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Chapter 7 Second Algebraization
CHAPTER 7
SECOND ALGEBRAIZATION
In our first method of algebraization we used the mapping U V P . which assigns to each F of a relational type the Iw-operation U U P F = & {SF: s < w } . In our second method of algebraization, which will now be presented, we shall use the mapping 3 of ch. 3, which assigns to each F of a relational type the w-operation dF. This mapping also commutes with substitutions. From theoperations icpiin /L,i,dleadstothesetTj" = {Gjcpj: icpi E [L,:}. which in turn gives rise to clG* :L,:, the closure of '*w 'j L, i under substitutions. We shall give two sets of generators for cl d*'i L, i. T h e first of these Y,D , Ci, Q, P : 0 d i < w } of w-operations, will be the set {+, *,-, where each operation in the set is defined as the corresponding Iwoperation was defined in ch. 2 except that in place o f t = '"U one now uses Y = '"U.The second set will be described presently. The correspondence between the function symbols of L,, and the operations in the first generating set induces a mapping of the set of terms of L, onto the set clD" / L, i. We let \Lp4\ be L,, together with this induced mapping and \ T\ the w-operation into which 7 is thus mapped. For the effective mapping tm of the formulas of L, into terms of L,, which was defined in ch. 2 we shall show that\rmcp\= &:pi. Hence, if icpi = i+[, then \tmcp\=\tm+\. T h e converse does not always hold since 3 is not 1- 1, nor even practically 1- 1. However, the weaker assertion holds that if F and G .are of the same relational type then GF = 3 C implies F = C . Thus, if cp and JI are of the same relational type, then\rmcp\=\rm+\if and only if icpi = i+i. Thus the mapping tm is an interpretation of i L,i in\ LPq\which is not fully faithful, but faithful enough to reduce problems of validity for i L,: to problems of validity for \ Lw\. For the second set of generators for c/G*i L, we shall define a mapping with similar properties. I
I
,
,
101
102
S t C O N D ALGEBRAIZATION
[CH.
7
We shall also characterize cI5;"i L a / more directly as Gr: red* ; L a ; , the set of w-successors of Iw-operations which are reducible to some icp i. Compactness and definability will be discussed briefly. We conclude ch. 7 by giving two sets of generators for IL,I which are closely related to the second set of generators for \ Lpq\. For any set I, a trunsformation (on I) shall be any function a such that dom a = I and ran a G I . Composition, or product formation, involving transformations will henceforth be often denoted by juxtaposition, although at times the product ( a ( p ( i ) ) :i E I ) = { ( a ( p ( i ) )i): , i E I } of the transformations a and p will also be crop. Thus, if f = (J;,,fr ,...) is in "U and a = ( a ( O ) , a ( ] ),...) - ( a o ,a I, ...) and p = ( p (0) ,p( 1 ) , ...) are transformations on w, thenfashall be(f,,,,,, f ( ~ ,andapshall ~ . . - )b e ( a ( P ( O ) ) , a ( P ( l ) ) , ...) = (am).a B ( 1 ) .
...>.
We let a be an ciirnost constunt s h i f , or a E Acs, if and only if a is a transformation on w and there is some k in w such that a ( k + n ) = a ( & ) n for every n in w . Let a be any transformation on w . We now introduce two l-ary w-operations. Of these, (Sa)has been investigated by Halmos [ 131, while ( T a ) is induced by the converse of the relation { ( f , f a ):f E X} which induces (Sa).H e r e 1 = " U and X C R:
+
( S a ) X = { f E &fa E X } . (Ta)X={ f a : f X ~ } = { g : g = f a f o r s o m e f E X}. The second set which will be shown to generate cl'Tj*iL,i is the , 9 = Acs. set {+, ., -,$, (Sa),( T a ) :a E Y } where We now proceed to give details. As in ch. 3, we let Z F = Ti tre F whenever n s w and F is of a relational type. It follows that for any F of type (rOis,,, one can characterize "wF as that rn-ary o-operation which for any argument ( X i )Isism satisfies: (13) For any h E " U , 'W
*
(('TjF)(Xi)lsis,,l)wh = F("U .X;h),~i~,,t.
Now let F and G be of the same type Consider any argu. any h E "U and, for 1 s i s rn, let X i = Yi h ment ( Y i ) , s i s m Take = { g - h : g E Y , } . Then YJ . X i - h = Y i , 1 d i s rn. Hence, for any g E ' " U , g-h E ( Z j F ) ( X i ) l s i s m if and only if g E F ( Y J l S i s m ; similarly for C. Thus we have shown: I
CH.
71
103
SECOND ALCEBRAIZATION
LEMMA 7.1: If F and G are of the same relutional type and if F = G.
TjF = "OG,then
T h e condition that F and G are of the same relational type cannot be omitted from 7.1. For example, if F if any of the operations C.,1 ,+I+!-' then & F is the w-operation Ci. Also, for each r < w, if F is t r , . t r , or r , then ZF is the w-operation ., or - respectively. We now let 12 be the n-ary w-operation satisfying for each argument (Xj)lcjsn the condition I>(Xj)lsjsn = X k and, for any S C " U , we let K.;!' be the n-ary w-operation satisfying for each argument (Xj)jsn the condition K,k!l(Xj)lsjsn = S . Note that if R G ' U is taken as a 0-ary operation of type ( r ) , then GR = {g-h: g E R and h E " U } . For w-operations and sets of these we define cl and related notions by making the obvious changes in the definitions given in ch. 1. T h e proof of part (a) of the following analogue of 2.2 and of 3.2 is similar to the proofs for these. The other parts are obvious.
+
+,
I
LEMMA7.2: Let F be of type (ri)ism,for 1 < i < m let G i be of " U , ..., Y,, G " U . Then: type (Ti, s,, ...,s,) = (ri)-f, and let Y , (a) ( " w ( F ( G i ( X j ) , s , , , ) , s i s , n : X jC "'Ufor 1 <.j d n))(Yj),sjs,e = (&F) ( G G i ) ( Y j )1s j s n ) I s i s m (b) For any s,, und R E "'U, sjKY( = dK::,., = K:it. (c) F o r uny k , 0 < k 6 n, GI:. = G I ; = I;.".
c
T h e condition that F , G I , ..., G,,, are of suitable relational type cannot be omitted from 7.2. For example, let F be tre-rr, and let G = G, be tre-rr+t. Then Z F and "OG are the w-operation -, so that ( G F ) { ( G G ) ( Y ) )= Yforeach Y C " U . O n t h e o t h e r h a n d . F ( G ( X ) ) - F(-,r+i("+'U . X ) ) = - t r ( " U . ("'U .-A')) = - t r @ = "U for '"U))(Y) = "U for each each X C '"U, so that ( G ( F ( G ( X ) ) : X Y "U. For w-operations we now define w-quusiterm, the m-ary and v-ary w-operation \ 6 \ denoted by an w-quasiterm 6 ,etc. as these notions were defined in ch. 2 for lo-operations and for operations of relational similarity types. Then a proof similar to that of 2.4, except that no analogue of 2.l(c) is needed and that 6 may be a quasivariable, yields the following theorem.
c
THEOREM 7.3: Let 6 be a quasiterm of a relational type and let 6'
104
[CH.
SECOND ALCEBRAIZATION
7
be the w-qircisiterm M?hichresirlts from 6 when euch occurrence of any G cfu relationul type is repluced byZG. Then\b'\ = Gj19 i.
Unless noted otherwise. we now let +, -,-,A', D , C j ,Q , P , D j j , and Si be those w-operations which are defined as the corresponding lo-operations were defined in ch. 2, except that k = "U is now used instead o f Y = '"U.
THEOREM 7.4: The set {+, .,-,4', D , Cj,P , Q : 0 d i < w } of woperations grnerutes cIG*: L 3i. Proo$ We already observed that .,-, C j are GF where for some r < W , F is + i r , . i r , - f r , or C,,iZ+l+,, respectively. One also observes that D is GD,),P is&P'tl, and Q is G Q [ o . Also,Y= - D + D. Hence, by 1.3, cl {+, .,--,A', D , C ; ,Q , P : i < W } is included in cl G*' La:. The other inclusion follows from these observations, from 7 . 3 , a n d f r o m 6 D j = Q'D, (GC,,,, ) X = X f o r i a r, ( G Q ' , r ) X = S ; - l . . . S: S,l)Q X . and ( G P ' , ) X = P S',' S,: ... S;:-I C,.X for r > 0 . 0 ~
+,
Next, we turn to the set Acs, which will be used for defining the second generating set. For any index set I and any i and j in I , we let (ilj), the rc~plticrmentof i by j, be that transformation on I which satisfies ( i / j ) n = n i f n # iand ( i / j ) n = j i f n = i.Wealsolet ( i , j ) i = j , ( i , j ) j = i , and ( i , j )n = n if i # n # j . The I concerned will often be taken as understood. We let (+ I ) be that transformation on w which satisfies ( + l ) n = n+ 1 for each n in o.We let (-1) be that transformation on w which satisfies (- 1) n = n - 1 if n > 0 and 1) n = 0 if n = 0. We let (0; 1 ) = I ) , and for any r in o we let ( r + 1 ; 1 ) betheproduct ( r ; - l ) ~ ( r + l / r + 2 ) . T h e n ( r ; - l ) n = n i f n s rand 1 ) = ~ n - 1 if I Z > r. Finally for any I , any transformation a on (r; 1. and any t in w we let a" = a and at' I = a' a. In the next lemma. cl is closure under composition. 0
LEMMA 7.5: Acs = ci { ( + I ) , (L I ) , (i/j) : i < w , j < w } . Pro($ Clearly, ( + l ) , ( L 1 ) . and all (ilj), where i < w and j < w, are in Acs. Now consider any a in Acs. There is some k < w such that a k t,, = ak++nfor every n < w and such that, in addition, if k' < k, then ak' < ak. Let t 2 ak and let p = (t/ao) ( ( t l ) / a , )0 ( ( t+k . . .) ) o (+ 1 1'. Then P = (Po,PI, . . . ,Pk-,)^(Pk, = ( a o ,a I , .. . , a ~ - ~ ) - ( k + t k , + t + 1 , . . .). Hence a is a product of transformations (+ I), 1) and (ilj),since a = ( a k - 1; l)k+r-akoB. 0 0
+
A
a e . 0
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I05
THEOREM 7.6: The set (+, *, - , X , (Sa), ( T a ): a E Acs} generates clZ* /La;. Proof: Consider any X C W. If i # j then ( S ( i / j ) ) X= C,(D,,. X ) and ( T ( i / j ) ) X= D,, * C,X, and if i = j then ( S ( i / j ) ) X= ( T ( i / j ) ) X =X.Also, ( S ( + I ) ) X = Q X a n d (T(+l))X=PX.Further,(S(-1))X = P(D . X) and ( T ( - 1 ) ) X= D . QX. Finally ( S a p ) X = (Sa)( S p ) X and ( T a p ) X = ( T p )( T a ) X . It follows from 7.5 that c l { + , .,-,A’, (Sa),( T a ) :a E Acs} is included in c l { + , D , C , , Q, P : i < a}. For the other inclusion we observe that D = (T(0/1)) ( S ( O / l ) ) d , C,X = ( S ( i / i + 1 ) ) ( T ( i / i + 1))X and again that QX = ( S ( + I ) ) X and PX = ( T ( + 1 ) ) X. T h e theorem then follows from 7 . 4 . 0 a,-,,?,
The proof also shows that another set of generators for c 1Z*;L, is (Sa),(Ta):a E cl{(+I),(i/i+i):i
{+;,-,A:
THEOREM 7.7: Let tm be one of the two mappings just described.
(a)\tm+\=G j+/. (b) lf cp and are of the same relational type, then\ tm cp\ =\tm +\ if and only if icp / = / i.
+
+
Part (b) of 7.7 allows us to reduce problems of validity for / L a / to problems of validity for\L,,\ or for \L4c5\.Neither for the language of
I Oh
SECOND ALCEBRAlZATlON
ICH.
7
w-dimensional cylindric algebras nor for the slightly more expressive language obtained from it by adding symbols for each (Sa)such that a has finite support (see [ 151 and [ 131) do we know of a similar reduction. In particular, the mapping of formulas in standard form which one is first inclined to use will not work. For example, x(v,,) would be but not mapped into x, and 3,,x(v0) into cIx.Now ix(vll)/= /3,,x(vo)~ \x\= \ c,x\. We now turn to the task of characterizing \ L,\ = \ L.4rs\ more directly. First, one can verify that for any Jw-operationF , Ej commutes as follows with substitution of variables for variables. LEMMA7.8: Let F be any m-ary Iw-operation and let Y , Y,, C " U . Then ( Z { F ( / [(
G '"Ufor I
~ i ) l s j s , , ) , ~ j ~ , J , : ~ j
G
=
THEOREM 7.9: Let S be
(I
j
G
n))(YJ)lsjs-n
( G F )( ( G I : , 1 ( Y j ) l s . i < n ) ~ s i c , r i .
set of operations of relational similarity
types. (a) 1fS contains @, then G* cl tre" S el G'j' S . (b) If S contriins each H such tlwt ire H = 33 f o r some s
c
G E S , then c l Z " S Proof:
u U , ...,
c Z'!' cl tre'"S .
< w and
(a) Assume that S contains 4. Consider any F € cltre" S . Then F = 16") for some lo-quasiterm 6"such that each lw-operation occurring in 6"is tre H for some H € s. Now suppose there is a subquasi, , , for term (tre H ) ^ 6 , - ... -6,,,of 8"such that N is of type ( T ~ ) ~ ~while some i, 1 d i d in, 6; begins with some tre G , where 15is of a type ( s ~ ) ~such ~ , , that ri # so. Then replacement in 6"of this occurrence of bjby @ = tre @ yields an Iw-quasiterm which also denotes F . Repeating this process one can obtain an lo-quasiterm 6 such that 161 = F and such that, for some quasiterm A of a relational type in which each G belongs to S, A yields 8 by the replacement of each G by tre G. Let 8' be the w-quasiterm which results from 6 when each t r e G is replaced by G C . Then \6'\ belongs to (.I G:::S. By 7.3, \8'\=W-i A = Z ( t r e / A : ) . By the analogue for tre of 2.3, tre / A : = 161. Hence \a'\ = GI61 = G F . Therefore G*:cltre4: S C c1G:;:S. (b) Assume that S contains each H such that tre H =SG for some s < w and G E S. First, consider any w-quasiterm 6' such that no
CH.
71
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SECOND ALGEBRAIZATION
quasivariable occurs in 6'more than once and such that each w-operation occurring in 6'is 3H for some H E S. By 3.l(b), 3H = d(3H) for each H E S and s < w . By induction one shows that to each occurrence of any GH in 6'one can assign an s < w such that, given any occurrence of a subquasiterm (DH)-b,^... -6, in a', if H is of type 1 S i S m,ai begins with TjG, G is of a type (s~)~~?,, r has been assigned to the occurrence of 3H being considered, and s has been assigned to the occurrence of G being considered, then s,, s = ri r. Let 6 be the Iw-quasiterm which results from 6'by replacing each particular occurrence of any G H by 3H, where s is the number that has been assigned to this occurrelice of &H. Since no quasivariable occurs in 6 more than once, there is a quasiterm A of a relational type such that A yields 6 by the replacement of each G by tre G. Then GI61 = G ( t r e / A j ) = \ 6 ' \ by the analogue for tre of 2.3, and by 7.3 and 3.l(b), respectively. By the assumption on S, each 2H in 6 is tre G for some G E S, so that 161 E cl ?re* S. Now consider any n-ary F E clZ*S. Then there is some w-quasiterm 6' of the kind just considered and there are Zi,, 1 i < m , such that F ( Yj)lsjsn = \6'\(Z~~~(Yj)lSjsn)lsism for each Y, C " U , ..., Y , C " U . If 6 is obtained from 6'as above, then
+
+
F(Y J ) ] <= J ~(Ol6l) ~ ((a]:) (yj)1SJS,)1s2s7rl
=(&(l8l(fz (Xj)lSJsn)lstsm:XJ c I"Ufor 1
4j G
n))(Y,),,,,,
by 7 . 2 ( c ) ,and 7.8, respectively. Since cf tre* S is closed under substitutions and since 161 is in cl tre* S , so is (ISl(l;( X j ) l s , s n ) l s t s , n : X, I"U for 1 s j C n). Hence F is in G* cl ?re* S . Therefore cl o'*S c G* cl tre* S . 0 Since /L3/is closed under substitutions, 7.9 and 3.8, together with 7.4 and 7.6, yield:
THEOREM 7.10: \Lw\ =\ LAC\\ = cf o'* i La i = G* red* :La[ Turning briefly to compactness, we should like to mention that Ralph McKenzie, in a work not yet published, has used ultraproducts to show that, if every finite subset of a set E of equalities has an wmodel, then E has an w-model. In contrast, the w-consequence relation E b T = T ' is not of finite character. For example, let E be the set
I ox
SECOND A1 CnEBRAIZATION
[Ch.7
q’d: 0 S i < w}. Then E ‘F qx = cox. but there is no finite subset E’ of E such that E’ ‘F qx = c,,x. Next. we shall give an example where one has an implicit definition but no explicit definability. Let any U # @ be given. The constcrnt functions shall be those j E “U such that J; =f, for each i < j < w. We let
{x
Q
11$ = { ( r i ) - j J’ E and eitherjis constant and 11 =A, o r j ’ i s not constant and II =A+, for the least i such that
./; + t ; I I *
‘
Then D$ may be regarded as a characteristic function of the set of constant functions., By 7.10. each \ T\ is G ( r 1 F ) for some F and Y < w . Hence. if U contains at least two objects, then there is no T which explicitly defines D$ in the sense that \T\ has D$ as its only value. Let E d + consist of the following five equalities: ( I ) c , , x = 1, (2) qx * c , ( d * qx) C d. ( 3 ) d * x Q qd * qx. ( 4 ) -qd - x c,(d qd), and ( 5 ) qd * x * cl(-d * qx) = 0. Let ( U , X 6 ) 6 , , be any w-structure. One verifies that if X,,= DS, then ( U , X 6 ) 6 c r )is an w-model of E d $ . Now assume that ( U ,X 8 ) 6 , , is an w-model of Ed*.Let X = XI).We shall show that X = D$ and hence that Ed+implicitly defines D$. Consider any .f E “ U . Suppose ( r r ) - f E X and (~i,)-fE X . Then ( i t , , u)-f E Q X . Also ( i t - , i t . ) - j E D . Q X and therefore (M*,rr>-f E C, ( D . Q X ) . By (21, i.e., since ( U , X6)s
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S E C O N D ALGEBRAIZATION
now the case where J;, # u. Iffo f fi, then, by (4), u =f, and hence u =A+1 for the least i such that A # Now assume as inductive hypothesis that if the least j such that gj # gj+lis i , then ( w ) ^ g E X if and only if w = gi+l.Suppose that the least j such that f, # f,+, is i + l . Then f o = f l and hence ( u ) ^ f E Q D . X . Let g = (fi,h,...). If ( w ) - g were in X and w # 11, then (11, w)^g would be in - D * QX, and hence ( u ) - f = (u,J;,)^g in C , ( - D * Q X ) , which is impossible by (5). Hence, by ( 1 ) , ( u ) ^ g E X. Hence, by the inductive hypothesis, u = gi+l
=A+2.
For the rest of ch. 7 we return to I LJ. Let Acs
t
=Acs
*
{a:a ( j ) 2 a ( i ) wheneverj 2 i } .
F o r each a in Acs t andfin I"U, the set { i : a ( i ) E d o m f } is an initial segment of w and hencefa is in IWU.Thus for each a in Acs t one can define lo-operations ('Sa)and ('Ta) as (Sa)and ( T a ) were defined at the beginning of ch. 7 , except that one now lets,= /' IWU. T h e next theorem gives a set of generators for I L,I, which is closely related to the set of generators for\ L,\ that was given in 7.6. The Boolean operations involved are, of course, lo-operations.
THEOREM 7.11: ILwI=cl{+,.,-,X, ( ' S a ) , ('Ta): a E A c s ? } ., -,I,('Sa), ( ' T a ) : a E { (+ 1) } + { ( i / i+ 1 ) : i E w } } . Proof. Note that f E ( ' S y ) ('Ty) X if and only if, for some h E X, dom f . r m y = dom h .run y and f ( j )= h ( j ) for every j in d o m f . run y. It follows that
= cl{+,
( ' S ( i / i + l ) ) ( ' T ( i / i +I ) ) k U = kU ifk $Z { i , i + 1 ) . ( ' S ( i / i + l ) ) ( ' T ( i / i +l ) ) k U = iCJ+i+lUifk E { i , i + I } .
Now let i < w and let Ci, P, Q, Djk be as in ch. 2. One verifies that ('T(i+ l / i + 2 ) ) ( ' T ( i / i +l))4'= ( D i ( i + lD) .( i + l H i + 2Ii+'U, ))+ ('S(i+ l / i + 2 ) ) ( ' T ( i + l / i + 2))((Di(i+1) . D(i+l)(i+2))+ "+'U) = Di(iC2) "+'U,
+
( ' S ( i / i + l))('T(i/i+1))(Di(i+2)+ li+lU) = - 1 i + 3 U + 1 i + 2= U -i+zU,
( ' T ( + 1 ) ) ( ' T ( + 1 ) ) (i+")
= 'U.
I10
Also,
SECOND ALGEBRAIZATION
[CH.
7
+
C,X = ('U . X ) (--IU. ( ' S ( i / i + 1 ) ) ( ' T ( i / i +l ) ) X ) , Q X = - " U . ('S(+l))X, . PX= ('T(+l))(-('U.X),
D =-"U. ('T(O/l))X. Hence the third set of the theorem includes the first. By 3. I I , the first \et includes the second. (A more direct proof can be given by showing, with methods similar to those in the proof of 7.5, thatAc.7 t = (.I{(+ l ) , (A I ) , ( i / i + I ) , ( i + l / i ) :0 d i < w } . ) Evidently, the second set of the theorem includes the third. 0 Unfortunately, an axiomatic approach to ILpqlwhich takes the operations ('Sa)and ( ' T a ) , a E Acs T , as primitive does not look promising. Mainly. ( ' S a ) ( Ta)( ' 'Sp)('T p ) and ('Sp)('T p ) (' S t y ) ( ' T a ) often differ. For example, if i > 0 then ( ' S ( i - l / i ) ) ( ' T ( i - l / i ) ) ( ' S ( i / i + 1 ) ) ( ' T ( i / i +l ) ) ' + ' U is ?-'I!) + ' U +'+'U whereas ( ' S ( i / i + 1 ) ) ( ' T ( i / i + 1 ) ) ( ' S ( i - I / i ) ) ( ' T ( i - l/i))I+'U is IU+'+'U. This failure of commutativity is caused by sequences f and g in IWUforwhichfo( i / i + I ) = g o ( i / i + l ) a n d y e t d o m f = { O , . . . , i - I } # (0, ..., i - I , i } = dorn g . This suggests that, given a transformation a in Acs, we restrict ourselves to certainfa. T h e restriction will be carried out by choosing a certain number t h r a , the threshold for a, and letting & be the function which satisfies (i) d o m & = C { " U : n 2 t h r a } , a n d (ii) if f E d o m n , then &( f ) =fa = ( ( 1 4 , i): a ( i ) E dom f and f ( a ( i ) )= 1 4 ) . It is desirable to choose the values thr a so as to satisfy the following four conditions: (iii) run & 1 " ~ ; (iv) if & ( f ) = & ( g ) ,then dom f = dqm g ; (v) iff E dorn & and & ( f ) E dom @, thenf E dom ap; (vi) i f f E dom & and if i # j but a ( i ) = a ( j ) ,then a ( ; ) E dom f a n d h e n c e i E d o m & ( f )a n d j E d o m & ( f ) . To carry out the proof of 8.6 below, we also want rhr a to satisfy: = (&( f))-h = (fa)^h for any (vii) if f E dorn &, then &vAh) h E IWU. Also, we should like h t o have the largest domain compatible with
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SECOND ALGEBRAIZATION
these conditions. For any a in Acs we therefore pick the least k which satisfies both (i)' a ( k + m ) = a ( k ) + m f o r e v e r y m < wand (ii)' a ( k ' ) < a ( k ) for every k' < k, and then let thr a = a ( k ) . One can verify that k satisfies (iii), ..., (vii). For any a in Acs we now introduce the following two lo-operations, distinguishing them from the corresponding o-operations of 7.6, which will be used much more often, by means of context: (Sa)X=
{fi&(f) E X } = { f E C { " U : n 2
rhra}:fa E X } ,
( T a ) X = { & ( f ) : f ~ X } = { f i E C { W : a ( m )2 t h r a } : f E X } .
To illustrate, let a = (+ 1 ) . Then thr a = 1, dom (x = C n b 1) = Q x a n d i f n 2 1 a n d f = (fo,fi, ...,f n - l ) i s i n " U t h e n & ( f ) = (fi, ..., f,-,).Then (Sa)is the lo-operation Q and ( T a ) the lo-operation P . Now let a = ( j / i ) where j # i. Then t h r a = mrtx(i,j) 1 , dom & = Qmfrsci~j'+l,K (Sa)is the lo-operation Sf, and (Ta).Y is the lo-operation Dji. Assigning the above lo-operations (Sa)and ( T a ) to ( s a ) and (ta) respectively, we obtain in the now familiar way for each term 7 of LACS an lo-operation 171. We let I LACS/ be LA,, together with this interpretation of its terms. Often, IL,,,I will be simply the set of these is the set of lo-operations lo-operations 171. I n the latter sense, lLAcSl which is generated by {+, *, -,X,(Sa),( T a ): a E Acs}.
+
THEOREM 7.12. I LAC,/= I L, I. Proofi One can verify that C , , X = - Q X . X + Q P X and C,+,X = --Q'+'X*X+ ( S ( i / i - 1 ) ) ( T ( i / i - -1 ) ) X . It follows from the illustra= cl{+, .,--,A', (Sa),( T a ) : a E Acs} tions given above that lLAcSl includes I L, I. T h e other inclusion follows from 3.1 I. 0 An axiomatization of lLAcSlusing the primitives suggested by the preceding discussion has not been undertaken but may be worthwhile. In applications, it would have the advantage over E,, that translations from L, would be simpler. Also, by 8.6 and 9.18 below, it would yield a subtheory, evidently proper, of the theory yielded by the axiom set TBACsof ch. 9, and thus might add to our understanding of it.
SECOND ALGEBRAIZATION
In our first method of algebraization we used the mapping U V P . which assigns to each F of a relational type the Iw-operation U U P F = & {SF: s < w } . In our second method of algebraization, which will now be presented, we shall use the mapping 3 of ch. 3, which assigns to each F of a relational type the w-operation dF. This mapping also commutes with substitutions. From theoperations icpiin /L,i,dleadstothesetTj" = {Gjcpj: icpi E [L,:}. which in turn gives rise to clG* :L,:, the closure of '*w 'j L, i under substitutions. We shall give two sets of generators for cl d*'i L, i. T h e first of these Y,D , Ci, Q, P : 0 d i < w } of w-operations, will be the set {+, *,-, where each operation in the set is defined as the corresponding Iwoperation was defined in ch. 2 except that in place o f t = '"U one now uses Y = '"U.The second set will be described presently. The correspondence between the function symbols of L,, and the operations in the first generating set induces a mapping of the set of terms of L, onto the set clD" / L, i. We let \Lp4\ be L,, together with this induced mapping and \ T\ the w-operation into which 7 is thus mapped. For the effective mapping tm of the formulas of L, into terms of L,, which was defined in ch. 2 we shall show that\rmcp\= &:pi. Hence, if icpi = i+[, then \tmcp\=\tm+\. T h e converse does not always hold since 3 is not 1- 1, nor even practically 1- 1. However, the weaker assertion holds that if F and G .are of the same relational type then GF = 3 C implies F = C . Thus, if cp and JI are of the same relational type, then\rmcp\=\rm+\if and only if icpi = i+i. Thus the mapping tm is an interpretation of i L,i in\ LPq\which is not fully faithful, but faithful enough to reduce problems of validity for i L,: to problems of validity for \ Lw\. For the second set of generators for c/G*i L, we shall define a mapping with similar properties. I
I
,
,
101
102
S t C O N D ALGEBRAIZATION
[CH.
7
We shall also characterize cI5;"i L a / more directly as Gr: red* ; L a ; , the set of w-successors of Iw-operations which are reducible to some icp i. Compactness and definability will be discussed briefly. We conclude ch. 7 by giving two sets of generators for IL,I which are closely related to the second set of generators for \ Lpq\. For any set I, a trunsformation (on I) shall be any function a such that dom a = I and ran a G I . Composition, or product formation, involving transformations will henceforth be often denoted by juxtaposition, although at times the product ( a ( p ( i ) ) :i E I ) = { ( a ( p ( i ) )i): , i E I } of the transformations a and p will also be crop. Thus, if f = (J;,,fr ,...) is in "U and a = ( a ( O ) , a ( ] ),...) - ( a o ,a I, ...) and p = ( p (0) ,p( 1 ) , ...) are transformations on w, thenfashall be(f,,,,,, f ( ~ ,andapshall ~ . . - )b e ( a ( P ( O ) ) , a ( P ( l ) ) , ...) = (am).a B ( 1 ) .
...>.
We let a be an ciirnost constunt s h i f , or a E Acs, if and only if a is a transformation on w and there is some k in w such that a ( k + n ) = a ( & ) n for every n in w . Let a be any transformation on w . We now introduce two l-ary w-operations. Of these, (Sa)has been investigated by Halmos [ 131, while ( T a ) is induced by the converse of the relation { ( f , f a ):f E X} which induces (Sa).H e r e 1 = " U and X C R:
+
( S a ) X = { f E &fa E X } . (Ta)X={ f a : f X ~ } = { g : g = f a f o r s o m e f E X}. The second set which will be shown to generate cl'Tj*iL,i is the , 9 = Acs. set {+, ., -,$, (Sa),( T a ) :a E Y } where We now proceed to give details. As in ch. 3, we let Z F = Ti tre F whenever n s w and F is of a relational type. It follows that for any F of type (rOis,,, one can characterize "wF as that rn-ary o-operation which for any argument ( X i )Isism satisfies: (13) For any h E " U , 'W
*
(('TjF)(Xi)lsis,,l)wh = F("U .X;h),~i~,,t.
Now let F and G be of the same type Consider any argu. any h E "U and, for 1 s i s rn, let X i = Yi h ment ( Y i ) , s i s m Take = { g - h : g E Y , } . Then YJ . X i - h = Y i , 1 d i s rn. Hence, for any g E ' " U , g-h E ( Z j F ) ( X i ) l s i s m if and only if g E F ( Y J l S i s m ; similarly for C. Thus we have shown: I
CH.
71
103
SECOND ALCEBRAIZATION
LEMMA 7.1: If F and G are of the same relutional type and if F = G.
TjF = "OG,then
T h e condition that F and G are of the same relational type cannot be omitted from 7.1. For example, if F if any of the operations C.,1 ,+I+!-' then & F is the w-operation Ci. Also, for each r < w, if F is t r , . t r , or r , then ZF is the w-operation ., or - respectively. We now let 12 be the n-ary w-operation satisfying for each argument (Xj)lcjsn the condition I>(Xj)lsjsn = X k and, for any S C " U , we let K.;!' be the n-ary w-operation satisfying for each argument (Xj)jsn the condition K,k!l(Xj)lsjsn = S . Note that if R G ' U is taken as a 0-ary operation of type ( r ) , then GR = {g-h: g E R and h E " U } . For w-operations and sets of these we define cl and related notions by making the obvious changes in the definitions given in ch. 1. T h e proof of part (a) of the following analogue of 2.2 and of 3.2 is similar to the proofs for these. The other parts are obvious.
+
+,
I
LEMMA7.2: Let F be of type (ri)ism,for 1 < i < m let G i be of " U , ..., Y,, G " U . Then: type (Ti, s,, ...,s,) = (ri)-f, and let Y , (a) ( " w ( F ( G i ( X j ) , s , , , ) , s i s , n : X jC "'Ufor 1 <.j d n))(Yj),sjs,e = (&F) ( G G i ) ( Y j )1s j s n ) I s i s m (b) For any s,, und R E "'U, sjKY( = dK::,., = K:it. (c) F o r uny k , 0 < k 6 n, GI:. = G I ; = I;.".
c
T h e condition that F , G I , ..., G,,, are of suitable relational type cannot be omitted from 7.2. For example, let F be tre-rr, and let G = G, be tre-rr+t. Then Z F and "OG are the w-operation -, so that ( G F ) { ( G G ) ( Y ) )= Yforeach Y C " U . O n t h e o t h e r h a n d . F ( G ( X ) ) - F(-,r+i("+'U . X ) ) = - t r ( " U . ("'U .-A')) = - t r @ = "U for '"U))(Y) = "U for each each X C '"U, so that ( G ( F ( G ( X ) ) : X Y "U. For w-operations we now define w-quusiterm, the m-ary and v-ary w-operation \ 6 \ denoted by an w-quasiterm 6 ,etc. as these notions were defined in ch. 2 for lo-operations and for operations of relational similarity types. Then a proof similar to that of 2.4, except that no analogue of 2.l(c) is needed and that 6 may be a quasivariable, yields the following theorem.
c
THEOREM 7.3: Let 6 be a quasiterm of a relational type and let 6'
104
[CH.
SECOND ALCEBRAIZATION
7
be the w-qircisiterm M?hichresirlts from 6 when euch occurrence of any G cfu relationul type is repluced byZG. Then\b'\ = Gj19 i.
Unless noted otherwise. we now let +, -,-,A', D , C j ,Q , P , D j j , and Si be those w-operations which are defined as the corresponding lo-operations were defined in ch. 2, except that k = "U is now used instead o f Y = '"U.
THEOREM 7.4: The set {+, .,-,4', D , Cj,P , Q : 0 d i < w } of woperations grnerutes cIG*: L 3i. Proo$ We already observed that .,-, C j are GF where for some r < W , F is + i r , . i r , - f r , or C,,iZ+l+,, respectively. One also observes that D is GD,),P is&P'tl, and Q is G Q [ o . Also,Y= - D + D. Hence, by 1.3, cl {+, .,--,A', D , C ; ,Q , P : i < W } is included in cl G*' La:. The other inclusion follows from these observations, from 7 . 3 , a n d f r o m 6 D j = Q'D, (GC,,,, ) X = X f o r i a r, ( G Q ' , r ) X = S ; - l . . . S: S,l)Q X . and ( G P ' , ) X = P S',' S,: ... S;:-I C,.X for r > 0 . 0 ~
+,
Next, we turn to the set Acs, which will be used for defining the second generating set. For any index set I and any i and j in I , we let (ilj), the rc~plticrmentof i by j, be that transformation on I which satisfies ( i / j ) n = n i f n # iand ( i / j ) n = j i f n = i.Wealsolet ( i , j ) i = j , ( i , j ) j = i , and ( i , j )n = n if i # n # j . The I concerned will often be taken as understood. We let (+ I ) be that transformation on w which satisfies ( + l ) n = n+ 1 for each n in o.We let (-1) be that transformation on w which satisfies (- 1) n = n - 1 if n > 0 and 1) n = 0 if n = 0. We let (0; 1 ) = I ) , and for any r in o we let ( r + 1 ; 1 ) betheproduct ( r ; - l ) ~ ( r + l / r + 2 ) . T h e n ( r ; - l ) n = n i f n s rand 1 ) = ~ n - 1 if I Z > r. Finally for any I , any transformation a on (r; 1. and any t in w we let a" = a and at' I = a' a. In the next lemma. cl is closure under composition. 0
LEMMA 7.5: Acs = ci { ( + I ) , (L I ) , (i/j) : i < w , j < w } . Pro($ Clearly, ( + l ) , ( L 1 ) . and all (ilj), where i < w and j < w, are in Acs. Now consider any a in Acs. There is some k < w such that a k t,, = ak++nfor every n < w and such that, in addition, if k' < k, then ak' < ak. Let t 2 ak and let p = (t/ao) ( ( t l ) / a , )0 ( ( t+k . . .) ) o (+ 1 1'. Then P = (Po,PI, . . . ,Pk-,)^(Pk, = ( a o ,a I , .. . , a ~ - ~ ) - ( k + t k , + t + 1 , . . .). Hence a is a product of transformations (+ I), 1) and (ilj),since a = ( a k - 1; l)k+r-akoB. 0 0
+
A
a e . 0
CH.
71
SECOND ALGEBRAIZATION
I05
THEOREM 7.6: The set (+, *, - , X , (Sa), ( T a ): a E Acs} generates clZ* /La;. Proof: Consider any X C W. If i # j then ( S ( i / j ) ) X= C,(D,,. X ) and ( T ( i / j ) ) X= D,, * C,X, and if i = j then ( S ( i / j ) ) X= ( T ( i / j ) ) X =X.Also, ( S ( + I ) ) X = Q X a n d (T(+l))X=PX.Further,(S(-1))X = P(D . X) and ( T ( - 1 ) ) X= D . QX. Finally ( S a p ) X = (Sa)( S p ) X and ( T a p ) X = ( T p )( T a ) X . It follows from 7.5 that c l { + , .,-,A’, (Sa),( T a ) :a E Acs} is included in c l { + , D , C , , Q, P : i < a}. For the other inclusion we observe that D = (T(0/1)) ( S ( O / l ) ) d , C,X = ( S ( i / i + 1 ) ) ( T ( i / i + 1))X and again that QX = ( S ( + I ) ) X and PX = ( T ( + 1 ) ) X. T h e theorem then follows from 7 . 4 . 0 a,-,,?,
The proof also shows that another set of generators for c 1Z*;L, is (Sa),(Ta):a E cl{(+I),(i/i+i):i
{+;,-,A:
THEOREM 7.7: Let tm be one of the two mappings just described.
(a)\tm+\=G j+/. (b) lf cp and are of the same relational type, then\ tm cp\ =\tm +\ if and only if icp / = / i.
+
+
Part (b) of 7.7 allows us to reduce problems of validity for / L a / to problems of validity for\L,,\ or for \L4c5\.Neither for the language of
I Oh
SECOND ALCEBRAlZATlON
ICH.
7
w-dimensional cylindric algebras nor for the slightly more expressive language obtained from it by adding symbols for each (Sa)such that a has finite support (see [ 151 and [ 131) do we know of a similar reduction. In particular, the mapping of formulas in standard form which one is first inclined to use will not work. For example, x(v,,) would be but not mapped into x, and 3,,x(v0) into cIx.Now ix(vll)/= /3,,x(vo)~ \x\= \ c,x\. We now turn to the task of characterizing \ L,\ = \ L.4rs\ more directly. First, one can verify that for any Jw-operationF , Ej commutes as follows with substitution of variables for variables. LEMMA7.8: Let F be any m-ary Iw-operation and let Y , Y,, C " U . Then ( Z { F ( / [(
G '"Ufor I
~ i ) l s j s , , ) , ~ j ~ , J , : ~ j
G
=
THEOREM 7.9: Let S be
(I
j
G
n))(YJ)lsjs-n
( G F )( ( G I : , 1 ( Y j ) l s . i < n ) ~ s i c , r i .
set of operations of relational similarity
types. (a) 1fS contains @, then G* cl tre" S el G'j' S . (b) If S contriins each H such tlwt ire H = 33 f o r some s
c
G E S , then c l Z " S Proof:
u U , ...,
c Z'!' cl tre'"S .
< w and
(a) Assume that S contains 4. Consider any F € cltre" S . Then F = 16") for some lo-quasiterm 6"such that each lw-operation occurring in 6"is tre H for some H € s. Now suppose there is a subquasi, , , for term (tre H ) ^ 6 , - ... -6,,,of 8"such that N is of type ( T ~ ) ~ ~while some i, 1 d i d in, 6; begins with some tre G , where 15is of a type ( s ~ ) ~such ~ , , that ri # so. Then replacement in 6"of this occurrence of bjby @ = tre @ yields an Iw-quasiterm which also denotes F . Repeating this process one can obtain an lo-quasiterm 6 such that 161 = F and such that, for some quasiterm A of a relational type in which each G belongs to S, A yields 8 by the replacement of each G by tre G. Let 8' be the w-quasiterm which results from 6 when each t r e G is replaced by G C . Then \6'\ belongs to (.I G:::S. By 7.3, \8'\=W-i A = Z ( t r e / A : ) . By the analogue for tre of 2.3, tre / A : = 161. Hence \a'\ = GI61 = G F . Therefore G*:cltre4: S C c1G:;:S. (b) Assume that S contains each H such that tre H =SG for some s < w and G E S. First, consider any w-quasiterm 6' such that no
CH.
71
I07
SECOND ALGEBRAIZATION
quasivariable occurs in 6'more than once and such that each w-operation occurring in 6'is 3H for some H E S. By 3.l(b), 3H = d(3H) for each H E S and s < w . By induction one shows that to each occurrence of any GH in 6'one can assign an s < w such that, given any occurrence of a subquasiterm (DH)-b,^... -6, in a', if H is of type 1 S i S m,ai begins with TjG, G is of a type (s~)~~?,, r has been assigned to the occurrence of 3H being considered, and s has been assigned to the occurrence of G being considered, then s,, s = ri r. Let 6 be the Iw-quasiterm which results from 6'by replacing each particular occurrence of any G H by 3H, where s is the number that has been assigned to this occurrelice of &H. Since no quasivariable occurs in 6 more than once, there is a quasiterm A of a relational type such that A yields 6 by the replacement of each G by tre G. Then GI61 = G ( t r e / A j ) = \ 6 ' \ by the analogue for tre of 2.3, and by 7.3 and 3.l(b), respectively. By the assumption on S, each 2H in 6 is tre G for some G E S, so that 161 E cl ?re* S. Now consider any n-ary F E clZ*S. Then there is some w-quasiterm 6' of the kind just considered and there are Zi,, 1 i < m , such that F ( Yj)lsjsn = \6'\(Z~~~(Yj)lSjsn)lsism for each Y, C " U , ..., Y , C " U . If 6 is obtained from 6'as above, then
+
+
F(Y J ) ] <= J ~(Ol6l) ~ ((a]:) (yj)1SJS,)1s2s7rl
=(&(l8l(fz (Xj)lSJsn)lstsm:XJ c I"Ufor 1
4j G
n))(Y,),,,,,
by 7 . 2 ( c ) ,and 7.8, respectively. Since cf tre* S is closed under substitutions and since 161 is in cl tre* S , so is (ISl(l;( X j ) l s , s n ) l s t s , n : X, I"U for 1 s j C n). Hence F is in G* cl ?re* S . Therefore cl o'*S c G* cl tre* S . 0 Since /L3/is closed under substitutions, 7.9 and 3.8, together with 7.4 and 7.6, yield:
THEOREM 7.10: \Lw\ =\ LAC\\ = cf o'* i La i = G* red* :La[ Turning briefly to compactness, we should like to mention that Ralph McKenzie, in a work not yet published, has used ultraproducts to show that, if every finite subset of a set E of equalities has an wmodel, then E has an w-model. In contrast, the w-consequence relation E b T = T ' is not of finite character. For example, let E be the set
I ox
SECOND A1 CnEBRAIZATION
[Ch.7
q’d: 0 S i < w}. Then E ‘F qx = cox. but there is no finite subset E’ of E such that E’ ‘F qx = c,,x. Next. we shall give an example where one has an implicit definition but no explicit definability. Let any U # @ be given. The constcrnt functions shall be those j E “U such that J; =f, for each i < j < w. We let
{x
Q
11$ = { ( r i ) - j J’ E and eitherjis constant and 11 =A, o r j ’ i s not constant and II =A+, for the least i such that
./; + t ; I I *
‘
Then D$ may be regarded as a characteristic function of the set of constant functions., By 7.10. each \ T\ is G ( r 1 F ) for some F and Y < w . Hence. if U contains at least two objects, then there is no T which explicitly defines D$ in the sense that \T\ has D$ as its only value. Let E d + consist of the following five equalities: ( I ) c , , x = 1, (2) qx * c , ( d * qx) C d. ( 3 ) d * x Q qd * qx. ( 4 ) -qd - x c,(d qd), and ( 5 ) qd * x * cl(-d * qx) = 0. Let ( U , X 6 ) 6 , , be any w-structure. One verifies that if X,,= DS, then ( U , X 6 ) 6 c r )is an w-model of E d $ . Now assume that ( U ,X 8 ) 6 , , is an w-model of Ed*.Let X = XI).We shall show that X = D$ and hence that Ed+implicitly defines D$. Consider any .f E “ U . Suppose ( r r ) - f E X and (~i,)-fE X . Then ( i t , , u)-f E Q X . Also ( i t - , i t . ) - j E D . Q X and therefore (M*,rr>-f E C, ( D . Q X ) . By (21, i.e., since ( U , X6)s
CH.
71
I09
S E C O N D ALGEBRAIZATION
now the case where J;, # u. Iffo f fi, then, by (4), u =f, and hence u =A+1 for the least i such that A # Now assume as inductive hypothesis that if the least j such that gj # gj+lis i , then ( w ) ^ g E X if and only if w = gi+l.Suppose that the least j such that f, # f,+, is i + l . Then f o = f l and hence ( u ) ^ f E Q D . X . Let g = (fi,h,...). If ( w ) - g were in X and w # 11, then (11, w)^g would be in - D * QX, and hence ( u ) - f = (u,J;,)^g in C , ( - D * Q X ) , which is impossible by (5). Hence, by ( 1 ) , ( u ) ^ g E X. Hence, by the inductive hypothesis, u = gi+l
=A+2.
For the rest of ch. 7 we return to I LJ. Let Acs
t
=Acs
*
{a:a ( j ) 2 a ( i ) wheneverj 2 i } .
F o r each a in Acs t andfin I"U, the set { i : a ( i ) E d o m f } is an initial segment of w and hencefa is in IWU.Thus for each a in Acs t one can define lo-operations ('Sa)and ('Ta) as (Sa)and ( T a ) were defined at the beginning of ch. 7 , except that one now lets,= /' IWU. T h e next theorem gives a set of generators for I L,I, which is closely related to the set of generators for\ L,\ that was given in 7.6. The Boolean operations involved are, of course, lo-operations.
THEOREM 7.11: ILwI=cl{+,.,-,X, ( ' S a ) , ('Ta): a E A c s ? } ., -,I,('Sa), ( ' T a ) : a E { (+ 1) } + { ( i / i+ 1 ) : i E w } } . Proof. Note that f E ( ' S y ) ('Ty) X if and only if, for some h E X, dom f . r m y = dom h .run y and f ( j )= h ( j ) for every j in d o m f . run y. It follows that
= cl{+,
( ' S ( i / i + l ) ) ( ' T ( i / i +I ) ) k U = kU ifk $Z { i , i + 1 ) . ( ' S ( i / i + l ) ) ( ' T ( i / i +l ) ) k U = iCJ+i+lUifk E { i , i + I } .
Now let i < w and let Ci, P, Q, Djk be as in ch. 2. One verifies that ('T(i+ l / i + 2 ) ) ( ' T ( i / i +l))4'= ( D i ( i + lD) .( i + l H i + 2Ii+'U, ))+ ('S(i+ l / i + 2 ) ) ( ' T ( i + l / i + 2))((Di(i+1) . D(i+l)(i+2))+ "+'U) = Di(iC2) "+'U,
+
( ' S ( i / i + l))('T(i/i+1))(Di(i+2)+ li+lU) = - 1 i + 3 U + 1 i + 2= U -i+zU,
( ' T ( + 1 ) ) ( ' T ( + 1 ) ) (i+")
= 'U.
I10
Also,
SECOND ALGEBRAIZATION
[CH.
7
+
C,X = ('U . X ) (--IU. ( ' S ( i / i + 1 ) ) ( ' T ( i / i +l ) ) X ) , Q X = - " U . ('S(+l))X, . PX= ('T(+l))(-('U.X),
D =-"U. ('T(O/l))X. Hence the third set of the theorem includes the first. By 3. I I , the first \et includes the second. (A more direct proof can be given by showing, with methods similar to those in the proof of 7.5, thatAc.7 t = (.I{(+ l ) , (A I ) , ( i / i + I ) , ( i + l / i ) :0 d i < w } . ) Evidently, the second set of the theorem includes the third. 0 Unfortunately, an axiomatic approach to ILpqlwhich takes the operations ('Sa)and ( ' T a ) , a E Acs T , as primitive does not look promising. Mainly. ( ' S a ) ( Ta)( ' 'Sp)('T p ) and ('Sp)('T p ) (' S t y ) ( ' T a ) often differ. For example, if i > 0 then ( ' S ( i - l / i ) ) ( ' T ( i - l / i ) ) ( ' S ( i / i + 1 ) ) ( ' T ( i / i +l ) ) ' + ' U is ?-'I!) + ' U +'+'U whereas ( ' S ( i / i + 1 ) ) ( ' T ( i / i + 1 ) ) ( ' S ( i - I / i ) ) ( ' T ( i - l/i))I+'U is IU+'+'U. This failure of commutativity is caused by sequences f and g in IWUforwhichfo( i / i + I ) = g o ( i / i + l ) a n d y e t d o m f = { O , . . . , i - I } # (0, ..., i - I , i } = dorn g . This suggests that, given a transformation a in Acs, we restrict ourselves to certainfa. T h e restriction will be carried out by choosing a certain number t h r a , the threshold for a, and letting & be the function which satisfies (i) d o m & = C { " U : n 2 t h r a } , a n d (ii) if f E d o m n , then &( f ) =fa = ( ( 1 4 , i): a ( i ) E dom f and f ( a ( i ) )= 1 4 ) . It is desirable to choose the values thr a so as to satisfy the following four conditions: (iii) run & 1 " ~ ; (iv) if & ( f ) = & ( g ) ,then dom f = dqm g ; (v) iff E dorn & and & ( f ) E dom @, thenf E dom ap; (vi) i f f E dom & and if i # j but a ( i ) = a ( j ) ,then a ( ; ) E dom f a n d h e n c e i E d o m & ( f )a n d j E d o m & ( f ) . To carry out the proof of 8.6 below, we also want rhr a to satisfy: = (&( f))-h = (fa)^h for any (vii) if f E dorn &, then &vAh) h E IWU. Also, we should like h t o have the largest domain compatible with
CH.
71
Ill
SECOND ALGEBRAIZATION
these conditions. For any a in Acs we therefore pick the least k which satisfies both (i)' a ( k + m ) = a ( k ) + m f o r e v e r y m < wand (ii)' a ( k ' ) < a ( k ) for every k' < k, and then let thr a = a ( k ) . One can verify that k satisfies (iii), ..., (vii). For any a in Acs we now introduce the following two lo-operations, distinguishing them from the corresponding o-operations of 7.6, which will be used much more often, by means of context: (Sa)X=
{fi&(f) E X } = { f E C { " U : n 2
rhra}:fa E X } ,
( T a ) X = { & ( f ) : f ~ X } = { f i E C { W : a ( m )2 t h r a } : f E X } .
To illustrate, let a = (+ 1 ) . Then thr a = 1, dom (x = C n b 1) = Q x a n d i f n 2 1 a n d f = (fo,fi, ...,f n - l ) i s i n " U t h e n & ( f ) = (fi, ..., f,-,).Then (Sa)is the lo-operation Q and ( T a ) the lo-operation P . Now let a = ( j / i ) where j # i. Then t h r a = mrtx(i,j) 1 , dom & = Qmfrsci~j'+l,K (Sa)is the lo-operation Sf, and (Ta).Y is the lo-operation Dji. Assigning the above lo-operations (Sa)and ( T a ) to ( s a ) and (ta) respectively, we obtain in the now familiar way for each term 7 of LACS an lo-operation 171. We let I LACS/ be LA,, together with this interpretation of its terms. Often, IL,,,I will be simply the set of these is the set of lo-operations lo-operations 171. I n the latter sense, lLAcSl which is generated by {+, *, -,X,(Sa),( T a ): a E Acs}.
+
THEOREM 7.12. I LAC,/= I L, I. Proofi One can verify that C , , X = - Q X . X + Q P X and C,+,X = --Q'+'X*X+ ( S ( i / i - 1 ) ) ( T ( i / i - -1 ) ) X . It follows from the illustra= cl{+, .,--,A', (Sa),( T a ) : a E Acs} tions given above that lLAcSl includes I L, I. T h e other inclusion follows from 3.1 I. 0 An axiomatization of lLAcSlusing the primitives suggested by the preceding discussion has not been undertaken but may be worthwhile. In applications, it would have the advantage over E,, that translations from L, would be simpler. Also, by 8.6 and 9.18 below, it would yield a subtheory, evidently proper, of the theory yielded by the axiom set TBACsof ch. 9, and thus might add to our understanding of it.