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Quantifier elimination in discriminator varieties
Annals of Pure and Applied Logic 32 (1986) 83-101 North-Holland
83
Q U A N T I F I E R E L I M I N A T I O N IN DISCRIMINATOR VARIETIES Francoise POINT Universit~ de l'Etat ~ Mons, Facult6 des Sciences, Mons, Belgique Communicated by Y. Gurevich Received 16 July 1984; revised 21 February 1985
0. Introduction In this paper, we want to classify those classes of elements of a discriminator variety V, which admit quantifier elimination (q.e.) in terms of classes of simple elements of V. Discriminator varieties come up in decidability questions for certain varieties. Burris and McKenzie [3] have shown that any finitely generated decidable modular variety of finite type is the product of a finitely generated discriminator variety and of a decidable abelian variety. Our starting point is a representation theorem due to Bulman-Fleming, Keimel and Werner: Any element of a discriminator variety is isomorphic to a boolean product of subdirectly irreducible members of that variety [7, Theorem 4.9, p. 56]. Then in Section 1 we analyze transfer of q.e. in finite direct products with projector. Here we could apply the Feferman-Vaught theorem on direct products, but we choose to give a direct proof of transfer of q.e. from the factors of a finite direct product to the product and vice versa. In Section 2, we prove our main result. Given a quantifier diminable element of a discriminator variety, we derive stringent conditions on its base space and on the class of its stalks. For proving that these conditions are indeed sufficient, we extend the result proved by Boffa and Cherlin in [2] to a non-compact situation. Finally we consider the effectiveness of the given procedures and we classify the classes of dements of discriminator varieties, with q.e.
Preliminaries To begin with, we recall some logic terminology. A basic formula is either an atomic formula or the negation of an atomic formula. A CB-formula is a conjunction of basic formulas. An open formula is a boolean combination of atomic formulas. 0168-0072/86/$3.50 (~) 1986, Elsevier Science Publishers B.V. (North-Holland)
84
F. Point
A positive formula is built up from atomic formulas using only the connectives A, V and the quantifiers V, =1.
A primitive formula cp(x) is an existential :iy 1 • • • :iynO(y, x) where 0(y, x) is a CB-formula.
formula
of
the
form
We need certain notions related to quantifier elimination in classes of .Y-structures. These appear in [2] and [1]. Let C be a class of -y-structures. (1) C admits quantifier elimination (q.e.) itt for any formula q ~ ( x l , . . . , xn) there exists an open formula O ( x l , . . . , x,) such that the formula Vx (q0(x) O(x)) holds in any element of C. (2) C admits connective elimination (c.e.) iff for any open formula O ( x l , . . . , x~) there exists an atomic formula % ( x ~ , . . . , x~) such that the formula Vx (O(x)<--~%(x)) holds in any element of C. (3) C admits strong quantifier elimination (s.e.) iff C admits q.e. and c.e. (4) C admits positive quantifier elimination (positive q.e.) iff for any positive existential formula q0(Xl,..., x~), there exists an open formula O ( X l , . . . , xn) such that the formula Vx (cp(x) ~ O(x)) holds in any element of C. Note that to show that C admits q.e., we need only to consider primitive formulas q9 with one existential quantifier. Now we recall some facts about discriminator varieties. Let V be a discriminator variety, i.e., an equationally defined class s.t. there is a term t(u~, u2, u3) with the class of subdirectly irreducible members of V satisfying
(,)
t(Ul , U2, U3) = U4 ~ ( u 1"- U2 • U3 = u 4 ) V (U l::t=u 2 & U 1"- U4).
If the equivalence ( * ) holds in an element of A of V, then t(-,-, .) is called a discriminator for A. For any non-trivial element of V (i.e. those containing more than one element), the properties of being simple, subdirectly irreducible are equivalent to the fact that t(.,., -) is a discriminator [7, Theorem 2.2, p. 22]. Before stating the boolean representation theorem for any element A of V, we define the support of an element a of A. From now on we suppose that the language -Y of V contains at least one constant 0. So *Y will always denote a language of algebras containing a ternary function t(., -, .) and at least a constant 0. Then the support of a in a sub-direct representation II~iAi of A by subdirectly irreducible members Ai of V, is the subset {i e l I Aj~a(i):/:O}. The set of supports (with respect to a fixed subdirect representation of A) is the domain of a generalized boolean algebra and if we choose another subdirect representation, we will obtain isomorphic generalized boolean algebras [4, Lemma 9.1]. Theorem 0 [7]. Let A be a non-trivial element of V. Then A is isomorphic to the
-y-structure F~(X, Ux~xAx) of all sections with compact clopen supports of a locally boolean sheaf (X, Ux ~x A x) of simple elements of V.
Quanttfier elimination in discriminator varieties
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Let r be a n-tuple of elements of A. Let q~ be a formula. We call the subset {x • X IAx ~ tp(r(x))} = [~(r)] the truth value of tp. Note that the truth value [r~:/:r2] is the support of the element t(n(rl, r2, O, rl), O, n(rl, rE, O, r2)), where n(ul, UE, Vl, rE) is defined as follows:
n(ul, UE, Vl, rE) = t(t(ul, U2, Vl), t(ul, U2, rE), rE). 1. Direct product with projector Let V be a discriminator variety. Let 1-li~xA~ be a direct product of elements Ai of V. Definition 1.1. A projector for 1-L~zAi is a sequence of terms (Pi)~z satisfying the following: for any element a = (a(i))i,z of the product, ~a(j) p,(a)(j) = (0(j)
if i = j , if i 4:j.
If the terms (P~)i~z are ~ O C-terms where C is a finite set of new constants, we say that the direct product IIi~zAi has a projector in ~ U C. In the following propositions, we show transfer of quantifier elimination from the factors of a finite direct product with projector to the direct product, and vice versa. In Section 2, we will see moreover that any q.e. element of V is isomorphic to a certain finite direct product with projector.
Definition 1.2. For (Pi)i~z a projector, we define on formulas q9 the transformation q9~',). Let 0 be the atomic formula s ( x , , . . . , x n ) = 0 where s is a term 0 ° ' ) = pi(s(x))=O. Let 0 be a boolean combination of atomic formulas Ok, say ~/ ( A Ok ^ /~ -nOk,). Define 0~, ) as V ( A O~ ') ^ / ~ ~0~')) • Let q9 be an existential formula of the form :ly 0(y, x), where 00', x) is an open formula. Define q0°') as :ly 00',). Notation 1.3. (i) If A , , A2 are two elements of V and r = ( r l , . . •, rn) is a n-tuple of elements of A1, we denote by r* the n-tuple ((r~, 0 ) , . . . , (r~, 0)) of elements of A~ x A2. (ii) If E is a finite set of sentences, we denote by A E the conjunction of elements of E. (iii) C = {c} is a finite set of new constants with respect to ~ and for 1 ~< i ~< 2, Ci = {el} is a finite set of new constants such that C / = {c(i) I (c(1), c(2)) e C}. In L e m m a 2.4, we will deal with the case where C = C1 and C-,z= (0}. L e m m a 1.4. For any open formula O(xl, . . . , xn, c) of &e U C, there exists a finite
86
F. Point
set Po of CB-sentences X of ~ U C2 and corresponding CB-formulas Ox(x, cl) of , ~ . U C 1 such that for any A1 and A2, Vrl,...,rneAl(AlxA2gO(r*,c) iff Vx~e (A, ~ Ox(r, cl) and A2 ~ X)).
Proof. First we put 0 in a disjunctive normal form Vi~z 0~, where each 0~ is of the form AI~L, tt = 0 A Ak,r, Sk :/: O, where tt and Sk are 3?-terms. (i) From each Oi, we construct a set of formulas as follows. For each subset K of Ki, let %K be Al~Zi tl = 0 A Ak~r Sk :/: O. (ii) Let r be a n-tuple of elements of A1. We have for any i belonging to I:
(,)
A1xA2~Oi(r*,c)
iff
V (Al~xr(r, cl) andA2~xr,-r(O, c2)).
K~_Ki
We have simply used the fact that an element in A, x A2 is not zero iff one of its coordinates is not zero. (iii) Let Po be the set of all open sentences Xr(0, c2) for all subsets K of Ki and all i of I. For each X of Po, we denote by Ox(.) the formula Xr,-r(', cl) where xK(O, C2) is X. With this notation, applying (ii) we get:
A1XAz~O(r*,c)
if
V (Al~Ox(r, cl) andAz~X).
[]
x cPo
In the following two propositions, we will use the same notation as in L e m m a 1.4. In addition, we suppose that A 1 X A2 is a direct product with projector in ~UC. Let (Pl, P2) be a projector for A1 x A2, where p~, P2 are ~ U C-terms. Note that p , ( y ) = t(y, P2(Y), 0). L e m m a 1.5. Let AI x A2 be a direct product with projector (p~, P2). Given any
primitive (respectively open) formula ¢p, there exists a primitive (respectively open) formula q90") such that: Ai~q)(r(i))
iff
A 1 xA2~qg(P0(r),
where r is a sequence of elements from A 1 x A2. Proof. It suffices to note that for any term s and any n-tuple r of elements of A, A i ~ s ( r ( i ) ) = O ¢ - - > A l X A 2 ~ p i ( s ( r ) ) = O , for 1<~i<~2. [] Proposition 1.6. Let A1 x A 2 be a direct product with projector (pl, P2). Then
Ax x A2 admits q.e. in .~ U C implies that A~ and A2 each admit q.e. in .~ O (71
(respectively in ~ U Cz). Proof. Given any primitive formula q~(x), we find an open formula ~,(x) such
Quantifier elimination in discriminator varieties
87
that (i) A~ ~ Vx (cp(x)~->~p(x)),
(ii) ~p(-) depends on the CB-sentences satisfied by A2. By Lemma 1.5~ for any n-tuple r of elements of A~, we have:
A~ ~ ~(r)
i~ A ~ ~ ) ( r * ) .
Since A admits q.e. in ~ U C, there exists an open formula O(x, c) such that for any n-tuple r of elements of AI we have: Al~qg(r)
iff A~O(r*,c).
By Lemma 1.4, there exists a set Po of CB-sentences of ~t.JC2 and CB-formulas Ox(.) of ~ U C1, where Z belongs to Po such that for any n-tuple r of A1,
A~O(r*,c)
V (Al~Ox(r) andA2~z ).
iff
z ~Po
Let E be the set {Z e Po [ A2 k Z}. Then g(x) is either V Ox(x),
if E is not empty,
or
if E is empty.
zeE
0 #: O,
[]
Proposition 1.7. If Ai admits q.e. in ~ tJ Ci, 1 <~i <<-2, then A admits q.e. in LUC. Proof. Let qg(x) be a primitive formula, ~(x) = 3yAtj(y, x) = o ^
/~ Sk(y, x) ~ O,
k~K
where t/, Sk are .T U C-terms. For any subset L of K, let XL be
3yAtj(y,x)=O ^ Ask(y,x)~O. j~J
k~L
For any n-tuple r = (r(1), r(2)) of elements of A1 x A2,
A~qg(r) iff
V (AI~ZL(r(1)) andA2~Zr_L(r(2))).
L~_K
For any subset L of K, let 0~(-) be an open formula equivalent in Ai to XL(')Then we have for any n-tuple r of elements of A:
A~q~(r) aft
V (Al~O~(r(1)) andA2~O2(r(2))). L~_K
To finish the proof, we apply Lemma 1.5. For any subset L of K, we have that
AigO~(r(i))
iff AgO~o'i)(r), 1 <~i<~2.
[]
88
F. Point Collecting Proposition 1.6 and 1.7, we have:
Theorem 1.8. If A1 x A2 is a direct product with projector (p,, P2), then A1 x A2 admits q.e. in . ~ U C iff A1 and A2 admit q.e. in .~UC1 (respectively in
LUG).
[]
2. Quantifier eliminable subdasses of a discriminator variety We preview the results up to Proposition 2.6 in this paragraph. First we recall a result of Burris and Werner that any open formula is equivalent to a basic formula and this uniformly in any simple elements of any discriminator variety V. Then we look at the relationship between the satisfaction of an open formula in a stalk Ax of an element A of V and the satisfaction of the same formula in A(U) x A ( X - U ) , where U is a neighborhood of x and A(U) (respectively A ( X - U)) is the L-structure whose domain is the set of restrictions of elements of A to U (respectively to X - U). We use the facts that each stalk Ax is the direct limit of the direct system {A(U) [ U is a clopen neighborhood of x}, and that there is for any open sentence Z, a neighborhood U of x such that for any U1, U2 included in U, we have
A ( X - UI) ~x iff A ( X - U2)~X. Further, we show that the class of stalks of any q.e. element of V admits positive q.e. Let us fix some notation. Let A be an element of V and F~(X,Ux~xA,) its representation. (I) For any ~-term t(x), any element r of A and any subset Y of X "t(r) - r/Y (30/X - Y" means that t(r) equals r on Y and 0 elsewhere. For ease of notation, we will denote it by r/Y U O. (2) For any n-tuple ( r l , . . . , rn) of elements of A, we denote by r. the n-tuple ( n ( u , O, O, r,), . . . , n ( u , O, O, rn)) = (r,/[u =t= 0] U 0 , . . . ,
r~/[u =/= 0] U 0).
(3) Capital U is a variable for compact clopen set. For the ease of notation, given an n-tuple r of elements of A~, we often denote by r* any n-tuple of elements of A which satisfy r*(x) = r. Proposition 2.1. Let ~ be a class of simple elements of a discriminator variety.
Then any non-positive (positive) CB-formula is equivalent to the negation of an 2tomic (atomic) formula and this uniformly in all elements of ~. Moreover, if there is a closed term different from 0 in all elements of ~, then ~dmits c.e. Proof. See [4, Lemma 9.1]. Note that this implies that any open formula is ~quivalent to a basic formula, uniformly in cg. []
Quantifier elimination in discriminator varieties
89
Proposition 2.2. Let V be a discriminator variety with O. Then every CB-formula is equivalent to a CB-formula with exactly one atomic conjunct:
t = 0 A A s j * 0. j~J
Proof. Given a CB-formula A i E i t i = 0 A A j e j S j :[= 0 we assume I =g=~ (otherwise add 0 = 0). By Proposition 2.1, in the class of the simple elements of V we have A~EI t~ = 0.-> t = 0. Let A belong to V and r to A. A ~ Ati(r) iEl
=
0 A Astr ) * 0 j~J
iff
[Ati(r)=O]=X
iff
[t(r)=0]=X
and
and
itI A ~ t(r) A Asj(r) jEJ
[si(r) 6:01 * ~
[sj(r)~=0]#=~
O.
forj~J
forjeJ
[]
In the next two lemmas, we arae going to look at how to express in A that an open formula is satisfied in a stalk A~.
Lemma 2.3. For any open formula O(xa, . . . , x~) and for any n-tuple (rl, • • •, r~) o f elements o f Ax, 3Uo (x ~ Uo A V U (x ~ U ~_ Uo)"->A ( U ) ~ O(r/U)) iff A~ ~ O(r).
Proof. We apply exercise 5.2.24 of [5] (see also [6]). A sentence tp is preserved under direct limits of ~-structures iff tp is equivalent to a sentence of the form m
AVI1,
i=1
" " " , I s (~li-"> 3 y ~ " " " y t Oi),
where the IPi and 0i are open positive formulas. We put s = t = 0 and we add to the language ~ new constants c l , . . . , c~ interpreted in A~ by r ~ , . . . , rn, respectively by rl/U, . . . , r,,/U in A ( U ) . Then A~ is still the direct limit of the direct system of ~ U {Cl, •. •, Cn}-Structures A ( U ) , where U is a clopen compact neighborhood of x. Therefore we can apply the result from Keisler. [] In the next lemma, c, a new constant added to L, will be interpreted in Ax by a non-zero element and in A by an element u such that u(x) = c.
Lemma 2.4. For any open formula O(xl, . . . , xn+O, there exists an open formula O'(xl, . . . , xn+O such that for any x o f X and any n-tuple r o f elements o f A , we have:
(.)
:IUo (x e Uo A VU (u ¢ A A u ( x ) = c A [u q:O] c_ Uo--~ a ~ O(r,,, u))) iff
Ax ~ O'(r(x), c).
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F. Point
Proof. Let U be a clopen subset of X. We consider A as the direct product A(U) x A ( X - U). Let c be a new constant added to Le such that the support of cA = u is U. Applying Lemma 1.4 to the open formula O ( X l , . . . , x,,, c), we get that there exists a finite set Po of CB-sentences X of L# and CB-formulas Ox(x ~, . . . , x,,, c) of &# t3 {c} such that for any n-tuple r of elements of A,
A ~ O(ru, u)
iff
V (A(U) ~ Ox(r/U, u/U) and A ( X -
U) ~ X).
x cPo
Given any element x of X, we are going to prove that there exists a neighborhood U, of x such that for any clopen compact subsets U1, U2 strictly included in Ux,
A ( X - U1) ~ X iff A ( X - U2) ~ X, for any X e Po. We proceed as follows: Let Fo be the set of truth values of the basic sentences occurring in each element of Po. Taking complement and finite intersections of these truth values, we get a partition of X whose elements Vii are of the form [ti = 0] or [ti 4: 0], 1 ~
r = O A ASk4=O, keK
where r and S k are closed ~-terms (see Proposition 2.2). If A ( X - U)~I, then A ( X - U)~ r = 0. Thus [r = 0] has a non-empty intersection with V~ and so [ r = 0 ] = X . Similarly for any k in K, A ( X - U)~Sk~O. SO either [Sk V~0] is disjoint from V~or [sk #=0] is included in V/. The assumption on U implies in the first case that A ( X - V ~ ) ~ s k 4=0 and in the second case that V~ equals [Sk ~ 0]. NOW we are ready to construct a formula 0'(.) such that the equivalence ( , ) holds. Let Ov,(', c) be 0 =/=0,
V Ox(', c),
if E~ is empty, otherwise.
Now, for 1 ~
or,(-, c) ^ t, = o,
if v / = [t, = o]
or
if E =
Quantifier elimination in discriminator varieties
91
Set O'(.,c) equal to Vm=10"(., c). Then we have 3 U o ( x e U o ^ V U ( u e A ^ u(x) = c ^ [u 4=O]~_ Uo-->A ~ O(r.,, u))) iff A~ k O'(r(x), c). Indeed, let x be any element of X. As the set { V~[ 1 <~i ~
Ax~VO'(r(x),c) i=1
iff Ax~Oi'o(r(x),c)
iff Ax~Ov, o(r(x),c ).
By Lemma 2.3,
Ax ~ Ov,o(r(x), c) if[
3U0(x ~ Uo ^ V U (x ~ U c_ Uo-->A(U)~ OV,o(r/U, u/U))),
where u is an element of A, whose support is U and such that u ( x ) = c. By definition of the subset E~ of Po, for any U included in V/0, we have:
V
(A(U) ~ Ox(r/U, u/U) and A ( X -
x~,
U) ~ X) iff
V (A(V) O (r/U, u/U)).
x ~F-~
Therefore by Lemma 1.4, we have for any n-tuple r of elements of A
3Uo (x e Uo A V U (x e U ~_ Uo-e A ( U ) ~ Ov,o(r/U, u/U)) )
iff
3U0 (x ~ Uo A VU (U e A a U(X) = C A [U ~ 0 ] c Uo'> A ~ O(ru, u))).
[]
Now we are ready to prove one of our main results on the class of the stalks of an q.e. element of V.
Proposition 2.5. I f an element A o f V admits q.e., then the class of its stalks admits positive q. e. Proof. By Proposition 2.1, the primitive positive formulas in the class of the simple elements of V are of the form 3y sO', z) = 0 where s is an ~-term. We will assume that y is a 1-tuple. By definition of the discriminator t(-), for any x of X, for any non-zero element c of Ax and any n-tuple r of elements of Ax, we have:
A~ ~ 3y s(y, r) = O~-> (1)
3Uo(xeUoAVU(u~AAu(x)=cA[u#O]cUo --->A ~ :ly t(0, u, s(y, r,,)) = s(y, r.,))).
By hypothesis, A admits q.e. Thus there exists an open formula O ( Z l , . . . , zn+l) equivalent in A to 3y t(zn+l, O, s(y, zl, . . . , z~)) = s(y, zl, , . . , zn). By Lemma 2.4, there exists an open formula O ' ( z l , . . . , z,+l) such that for any x of X and for any n-tuple r of elements of A,,:
(2)
3Uo(x e Uo^ Vu (u e A A u ( x ) = c ^ [u:/:O]c_ Uo--~A~O(r.,, u))) ~-~Ax ~ O' (r, c).
F. Point
92
Now, we are going to show for any x of X that we can get rid of the non-zero element c of Ax. Indeed, we have the following equivalence:
Ax ~ 3y s(y, z) = O
~">mx~[(:iys(y'z)=O^~/zi=/=O (3) v
=lys(y,O)=O^ A z ~ = O A s ( o , o ) ~ o i=l
v(3ys(y,O)=O^A(zi=O^s(O,O)=O))].i__l Applying equivalences (1) and (2), together with (3), we obtain for any point x of X:
Ax ~ 3y s(y, z) = O ~-->Ax~
O'(z, zi) ^ zi:/:O) v
( o'(o,s(O,O))^Azi=O^s(O,O) n ) O i=1
v
(A
z~ =0
As(0, 0)=0
)]
[]
The next proposition will show that any q.e. element of V is a certain finite direct product with projector. The factors have a base space w h i c h either is discrete with one or two points or has no isolated points. Thus by Propositions 1.6 and 1.7, characterizing q.e. elements of V reduces to characterizing (1) those with compact base spaces either discrete with exactly two points or without isolated points, (2) those with non-compact base space without isolated points. In cases (1) Boffa and Cherlin [2] have given sufficient conditions on the class of stalks for the global structure to admit q.e.
Proposition 2.6. If A admits (effective) q.e. then there is an (effective) procedure to decompose A into a finite direct product with projector (I[~=oAi, (ti)i=o..... n) such that (i) Each
admits (effective) q.e. (ii) The base space Xo of Ao has no isolated points and the base spaces Xi of the Ai, i >~1, are discrete with one or two points. Moreover, if Xo is compact and if any X~'s, i >~1, contain two points, then they form the support of closed terms. A i
Quannfier elimination in discriminator varieties
93
Proof. Step One. First we prove (a) and (b). (a) X has finitely many isolated points. (b) Moreover, the largest discrete subspace of X is the union of discrete subspaces X1, • • •, Xn of X, each of which contain at most two points. Moreover, for each X~, there is an &e-term ti with 6(r) = r/Xi t2 O, for any element r of A; and in case X~ contains two points, ti(r) = n(ti, O, O, r), where 6 is a closed &e-term. Note that with these &e-terms tl(.), . . . , t,(-), we can construct to('). Let
(*)
rl=n(r, tl(r),O,r);
ri+a=n(r,t~(r),O,r),
l<~i<~n.
Then to(r) = rn+l. Proof of (a). Let O(z) be an open formula expressing that the support of z is minimal, i.e., the support of z is a singleton (O(z) is equivalent to Vr (r =/=0 and t(O, z, r)=O--.t(O, r, z)=0). By L e m m a 1.4, there exists a finite set Po of CB-sentences of L such that for any element u of A whose support is U we have:
A ~ O(u) iff
V (A(U) ~ Ox(u ) and A ( X -
x ePo
U) ~X).
Suppose that there is a minimal clopen subset U. For any element r of A whose support is U, A ~ O(r). Let p v be {2' e Po IA(U) ~ 3u O~(u) and A ( X - U) ~ X}. Let X be of the form (tx = 0 ^/~J~Jx sj =/=0) with tx and sj closed L-terms (see Proposition 2.2). There are two possibilities: (1) There exists X belonging to Pov such that [tx = 0 ] ¢ : X , in this case U= [tx 4=0]. (2) For all X of pv, A ~ tx = O. In the latter case, we consider for any element r of A the following truth values for X e poV: Jr=/=0 ^ Ox(r) ^ A s j = O ^ j elo
/~ sj¢O] = [vx(r)] ,
Jelx -Jo
where Jo = {j eJ x IA(U)~sj=O } and 1:x is a basic formula (see Proposition 2.1). Note that for any element r of A whose support is U there exists X in Pov such that
[,x(r)l = [r/U ¢O]. Two subcases may arise: (2') For any element r of A, I,.Jx~e~, [vx(r)] is included in U, which implies that it equals to [r/U ¢ 0]. (2") There is some r and some X e Pov such that [vx(r)] is not included in U. In this latter case, we are going to show that there exists a closed term whose support is a discrete space with two points which includes U. Let v be r/[~x(r)] t'l ( X - U). We show that the support of v is minimal. First A([v ¢ 01) ~ Ox(v ). Then
A(X-[v¢O])~
A s j :/: 0 I EJo
94
F. P o i n t
(for [/~J~Jo sj = O] _~ U U [v 4= O] and A ( X - U) ~ / ~ 4
sj :/: 0 since X belongs to
Pg'); A(X-[v=/:O])~
A
si:#O
j E.lx - Jo
(because A ( U ) ~ AJ~4-Jo sj :/: 0 and U tq [v 4= 0] = 9)Let u0 be an element of A whose support is U such that A ( U ) ~ Ox(uo)O(" ~ PoV); and let W
j'Uo on U, [ v onX-U.
Since A([x #:01) ~ Ox(w ) (0 x is a CB-formula), A ( X - [ w : k O ] ) ~ Aj~joSj:/=O and A ~t O(w), there exists j ~ Jx - Jo such that [sj :/: 0] ~_ [w 4= 0]. Since for every j e Jx - Jo, [sj 4=0] _~ U t.J [v 4= 0], we have that [sj :/: 0] = [w 4= 0]. To summarize, any minimal clopen subset U is determined by 0 in the following way: (we use the above notation) -C ase (1): for one Z ~ Pov, U = [tx = 0]. -C ase (2"): for some Z e Pov and some j e Jx - J o, [si 4=0] is a discrete space with two points which includes U. -C ase (2'): for any element r of A, [r/U:/:O]=Ux~euo[rx(r)], where r x is a basic formula constructed from Z and 0 x. From the rx's, we can construct a basic formula r such that [r(r)] = [_Jx~e~[rx(r)] (see Proposition 2.1). Since there are only finitely many X, there are only finitely many minimal clopen subsets. Proof of (b). Suppose the discrete subspace Xk, 1 <-k <~n, is of the form Is 4= 0] where s is a closed term. Then tk(r) = n(s, O, O, r). Suppose that the discrete subspace Xk is the truth value of a basic formula r. If r is of the form s(-) = 0, then tk(r) = n(s(r), 0, r, 0). If the basic formula r is of the form s(.) #: 0, then tk(r) = n(s(r), O, O, r).
Step Two. Let Xo be the complement in X of the set of isolated points. We will show that Xo is compact iff it is the support of a closed &P-term. Let O(z) be an open formula expressing [z :/: 0] = X, say
O(z) = V tk(z) = o ^ A skAz) * o. jeJk
k
If A ~ O(r), say A satisfies the disjunct
t(r) = 0 A A sj(r) :/=O, J
then we claim either (i) X is finite or (ii) [t(0) = 0] is finite. Proof of claim. If X is infinite, let J = { 1 , . . . , n} and choose X l , . . . , xn e X such that sj(r(xj)) 4=O. If [t(0) = 0] is not finite, then we can find a clopen subset M of [ t ( O ) = O ] - { x l , . . . , x n } and create l e A such that [ f = r ] = X - M , [f = 0] = M. But then A ~ t(f) = 0 ^ / ~ sj(f) ~ 0, which contradicts the fact that [f 4: 0] 4: X. Thus [t(0) = 0] is finite, proving the claim.
Quantifier elimination in discriminator varieties
95
In case (i), Xo = t~, and in case (ii) we have a closed term t(0) whose support contains Xo. [] Before stating the corollary, we recall a notation used in [2]. If ~ is a class of ~-structures, then (i) c¢2 is the class of all direct products of two elements of c¢. (ii) ~ n t o r is the class of all structures of sections of sheaves of elements of c~ over a Cantor space. Corollary 2.7 (to Propositions 2.5 and 2.6). An element A of V admits q.e. iff A is a finite direct product with projector such that for each of its factors A i , 1 <~i ~ n, one of the following holds: (1) Ai is a simple element of V which admits q.e. (2) Ai belongs to ( ~i)2 where ~i is a class of simple elements of V admitting s.c. (3) A i belongs to (~i)cantor where ~i is a class of simple elements of V admitting s.e.
(4)
is an element of V admitting q.e., whose base space is not compact and has no isolated points. A i
Proof. (--->) Applying Proposition 2.6, we get that any quantifier eliminable element of V is a finite direct product with projector: [I7=1Ai with specific base spaces. Applying Proposition 1.6, we know that each factor of A admits q.e. Thus, if the base space of the factor Ai is either discrete with one point, or non-compact without isolated points, then we are in Case (1) or (4). If the base space of the factor Ai is compact either discrete with two points or without isolated points, then we know that it is the support of a closed ~-term ti. Thus the Ai have a closed term ti with ti ~ 0, so the class ~i = {Ax [x e X/} admits c.e., by Proposition 2.1. Since by Proposition 2.5, the class of stalks of A admits positive q.e., the class ~gi admits s.e. (~---) By Proposition 1.7, if each factor of a direct product A with projector admits q.e., then A admits q.e. Thus we have to prove that, if ~ is a class of ~-structures which admits s.e., then the classes ~2 and ~ t o r admit q.e. This is exactly the Boffa-Cherlin theorem in [2]. [] This corollary reduces the problem of characterizing the elements of V which admit q.e. to characterizing those whose base space is not compact and has no isolated points. This will be undertaken in Propositions 2.8 and 2.9. In Proposition 2.8, we will prove that the class of the stalks of such elements of V admits not only positive q.e. (Proposition 2.5) but that a certain kind of elimination holds in the class with respect to negative existential formulas. In Proposition 2.9, we will show the sufficiency of these conditions.
Proposition 2.8. Suppose A, an element of V, admits (effective) q.e. Suppose that
96
F. Point
its base space X has no isolated points and is not compact. Then A has the (effective) local negative elimination property, i.e., for any non-positive primitive formula cp there exists an open formula 0 such that for any n-tuple r of elements of A: [tp(r)]~
~
[0(r)]~.
Moreover, the procedure to find 0 is effective if the quantifier elimination procedure in A is effective. Proof. For convenience of the reader, we prove the proposition for 1-tuples. To prove the proposition, we need only to consider non-positive existential formulas tp of the form :ly t(y, .) ~ 0, where t is an L-term (see Proposition 2.1). In case A~qg(r), we have [ q g ( r ) ] ~ - > [ 0 = 0 ] ~ . So from now on we can assume that A It Vr tp(r). The open subset B = [tp(0)] is included in a compact subset of X. [If not, then for any element r of A the subset [r = 0] would have a nonempty intersection with B. As [r = 0] f7 B c_ [qg(r)] one would conclude [qg(r)] :/: ~, so A ~ tp(r). This contradicts the assumption that A lt'Cr tp(r). ] We call f a restriction of r if (i) [ f ~ 0 l ~ _ [ r ~ 0 ]
and
(ii) ~/[f=/:Ol=r/[f~O].
Let AkEr Ok be an open formula equivalent to tp in A, where Ok= V 6 = 0
vsk~:0.
iElk
(For the fact that Ok can be put in this form see Proposition 2.2). We want to prove that there exist k in K such that [Sk(0)#: 0] is equal to the closure/} of B. To prove this fact, we claim that if for any given k in K, (1)
either [Sk(0) ~ 0]~/~,
(2)
or
[Sk(O)~ O] ~/~,
then for any element r whose support includes B there exists a restriction f whose support also includes B such that all restrictions of f whose supports include B satisfy Ok. Suppose the claim is true. Then the assumption that (1) or (2) holds for every k in K leads to a false conclusion, namely A ~'¢r tp(r), as follows. First note that A~qg(r) for all r with [ r = 0 ] N B # = ~ for then ~[r~O]f7B~_[qg(r)] yields A ~ qg(r). So let us now consider an element r of A whose support includes B. By assumption there exists ~k a restriction of r such that all restrictions of ~k whose supports include B satisfy Ok. Let ~ be the restriction of r such that [~ ~ 0] = ['~k~K[rk:~:0]. So we have A ~ AkErOk(r). As [ F = 0 ] A B = ~ , one has [tp(f)]_= [ f ~ 0 ] . Thus for x e [tp(f)] we have f ( x ) = r ( x ) , using the fact [f #: 0] c_ [r #: 0] , and hence x e [qg(r)]. This suffices to prove A ~ qg(r), and the contradiction follows.
Quantifier elimination in discriminator varieties
97
Thus, if we prove the claim, then we can conclude that/~
= [Sk(0 ) ~f: 0] for some
kinK. Proof of the claim. Let r be an element of A whose support includes B. (1) Suppose that [Sk(0)S0]$/~. Let N be a non-empty clopen subset of [Sk(O) :#: O] disjoint from/~. Let f be equal to zero on N and to r elsewhere. Then A ~Sk(F)S0 and If $ 0 ] _~/~. Moreover, all restrictions of f satisfy Ok, since their supports are disjoint from N. (2] Suppose that [Sk(0) S 0] ~ B. Then there exists x0 in B - [sk(0) S 0] since [Sk(0) S 0] is a closed subset. Let ro be the restriction of r, whose support is [Sk(0) S 0]. Since [ro = 0] N B S ~, A ~ qg(r0). Either (i) Sk(ro)S 0 and this implies that there exists an x in [Sk(0)S0] such that sk(ro(x))SO. Since ro(x)=r(x), we have sk(r(x))SO. ThUS Sk(F(X))*O for any restriction F of r whose support contain B. (Note that [F S 0] ~ B implies that [ f S 0 ] _ D B ~x.) Or (ii) There exists i e Ik such that ti(ro)=0 and so the set ~ = {i e Ik: [ti(0) :/: 0] ~ /~} is not empty. Either there exists i e ~ such [ti(r) S 0] N B = 9, or for all i e ~ , [ti(r) ~ 0] N B ~ ~. In the first case, for any restriction f of r whose support includes B, equal to zero on [ti(r) ~ 0] and to r elsewhere, we have that A ~ ti(F) = 0. In the latter case, for any i in ~ let Ni be a non-empty clopen subset in [t~(r) #:0] A B. If all N~ are not included in [Sk(0)S0], pick a point z in some Nj - [Sk(0) ~ 0] and choose in each N~ a subset Mi not containing z. Let rl be equal to r on [Sk(0) :/: 0] f3 I,.Ji~zOMi and equal to zero elsewhere. Since [rl = 0] N B S ~, we have that A ~ qg(r~). In particular, A ~ 0k(rl). Therefore, either A ~Sk(rl) SO and so by the same argument as in (i) above, the claim is verified or A ~ ti(r~) = O, so i e ~ (indeed the support of r~ is included in/~) and since on M~, !'1 equals r, this leads to a contradiction. Now we are in a position to finish the proof. The class of stalks of a q.e. element of V admits positive q.e. So let ~p(s) be an open ~-formula equivalent to 3y n(t(y, s), O, s, O)= O. Let Sko(O) be an ~ - t e r m such that [Sko(O) ~/:0]--n. Let r be an element of A. For x e X
A~ ~ 3y tO,, r(x)) q~0 <-->A~ ~ (3y t(y, r(x)) q: 0 ^ r(x) * O) v (=iyt(y, r(x))SO A r(x)=O). Thus [qo(r)] ~ ~<-->[=ly t(y, r) #: 0 ^ r * 0] ~ ~ or [3y t(y, r) S 0 ^ r = 01S <-->[W(r) ^ r#:0] q:0 or [=ly tO', r) q:0 ^ r=O] q~O Now [3y tO', r) #=0 A r =01 #: ~ <-->B rl [r = 0 1 S ~ (-->/~ N [r = 01 :# I~ <-->[Sko(0) q: 01CI [r = 01 q: I~
98
F. Point
Thus [qg(r)]*~
iff [(~(r) A r , O ) V (Sko(O),O A r = O ) ] , ~ .
The terms Sk(0) for which [Sk(0) 4=0] =/~ are those among the terms Sk(0), k ~ K, which satisfy the open formula X(u) equivalent in A to
(*)
Vy ([t(y, O) #: O] ~ [u :/: 0]),
(1)
Vv ((Vy [t(y, O) 4=O] ~ [v :/: 0])---> ([u 4= O] _ [v 4= 0])).
(2)
Indeed Vy [t(y, O) 4=O] ~ [u 4=O] is equivalent to [3y t(y, O) 4=O] ~ [u 4= O] which holds iff / ~ = [ u 4 : 0 ] . So (2) says Vv(B=_[v:/:O]--->[u:/:O]c_[v:/:O]). One can rewrite ( * ) as Vy n(u, O, t(y, 0), O)= 0
and
Vv (Vy n(v, O, t(y, 0), O) = 0--> n(v, O, u, O) = 0).
[]
In the setting of discriminator varieties, the next proposition generalizes to the non-compact case the following result of [2]: "If a class c~ of .T-structures admits s.e., then Cgca~toradmits q.e." Proposition 2.9. Let ~ be a class of simple elements of V and consider a class ~ of
elements of V whose stalks belong to ~ and whose base space is not compact and without isolated points. Suppose (1) ~ admits positive q.e., (2) ~ has the local negative elimination property. Then ~ admits q.e. in .T. Proof. Let q9( Z 1 ,
. . • , Zn)
be a primitive formula of the form:
3yAt,(y, z)=O A Asj(Y, z)•O, i~l
j~J
where tj and sj are .T-terms. Let A be an element of ~ and let r be a n-tuple of elements of A. First, we express that the existence of a global solution to qg(r) in A is equivalent to the existence of local solutions for certain existential formulas in
{A,,lxeX}. A ~ ¢p(r) *->Vx
e
X A x g 3 y A t i ( y , r(x)) = 0 i~l
(1)
^ A3xjeXA~,~ 3yAt~(y, r(xj)) = 0 ^ s~(y, r ( x j ) ) * O. j~J
iEl
(See [8, proof of part 1 Theorem 2].) The implication (--->) is obvious as A is a subdirect product of the A . , x e X. To prove the implication (*-), we are going to build from local solutions, a global one. Let yj be an element of A such that the truth value [Ai~ ti(yj, r) = 0 ^
Quantifier elimination in discriminator varieties
99
sj(yj, r) ~ O] contains the point xj, for each j ~ J. As X has no isolated points, we can choose non-empty disjoint clopen (compact) subsets Uj, j E J, included in each of the above truth values. Let E be the following compact clopen subset of X: n
E = U [rk =/=O] U U[t,(O, O) #: O] - UU/. i~l
k=l
j~J
For each x E E, let y~ be an element of A such that [Ai~1 ti(y~, r) = 0] contains x. This set of truth values forms a clopen covering of E. As E is compact, there exists a finite sub-covering which can be refined to a finite covering { Vm [ m ~ M} of E. Let Ym be elements of A such that [/~i~I ti(Ym, r) = 0] = Vm, m ~ M. Now we define a global solution a for q0(r) as follows:
a/U/=yj/U/,
j EJ,
a/VmNE=ym/VmNE,
meM,
a/X-(UUif')E)=O. \jEJ
It is straightforward to verify that A ~ Ati(a, r) = 0 ^ Asj(a, r) =/=O. i$l
j~J
Second, we show that the existential local conditions in {A~ Ix ~X} are equivalent to open formulas, whether the existential formulas are positive or not. (2) By assumption, the existential positive formula 3y Ai~iti(y, z ) = 0 is equivalent in any element of :£ to an open formula. This open formula is equivalent, by Proposition 2.1, either to z = 0 or to ~ ~ 0, where z is an ~-term. Thus either
Vx E X A x k 3 y A t A y , r(x)) = o o Vx ~ X A x k z(r(x)) = 0 iEl
<-->A g z(r) = 0, or
Vx E X A x k 3yAti(y,r(x))=O<->Vx E X A x k ~ ( r ( x ) ) * O i~l
*->A ~O #=O, as X is not compact and the support of ~(r) is a clopen compact subset of X. (3) By Proposition 2.1, in any simple element of V, the CB-formula Ai~l ti(y, z) = 0 A sj(y, Z) ~ 0 is equivalent to the negation of an atomic formula,
i.e., rj(y, z) ~ 0, where rj is an ~-term. By hypothesis on the class ~, there exist open formulas 0j, j ~ J, such that
3xjA,,jk 3y~j(y,z),PO<->3xjAxj~ Oj(z),
for any j eJ.
By Proposition 2.1, we can assume that, in any simple element of V, the open formulas Oj are equivalent either to tj(.)= 0 or to tj(.)=/=O, where tj are ~-terms,
j~J.
100
F. Point
In both cases, we can express in A that 3x s Axj ~ Os(r(xs)):
3x s Ax, ~ ts(r(xs) ) = 0 ~ A
~ (0 = 0),
3x/ Ax ~ ts(r(x/)) =/=O~-->A ~ ts(r) ~ O. Thus, putting (1), (2), (3) together we have proved that we can construct an open formula equivalent in A to tp(z). Moreover this open formula depends only on ~ and the fact that the base space of A is not compact, without isolated points. Therefore we may conclude that ~ admits q.e. [] Now we are in a position to state the main results.
Theorem 2.10. Let V be a discriminator variety. Let A be an element of V. Then A admits (effective) q.e. iff there is an (effective)procedure to decompose A into a finite direct product with projector such that for each o f its factor A~, 1 <~i <~n, one o f the following holds: (1) Ai is a simple element o f V which admits (effective) q.e., (2) A i belongs to (~i)2 where ~i is a class of simple elements of V, admitting (effective) s.e., (3) Ai belongs to ( ~i)~tor where cgi is a class of simple elements of V, admitting (effective) s.e., (4) Ai is isomorphic to F~(Xi, [,.Jx~x, Ax), where Xi is a locally boolean space, non-compact, without isolated points and the class {Ax Ix • Xi} admits (effective) positive q.e. and A i has the (effective) local negative elimination property.
Theorem 2.11. Let V be a discriminator variety. Let ~r be a subclass of V. Then admits q.e. iff (1) There exist open sentences It1, • • •, ~.Lms.t. m -
j=l
A ~ Its implies that the base space o f A is the topological sum o f a subspace Xo without isolated points and finitely many discrete subspaces Xz, • • •, Xns with one or two points s.t. (i) if Xo is non-empty compact, then it is the support o f a closed &P-term and if X~ contains two points, i >~1, then it is the support o f a closed .~-term; (ii) for any Xi, i >1O, there exists .fC-terms ti s.t. for any element r o f A, ti(r) = r/X~ U O / X - X~. (2) For every positive open formula qg(z,y), there exist open sentences vz, . . . , VmC~) and open formulas q91(y), . . . , qgm(y) s.t. -
~
Vi, i=1
ViVA(A~--Fc(X, xUxAx))^A~v,-->(VxeX,
Ax~3zqg(z,Y)<->qg,(Y)).
101
Quantifier elimination in discriminator varieties
(3)
For every non-positive
open formula
v'l, . . . , v ' ( ~ , ) a n d o p e n f o r m u l a s m(~.)
~(z, y),
there exist open sentences
~ I ( Y ) , . . . , ~ m ( Y ) s.t.
,
Vv ,
i=1
Vi Vii A ~ v; ~ (3x e XAx ~ 3z g,(z, y ) ~ 3x e XAx ~ ~])i@))"
[]
Acknowledgments This article is a slight generalization of the chapter of my thesis on projectable L-groups. I warmly thank Professor A. Macintyre, A.M.W. Glass, V. Weispfenning and C. Wood for their help and advice. I am indebted to the referee for his thorough reading of this paper and his numerous suggestions. I would like to mention his simplification of part (b) of Proposition 2.6 and his discovery of errors in the proof of Propositions 2.6 and 2.8.
References [1] M. Boffa, Quantifier elimination in boolean sheaves, Preprint. [2] M. Boffa and G. Cherlin, Elimination des quantificateurs dans les faisceaux, C. R. Acad. Sci. Pads Set. A-B 290, (1980) A355-357. [3] S. Burris and R. McKenzie, Decidable varieties with modular congruence lattices, Bull. Amer. Math. Soc. 4 (3) (1981) 350-352. [4] S. Burris and H. Wemer, Sheaf constructions and their elementary properties, Trans. Amer. Math. Soc. 248 (1979) 269-309. [5] C.C. Chang and H.J. Keisler, Model Theory (North-Holland, Amsterdam, 1973). [6] H.J. Keisler, Some applications of infinitely long formulas, J. Symbolic Logic 30, 339-349. [7] H. Wemer, Discriminator Algebras, Studien zur Algebras und ihre Anwendungen, Band 6 (Akademie-Vedag, Berlin, 1978). [8] A. Macintyre, Model completeness for sheaves of structures, Fund. Math. 81 (1973) 73-89.
83
Q U A N T I F I E R E L I M I N A T I O N IN DISCRIMINATOR VARIETIES Francoise POINT Universit~ de l'Etat ~ Mons, Facult6 des Sciences, Mons, Belgique Communicated by Y. Gurevich Received 16 July 1984; revised 21 February 1985
0. Introduction In this paper, we want to classify those classes of elements of a discriminator variety V, which admit quantifier elimination (q.e.) in terms of classes of simple elements of V. Discriminator varieties come up in decidability questions for certain varieties. Burris and McKenzie [3] have shown that any finitely generated decidable modular variety of finite type is the product of a finitely generated discriminator variety and of a decidable abelian variety. Our starting point is a representation theorem due to Bulman-Fleming, Keimel and Werner: Any element of a discriminator variety is isomorphic to a boolean product of subdirectly irreducible members of that variety [7, Theorem 4.9, p. 56]. Then in Section 1 we analyze transfer of q.e. in finite direct products with projector. Here we could apply the Feferman-Vaught theorem on direct products, but we choose to give a direct proof of transfer of q.e. from the factors of a finite direct product to the product and vice versa. In Section 2, we prove our main result. Given a quantifier diminable element of a discriminator variety, we derive stringent conditions on its base space and on the class of its stalks. For proving that these conditions are indeed sufficient, we extend the result proved by Boffa and Cherlin in [2] to a non-compact situation. Finally we consider the effectiveness of the given procedures and we classify the classes of dements of discriminator varieties, with q.e.
Preliminaries To begin with, we recall some logic terminology. A basic formula is either an atomic formula or the negation of an atomic formula. A CB-formula is a conjunction of basic formulas. An open formula is a boolean combination of atomic formulas. 0168-0072/86/$3.50 (~) 1986, Elsevier Science Publishers B.V. (North-Holland)
84
F. Point
A positive formula is built up from atomic formulas using only the connectives A, V and the quantifiers V, =1.
A primitive formula cp(x) is an existential :iy 1 • • • :iynO(y, x) where 0(y, x) is a CB-formula.
formula
of
the
form
We need certain notions related to quantifier elimination in classes of .Y-structures. These appear in [2] and [1]. Let C be a class of -y-structures. (1) C admits quantifier elimination (q.e.) itt for any formula q ~ ( x l , . . . , xn) there exists an open formula O ( x l , . . . , x,) such that the formula Vx (q0(x) O(x)) holds in any element of C. (2) C admits connective elimination (c.e.) iff for any open formula O ( x l , . . . , x~) there exists an atomic formula % ( x ~ , . . . , x~) such that the formula Vx (O(x)<--~%(x)) holds in any element of C. (3) C admits strong quantifier elimination (s.e.) iff C admits q.e. and c.e. (4) C admits positive quantifier elimination (positive q.e.) iff for any positive existential formula q0(Xl,..., x~), there exists an open formula O ( X l , . . . , xn) such that the formula Vx (cp(x) ~ O(x)) holds in any element of C. Note that to show that C admits q.e., we need only to consider primitive formulas q9 with one existential quantifier. Now we recall some facts about discriminator varieties. Let V be a discriminator variety, i.e., an equationally defined class s.t. there is a term t(u~, u2, u3) with the class of subdirectly irreducible members of V satisfying
(,)
t(Ul , U2, U3) = U4 ~ ( u 1"- U2 • U3 = u 4 ) V (U l::t=u 2 & U 1"- U4).
If the equivalence ( * ) holds in an element of A of V, then t(-,-, .) is called a discriminator for A. For any non-trivial element of V (i.e. those containing more than one element), the properties of being simple, subdirectly irreducible are equivalent to the fact that t(.,., -) is a discriminator [7, Theorem 2.2, p. 22]. Before stating the boolean representation theorem for any element A of V, we define the support of an element a of A. From now on we suppose that the language -Y of V contains at least one constant 0. So *Y will always denote a language of algebras containing a ternary function t(., -, .) and at least a constant 0. Then the support of a in a sub-direct representation II~iAi of A by subdirectly irreducible members Ai of V, is the subset {i e l I Aj~a(i):/:O}. The set of supports (with respect to a fixed subdirect representation of A) is the domain of a generalized boolean algebra and if we choose another subdirect representation, we will obtain isomorphic generalized boolean algebras [4, Lemma 9.1]. Theorem 0 [7]. Let A be a non-trivial element of V. Then A is isomorphic to the
-y-structure F~(X, Ux~xAx) of all sections with compact clopen supports of a locally boolean sheaf (X, Ux ~x A x) of simple elements of V.
Quanttfier elimination in discriminator varieties
85
Let r be a n-tuple of elements of A. Let q~ be a formula. We call the subset {x • X IAx ~ tp(r(x))} = [~(r)] the truth value of tp. Note that the truth value [r~:/:r2] is the support of the element t(n(rl, r2, O, rl), O, n(rl, rE, O, r2)), where n(ul, UE, Vl, rE) is defined as follows:
n(ul, UE, Vl, rE) = t(t(ul, U2, Vl), t(ul, U2, rE), rE). 1. Direct product with projector Let V be a discriminator variety. Let 1-li~xA~ be a direct product of elements Ai of V. Definition 1.1. A projector for 1-L~zAi is a sequence of terms (Pi)~z satisfying the following: for any element a = (a(i))i,z of the product, ~a(j) p,(a)(j) = (0(j)
if i = j , if i 4:j.
If the terms (P~)i~z are ~ O C-terms where C is a finite set of new constants, we say that the direct product IIi~zAi has a projector in ~ U C. In the following propositions, we show transfer of quantifier elimination from the factors of a finite direct product with projector to the direct product, and vice versa. In Section 2, we will see moreover that any q.e. element of V is isomorphic to a certain finite direct product with projector.
Definition 1.2. For (Pi)i~z a projector, we define on formulas q9 the transformation q9~',). Let 0 be the atomic formula s ( x , , . . . , x n ) = 0 where s is a term 0 ° ' ) = pi(s(x))=O. Let 0 be a boolean combination of atomic formulas Ok, say ~/ ( A Ok ^ /~ -nOk,). Define 0~, ) as V ( A O~ ') ^ / ~ ~0~')) • Let q9 be an existential formula of the form :ly 0(y, x), where 00', x) is an open formula. Define q0°') as :ly 00',). Notation 1.3. (i) If A , , A2 are two elements of V and r = ( r l , . . •, rn) is a n-tuple of elements of A1, we denote by r* the n-tuple ((r~, 0 ) , . . . , (r~, 0)) of elements of A~ x A2. (ii) If E is a finite set of sentences, we denote by A E the conjunction of elements of E. (iii) C = {c} is a finite set of new constants with respect to ~ and for 1 ~< i ~< 2, Ci = {el} is a finite set of new constants such that C / = {c(i) I (c(1), c(2)) e C}. In L e m m a 2.4, we will deal with the case where C = C1 and C-,z= (0}. L e m m a 1.4. For any open formula O(xl, . . . , xn, c) of &e U C, there exists a finite
86
F. Point
set Po of CB-sentences X of ~ U C2 and corresponding CB-formulas Ox(x, cl) of , ~ . U C 1 such that for any A1 and A2, Vrl,...,rneAl(AlxA2gO(r*,c) iff Vx~e (A, ~ Ox(r, cl) and A2 ~ X)).
Proof. First we put 0 in a disjunctive normal form Vi~z 0~, where each 0~ is of the form AI~L, tt = 0 A Ak,r, Sk :/: O, where tt and Sk are 3?-terms. (i) From each Oi, we construct a set of formulas as follows. For each subset K of Ki, let %K be Al~Zi tl = 0 A Ak~r Sk :/: O. (ii) Let r be a n-tuple of elements of A1. We have for any i belonging to I:
(,)
A1xA2~Oi(r*,c)
iff
V (Al~xr(r, cl) andA2~xr,-r(O, c2)).
K~_Ki
We have simply used the fact that an element in A, x A2 is not zero iff one of its coordinates is not zero. (iii) Let Po be the set of all open sentences Xr(0, c2) for all subsets K of Ki and all i of I. For each X of Po, we denote by Ox(.) the formula Xr,-r(', cl) where xK(O, C2) is X. With this notation, applying (ii) we get:
A1XAz~O(r*,c)
if
V (Al~Ox(r, cl) andAz~X).
[]
x cPo
In the following two propositions, we will use the same notation as in L e m m a 1.4. In addition, we suppose that A 1 X A2 is a direct product with projector in ~UC. Let (Pl, P2) be a projector for A1 x A2, where p~, P2 are ~ U C-terms. Note that p , ( y ) = t(y, P2(Y), 0). L e m m a 1.5. Let AI x A2 be a direct product with projector (p~, P2). Given any
primitive (respectively open) formula ¢p, there exists a primitive (respectively open) formula q90") such that: Ai~q)(r(i))
iff
A 1 xA2~qg(P0(r),
where r is a sequence of elements from A 1 x A2. Proof. It suffices to note that for any term s and any n-tuple r of elements of A, A i ~ s ( r ( i ) ) = O ¢ - - > A l X A 2 ~ p i ( s ( r ) ) = O , for 1<~i<~2. [] Proposition 1.6. Let A1 x A 2 be a direct product with projector (pl, P2). Then
Ax x A2 admits q.e. in .~ U C implies that A~ and A2 each admit q.e. in .~ O (71
(respectively in ~ U Cz). Proof. Given any primitive formula q~(x), we find an open formula ~,(x) such
Quantifier elimination in discriminator varieties
87
that (i) A~ ~ Vx (cp(x)~->~p(x)),
(ii) ~p(-) depends on the CB-sentences satisfied by A2. By Lemma 1.5~ for any n-tuple r of elements of A~, we have:
A~ ~ ~(r)
i~ A ~ ~ ) ( r * ) .
Since A admits q.e. in ~ U C, there exists an open formula O(x, c) such that for any n-tuple r of elements of AI we have: Al~qg(r)
iff A~O(r*,c).
By Lemma 1.4, there exists a set Po of CB-sentences of ~t.JC2 and CB-formulas Ox(.) of ~ U C1, where Z belongs to Po such that for any n-tuple r of A1,
A~O(r*,c)
V (Al~Ox(r) andA2~z ).
iff
z ~Po
Let E be the set {Z e Po [ A2 k Z}. Then g(x) is either V Ox(x),
if E is not empty,
or
if E is empty.
zeE
0 #: O,
[]
Proposition 1.7. If Ai admits q.e. in ~ tJ Ci, 1 <~i <<-2, then A admits q.e. in LUC. Proof. Let qg(x) be a primitive formula, ~(x) = 3yAtj(y, x) = o ^
/~ Sk(y, x) ~ O,
k~K
where t/, Sk are .T U C-terms. For any subset L of K, let XL be
3yAtj(y,x)=O ^ Ask(y,x)~O. j~J
k~L
For any n-tuple r = (r(1), r(2)) of elements of A1 x A2,
A~qg(r) iff
V (AI~ZL(r(1)) andA2~Zr_L(r(2))).
L~_K
For any subset L of K, let 0~(-) be an open formula equivalent in Ai to XL(')Then we have for any n-tuple r of elements of A:
A~q~(r) aft
V (Al~O~(r(1)) andA2~O2(r(2))). L~_K
To finish the proof, we apply Lemma 1.5. For any subset L of K, we have that
AigO~(r(i))
iff AgO~o'i)(r), 1 <~i<~2.
[]
88
F. Point Collecting Proposition 1.6 and 1.7, we have:
Theorem 1.8. If A1 x A2 is a direct product with projector (p,, P2), then A1 x A2 admits q.e. in . ~ U C iff A1 and A2 admit q.e. in .~UC1 (respectively in
LUG).
[]
2. Quantifier eliminable subdasses of a discriminator variety We preview the results up to Proposition 2.6 in this paragraph. First we recall a result of Burris and Werner that any open formula is equivalent to a basic formula and this uniformly in any simple elements of any discriminator variety V. Then we look at the relationship between the satisfaction of an open formula in a stalk Ax of an element A of V and the satisfaction of the same formula in A(U) x A ( X - U ) , where U is a neighborhood of x and A(U) (respectively A ( X - U)) is the L-structure whose domain is the set of restrictions of elements of A to U (respectively to X - U). We use the facts that each stalk Ax is the direct limit of the direct system {A(U) [ U is a clopen neighborhood of x}, and that there is for any open sentence Z, a neighborhood U of x such that for any U1, U2 included in U, we have
A ( X - UI) ~x iff A ( X - U2)~X. Further, we show that the class of stalks of any q.e. element of V admits positive q.e. Let us fix some notation. Let A be an element of V and F~(X,Ux~xA,) its representation. (I) For any ~-term t(x), any element r of A and any subset Y of X "t(r) - r/Y (30/X - Y" means that t(r) equals r on Y and 0 elsewhere. For ease of notation, we will denote it by r/Y U O. (2) For any n-tuple ( r l , . . . , rn) of elements of A, we denote by r. the n-tuple ( n ( u , O, O, r,), . . . , n ( u , O, O, rn)) = (r,/[u =t= 0] U 0 , . . . ,
r~/[u =/= 0] U 0).
(3) Capital U is a variable for compact clopen set. For the ease of notation, given an n-tuple r of elements of A~, we often denote by r* any n-tuple of elements of A which satisfy r*(x) = r. Proposition 2.1. Let ~ be a class of simple elements of a discriminator variety.
Then any non-positive (positive) CB-formula is equivalent to the negation of an 2tomic (atomic) formula and this uniformly in all elements of ~. Moreover, if there is a closed term different from 0 in all elements of ~, then ~dmits c.e. Proof. See [4, Lemma 9.1]. Note that this implies that any open formula is ~quivalent to a basic formula, uniformly in cg. []
Quantifier elimination in discriminator varieties
89
Proposition 2.2. Let V be a discriminator variety with O. Then every CB-formula is equivalent to a CB-formula with exactly one atomic conjunct:
t = 0 A A s j * 0. j~J
Proof. Given a CB-formula A i E i t i = 0 A A j e j S j :[= 0 we assume I =g=~ (otherwise add 0 = 0). By Proposition 2.1, in the class of the simple elements of V we have A~EI t~ = 0.-> t = 0. Let A belong to V and r to A. A ~ Ati(r) iEl
=
0 A Astr ) * 0 j~J
iff
[Ati(r)=O]=X
iff
[t(r)=0]=X
and
and
itI A ~ t(r) A Asj(r) jEJ
[si(r) 6:01 * ~
[sj(r)~=0]#=~
O.
forj~J
forjeJ
[]
In the next two lemmas, we arae going to look at how to express in A that an open formula is satisfied in a stalk A~.
Lemma 2.3. For any open formula O(xa, . . . , x~) and for any n-tuple (rl, • • •, r~) o f elements o f Ax, 3Uo (x ~ Uo A V U (x ~ U ~_ Uo)"->A ( U ) ~ O(r/U)) iff A~ ~ O(r).
Proof. We apply exercise 5.2.24 of [5] (see also [6]). A sentence tp is preserved under direct limits of ~-structures iff tp is equivalent to a sentence of the form m
AVI1,
i=1
" " " , I s (~li-"> 3 y ~ " " " y t Oi),
where the IPi and 0i are open positive formulas. We put s = t = 0 and we add to the language ~ new constants c l , . . . , c~ interpreted in A~ by r ~ , . . . , rn, respectively by rl/U, . . . , r,,/U in A ( U ) . Then A~ is still the direct limit of the direct system of ~ U {Cl, •. •, Cn}-Structures A ( U ) , where U is a clopen compact neighborhood of x. Therefore we can apply the result from Keisler. [] In the next lemma, c, a new constant added to L, will be interpreted in Ax by a non-zero element and in A by an element u such that u(x) = c.
Lemma 2.4. For any open formula O(xl, . . . , xn+O, there exists an open formula O'(xl, . . . , xn+O such that for any x o f X and any n-tuple r o f elements o f A , we have:
(.)
:IUo (x e Uo A VU (u ¢ A A u ( x ) = c A [u q:O] c_ Uo--~ a ~ O(r,,, u))) iff
Ax ~ O'(r(x), c).
90
F. Point
Proof. Let U be a clopen subset of X. We consider A as the direct product A(U) x A ( X - U). Let c be a new constant added to Le such that the support of cA = u is U. Applying Lemma 1.4 to the open formula O ( X l , . . . , x,,, c), we get that there exists a finite set Po of CB-sentences X of L# and CB-formulas Ox(x ~, . . . , x,,, c) of &# t3 {c} such that for any n-tuple r of elements of A,
A ~ O(ru, u)
iff
V (A(U) ~ Ox(r/U, u/U) and A ( X -
U) ~ X).
x cPo
Given any element x of X, we are going to prove that there exists a neighborhood U, of x such that for any clopen compact subsets U1, U2 strictly included in Ux,
A ( X - U1) ~ X iff A ( X - U2) ~ X, for any X e Po. We proceed as follows: Let Fo be the set of truth values of the basic sentences occurring in each element of Po. Taking complement and finite intersections of these truth values, we get a partition of X whose elements Vii are of the form [ti = 0] or [ti 4: 0], 1 ~
r = O A ASk4=O, keK
where r and S k are closed ~-terms (see Proposition 2.2). If A ( X - U)~I, then A ( X - U)~ r = 0. Thus [r = 0] has a non-empty intersection with V~ and so [ r = 0 ] = X . Similarly for any k in K, A ( X - U)~Sk~O. SO either [Sk V~0] is disjoint from V~or [sk #=0] is included in V/. The assumption on U implies in the first case that A ( X - V ~ ) ~ s k 4=0 and in the second case that V~ equals [Sk ~ 0]. NOW we are ready to construct a formula 0'(.) such that the equivalence ( , ) holds. Let Ov,(', c) be 0 =/=0,
V Ox(', c),
if E~ is empty, otherwise.
Now, for 1 ~
or,(-, c) ^ t, = o,
if v / = [t, = o]
or
if E =
Quantifier elimination in discriminator varieties
91
Set O'(.,c) equal to Vm=10"(., c). Then we have 3 U o ( x e U o ^ V U ( u e A ^ u(x) = c ^ [u 4=O]~_ Uo-->A ~ O(r.,, u))) iff A~ k O'(r(x), c). Indeed, let x be any element of X. As the set { V~[ 1 <~i ~
Ax~VO'(r(x),c) i=1
iff Ax~Oi'o(r(x),c)
iff Ax~Ov, o(r(x),c ).
By Lemma 2.3,
Ax ~ Ov,o(r(x), c) if[
3U0(x ~ Uo ^ V U (x ~ U c_ Uo-->A(U)~ OV,o(r/U, u/U))),
where u is an element of A, whose support is U and such that u ( x ) = c. By definition of the subset E~ of Po, for any U included in V/0, we have:
V
(A(U) ~ Ox(r/U, u/U) and A ( X -
x~,
U) ~ X) iff
V (A(V) O (r/U, u/U)).
x ~F-~
Therefore by Lemma 1.4, we have for any n-tuple r of elements of A
3Uo (x e Uo A V U (x e U ~_ Uo-e A ( U ) ~ Ov,o(r/U, u/U)) )
iff
3U0 (x ~ Uo A VU (U e A a U(X) = C A [U ~ 0 ] c Uo'> A ~ O(ru, u))).
[]
Now we are ready to prove one of our main results on the class of the stalks of an q.e. element of V.
Proposition 2.5. I f an element A o f V admits q.e., then the class of its stalks admits positive q. e. Proof. By Proposition 2.1, the primitive positive formulas in the class of the simple elements of V are of the form 3y sO', z) = 0 where s is an ~-term. We will assume that y is a 1-tuple. By definition of the discriminator t(-), for any x of X, for any non-zero element c of Ax and any n-tuple r of elements of Ax, we have:
A~ ~ 3y s(y, r) = O~-> (1)
3Uo(xeUoAVU(u~AAu(x)=cA[u#O]cUo --->A ~ :ly t(0, u, s(y, r,,)) = s(y, r.,))).
By hypothesis, A admits q.e. Thus there exists an open formula O ( Z l , . . . , zn+l) equivalent in A to 3y t(zn+l, O, s(y, zl, . . . , z~)) = s(y, zl, , . . , zn). By Lemma 2.4, there exists an open formula O ' ( z l , . . . , z,+l) such that for any x of X and for any n-tuple r of elements of A,,:
(2)
3Uo(x e Uo^ Vu (u e A A u ( x ) = c ^ [u:/:O]c_ Uo--~A~O(r.,, u))) ~-~Ax ~ O' (r, c).
F. Point
92
Now, we are going to show for any x of X that we can get rid of the non-zero element c of Ax. Indeed, we have the following equivalence:
Ax ~ 3y s(y, z) = O
~">mx~[(:iys(y'z)=O^~/zi=/=O (3) v
=lys(y,O)=O^ A z ~ = O A s ( o , o ) ~ o i=l
v(3ys(y,O)=O^A(zi=O^s(O,O)=O))].i__l Applying equivalences (1) and (2), together with (3), we obtain for any point x of X:
Ax ~ 3y s(y, z) = O ~-->Ax~
O'(z, zi) ^ zi:/:O) v
( o'(o,s(O,O))^Azi=O^s(O,O) n ) O i=1
v
(A
z~ =0
As(0, 0)=0
)]
[]
The next proposition will show that any q.e. element of V is a certain finite direct product with projector. The factors have a base space w h i c h either is discrete with one or two points or has no isolated points. Thus by Propositions 1.6 and 1.7, characterizing q.e. elements of V reduces to characterizing (1) those with compact base spaces either discrete with exactly two points or without isolated points, (2) those with non-compact base space without isolated points. In cases (1) Boffa and Cherlin [2] have given sufficient conditions on the class of stalks for the global structure to admit q.e.
Proposition 2.6. If A admits (effective) q.e. then there is an (effective) procedure to decompose A into a finite direct product with projector (I[~=oAi, (ti)i=o..... n) such that (i) Each
admits (effective) q.e. (ii) The base space Xo of Ao has no isolated points and the base spaces Xi of the Ai, i >~1, are discrete with one or two points. Moreover, if Xo is compact and if any X~'s, i >~1, contain two points, then they form the support of closed terms. A i
Quannfier elimination in discriminator varieties
93
Proof. Step One. First we prove (a) and (b). (a) X has finitely many isolated points. (b) Moreover, the largest discrete subspace of X is the union of discrete subspaces X1, • • •, Xn of X, each of which contain at most two points. Moreover, for each X~, there is an &e-term ti with 6(r) = r/Xi t2 O, for any element r of A; and in case X~ contains two points, ti(r) = n(ti, O, O, r), where 6 is a closed &e-term. Note that with these &e-terms tl(.), . . . , t,(-), we can construct to('). Let
(*)
rl=n(r, tl(r),O,r);
ri+a=n(r,t~(r),O,r),
l<~i<~n.
Then to(r) = rn+l. Proof of (a). Let O(z) be an open formula expressing that the support of z is minimal, i.e., the support of z is a singleton (O(z) is equivalent to Vr (r =/=0 and t(O, z, r)=O--.t(O, r, z)=0). By L e m m a 1.4, there exists a finite set Po of CB-sentences of L such that for any element u of A whose support is U we have:
A ~ O(u) iff
V (A(U) ~ Ox(u ) and A ( X -
x ePo
U) ~X).
Suppose that there is a minimal clopen subset U. For any element r of A whose support is U, A ~ O(r). Let p v be {2' e Po IA(U) ~ 3u O~(u) and A ( X - U) ~ X}. Let X be of the form (tx = 0 ^/~J~Jx sj =/=0) with tx and sj closed L-terms (see Proposition 2.2). There are two possibilities: (1) There exists X belonging to Pov such that [tx = 0 ] ¢ : X , in this case U= [tx 4=0]. (2) For all X of pv, A ~ tx = O. In the latter case, we consider for any element r of A the following truth values for X e poV: Jr=/=0 ^ Ox(r) ^ A s j = O ^ j elo
/~ sj¢O] = [vx(r)] ,
Jelx -Jo
where Jo = {j eJ x IA(U)~sj=O } and 1:x is a basic formula (see Proposition 2.1). Note that for any element r of A whose support is U there exists X in Pov such that
[,x(r)l = [r/U ¢O]. Two subcases may arise: (2') For any element r of A, I,.Jx~e~, [vx(r)] is included in U, which implies that it equals to [r/U ¢ 0]. (2") There is some r and some X e Pov such that [vx(r)] is not included in U. In this latter case, we are going to show that there exists a closed term whose support is a discrete space with two points which includes U. Let v be r/[~x(r)] t'l ( X - U). We show that the support of v is minimal. First A([v ¢ 01) ~ Ox(v ). Then
A(X-[v¢O])~
A s j :/: 0 I EJo
94
F. P o i n t
(for [/~J~Jo sj = O] _~ U U [v 4= O] and A ( X - U) ~ / ~ 4
sj :/: 0 since X belongs to
Pg'); A(X-[v=/:O])~
A
si:#O
j E.lx - Jo
(because A ( U ) ~ AJ~4-Jo sj :/: 0 and U tq [v 4= 0] = 9)Let u0 be an element of A whose support is U such that A ( U ) ~ Ox(uo)O(" ~ PoV); and let W
j'Uo on U, [ v onX-U.
Since A([x #:01) ~ Ox(w ) (0 x is a CB-formula), A ( X - [ w : k O ] ) ~ Aj~joSj:/=O and A ~t O(w), there exists j ~ Jx - Jo such that [sj :/: 0] ~_ [w 4= 0]. Since for every j e Jx - Jo, [sj 4=0] _~ U t.J [v 4= 0], we have that [sj :/: 0] = [w 4= 0]. To summarize, any minimal clopen subset U is determined by 0 in the following way: (we use the above notation) -C ase (1): for one Z ~ Pov, U = [tx = 0]. -C ase (2"): for some Z e Pov and some j e Jx - J o, [si 4=0] is a discrete space with two points which includes U. -C ase (2'): for any element r of A, [r/U:/:O]=Ux~euo[rx(r)], where r x is a basic formula constructed from Z and 0 x. From the rx's, we can construct a basic formula r such that [r(r)] = [_Jx~e~[rx(r)] (see Proposition 2.1). Since there are only finitely many X, there are only finitely many minimal clopen subsets. Proof of (b). Suppose the discrete subspace Xk, 1 <-k <~n, is of the form Is 4= 0] where s is a closed term. Then tk(r) = n(s, O, O, r). Suppose that the discrete subspace Xk is the truth value of a basic formula r. If r is of the form s(-) = 0, then tk(r) = n(s(r), 0, r, 0). If the basic formula r is of the form s(.) #: 0, then tk(r) = n(s(r), O, O, r).
Step Two. Let Xo be the complement in X of the set of isolated points. We will show that Xo is compact iff it is the support of a closed &P-term. Let O(z) be an open formula expressing [z :/: 0] = X, say
O(z) = V tk(z) = o ^ A skAz) * o. jeJk
k
If A ~ O(r), say A satisfies the disjunct
t(r) = 0 A A sj(r) :/=O, J
then we claim either (i) X is finite or (ii) [t(0) = 0] is finite. Proof of claim. If X is infinite, let J = { 1 , . . . , n} and choose X l , . . . , xn e X such that sj(r(xj)) 4=O. If [t(0) = 0] is not finite, then we can find a clopen subset M of [ t ( O ) = O ] - { x l , . . . , x n } and create l e A such that [ f = r ] = X - M , [f = 0] = M. But then A ~ t(f) = 0 ^ / ~ sj(f) ~ 0, which contradicts the fact that [f 4: 0] 4: X. Thus [t(0) = 0] is finite, proving the claim.
Quantifier elimination in discriminator varieties
95
In case (i), Xo = t~, and in case (ii) we have a closed term t(0) whose support contains Xo. [] Before stating the corollary, we recall a notation used in [2]. If ~ is a class of ~-structures, then (i) c¢2 is the class of all direct products of two elements of c¢. (ii) ~ n t o r is the class of all structures of sections of sheaves of elements of c~ over a Cantor space. Corollary 2.7 (to Propositions 2.5 and 2.6). An element A of V admits q.e. iff A is a finite direct product with projector such that for each of its factors A i , 1 <~i ~ n, one of the following holds: (1) Ai is a simple element of V which admits q.e. (2) Ai belongs to ( ~i)2 where ~i is a class of simple elements of V admitting s.c. (3) A i belongs to (~i)cantor where ~i is a class of simple elements of V admitting s.e.
(4)
is an element of V admitting q.e., whose base space is not compact and has no isolated points. A i
Proof. (--->) Applying Proposition 2.6, we get that any quantifier eliminable element of V is a finite direct product with projector: [I7=1Ai with specific base spaces. Applying Proposition 1.6, we know that each factor of A admits q.e. Thus, if the base space of the factor Ai is either discrete with one point, or non-compact without isolated points, then we are in Case (1) or (4). If the base space of the factor Ai is compact either discrete with two points or without isolated points, then we know that it is the support of a closed ~-term ti. Thus the Ai have a closed term ti with ti ~ 0, so the class ~i = {Ax [x e X/} admits c.e., by Proposition 2.1. Since by Proposition 2.5, the class of stalks of A admits positive q.e., the class ~gi admits s.e. (~---) By Proposition 1.7, if each factor of a direct product A with projector admits q.e., then A admits q.e. Thus we have to prove that, if ~ is a class of ~-structures which admits s.e., then the classes ~2 and ~ t o r admit q.e. This is exactly the Boffa-Cherlin theorem in [2]. [] This corollary reduces the problem of characterizing the elements of V which admit q.e. to characterizing those whose base space is not compact and has no isolated points. This will be undertaken in Propositions 2.8 and 2.9. In Proposition 2.8, we will prove that the class of the stalks of such elements of V admits not only positive q.e. (Proposition 2.5) but that a certain kind of elimination holds in the class with respect to negative existential formulas. In Proposition 2.9, we will show the sufficiency of these conditions.
Proposition 2.8. Suppose A, an element of V, admits (effective) q.e. Suppose that
96
F. Point
its base space X has no isolated points and is not compact. Then A has the (effective) local negative elimination property, i.e., for any non-positive primitive formula cp there exists an open formula 0 such that for any n-tuple r of elements of A: [tp(r)]~
~
[0(r)]~.
Moreover, the procedure to find 0 is effective if the quantifier elimination procedure in A is effective. Proof. For convenience of the reader, we prove the proposition for 1-tuples. To prove the proposition, we need only to consider non-positive existential formulas tp of the form :ly t(y, .) ~ 0, where t is an L-term (see Proposition 2.1). In case A~qg(r), we have [ q g ( r ) ] ~ - > [ 0 = 0 ] ~ . So from now on we can assume that A It Vr tp(r). The open subset B = [tp(0)] is included in a compact subset of X. [If not, then for any element r of A the subset [r = 0] would have a nonempty intersection with B. As [r = 0] f7 B c_ [qg(r)] one would conclude [qg(r)] :/: ~, so A ~ tp(r). This contradicts the assumption that A lt'Cr tp(r). ] We call f a restriction of r if (i) [ f ~ 0 l ~ _ [ r ~ 0 ]
and
(ii) ~/[f=/:Ol=r/[f~O].
Let AkEr Ok be an open formula equivalent to tp in A, where Ok= V 6 = 0
vsk~:0.
iElk
(For the fact that Ok can be put in this form see Proposition 2.2). We want to prove that there exist k in K such that [Sk(0)#: 0] is equal to the closure/} of B. To prove this fact, we claim that if for any given k in K, (1)
either [Sk(0) ~ 0]~/~,
(2)
or
[Sk(O)~ O] ~/~,
then for any element r whose support includes B there exists a restriction f whose support also includes B such that all restrictions of f whose supports include B satisfy Ok. Suppose the claim is true. Then the assumption that (1) or (2) holds for every k in K leads to a false conclusion, namely A ~'¢r tp(r), as follows. First note that A~qg(r) for all r with [ r = 0 ] N B # = ~ for then ~[r~O]f7B~_[qg(r)] yields A ~ qg(r). So let us now consider an element r of A whose support includes B. By assumption there exists ~k a restriction of r such that all restrictions of ~k whose supports include B satisfy Ok. Let ~ be the restriction of r such that [~ ~ 0] = ['~k~K[rk:~:0]. So we have A ~ AkErOk(r). As [ F = 0 ] A B = ~ , one has [tp(f)]_= [ f ~ 0 ] . Thus for x e [tp(f)] we have f ( x ) = r ( x ) , using the fact [f #: 0] c_ [r #: 0] , and hence x e [qg(r)]. This suffices to prove A ~ qg(r), and the contradiction follows.
Quantifier elimination in discriminator varieties
97
Thus, if we prove the claim, then we can conclude that/~
= [Sk(0 ) ~f: 0] for some
kinK. Proof of the claim. Let r be an element of A whose support includes B. (1) Suppose that [Sk(0)S0]$/~. Let N be a non-empty clopen subset of [Sk(O) :#: O] disjoint from/~. Let f be equal to zero on N and to r elsewhere. Then A ~Sk(F)S0 and If $ 0 ] _~/~. Moreover, all restrictions of f satisfy Ok, since their supports are disjoint from N. (2] Suppose that [Sk(0) S 0] ~ B. Then there exists x0 in B - [sk(0) S 0] since [Sk(0) S 0] is a closed subset. Let ro be the restriction of r, whose support is [Sk(0) S 0]. Since [ro = 0] N B S ~, A ~ qg(r0). Either (i) Sk(ro)S 0 and this implies that there exists an x in [Sk(0)S0] such that sk(ro(x))SO. Since ro(x)=r(x), we have sk(r(x))SO. ThUS Sk(F(X))*O for any restriction F of r whose support contain B. (Note that [F S 0] ~ B implies that [ f S 0 ] _ D B ~x.) Or (ii) There exists i e Ik such that ti(ro)=0 and so the set ~ = {i e Ik: [ti(0) :/: 0] ~ /~} is not empty. Either there exists i e ~ such [ti(r) S 0] N B = 9, or for all i e ~ , [ti(r) ~ 0] N B ~ ~. In the first case, for any restriction f of r whose support includes B, equal to zero on [ti(r) ~ 0] and to r elsewhere, we have that A ~ ti(F) = 0. In the latter case, for any i in ~ let Ni be a non-empty clopen subset in [t~(r) #:0] A B. If all N~ are not included in [Sk(0)S0], pick a point z in some Nj - [Sk(0) ~ 0] and choose in each N~ a subset Mi not containing z. Let rl be equal to r on [Sk(0) :/: 0] f3 I,.Ji~zOMi and equal to zero elsewhere. Since [rl = 0] N B S ~, we have that A ~ qg(r~). In particular, A ~ 0k(rl). Therefore, either A ~Sk(rl) SO and so by the same argument as in (i) above, the claim is verified or A ~ ti(r~) = O, so i e ~ (indeed the support of r~ is included in/~) and since on M~, !'1 equals r, this leads to a contradiction. Now we are in a position to finish the proof. The class of stalks of a q.e. element of V admits positive q.e. So let ~p(s) be an open ~-formula equivalent to 3y n(t(y, s), O, s, O)= O. Let Sko(O) be an ~ - t e r m such that [Sko(O) ~/:0]--n. Let r be an element of A. For x e X
A~ ~ 3y tO,, r(x)) q~0 <-->A~ ~ (3y t(y, r(x)) q: 0 ^ r(x) * O) v (=iyt(y, r(x))SO A r(x)=O). Thus [qo(r)] ~ ~<-->[=ly t(y, r) #: 0 ^ r * 0] ~ ~ or [3y t(y, r) S 0 ^ r = 01S <-->[W(r) ^ r#:0] q:0 or [=ly tO', r) q:0 ^ r=O] q~O Now [3y tO', r) #=0 A r =01 #: ~ <-->B rl [r = 0 1 S ~ (-->/~ N [r = 01 :# I~ <-->[Sko(0) q: 01CI [r = 01 q: I~
98
F. Point
Thus [qg(r)]*~
iff [(~(r) A r , O ) V (Sko(O),O A r = O ) ] , ~ .
The terms Sk(0) for which [Sk(0) 4=0] =/~ are those among the terms Sk(0), k ~ K, which satisfy the open formula X(u) equivalent in A to
(*)
Vy ([t(y, O) #: O] ~ [u :/: 0]),
(1)
Vv ((Vy [t(y, O) 4=O] ~ [v :/: 0])---> ([u 4= O] _ [v 4= 0])).
(2)
Indeed Vy [t(y, O) 4=O] ~ [u 4=O] is equivalent to [3y t(y, O) 4=O] ~ [u 4= O] which holds iff / ~ = [ u 4 : 0 ] . So (2) says Vv(B=_[v:/:O]--->[u:/:O]c_[v:/:O]). One can rewrite ( * ) as Vy n(u, O, t(y, 0), O)= 0
and
Vv (Vy n(v, O, t(y, 0), O) = 0--> n(v, O, u, O) = 0).
[]
In the setting of discriminator varieties, the next proposition generalizes to the non-compact case the following result of [2]: "If a class c~ of .T-structures admits s.e., then Cgca~toradmits q.e." Proposition 2.9. Let ~ be a class of simple elements of V and consider a class ~ of
elements of V whose stalks belong to ~ and whose base space is not compact and without isolated points. Suppose (1) ~ admits positive q.e., (2) ~ has the local negative elimination property. Then ~ admits q.e. in .T. Proof. Let q9( Z 1 ,
. . • , Zn)
be a primitive formula of the form:
3yAt,(y, z)=O A Asj(Y, z)•O, i~l
j~J
where tj and sj are .T-terms. Let A be an element of ~ and let r be a n-tuple of elements of A. First, we express that the existence of a global solution to qg(r) in A is equivalent to the existence of local solutions for certain existential formulas in
{A,,lxeX}. A ~ ¢p(r) *->Vx
e
X A x g 3 y A t i ( y , r(x)) = 0 i~l
(1)
^ A3xjeXA~,~ 3yAt~(y, r(xj)) = 0 ^ s~(y, r ( x j ) ) * O. j~J
iEl
(See [8, proof of part 1 Theorem 2].) The implication (--->) is obvious as A is a subdirect product of the A . , x e X. To prove the implication (*-), we are going to build from local solutions, a global one. Let yj be an element of A such that the truth value [Ai~ ti(yj, r) = 0 ^
Quantifier elimination in discriminator varieties
99
sj(yj, r) ~ O] contains the point xj, for each j ~ J. As X has no isolated points, we can choose non-empty disjoint clopen (compact) subsets Uj, j E J, included in each of the above truth values. Let E be the following compact clopen subset of X: n
E = U [rk =/=O] U U[t,(O, O) #: O] - UU/. i~l
k=l
j~J
For each x E E, let y~ be an element of A such that [Ai~1 ti(y~, r) = 0] contains x. This set of truth values forms a clopen covering of E. As E is compact, there exists a finite sub-covering which can be refined to a finite covering { Vm [ m ~ M} of E. Let Ym be elements of A such that [/~i~I ti(Ym, r) = 0] = Vm, m ~ M. Now we define a global solution a for q0(r) as follows:
a/U/=yj/U/,
j EJ,
a/VmNE=ym/VmNE,
meM,
a/X-(UUif')E)=O. \jEJ
It is straightforward to verify that A ~ Ati(a, r) = 0 ^ Asj(a, r) =/=O. i$l
j~J
Second, we show that the existential local conditions in {A~ Ix ~X} are equivalent to open formulas, whether the existential formulas are positive or not. (2) By assumption, the existential positive formula 3y Ai~iti(y, z ) = 0 is equivalent in any element of :£ to an open formula. This open formula is equivalent, by Proposition 2.1, either to z = 0 or to ~ ~ 0, where z is an ~-term. Thus either
Vx E X A x k 3 y A t A y , r(x)) = o o Vx ~ X A x k z(r(x)) = 0 iEl
<-->A g z(r) = 0, or
Vx E X A x k 3yAti(y,r(x))=O<->Vx E X A x k ~ ( r ( x ) ) * O i~l
*->A ~O #=O, as X is not compact and the support of ~(r) is a clopen compact subset of X. (3) By Proposition 2.1, in any simple element of V, the CB-formula Ai~l ti(y, z) = 0 A sj(y, Z) ~ 0 is equivalent to the negation of an atomic formula,
i.e., rj(y, z) ~ 0, where rj is an ~-term. By hypothesis on the class ~, there exist open formulas 0j, j ~ J, such that
3xjA,,jk 3y~j(y,z),PO<->3xjAxj~ Oj(z),
for any j eJ.
By Proposition 2.1, we can assume that, in any simple element of V, the open formulas Oj are equivalent either to tj(.)= 0 or to tj(.)=/=O, where tj are ~-terms,
j~J.
100
F. Point
In both cases, we can express in A that 3x s Axj ~ Os(r(xs)):
3x s Ax, ~ ts(r(xs) ) = 0 ~ A
~ (0 = 0),
3x/ Ax ~ ts(r(x/)) =/=O~-->A ~ ts(r) ~ O. Thus, putting (1), (2), (3) together we have proved that we can construct an open formula equivalent in A to tp(z). Moreover this open formula depends only on ~ and the fact that the base space of A is not compact, without isolated points. Therefore we may conclude that ~ admits q.e. [] Now we are in a position to state the main results.
Theorem 2.10. Let V be a discriminator variety. Let A be an element of V. Then A admits (effective) q.e. iff there is an (effective)procedure to decompose A into a finite direct product with projector such that for each o f its factor A~, 1 <~i <~n, one o f the following holds: (1) Ai is a simple element o f V which admits (effective) q.e., (2) A i belongs to (~i)2 where ~i is a class of simple elements of V, admitting (effective) s.e., (3) Ai belongs to ( ~i)~tor where cgi is a class of simple elements of V, admitting (effective) s.e., (4) Ai is isomorphic to F~(Xi, [,.Jx~x, Ax), where Xi is a locally boolean space, non-compact, without isolated points and the class {Ax Ix • Xi} admits (effective) positive q.e. and A i has the (effective) local negative elimination property.
Theorem 2.11. Let V be a discriminator variety. Let ~r be a subclass of V. Then admits q.e. iff (1) There exist open sentences It1, • • •, ~.Lms.t. m -
j=l
A ~ Its implies that the base space o f A is the topological sum o f a subspace Xo without isolated points and finitely many discrete subspaces Xz, • • •, Xns with one or two points s.t. (i) if Xo is non-empty compact, then it is the support o f a closed &P-term and if X~ contains two points, i >~1, then it is the support o f a closed .~-term; (ii) for any Xi, i >1O, there exists .fC-terms ti s.t. for any element r o f A, ti(r) = r/X~ U O / X - X~. (2) For every positive open formula qg(z,y), there exist open sentences vz, . . . , VmC~) and open formulas q91(y), . . . , qgm(y) s.t. -
~
Vi, i=1
ViVA(A~--Fc(X, xUxAx))^A~v,-->(VxeX,
Ax~3zqg(z,Y)<->qg,(Y)).
101
Quantifier elimination in discriminator varieties
(3)
For every non-positive
open formula
v'l, . . . , v ' ( ~ , ) a n d o p e n f o r m u l a s m(~.)
~(z, y),
there exist open sentences
~ I ( Y ) , . . . , ~ m ( Y ) s.t.
,
Vv ,
i=1
Vi Vii A ~ v; ~ (3x e XAx ~ 3z g,(z, y ) ~ 3x e XAx ~ ~])i@))"
[]
Acknowledgments This article is a slight generalization of the chapter of my thesis on projectable L-groups. I warmly thank Professor A. Macintyre, A.M.W. Glass, V. Weispfenning and C. Wood for their help and advice. I am indebted to the referee for his thorough reading of this paper and his numerous suggestions. I would like to mention his simplification of part (b) of Proposition 2.6 and his discovery of errors in the proof of Propositions 2.6 and 2.8.
References [1] M. Boffa, Quantifier elimination in boolean sheaves, Preprint. [2] M. Boffa and G. Cherlin, Elimination des quantificateurs dans les faisceaux, C. R. Acad. Sci. Pads Set. A-B 290, (1980) A355-357. [3] S. Burris and R. McKenzie, Decidable varieties with modular congruence lattices, Bull. Amer. Math. Soc. 4 (3) (1981) 350-352. [4] S. Burris and H. Wemer, Sheaf constructions and their elementary properties, Trans. Amer. Math. Soc. 248 (1979) 269-309. [5] C.C. Chang and H.J. Keisler, Model Theory (North-Holland, Amsterdam, 1973). [6] H.J. Keisler, Some applications of infinitely long formulas, J. Symbolic Logic 30, 339-349. [7] H. Wemer, Discriminator Algebras, Studien zur Algebras und ihre Anwendungen, Band 6 (Akademie-Vedag, Berlin, 1978). [8] A. Macintyre, Model completeness for sheaves of structures, Fund. Math. 81 (1973) 73-89.