a remark on boolean algebras with operators

A Remark on Boolean Algebras with OperatorsAuthor(s): Hugo RibeiroSource: American Journal of Mathematics, Vol. 74, No. 1 (Jan., 1952), pp. 163-167Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/2372075 .

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A REMARK ON BOOLEAN ALGEBRAS WITH OPERATORS.*

By HUcGO RIBEIRO.

Bjarni Jonsson and Alfred Tarski, in their paper Boolean algebras with operators (73, 891 and 74, 127 of this JOURNAL), call a Boolean algebra B a regular subalgebra of a Boolean algebra A if A is complete and atomistic and B is a subalgebra of A for which: i) if I is an arbitrary set and if the elements xi? B with i s I are such that N xi= 1, then there exists a finite

i eI

subset J of I such that N xi 1, ii) if u and v are distinlct atoms of A then ieJ

there exists an element b s B such that u < b and v b = 0 (Definition 1. 19). The set C of all "closed" elements of A is then defined as the set of all elements x ? A such that x U ] y (Definition 1. 20); and Am, Bin, C,n

x?yeB

designate the sets of all mn-termed sequences, x <xo* , Xmi>, of elements of A, B, C respectively. Furthermore, a function f on B"17 to B is called monotonic if given two sequences x < y s Bn& (that is xi ? yi c B for any 3-0i * 0 n- m 1) we always have f(x) < f(y), additive if given any j < in and x, y s B'm such that xp = y, whenever i ? p < mn we always have f(x + y) = f(x) + f(y) (Definition 1. 1); f+/C,n designates the restriction of the function f+ to Cm and by the composition f[go, , * * gmn-1] of f, on Bml to B, with go, . . . , gm-,, on Bn to B, it is understood the function h on Bn to B such that h (x) = f(go(x), gm l(x) ) whenever x s Bn. Finally, to any function f on Bm to B an extension, f+, on Am to A is defined by f (X) = > 11 f(z) for any x gAmn (Definition 2. 1), and it is shown

x?y e Cy, yz<ze Bm

(as an immediate consequence of Theorem 2. 10) that if an equation involving additive functions on Bm to B is identically satisfied, then the correspondilng equation involving their extensions is also identically satisfied. Such a statement is also true (Theorem 2. 11) of certain implications between two equations, aiid it yields several interesting results.

The purpose of the present note is to give a direct proof of an extension of that Jonsson-Tarski's Theorem 2. 10. This extension (Theorem II) con- sists in gettinig the conclusion under a weaker hypothesis on the functions go , , gm1-, namely the monotonicity instead of membership in the set q.

* Received March 31, 1950. 163



164 HUGO RIBEIRO.

Otherwise our statement (Theorem II) is as Jonsson-Tarski's Theorein 2. 10. (It must be pointed out that in that same paper it is also sliowln: Theorem 2. 10 does not hold with the hypothesis that f is monotonic even wheii

,q g, ,gm-i are additive, and on the other hand, it holds whenever f is monotonic and go, , g,,, are " identity functions " (Tlieorem 2. 9)). Throughout our proof we shall make free use of many of the terminology and notation in Jonsson-Tarski's paper, and we shall continue to refer to its definitions and theorems by using the reference numbers therein.1

THEOREM I. Let B <Bo, +, 0, ,1> be a regular subalgebra of a Booleanm algebra A = <Ao, 4, O, ,1> and let m, and n be positive integers. Then, if f is an additive function orn Bm to B and go, . . . , gin-, are monotonic functions on Bn to B we have

(f [go, . ~gm-l])+ ==ft+[go+, * * , g-l+]

Proof. First we remark that f+/C$1 is on Cm to C and gj+/Cn (j 0, m, -n 1) are on Cn to C. The inclusion

1) f+[go+, * gm-,+] ? (f[go, *, g,1-,])+ is easily checked: Using the definition of composition, 2. 1 and 2. 2, the hypothesis on f together with 2. 4, and then the remark that x ? y?, , ym-1 ? Cn implies x ? yo +

+ ym1 C cn together with the monotonicity of f+ and gj+ (j = 0, m, in 1), we have for every x e An

f+ [go+, . , gm-I+] (x) = f+ (go+ (x), , gnI(x) )

f+ ( E go+(y?), * * gm1+(yfml-) X?-y0e Cn w>ynt-l e Cn

- Y f+(go+(y0), gm-,(Y ) $28?o C cn $2:yin-l C cn

E f+ (go+ (Y), * gm-,+ (Y)) aj,yOeCf Cn -lC

By 2. 2 the last sum is z H f(z) and it is included X?-yeCn <go+(y),..m.,gml+(y)>?<zcBm

in E H I (go(z), * gm-(z) ), since every factor of each product Xsy ecCn V-?-zeBri

of this sum is a factor of the corresponding product of the above sum: f is monotonic, and <go+ (y),* * gm-+ (y) > ? <go (z) gm-, (z) > 8 B1m when-

ever y < z e Bn because of the monotonicity of gj+ (j 0 * n m -1). Now

1 The results of the present note were originated from and reported to Professor Tarski's seminar on Topics in algebra and metamathematics at the University of California, Berkeley.



A REMARK ON BOOLEAN ALGEBRAS WITH OPERATORS. 165

z I f f(go(Z), ,gin-i(Z)) x2/y C" y-z y ?B"1

- l Hi f[g0o * gm-] (z) = (f[go, gm-d )+(x) x?2Y c CC ' -z ? Bn

by the definition of composition and then by 2. 1. From this proof of 1) it follows that, for every X ? Cn such an inclusion

holds even if f is monotonic not additive. In this case the sequence of equalities and inclusions yielding 1) will, essentially, begin after the first inclusion above, and there the additivity of f does not play any role.

Next, we prove for f monotonic (not necessarily additive) the inclusion

2) (f( [g0, * * gm-1D`f] ) g+ ? fg[g0+, ,

From the definition of composition, 2. 1 and 2. 2 it is clear that it will be sufficient to show that for every y s Cn we have

(f [go) . . . n gm-d ) +(y) C-- f (go+ (y), gm-,+ (y) )

By 2. 2 and, again by the definition of composition, this inclusion will be established, for any y e Cn, if we prove that to each factor of

I f (ZI) <go+(yl),...,gm_j+(y/)>-<z c Bm

there is at least one factor of II f(go(z"), , g1(z") ) which is y?z" e Bn

included in it. Since f is monotonic it is now sufficient to show that to each z' F Bm

for which <go+(y), - * * , gm-,+(y)> z', there is some z" ? Bn having the properties I) y ? z", II) <g0(z"), , gm-1(z")> < z'.

To do this let z'~ <z'o,* *, e B'vl and let us remark that, by 2. 2 our hypothesis means

H gj(z) ?z'j1B (1=0, *,m -). V?<z e B"l

First, we have that for any j = 0, , n - 1 there is z"i ? Bn such that at same time y ?< zji and gj(z"i) ? z'j. This is true since z'j being openl and including a product of closed elements, it will include (by 1. 21, (iv) ) some

r, finite product H gj(zk) of such closed (and open) factors:

k=O

ri H gj (Zk,) _Zfj (k-== 0* nri) k=o

ri with y ? zk ? B8. Now, putting z1 - II zk, we will have not only y ? z"j ? B

k=O



166 iHUGO RIBEIRO.

ri but also gj (z"i) < z'j, since g1 (z"i) < rI gj (zk) because of the monotonicity

k=O rn-

of gj. As second and final step it is easily seen that z" =I Hz"j is an j=O

element of Bn having the properties I) and II) above: y < z" since y ? z"i for every j =0, ,m - 1; and gj(z") ?z'j for every j =0, since gj being monotonic we have gj(z") < g (z"') ? z';.

The proof of 2) is now complete. From 1) and 2) the theorem follows.

Remark. As a consequence of the preceding proof of 2) and of the comment at the end of the proof of 1) we have

(f [g0~ . . . gmn-d ) +(1) -= f+ [gO+n** gmn-,+] ( Y)

wheliever f, go, * gm-i are monotonic and y c C'".

THEOREM II. Let B = <Bo, 4, O, ,1> be a regular subalgebra of a Boolean, algebra, A <Ao, +, O,, 1>, let rn and n be positive integers and let (p be the smallest set having the two properties:

i) to include all addittve functions on Bt to B for any t (integer positive) ii) to be closed tn respect to the operation of composition (of ftnctions).I Then, if go, , gm-i are monotonic functions on Bn to B and f e 0 is

a function on B'r to B, we have

(f [go, , g- -1] ) = f [go+, * gm-,+].

Proof. Remark first that the operation of composition is associative and that for a function f on Bnm to B to verify the hypothesis it is necessary (and sufficient) that non negative integers lk and r70 =0 r0, , 1 r7' exist such that

f ho0[hol. . hr,l] [hokn , hrkk]

for some additive functions h1k (1 0, , rk) on B"m to B and hji (i== O, , k- 1; j 0, ,ri) on Br,iL to B.

Put h ho?. If we have k = 0 in the equality above, then f is just the additive function It, and we have the desired conclusion from Theorem I. We prove by iniduction for k 0 0. Putting

f=j hjl [ho20 hr22] [hok, . h.kk] (j= 0, , ri)

we have

(f[go,. -, gm-1]J)Y f1+[go+- , gn-1] (1 0, , ri)



A REMARK ON BOOLEAN ALGEBRAS WITH OPERATORS. 167

as induction hypothesis. On the other hand, since f'j and also f'j[go0,* g ] (j 0, ,r1) are monotonic and h is additive, we have

h+ [f`o+...f' "r] - (h [f`o. . f<r, )+ and also

(h [f' [go, * n gm-,], ftr:[go) .., g(f-1])+

=--h+ [ (f'to [gon . . .n gm-1 )+n *n(ftr: [go, 5 gm-1] ) +] ,

by Theorem I. Hence,

(f[go, ,gm-l])+ ((h [fo , f'rj) [go, g.-1] ) +

(h[f'o [go, , gm-,], ,f'r1_[go, gm-d]I

h+ [(f'o[go, , gm-,])+, , (f'r1[go, ,gm-d )+]

h+[f'o+[go+, , gm-1+]n . , f'r,+[go+n gm-,+]]

= (h+[f'0o+, , f'+]) [go+, ,gin-,+]

(h [(f'o, , f'rl] )+ [go+, , gm-l+,] f+ [go+0 . . .m-11,

Thus Theorem II is proved.

UNIVERSITY OF CALIFORNIA,

BERKELEY, CALIFORNIA.



a remark on boolean algebras with operators

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