 is a projective algebra of all subsets of a set i of points p. We prove first two preliminary lemmas.

86 0. J. EVERETT AND S. ULAM.

LEMMA 1. If N is any cardinal number, there exists a (commutative) group containing exactly N elements.

For a finite N we may use the cyclic group of order N. Now let N be infinite, and let V be a vector space of N basis elements over the galois field GF(2). The elements of V may be considered as functions on a set of power N to the set (0, 1) where addition is component-wise, mod 2. Denote by W the subgroup of all elements of V with only a finite number of 1-components, together with the zero element. W consists of the mutually exclusive subsets

0o

WFj of elements containing exactly j 1-components. Thus W = l W1. The 00

power of TV is therefore 1 f Ni N. 1

LEMIMA 2. If i is any (additive) group of elements p, the correspondence

p -> p + q, q fixed, p arbitrary in i, defines a permutation Wrq (one-onte trans- formation) of i onto all of i, and the correspondence q -*rq is an isomnorphism of i onto a group 1I of permutations of the elements of i. In particular (a) the powers of Il and of i are equal. (b) for every p, q E i there exists a 7r EH

such that 7rw,(p) = q, (c) if 7rV-(so) = rq(So) for some soE i, then rp 7Tq.

This is only a statement of the familiar Cayley representation. of a group by permlutations.

Now let i be a projective algebra of all subsets of a set i. We recall that

PO = /\ A iy so that all points of i are in one of the following disjoint classes:

(a) po, (b) pEiv p #po, (e) pEiy p#po, (d) p 'ixj V iy-

Moreover it is clear (C. 30) that the points in i, are precisely all px where p ranges over i; similarly for ii,. Indeed, from C. 31, it follows that a,, is exactly the set of all p. with p in a.

Let X be the class consisting of po and of all pairs (px, q), px ranging over all points p po in ix, q over all points in i. Let Y be the class consisting of po and all pairs (pv, q), py ranging over all points + po in iy, q over all points in i.

Defi-ne the relation p q on points p, q of i to mean p$ q= py = qy.

This is an equivalence relation splitting i into mutually exclusive sets (p) of

points. In each class we distinguish a special representative Li (Zermelo). The points in any class (p-) are thus all points of i of equal closure P$X O fn.

We define a group on the points of i, and set up the correspondence q 7rq

of Lemma 2, and set up the fqllowing correspondence on points of i to subsets of [X;}Y].



(a) po --> [po; po].

(b) for p e i*, p'=Po, p {[(p, q); po], q ei}

(c) for P e i, P #,Po, p > { [po; (py, q) ], q }

(d) for ptix /V iy, and p (E7), if

p =/A p >{ Sq) ; (py,7p (q) ]q Ei}

But

) {[X) q) ; (&U,r(q))] q Et

and

{ [(sx q); ( t (q))], q e i, t an arbitrary point in the i-complement of (p) }. For anv a A, where A is the set of all images of all points

of a under (a-d).

LEMMA 3. If s -> [x;y] and t-- [x;y] under (a-d) then s t.

Different points of the type (b) map into different [x; y], since the point itself appears as one of the coordinates, and similarly for type (c). If [ (Pj0 q); (py, rs (q))] is the map of two different points, they would be points with the same closure, and hence in the same (p). But the pair of elements q, r, (q) can arise only from a unique s (Lemma 2c), and the two points would then both be p or both be pj, according to (d).

LEMIMIA 4. Every point [x; y] of [X; Y] occurs as an image of some point p of i.

For the poi'nits of [X; Y] fall into the following classes:

(a) [x; y] [po; po] which is the map of p. (b) [x; y] (px, q) ; po] which is the map of px ? ix, px = pO.

(C) [x; y] [po; (py, q)] which is the map of py yY, Py #P po. (d) [x; y] [(px, q); (ry, s) ] p po, ry =7 p,. ln i, form the product

px El r. 7 0. There exists at least one point t ? px D ry and 0 < t0 < pr, 0 < tq, ? ry. Since t > 0, tx = pxs, ty ,p,. EIence [x;,y] [(is, q); (t,,s)], tz =;po, ty ,# po. But there exists an element n E i (Lemma 2b) such that wn (q) -- s. Thus [x; y] [(n, q) ; (y,nr.(q)] is the map of some point in (t).

THEOREM 2. A projective algebra defined on all subsets of a set i of points p is isomorphic to a projective algebra of certain subsets of a direct product [X; Y], with preservation of unit (i - [X; Y]), union, intersection, complement, projection, and W product.

The correspondence is that already defined. If now a -> A, b --> B,

a V b -- S, a A b -> P, one readily verifies that S = A u B, P = A n B, since each member includes the other. For the latter equality one uses Lemma 3.

Now let a -> A, a' -* B. Then B = complement of A in [X; Y]. Sup- pose [x; y] EB, i. e., s-> [x; y] where sEa'. Then [x; y] EA', for if [x; y] EA one has t [x;y], tEa. By Lemma 3, s t. Hence BC A'. Now let [x;y] EA'. By Lemma 4, s-> [x;y], for some sEi. Hence sEa', and

[x;y] EB. Next let a -> A, a,, -> B. Then (A) x ,B. Let [x; p0] EAx. For some

y, [x; y] E A, hence p -> [x; y] for p E a. By examining the correspondences (a-d) one sees that p -* [x; y] implies px -> [x; PG] in all cases. For example, if p is of the type (d), px i,, and px #= pO (for if so, p ?< i,, cf. (C. 1 8) ). Thus p, -> [x; p0] under (b). Hence in general [x; po EB. Similarly, if a --> A, a ->C, then (A)= C.

Finally, let O < a ? ix, O A, b ->B, a b --> C. , Then A O B C. For, let [x; y] EC. Then p- [x;y], pEa, b, and px a,

Pv ? b. But as poiiited out above, px -> [x; po] and pv, > [po; y], hence [x;po]EA, [po;y] EB. Thus [x;y]EALIB.

Conversely 'if [x; y] EA [ B, [x, p0] EA, [p0; y] EB. Then for p _?a, q b, p -> [x;po], q ->[p0;y]. But then O [x; y]. Hence [x; y] E C.

UNIVERSITY OF WISCONSIN, AND

UNIVERSITY OF SOUTHERN CALIFORNIA.

Projective Algebra I

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