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On the reducibility order between borel equivalence relations
Logic, Methodology and Philosophy of Science IX D. Prawitz, B. Skyrms and D. Westerst~ihl (Editors) 9 1994 Elsevier Science B.V. All rights reserved.
151
ON THE REDUCIBILITY ORDER BETWEEN BOREL EQUIVALENCE RELATIONS
ALAIN LOUVEAU Equipe d'Analyse, Universitd Paris VI, Paris, France
Introduction
An equivalence relation E on a set X is a Borel equivalence relation if both X and E are Borel, in some Polish space ( which can always be taken to be the space 2~), and its square, respectively. We denote by B O R E Q the class of all Borel equivalence relations. We say that (X, E) is reducible to (Y, F) if there is a Borel function f " X ---, Y such that
Vx e X Vy e X (xEy ~ f ( x ) F f(y)). This defines a quasi-ordering _< on B O R E Q , with associated equivalence _=. For more information about this quasi-ordering, see the paper of Kechris [6]. In [2], H. Friedman and L. Stanley prove: F a c t 1.
(BOREQ, <_) has no maximum element.
In fact, they introduce a "jump" operator, a version of which is defined as follows: To each E, associate E + on X ~, defined by (Xn) E+ (Yn) e--->VTt~m (xnEym) A V n ~ m (xmEyn). Fact 1 then follows from: THEOREM (Friedman-Stanley). For all E in B O R E Q with at/east two
classes, E < E +. The original proof of this theorem used the deep results of H. Friedman on Borel diagonalizations, and in particular was not elementary (i.e. in second-order arithmetics). In [3], Harrington gives an elementary proof of this theorem. For each countable ordinal ~, let E(~) be the ~th iterated jump, using the above operator +, of (2 "~, =). Harrington proves"
152 THEOREM
BOREQ.
(Harrington). The family E(~), ~CWl
iS unbounded in In fact for each ~, and each E~ equivalence relation E,
E ( ~ + 1) 2~ E. This result easily implies the Friedman-Stanley theorem, hence Fact 1, but also F a c t 2.
B O R E Q N E~ is not cofinal in B O R E Q .
Harrington's proof is elementary, but uses a delicate forcing argument. The aim of this paper is to give a different, and much simpler, proof of Facts 1 and 2, based on a different "jump" operator, for which we will prove analogs of the two theorems above. This proof also brings in an interesting invariant of the reducibility equivalence relation =, the potential Wadge class of a Borel equivalence relation. I would like to thank R. Sami and J. Saint Raymond for the discussions we had on the subject. 1. P o t e n t i a l W a d g e classes DEFINITION 1. Let F be a Wadge class, and X a Borel set. A subset A of X 2 is potentially of class F, written A E potF, if for some finer Polish topology T on X, A is in F in (X, T) 2. One can define the potential Wadge class of a Borel set A c_ X 2,
potF(A), as the least F such that A E potF (this is clearly well-defined). Now if A C X 2 and B C y2 are such that there exists a Borel function f " X ~ Y with (x, y) e A ~ (f(x), f(y)) e B, then potr(A) c_ potr(B). We will apply this remark to Borel equivalence relations. Note that the notion of potential Wadge classes is non trivial: For each non self dual Borel Wadge class F, with dual class F, there is in (2w) 2 a set in F which is not in potF, namely any F-universal set. To see this, note that any two Polish topologies, with one finer than the other, coincide on a dense G6 set, hence on a perfect set, which contains a set in F\F. It is usually hard to compute the exact potential Wadge class of a Borel equivalence relation. However, we will be able to do it in enough particular cases. Let 9~ be a filter on w. We
2
define the relation 2 T on 2 w by
e 7.
THEOREM 2. Let F be a Wadge class dosed under intersections. If 27 is in potF, then ~Y is in F. PROOF" Let T be the finer Polish topology on 2 W for which 2J: is in F, and H C_ 2 ~ be a dense G6 set on which the two topologies coincide. We
153 claim that there is a partition of w into two sets Ao, A1, and two sets Bo, B1, with for i - 0, 1 Bi C_ Ai, such that for i - 0 or 1, if A C_ w satisfies A N A i - Bi, then A C H. This claim will finish the proof, for one has, for A C a;" A c ~ ~ (A N Ao) t2 B 1 25-B1 A (A N A1) [-J Bo 2~:Bo (here and below, we identify a subset of ~ with its characteristic function). And as 2 7 is in F on H 2, this gives a F definition of $', as desired. To prove the claim, note first that for any dense open set G in 2 ~, and any k, there is an 1 > k and a subset S of [k,l[ such that any A c_ w with A N [k,/[= S is in G" E n u m e r a t e all subsets of [0, k[ as (Sn)n<2k, and build inductively kn and Tn C_ [kn, kn+l[, starting with ko - k, so that for each n < 2 k, if A N [ki, ki+l[= T~ for all i < n and A N [0, k[= Sn, then A E G, using the density of G. Then 1 - k2k and S - UnTn work. Applying the subclaim successively to a decreasing sequence ( G , ~ ) , ~ of dense open sets with intersection H gives a sequence k,~ with ko - 0, and sets Sn c_ [kn, kn+l[ such that if A N [kn, kn+l[-- Sn, A E Gn. T h e n Ai - I,Jn[k2n+i, k2n+i+l[ and Bi - U,~ S2n+i, for i - 0,1, satisfy the claim. -t Remark. The claim used in the previous proof is a folklore result. It can be used e.g. to show that a free Borel filter 9r on w is meager, or that there exists a finite-to-one function ~a 9 r - . w with ~a(9r) - A/'. More interestingly, W. Just uses it in [5] to prove that there are in (BOREQ, <_) antichains of a r b i t r a r y finite cardinality. By the previous result, the c o m p u t a t i o n of the potential Wadge class of 2 7 is reduced to the c o m p u t a t i o n of the Wadge class of 7 . We do not know exactly which Wadge classes are Wadge classes of filters on w (Easily, A ~ II ~ and E ~ are such classes, and by a Baire category argument, any II ~ filter is II ~ Calbrix [1] has exhibited filters of Wadge class II~ and E~ for all ~ > 2). Nevertheless, it is easy to check that Borel filters have Wadge classes unbounded in A~. In fact if we let Af be the Fr~chet filter, A; - {A C_ w " A is cofinite}, and if we define its iterates ( A f ~ ) ~ I by induction by
2r A c X +I
A} c
c N
where ~ is a bijection between w and w 2, and for limit A
Ac
A} c He(n)} e
154 where r is a bijection between w and A, then one easily checks that all Borel sets are obtained from the clopen sets by the operation of liminf along one of these iterates. So their Wadge classes are unbounded, and by the theorem above, we get:
COROLLARY 3. (a) The sequence ( 2 A r ~ ) ~ is unbounded in B O R E Q . (b) Given any countable ~, there is a ~ such that for any ~o equivalence relation E, 2Are ~ E (In fact, by the exact computations of Calbrix [1], one can take ~ = ~ ). R e m a r k s . 1. Theorem 2 has another nice consequence. In [4], Harrington-Kechris-Louveau prove that any Borel equivalence relation either is smooth, i.e. reducible to (2 ~, =), or else reduces 2Ar. From this, they infer, using a measure theoretic argument, that every G~ equivalence relation is smooth. This can also be derived from Theorem 2, by noting that otherwise 2Ar would be potentially II ~ hence jkf would be II ~ in 2 ~, a clear contradiction. 2. In the proofs above, the only property used of the reducing function was the Baire Property, so that our arguments would apply, using the appropriate level of determinacy, to more general notions of reducibility, up to reducibility by arbitrary functions in the context of AD, as noticed by A.S. Kechris. 2. A j u m p o p e r a t o r in B O R E Q DEFINITION 4. Let $" be a Borel filter on w, and E a Borel equivalence relation on some Borel X. We define the relation E J: on X ~ by
(xn)E ~(yn) ~ {It" xnZYn} e jz. Note that with this notation, 2~: is just (2, =)J:. THEOREM 5. The operator E H E Ar is a j u m p operator in B O R E Q " For every Borel E with at least two classes, E < EAr. PROOF: Clearly, E _< E X, by sending any x E X to the constant sequence (x). Assume that E X <_ E. We claim that for every ~ < Wl, E Xr _< E. This is proved by induction on ~. Suppose first ~ - ~?+ 1 is successor, and let f be a Borel reduction of E X, to E. Define F " X • ~ X ~ by: =
155 One gets
(xk)eZ,(yk)
{k" xkEyk} e
F( (xk ) )EArF( (yk ) ) So E Xr _< E X _ E, as desired. The proof for limit ~ is similar: Let for each ~ < ~ fv reduce EAr, to E, and set F((xk)k)
- (fr
By a c o m p u t a t i o n similar to the one above, one checks t h a t F reduces FAre to EAr, and as above we get the claim. Suppose now t h a t (2, = ) < E. Then easily for any $-, one gets 2 f _~ E 7, hence by the previous claim, for all ~ < Wl, we get 2Arc _< E, contradicting T h e o r e m 2. -~
REFERENCES
[1] J. CALBRIX, Classes de Baire et espaces d'applications continues, Note aux C. R. Acad. SC. Paris, 301, 1985, 759-762. [2] H. FRIEDMAN, L. STANLEY, A Borel reducibility theory for classes of countable structures, J. Symb. Logic 54 (1989), 894-914. [3] L. HARRINGTON, On the complexity of Borel equivalence relations, abstract, International Workshop on Set Theory, Marseille-Luminy, 1990. [4] L. HARRINGTON, A. S. KECHRIS, A. LOUVEAU, A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the A.M.S.4(3),1990,903-928. [5] W. JUST, More mutually irreducible ideals, preprint, 1990.
[6] A.S. KECHRIS, The structure of Borel equivalence relations in Polish spaces, to appear in the Proceedings of the Workshop on Set Theory and the Continuum, MSRI, Berkeley 1989.
151
ON THE REDUCIBILITY ORDER BETWEEN BOREL EQUIVALENCE RELATIONS
ALAIN LOUVEAU Equipe d'Analyse, Universitd Paris VI, Paris, France
Introduction
An equivalence relation E on a set X is a Borel equivalence relation if both X and E are Borel, in some Polish space ( which can always be taken to be the space 2~), and its square, respectively. We denote by B O R E Q the class of all Borel equivalence relations. We say that (X, E) is reducible to (Y, F) if there is a Borel function f " X ---, Y such that
Vx e X Vy e X (xEy ~ f ( x ) F f(y)). This defines a quasi-ordering _< on B O R E Q , with associated equivalence _=. For more information about this quasi-ordering, see the paper of Kechris [6]. In [2], H. Friedman and L. Stanley prove: F a c t 1.
(BOREQ, <_) has no maximum element.
In fact, they introduce a "jump" operator, a version of which is defined as follows: To each E, associate E + on X ~, defined by (Xn) E+ (Yn) e--->VTt~m (xnEym) A V n ~ m (xmEyn). Fact 1 then follows from: THEOREM (Friedman-Stanley). For all E in B O R E Q with at/east two
classes, E < E +. The original proof of this theorem used the deep results of H. Friedman on Borel diagonalizations, and in particular was not elementary (i.e. in second-order arithmetics). In [3], Harrington gives an elementary proof of this theorem. For each countable ordinal ~, let E(~) be the ~th iterated jump, using the above operator +, of (2 "~, =). Harrington proves"
152 THEOREM
BOREQ.
(Harrington). The family E(~), ~CWl
iS unbounded in In fact for each ~, and each E~ equivalence relation E,
E ( ~ + 1) 2~ E. This result easily implies the Friedman-Stanley theorem, hence Fact 1, but also F a c t 2.
B O R E Q N E~ is not cofinal in B O R E Q .
Harrington's proof is elementary, but uses a delicate forcing argument. The aim of this paper is to give a different, and much simpler, proof of Facts 1 and 2, based on a different "jump" operator, for which we will prove analogs of the two theorems above. This proof also brings in an interesting invariant of the reducibility equivalence relation =, the potential Wadge class of a Borel equivalence relation. I would like to thank R. Sami and J. Saint Raymond for the discussions we had on the subject. 1. P o t e n t i a l W a d g e classes DEFINITION 1. Let F be a Wadge class, and X a Borel set. A subset A of X 2 is potentially of class F, written A E potF, if for some finer Polish topology T on X, A is in F in (X, T) 2. One can define the potential Wadge class of a Borel set A c_ X 2,
potF(A), as the least F such that A E potF (this is clearly well-defined). Now if A C X 2 and B C y2 are such that there exists a Borel function f " X ~ Y with (x, y) e A ~ (f(x), f(y)) e B, then potr(A) c_ potr(B). We will apply this remark to Borel equivalence relations. Note that the notion of potential Wadge classes is non trivial: For each non self dual Borel Wadge class F, with dual class F, there is in (2w) 2 a set in F which is not in potF, namely any F-universal set. To see this, note that any two Polish topologies, with one finer than the other, coincide on a dense G6 set, hence on a perfect set, which contains a set in F\F. It is usually hard to compute the exact potential Wadge class of a Borel equivalence relation. However, we will be able to do it in enough particular cases. Let 9~ be a filter on w. We
2
define the relation 2 T on 2 w by
e 7.
THEOREM 2. Let F be a Wadge class dosed under intersections. If 27 is in potF, then ~Y is in F. PROOF" Let T be the finer Polish topology on 2 W for which 2J: is in F, and H C_ 2 ~ be a dense G6 set on which the two topologies coincide. We
153 claim that there is a partition of w into two sets Ao, A1, and two sets Bo, B1, with for i - 0, 1 Bi C_ Ai, such that for i - 0 or 1, if A C_ w satisfies A N A i - Bi, then A C H. This claim will finish the proof, for one has, for A C a;" A c ~ ~ (A N Ao) t2 B 1 25-B1 A (A N A1) [-J Bo 2~:Bo (here and below, we identify a subset of ~ with its characteristic function). And as 2 7 is in F on H 2, this gives a F definition of $', as desired. To prove the claim, note first that for any dense open set G in 2 ~, and any k, there is an 1 > k and a subset S of [k,l[ such that any A c_ w with A N [k,/[= S is in G" E n u m e r a t e all subsets of [0, k[ as (Sn)n<2k, and build inductively kn and Tn C_ [kn, kn+l[, starting with ko - k, so that for each n < 2 k, if A N [ki, ki+l[= T~ for all i < n and A N [0, k[= Sn, then A E G, using the density of G. Then 1 - k2k and S - UnTn work. Applying the subclaim successively to a decreasing sequence ( G , ~ ) , ~ of dense open sets with intersection H gives a sequence k,~ with ko - 0, and sets Sn c_ [kn, kn+l[ such that if A N [kn, kn+l[-- Sn, A E Gn. T h e n Ai - I,Jn[k2n+i, k2n+i+l[ and Bi - U,~ S2n+i, for i - 0,1, satisfy the claim. -t Remark. The claim used in the previous proof is a folklore result. It can be used e.g. to show that a free Borel filter 9r on w is meager, or that there exists a finite-to-one function ~a 9 r - . w with ~a(9r) - A/'. More interestingly, W. Just uses it in [5] to prove that there are in (BOREQ, <_) antichains of a r b i t r a r y finite cardinality. By the previous result, the c o m p u t a t i o n of the potential Wadge class of 2 7 is reduced to the c o m p u t a t i o n of the Wadge class of 7 . We do not know exactly which Wadge classes are Wadge classes of filters on w (Easily, A ~ II ~ and E ~ are such classes, and by a Baire category argument, any II ~ filter is II ~ Calbrix [1] has exhibited filters of Wadge class II~ and E~ for all ~ > 2). Nevertheless, it is easy to check that Borel filters have Wadge classes unbounded in A~. In fact if we let Af be the Fr~chet filter, A; - {A C_ w " A is cofinite}, and if we define its iterates ( A f ~ ) ~ I by induction by
2r A c X +I
A} c
c N
where ~ is a bijection between w and w 2, and for limit A
Ac
A} c He(n)} e
154 where r is a bijection between w and A, then one easily checks that all Borel sets are obtained from the clopen sets by the operation of liminf along one of these iterates. So their Wadge classes are unbounded, and by the theorem above, we get:
COROLLARY 3. (a) The sequence ( 2 A r ~ ) ~ is unbounded in B O R E Q . (b) Given any countable ~, there is a ~ such that for any ~o equivalence relation E, 2Are ~ E (In fact, by the exact computations of Calbrix [1], one can take ~ = ~ ). R e m a r k s . 1. Theorem 2 has another nice consequence. In [4], Harrington-Kechris-Louveau prove that any Borel equivalence relation either is smooth, i.e. reducible to (2 ~, =), or else reduces 2Ar. From this, they infer, using a measure theoretic argument, that every G~ equivalence relation is smooth. This can also be derived from Theorem 2, by noting that otherwise 2Ar would be potentially II ~ hence jkf would be II ~ in 2 ~, a clear contradiction. 2. In the proofs above, the only property used of the reducing function was the Baire Property, so that our arguments would apply, using the appropriate level of determinacy, to more general notions of reducibility, up to reducibility by arbitrary functions in the context of AD, as noticed by A.S. Kechris. 2. A j u m p o p e r a t o r in B O R E Q DEFINITION 4. Let $" be a Borel filter on w, and E a Borel equivalence relation on some Borel X. We define the relation E J: on X ~ by
(xn)E ~(yn) ~ {It" xnZYn} e jz. Note that with this notation, 2~: is just (2, =)J:. THEOREM 5. The operator E H E Ar is a j u m p operator in B O R E Q " For every Borel E with at least two classes, E < EAr. PROOF: Clearly, E _< E X, by sending any x E X to the constant sequence (x). Assume that E X <_ E. We claim that for every ~ < Wl, E Xr _< E. This is proved by induction on ~. Suppose first ~ - ~?+ 1 is successor, and let f be a Borel reduction of E X, to E. Define F " X • ~ X ~ by: =
155 One gets
(xk)eZ,(yk)
{k" xkEyk} e
F( (xk ) )EArF( (yk ) ) So E Xr _< E X _ E, as desired. The proof for limit ~ is similar: Let for each ~ < ~ fv reduce EAr, to E, and set F((xk)k)
- (fr
By a c o m p u t a t i o n similar to the one above, one checks t h a t F reduces FAre to EAr, and as above we get the claim. Suppose now t h a t (2, = ) < E. Then easily for any $-, one gets 2 f _~ E 7, hence by the previous claim, for all ~ < Wl, we get 2Arc _< E, contradicting T h e o r e m 2. -~
REFERENCES
[1] J. CALBRIX, Classes de Baire et espaces d'applications continues, Note aux C. R. Acad. SC. Paris, 301, 1985, 759-762. [2] H. FRIEDMAN, L. STANLEY, A Borel reducibility theory for classes of countable structures, J. Symb. Logic 54 (1989), 894-914. [3] L. HARRINGTON, On the complexity of Borel equivalence relations, abstract, International Workshop on Set Theory, Marseille-Luminy, 1990. [4] L. HARRINGTON, A. S. KECHRIS, A. LOUVEAU, A Glimm-Effros dichotomy for Borel equivalence relations, Journal of the A.M.S.4(3),1990,903-928. [5] W. JUST, More mutually irreducible ideals, preprint, 1990.
[6] A.S. KECHRIS, The structure of Borel equivalence relations in Polish spaces, to appear in the Proceedings of the Workshop on Set Theory and the Continuum, MSRI, Berkeley 1989.