Theories with Equational Forking

Theories with Equational ForkingAuthor(s): Markus Junker and Ingo KrausSource: The Journal of Symbolic Logic, Vol. 67, No. 1 (Mar., 2002), pp. 326-340Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2695013 .

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 67, Number 1, March 2002

THEORIES WITH EQUATIONAL FORKING

MARKUS JUNKER AND INGO KRAUS

Abstract. We show that equational independence in the sense of Srour equals local non-forking. We

then examine so-called almost equational theories where equational independence is a symmetric relation.

Many complete first order theories admit nice independence relations, for example simple theories and o-minimal theories. In the classical examples, these independence relations are controlled by rather simple families of formulae. It has been observed by Srour (and others) that these formulae are usually "positive" in some sense. This yields the possibility to define locally noetherian topologies in a model that are closely linked to the independence relation.

In this paper, we recall Srour's definition of equational independence (as in [4]), and we show that it is the same as local non-forking with respect to equations. Pushing further the development of [15], we then give a comprising topological characterization, generalising the definition of independence in algebraically closed fields via irreducibility and varieties.

Following [4], a stable theory is said to be almost equational if equational independence is non-forking. We generalise this concept to arbitrary theories and suggest that it may be an interesting setting to work in. Presently, no simple theory is known which is not almost equational. We conclude with some criteria for almost equationality and show many classical structures to be almost equational.

The main purpose of this article is to shed new light on old things and to put different aspects together. It is based on work of Hrushovski, Pillay, and Srour.

?1. Equational independence. Throughout the paper, T is a complete first order theory and it a monster model of T.

Notations are quite standard. D( U) denotes the realisation set in it of a partial type (. "Definable" means definable with parameters, unless otherwise specified. -A stands for "conjugate over A" (in the monster model), pJA for the unique non-forking extension of a stationary type p onto A.

As usual in stability theory, we consider formulae without parameters (i;j) with a fixed partition of the free variables into two different sorts: "special variables" x and "parameter variables" 5. We denote by - I the formula p where the roles of special variables and parameter variables are exchanged. An instance of ' is a formula 9 (X, b) where the parameter variables are replaced by parameters and the

Received March 17, 2000; revised February 14, 2001.

( 2002, Association for Symbolic Logic 0022-4812/02/6701-0022/$2.50

326



THEORIES WITH EQUATIONAL FORKING 327

special variables remain free. A boolean combination of formulae is always meant to respect the partition and the roles of the variables.

Recall the definition and the basic properties of an equation in the sense of Srour

([15], [6]): * A formula 97(X; j) is an equation (in the special variables x) if every conjunction

of instances equals a finite sub-conjunction, i.e., for all parameters (bi)iEI there is a finite Io C I such that t AiE, (p(; bi) +-> AiEo w(; bi).

* Equivalently, there are no (di bi)iE, such that di, by) for all j < i E co and t- i (ad, by) for all j .E co.

* Hence equations are stable (with respect to the same partition of the variables). * ? is an equation iff >-1 is an equation. * Positive boolean combinations of equations are equations. Let X C Un be any set and 9(X; j) an equation with lg(i) n. We put

Wx(x) := A {9(x, c) IX C w(pUc)}

Note that A 0 T in the corresponding boolean algebra, i.e., px (U) = Un if X is not covered by any (-instance. (px( U) will be the closure of X in the Qa }-topology to be defined below. Since Wox equals a finite conjunction of (p-instances, it is a first order formula. Moreover, if X is A-invariant, then px is A-definable. If is a (partial) type, we put p7^ := (pz(u).

Now we fix a family E of equations. One should understand E as a sorted family, i.e., as the disjoint union of some Ex where all formulae in Ej are equations in x. Let bc+(E) stand for the closure of E under positive boolean combinations and substitution of variables, both respecting the roles of the variables. For example

p (x; y) A W(px; z-) EE bc + (W(p_; y)), but 9(X _; z) /\ (p(Z_; y)fbc + (Wpx

DEFINITION 1.1. E-independence If is defined by

A Hf B | ftp(a/B,C) is almost over C for all a C A

c and all equations ( c Ex with lg(ix) = lg(a)

The corresponding notion of "free extension of types" is denoted by IE, i.e.,

tp(d/C) 1-E tp(d/B, C) 5 Ii Jf B C

If E is the set of all equations, we denote the relations by 1S and ze and speak of equational independence. Definition and notation are coming from [4].

REMARK 1.2.

(a) Since (( A 0l)x = Sx A VIx and ((p V q4)x = (x V V'x, we have J5 - c +

(E)

(b) If E' C E are two families of equations, then Jf implies dE.

(c) If p E S(A), then the union of canonical parameters of the (pp for (p E E form the smallest set AO C acle (A), up to inter-algebraicity, such that p [Ao oE p. In particular LIE satisfies the local character axiom and admits "weak canonical bases" (cf., Fact 2.2).



328 MARKUS JUNKER AND INGO KRAUS

(d) Assume C = acleq(C) C B. Then A I, B iff C

whenever tp(d/B) kF (p, d) for some W E E, then tp(d /C) F f (x d).

Because of (a) we may always assume E = bc+ (E) in the following considerations. Also we always assume that E is closed under adding dummy variables on both sides (technically, that x = x and y = y, as equations in x, are in E). This implies some obvious monotonicity properties like (AO C A and A IE B) X- Ao I5 B.

C C

Local forking. For the definition of local forking with respect to a family F of stable formulae (F-forking for short) and all the notions around we follow and refer to [13] chapter 1:

By definition, an F-formula over A is a boolean combination of F-instances that is over A. Note that the parameters of the instances may come from outside A. An F-type over A is a maximal consistent set of F-formulae over A. For a complete type p, its F-part PF is the unique F-type included in p. If F consists of stable formulae, every type has an F-definition, and a type (over a model) does not F-fork over A if the (unique) F-definition is almost over A.

Recall that E is a fixed family of equations. Our aim is to show:

THEOREM 1.3. E-independence is exactly non-E-forking.

Let p E S(M) be a type over a model and W an equation. It is obvious from the properties of 9p that Vi (Wp (i) 97X )) is a (p-definition of p. Therefore we know from the general theory:

PROPOSITION 1.4. Let 9t be a model, p E S(M) and A C M. Then

(a) PE is definable over A iffWpo is over A for every ( p E.

(b) p does not E-fork over A iffsp is over acleq (A) for every y, E E and iffp [A FE P.

This shows Theorem 1.3 for types over models. The missing part follows once we know monotonicity and transitivity of CIE, because then we can lift types to types over models. We will get this easily from the topological interpretation of E-independence. The proof of Theorem 1.3 will be completed with Corollary 1. 13. Equivalently, what remains to be shown is that Vi (Wp (i) --(, j)) is a good (p-definition of an arbitrary type p c S (A) in the sense of [12].

The E-topology. This section is a reworking of parts of [15], where only noetherian topologies were considered. This topological point of view is not necessary to prove Theorem 1.3, but it emphasises the analogy to algebraic geometry and gives a geometric interpretation. A direct proof without topological considerations is presented in [10].

DEFINITION 1.5. The subsets of Un defined by E-instances form a sub-basis of closed sets of a topology which we call the E-topology. An E-closed set is a closed set in the E-topology. We denote by clE (X) the topological closure of a set X in the E-topology. We also write clE(p) or CIE(d/B) if X is the realisation set of the

type p =tp(d/B).

If E is the set of all equations, we speak of the Srour topology. Obviously, the E-topology and the bc+(E)-topology coincide. Since clE(X)

n {(px (u) ( W c E }, an E-closed set is type-definable by at most I T many formulae




and the (infinite) tuple of canonical parameters of the SWx provides a canonical base. It follows in particular that a definable E-closed set is exactly a set defined by an instance of bc+ (E).

LEMMA 1.6. A JE B iffclE(1/B, C) is almost over C for all a C A. C

PROOF. Just a reformulation of the definition since clE(d/B, C) is almost over C iffcb(clE(a/B, C)) is almost over C iff ,tp(a/Bc) is almost over C for all W E? E. -1

DEFINITION 1.7. A subset of a topological space is called ,-irreducible if it is not the union of less than i, many relatively closed proper subsets. A ,-irreducible component is a maximal ,-irreducible subset.

Let in the sequel , := I U1 be the cardinality of the monster model. "Small" will mean less than r,.

PROPOSITION 1.8. If p E S(acleq(A)), then ClE(P) is ,-irreducible.

PROOF. For a global type, we let clE(p) be the realisation set of A{wP I W E E}. Suppose ClE (P) C Usmall Ci with E-closed Ci. We may assume the Ci to be definable. By compactness, p implies a finite sub-disjunction of the Ci and hence, being a complete type, one Ci0. This set being E-closed, clE (p) C Ci0 follows. Thus clE (p) is ,-irreducible.

Now choose p to be a global non-E-forking extension of p. Then 97p is over aCleq (A) for every equation W E E, whence 7p = 9p and clE (p) = clE(p). -

LEMMA 1.9. "ClE commutes with small unions": Let I be small and let X and Xi (i E I) be type-definable sets with X = Uic1 Xi. Then CIE (X) = UiCI CIE (Xi) -

PROOF. We may suppose that all the sets are defined over B. Let c V UiEI ClE (Xi)-

Thus for every i E I, there is a definable E-closed set Ci D Xi with c V Ci. Then the sets [Ci] form a clopen covering of the closed subset [X] in the Stone space S(B). By compactness, there are i1. . ,ik E I such that X C Cil U ... U Ci, i.e., c V clE(X). The other inclusion is trivial. I

COROLLARY 1.10. Let p E S(A). Then ClE(P) is the union of its ,-irreducible components which are given by ClE(Pi) for the extensions pi E S(acleq(A)) of p. In particular the ,-irreducible components are almost over A and conjugate over A.

PROOF. The ClE(pi) are E-closed ,-irreducible subsets of clp(E), they cover clE(p) by Lemma 1.9. If X is a r,-irreducible subset of clE(p), then X = Ui X n ClE (Pi) implies X C clE (pi) for some i. Moreover, the ClE (pi) are conjugate over A since the pi are. Therefore two of them are either equal or not included one in the other, hence they are maximal r,-irreducible subsets. -

LEMMA 1.11. Let p E S(A), p C q E S(B). Then ClE(q) contains a ,-irreducible component of ClE (p) iff clE(q) is a union of K-irreducible components of CE (P).

PROOF. One direction is trivial. Thus suppose clE (q) contains a K-irreducible component of clE (p), which is also a ,-irreducible component of clE (q). Let pi and qj be the extensions of p and q onto aCI q(A), aCI q(B) respectively. We may suppose clE (PI) = clE(qI). Let CIE (qj) be a ,-irreducible component of CIE (q). Being conjugate over B D A to clE(ql) clE(pI), it has to be itself a Ki-irreducible component of clE (p).




THEOREM 1.12. p -E q iffclE(q) is a union of ,-irreducible components of ClE(p).

PROOF. Let p = tp(i/A) and q = tp(d/B). "< " is clear with Lemma 1.6 and Corollary 1.10 because ClE (/B) as a union of sets over acle q(A) is also over acle q(A). "X" Each C E CIE(d/B) lies in a r,-irreducible component C(E W/acleq (B)) for some a' B aB. Since clE(J'/B) is over acleq(A) we get tp(5'/acleq(A)) C clE(a'/B) - clE(a/B). Then clE(a'/acleq(B)) C cla'/acleq(A)) C clE(7/B). -

COROLLARY 1.13. Let p C q C r. Then p -E r iff p [E q and q EE r.

PROOF. Easy with Lemma 1.1 1. A

Another direct way of proving Corollary 1.13 without topological considerations was suggested by the referee: show that E-independence implies non-E-forking using Shelah's original definition of forking; then show right monotonicity and deduce the corollary.

Transitivity and monotonicity now allow to lift types to types over models such that E-independence is preserved:

PROOF OF THEOREM 1.3. Let p E S(A) and q E S(B) an extension of p. Choose a non-E-forking extension r c S(M) of q onto a model M. Then q -E r by Proposition 1.4 and by transitivity of local forking we get

q is a non-E-forking extension of p

r is a non-E-forking extension of p

* x 1E r (Proposition 1.4) p zfE q (Corollary 1.13) A

Independence relations. An abstract independence relation in the sense of Kim and Pillay [9] is a ternary relation between parameter sets satisfying the properties: invariance under automorphisms; finite character; monotonicity and transitivity; symmetry; existence of free extensions; local character. (These axioms do not include the so-called independence theorem!)

From Theorem 1.3 we get immediately many properties of equational independence and of 1H showing that 1H satisfies all the axioms of an abstract independence relation except symmetry. (The "local forking symmetry" [13] 1.2.8 does not in general imply the global symmetry of the relation J5.) As an immediate corollary of (the proof of Claim 1 of) Theorem 4.2 in [9] we also get:

PROPOSITION 1.14. Non-dividing implies IE in an arbitrary theory T. In particular, if T is simple, then non-forking implies iE.

Finally we show a property of equational independence that is not obvious from the local forking interpretation.

DEFINITION 1. 15. Let EaCl be the smallest family F of formulae containing E and closed under the following operation: if 97(x1, x~2; Y) c F and %x1I, i2) isolates the algebraic type of x2 over x1, then 3x2 (W(x1, x2, Y) A % X2)) c F.

LEMMA 1.16. Eadl is a family of equations.

PROOF. Let by(xI, x~2; y) be p(xl, x~2, y) AX%(xl, x~2). It is sufficient to show that all 3x2 V ( , x2, 5) define Srour closed sets. We use the indiscernible criterion, see [4]




or [6]: let (Ji),, be an indiscernible sequence with - 3x2 V'(a, x2, Ji) for i > 1. We have to show k 3x2 qg(c, x2, co

Since x2 has to be algebraic over a, we have t y (a, b, cij) for some b and some infinite subsequence iij. Now k tg(7, b, co) since v'(5-, b, j) defines a closed set, and we are done. -

PROPOSITION 1.17. If A JE B then acl(A) Jf5Cl B C C

PROOF. We may assume C to be algebraically closed and C B. Let d E acl(A) with conjugates d = d1d, k, and let x(a, x) be an isolating formula in tp(d/A).

Suppose tp(d/B) F (i, b) for some ( EE E 1, say (p, b) = tp(J/B)X and we

have to show that tp(d/C) F (x, b). Since A I B and tp(7/B) H x(Z(5, i) A C

W(, b)), we get tp(d/C) H (Z ((d,Jx) A (x, b)). Hence tp(d/C) F w(i, b1) V ... V W(x, bk) for some b = 1, -. . , bk conjugate over a. It follows that tp(dji/C) H

W(x, b) for some i by irreducibility (Proposition 1.8 for the Qp}-topology). But then (x, b) Ptp(j, /c) is over C and thus tp(d /C) F- p(i, b).

The finitely generated case. To finish this section, we will examine further the case that E is finitely generated, i.e., sort by sort the positive boolean combinations of finitely many equations. We may as well assume that E = bc+ ((p) for one fixed equation (.

LEMMA 1.18. Let p be a stationary {(p}-type over A and q a positive boolean combination of f W}-instances. Then p U { qg} does not divide over A iff p F- q.

PROOF. Suppose pU { (x, 5) } does not divide over A. Since ( is stable, Pstp(C/A) iS

consistent with p I U, and moreover it is a {( }-formula over A. Hence p F- Wstp(E/A).

This direction then extends immediately to positive boolean combinations of {(p}- instances. The other direction is clear. A

From Proposition 1.4 we know that a stationary {f }-type p does not divide over A iff (pp is over aClq (A). But it is not sufficient that (pp does not divide over A; dividing is not necessarily witnessed by the positive part. Consider ACF and a line L through 0 with transcendental slope, defined by an instance of a polynomial (.

Then the generic type 1 of L divides over 0, but not L = W, because it contains 0. Finally we want to show that Lemma 1.18 characterises closed sets: PROPOSITION 1.19. A definable set X is closed in the Srour topology iff

(1) X is defined by an instance W (x~, c) of a stable formula and

(2) for all types p over algebraically closed sets A:

if p U {W (x,J)} does not divide over A, then p (Wx,c)

PROOF. Let W (xi; j) be stable. If W(pU, 5) is closed, then (2) holds for stationary {(p}-types by Lemma 1.18 and "=" follows.

Suppose that (p ( U, c) is not Srour closed. Then there is an indiscernible sequence (ji)i<. in tp(c/0) such that ni<, (p(U, ci) does not equal a finite sub-intersection. The sequence (si) is {(W1 }-independent over some initial segment, hence we may suppose that it is a {fW-1}-Morley sequence over some algebraically closed set A.

Claim: there is p e S(A) that is consistent with D := Ai<.W (px, Si), but does not imply (.




Suppose not. Then (D is A-invariant and D +-> Vjej pj for pj E S(A) n [o ] Then t -p; > D and hence t Vj1j p *-> (. By compactness, (D is a finite conjunction: contradiction.

Now we need the local analogue of proposition 3.6 in [7], to appear in [10]: non- dividing is already witnessed by one Morley sequence (we only need the consistency of ( for this). Therefore pSL U W (x, c) does not divide over A, where pSL denotes the Qp}-part of p. But then p U W(x, 5) does not divide over A either. -

?2. Almost equational theories. In [4] a stable theory is defined to be a-equational if 1 = 1. All presently known stable theories are a-equational. We interpret the "a" as "almost" and suggest several ways to extend this definition to arbitrary theories:

DEFINITION 2. 1. * A theory is said to have equationalforking if non-forking equals equational inde-

pendence, and equational dividing if non-dividing equals equational independence. * A theory is said to be almost equational if JS is an independence relation, i.e., if

,S is symmetric. * Analogously, we speak of almost E-equationality if JE is symmetric, and of an

E-equational theory if every definable set is a boolean combination of definable E-closed sets.

(As in stability theory, we call the second a global, the third a local aspect.)

Clearly, a simple theory has equational dividing if it has equational forking, and such a theory is almost equational. (Moreover, if T has equational dividing or forking and is almost equational, then T is simple by a result of Kim [8].) A partial converse is implied by a result of Hans Scheuermann.

Following Scheuermann, an abstract independence relation L: is said to have weak canonical bases if for every A, B there is a smallest, up to inter-algebraicity, BA C B such that A 12 B. The relation is called strict if A L: B implies A n B C

BA C

acleq (C) )

FACT 2.2 ([17] Satz 1.7.4). There is at most one strict independence relation with weak canonical bases in Te .

SKETCH OF THE PROOF. (The result will be published in [16].) Step 1: an independence relation is characterized by the 1 -Morley sequences:

a 1, B every Morley sequence in tp(a/B) is also Morley in tp(a/A) A A

there is a sequence that is Morley in tp(a/B) and in tp(a/A).

Step 2: a sequence of A-indiscernibles (ai)iei is a Morley sequence over A if acl ((ai)ieII, A) n acl ((aiu i,,2, A) = acl(A) for all II, 12 c I with I1 < I2.

We will always assume that E "contains equality", i.e., that the formula x = y, as an equation in x, is in E. Then JE is strict. Recall that J5 always admits weak canonical bases (Remark 1.2). Hence:

COROLLARY 2.3. Suppose E contains equality. If T is simple with elimination of

hyper-imaginaries and almost E-equational, then T has equational forking.




In particular, all stable theories and all supersimple theories are almost equational if they have equational forking, since these theories admit elimination of hyper- imaginaries (see [14] for the stable case and [1] for the supersimple case).

It is open whether Corollary 2.3 applies to all simple theories. Since there are non-simple almost equational theories, Kim's result shows that Corollary 2.3 is wrong for arbitrary theories. Still it seems plausible that a corollary of Fact 2.2 holds for all theories, namely:

PROBLEM 1. Assume E' C E. Does almost E'-equationality imply almost E- equationality?

This would be quite useful since often one has a good control over some equations, for example polynomial equations, but not all. We would expect this to hold for a more general reason: if J< is a strict independence relation with weak canonical bases, then we conjecture that L: implies local non-forking with respect to all stable formulae.

At present, no simple theory is positively known not to be almost equational. In analogy with the stable forking conjecture one might ask for the stronger Equational Forking Conjecture, whether all simple theories are almost equational, but we don't believe in it.

PROBLEM 2. Construct a simple theory which is not almost equational.

Of course there are numerous variations of the problem, and it would be interesting to know which structural properties of a theory do imply almost equationality. The following cases seem particularly interesting:

QUESTION 2.4. Are allfinite rank theories almost equational? Are o-minimal theories almost equational?

There are complete theories that are not almost equational for general reasons:

EXAMPLE 2.5 (communicated by H. Scheuermann). Consider a three-sorted structure: a sort X of elements, a sort C of classes and a sort R of equivalence relations. A ternary relation E (x, y, z) C X x X x R specifies for each fixed z E R to an equivalence relation E on the sort of elements, and such that these equivalence relations are free. Finally there is a surjection f : X x R -4 C with pre-images xEz x {z}, i.e., C contains an element for each equivalence class. If T is the complete theory of such a structure, then T does not admit any strict independence relation (satisfying local character), in particular it can't be almost equational.

A topological condition for the stable case. Let T be stable. Then 1E satisfies all the axioms of Harnik and Harrington in [2] except possibly the boundedness axiom. Therefore we get a nice (but probably not very useful) characterization of almost E-equational stable theories by topological means:

THEOREM 2.6. Let T be stable. Then T is almost E-equational if there is a cardinal n such that the following holds for the E-topology: for all types p E S(A) and all B D A, at most K many extensions q E S(B) of p are dense inC1E (P))

In E-equational theories types are determined by its E-part, hence only one extension q can be dense in p. As a corollary we get the well known result:

COROLLARY 2.7 ([4]). E-equational theories are almost E-equational.




The so far "only" example of a stable non-equational theory is a coloured free pseudo-space constructed by Hrushovski in [4]. It is however almost equational.

EXAMPLE 2.8. Positively known to be equational are * weakly normal, i.e., stable and one-based, theories (by definition, cf., [13] chap-

ter 4) * CM-trivial theories of finite continuous Lascar rank (in [3] and [4]; cf., [6]) * algebraically closed fields, differentially closed fields (clear), separably closed

fields of finite degree of imperfection (in [18]).

It is an open problem whether there is a non-equational superstable theory of finite rank.

Equational reducts. Any theory T has an equational reduct, namely pure equality, but T itself needs not to be almost E=-equational where E= := bc+(x = y) denotes the family of equations in pure equality language. Suppose for instance that a E acl(b) \ acl(0) and b V acl(a). Then ClE=(b/a) = CE=(b/0), ClE= (a/0) is infinite and ClE= (a/b) equals the set of conjugates of a over b. Hence almost- E-equationality can't be read off from the topology alone, the problem is that the notion of invariance over a parameter set is not the same in the reduct.

Still it is often possible to deduce almost equationality from the existence of an (almost) equational reduct. Let us fix the following setting: T' is an 2'-theory, T := T' [y its reduct to some Y C "', and E is a family of 2-equations. Note that an 2-formula is an equation in T iff it is an equation in T'.

PROPOSITION 2.9.

(a) Suppose T is almost E-equational, and for every parameter set A, a definable E-closed set is over A in T iff it is over A in T'. Then T' is almost E-equational.

(b) Suppose T' is simple with elimination of hyper-imaginaries. If T' has an acl- preserving reduct T that is almost equational, then T' is almost equational (and in fact almost E-equational with E = all equations in T).

PROOF. Part (a) is obvious. (b): if E denotes the family of all equations in T, then If remains a strict independence relation with canonical bases in T'. Then apply Fact 2.2. -

EXAMPLE 2.10. Th(Q, <) is almost E=-equational. The theory of an ordered vector space is almost E-equational where E consists

of the quantifier-free positive formulae in the vector space language.

In some cases we can show that a theory is almost equational if some reduct is, but we have to allow equations in the bigger language X?'. The guiding example is ACFA:

EXAMPLE 2.1 1. Consider ACFA, where a denotes the generic automorphism. Let X' := fa{&(x) I n E 2, x E X} and let Poly stand for the family of polynomial equations. A quantified c-polynomial equation is a formula in the smallest family F containing Poly and closed under the following operation: if ~o(i; Y) E F, then Exi(xi = a(xj) A &(x;iY)) E F. These are equations by Lemma 1.16; call this family Poly'. Now ACFA is almost Poly'-equational since

A IB in ACFA A## A , BI in ACF < # Au Qoly Bu , A Joy' B C Ca ca C




If E consists of all equations in ACF, then the corresponding family EU contains (up to equivalence) all equations in ACFA, hence the analogous argument shows directly that ACFA has equational forking.

Generalising this example we get:

PROPOSITION 2.12.

(a) Assume T' is simple, T has equational forking andfor all A, B, C the following holds:

A IBin T' 4#z#. acl(A) 1, acl(B) in T C acl(C)

Then T' has equational forking.

(b) If T in (a) is almost E-equational, then T' is almost Eacl-equational.

PROOF. Assume T is almost E-equational and let EaCl be as in Definition 1.15. Then

ael I 17 and a AlE B =. A acl(A) lE' acl(B)

C monotonicity acl(C)

A acl(A) J, acl(B) in T acl(C)

Aj-,B in T' C

and since always A JB iA t B =- A J ael B we are done. C C C

In analogy with the question about stable reducts of simple theories, one might ask for equational reducts of almost equational theories. The example of ACFA shows that, as in the case of stable reducts of simple theories, one has to enlarge the concept of reducts. One could think of introducing more general closure operators that are still controlled by equations in the manner of Proposition 1.17.

QUESTION 2.13. Which are the almost equational theories T' having an equational reduct T such that A J B in T' iffacl(A) Ji acl(B) in T?

C acl(C)

This seems to be wrong for unstable fields like pseudo-finite or real closed fields.

Interpretations.

PROBLEM 3. Is a theory interpretable in an almost equational theory again almost equational?

As often with this type of questions, reducts seem to be difficult to handle. Here we present some immediate approaches to the problem.

Obviously, if T is almost E-equational and C some set of parameters, then T (C) is almost E-equational. Hence:

LEMMA 2.14. If T is almost equational and simple with elimination of hyper- imaginaries, then T(C) is almost equational.

PROOF. This does not follow directly from the first remark since there are more equations in T(C) than in T, but with Fact 2.2. -i

QUESTION 2.15. For any T and parameter set C, is T almost equational iff T(C) is?




Concerning Teq, suppose we have imaginary sorts Vi := /Ri with projection maps ri: Ui -4 Vi. If x E VI and y E V2, it is immediate to check that (x; y) is an equation iff (E I(x); 72(Y)) is an equation. Thus for a E V1, b E V2 we get a Jib 7 i-'(a) >J n 1(b), whence

C C

LEMMA 2.16. T is almost equational ifTeq is almost equational.

There is no difficulty in figuring out a corresponding local result. Now suppose V is some 0-definable set in it and consider the induced structure

S. The types in 93 are the traces of types in it. If moreover V is stably embedded, then the Srour topologies in Q2 are the trace topologies of the Srour topologies in i (see [5]). This proves

LEMMA 2.17. If V is stably embedded in i and T is almost equational, then Th(QI) is almost equational.

Here a local result seems to be delicate. The advantage of the whole Srour topology is that it only depends on the automorphism group and can be defined in a language free way.

Putting everything together we get:

THEOREM 2.18. Assume T is almost equational and Q2 is interpretable in it. If C2

carries the whole induced structure of t, if the base set V is stably embedded and either the interpretation does not need additional constants, or T is simple with elimination of hyper-imaginaries, then C2 is almost equational.

?3. Fibres. An independence relation consists in fact of a family of relations a, ... an J B, one for each arity n, and symmetry is the only axiom linking these

A relations. Therefore we can't expect symmetry of E-independence without some links between the various E-topologies on the U'.

Since we always assume that E is stable under adding dummy variables, pro- jections Un+k > Un along coordinates are continuous. A crucial property for symmetry seems to be the following:

DEFINITION 3. 1. The family of E-topologies is calledfibrous if every fibre of an E- closed sets is again E-closed. More precisely: for all n and k, whenever X C Un+k is E-closed and a E Un, then Xa := {x E Uk < (x, a) E X} is E-closed.

We will present two natural fibrous E-topologies coming from perfect and from semi-perfect equations.

Perfect equations.

DEFINITION 3.2. * E is symmetric if p E E implies p' E E. * A formula W is a perfect equation if it is an equation with respect to any partition

of the free variables. E is perfect if it consists of perfect equations with all possible partitions of the variables. In particular, E is symmetric. Let EP be the set of all perfect equations.

* E is (almost) strict if any instance of a formula in E which is over A is equivalent to an A-instance (acl(A)-instance) of some formula in E.




The family of all equations is symmetric and strict (by a straightforward compactness argument), but not perfect (consider x z8 y A z - z). E is a perfect family, and more generally the positive formulae in a language of equivalence relations. Another example of a perfect family are polynomial equations over any field.

REMARK 3.3. If E is an almost strict family of equations, then Ea"' is a strict family of equations.

LEMMA 3.4. Suppose E is symmetric and strict, and let C C A n B. Then

AJFB ' '* BJFA C C

PROOF. Suppose B JE A and let a E A, o E E. C

As E is strict, t (Otp(5/B) <-4 b(J,b) for some equation Vt E E and b E B.

Hence t Vi(j, b), i.e., tp(b/A) F- Vg(-, j). Since E is symmetric, B AJE A implies C

tp(b//acleq(C)) F ig(7,j) for some b' --c b. This means t VI(a,b") for all b lacleq(C) b'. Hence also k i(-", b') for some a' -c a and all a-" -cleq(C) a' and we are done. -

LEMMA 3.5. Suppose E is perfect. If A JE B, then A U C Jf B. C C

PROOF. Wl.o.g. C is algebraically closed and C C B. Suppose tp(J, 5/B) F

(x,ymi) for some equation (xy;z) E E. Then tp(d /B) F- (,Eji). But o(x; ,2) is also in E, whence A J B implies tp(a/C) F- (, c, fm) and thus

C

tp(a, c/C) F- 0 (ix, jI). -

Remark: the proof only needs E to be semi-perfect; see Definition 3.10.

THEOREM 3.6. If some perfect family E is strict, then T is almost E-equational. In particular, if EP is strict, then T is almost EP-equational.

PROOF.

A A B A BU C monotonicity ofjH C C

=> AUCJFBUC byLemma3.5 C

> Z oBUCIfAUC by Lemma 3.4 C

B J, A monotonicity of JIf C

EXAMPLE 3.7. E is strict iff acl(A) = A for all A.

PROOF. "X" If b E acl(A) with conjugates bl,..., bk, then x = bl V.. V x bk is an E=-formula over A. This is only possible if bi E A.

"6" Any E=-instance o is a positive boolean combination of formulae xi = xj and Xk = b. These b provide a canonical parameter of o (a tuple of sets of the b) which is over A = acl(A) iff all b E A. -i

Hence the infinite set without structure, the random graph, and Th(Q, <) are almost E=-equational. Note that E= = EP in these examples (easy to check with quantifier elimination).




In fact,AJF B A A acl(BC) Cacl (C). C

EXAMPLE 3.8. Let T be a theory of fields and recall that Poly stands for the (perfect) family of polynomial equations. Poly is strict in algebraically closed fields, hence algebraically closed fields are Poly-equational. This is, of course, nothing new since the Poly-topology is exactly the (trace) of the Zariski topology. Are there possibly are other fields where Poly is strict?

If Poly is strict, then model theoretic algebraic closure equals the (relative) field theoretic algebraic closure. For suppose a,, . . ., ak are the conjugates of an element a E acl(B). Then x = al V ... V x = ak is over B, therefore by strictness equivalent to some polynomial system with coefficients in B.

If the two notions of algebraic closure coincide, then J>'oY implies algebraic independence, i.e.,

(*) A J 01y B =i td(A/acl(B, C)) = td(A/acl(C)) C

where td = transcendence degree.

PROOF. Wl.o.g. we may suppose C = acl(C) C B = acl(B). Let A' C A be a transcendence basis of A over C. If it is not algebraically independent over B, then there is some polynomial P, not identical 0, with coefficients b E B, and some a E A' such that P(j, b) 0. Hence tp(a/B) F- P(J, b) 0.

Consider the "minimal" formula (P(x, Y) = 0)tp(j/B), which is of the form

Ak=1 P(J, b 0) - 0 and over B. By almost Poly-equationality, it is almost over C. Hence tp(a/acleq (C)) F A =1 P (J, b ) 0. In particular, we get that tp (a /acle (C)) F- "some xi has finite orbit over x \ xi". Because model theoretic algebraic closure equals the relative algebraic closure in T, this implies that x is not algebraically independent over acleq (C).

If Poly is in addition almost strict, the converse of (*) holds:

PROOF. The right side yields A I"lY B in ACF, w.l.o.g. C C B = acl(B). Suppose C

tp(7/B) F- P(x, b) = 0 for some polynomial P, we may assume b E B by almost strictness. It follows that P(J b) = e C tPACF(j/B) because the formula is quantifier-free. Then tPACF (j / C) FACF P (X, b) = 0 by almost Poly-equationality (and elimination of imaginaries) of ACF, whence tp(J / C) F P(x, b) = 0, again since the formula is quantifier-free. -i

In real closed fields Poly is almost strict, but not strict: x V=X is over 0, but not equivalent to a polynomial equation over 0. Thus the condition of Theorem 3.6 is not necessary. More generally, Poly is almost strict if the two notions of algebraic closure coincide and if the extension i1/A is regular for every A = acl(A) (this follows from results in [10]). These theories of fields then are almost Poly-equational because algebraic independence is symmetric. Separably closed fields are examples of (almost) equational fields where the two notions of algebraic closure differ.

QUESTION 3.9. Is a theory T almost equational iff it is almost EP-equational?

We would rather expect a negative answer to this question.




The fibrous Srour topology. Strictness of the family of all equations allows to define Srour closed sets intrinsically: X is Srour closed iff the set of conjugates of X under the automorphism group satisfies the descending intersection condition. We don't have an analogous characterisation for EP-closed sets, and this seems to be an important point in trying to find topological conditions for almost equationality.

There is a largest family of equations defining a fibrous topology. For several reasons this seems to be a good topology to work with and a natural candidate to check almost equationality for.

DEFINITION 3.10. A formula ( (x; y) is a semi-perfect equation iff ( is an equation with respect to all sub-tuples of x (as special variables). Let E f be the family of all semi-perfect equations.

LEMMA 3.11.

(a) A definable set is Ef-closed (i.e., defined by an instance of a semi-perfect equation) iff X and all itsfibres are Srour closed.

(b) The family of semi-perfect equations is strict.

PROOF. "X" in (a) is obvious. For the other direction and (b), suppose the Ef-closed set X is defined by o(J, b). It is sufficient to show that there is a 5c(y)

with t 8(b) and such that p (x; j) A 8 (Y) is a semi-perfect equation. Given a projection, there is a uniform bound for the DIC on fibres of X corresponding to that projection: otherwise compactness would yield a fibre not satisfying the DIC. Hence to be Ef-closed is expressible by a small infinite disjunction of formulae implied by tp(b). Then 8 is given by compactness. -i

The Ef-topology, or fibrous Srour topology, is clearly the largest fibrous E-topology. We finish with a purely topological criterion for almost equationality with respect to a fibrous topology.

LEMMA 3.12. Let E be such that the E-topology isfibrous. If a- I b, then A

CIE (a, b/A)= CIE (a /A) X CIE(b/A).

Note that ClE (a, b/A)b D cIE ((/b, A) always holds because of the hypothesis on E. This is not always true for the ordinary Srour topology!

PROOF. "C" is obvious. By the hypothesis a J b, we have clE (a/b, A)= clE(a /A). Then we get

A

CIE(tp(j'/b', A) x {b'}) = CIE ( /A) x {b'}

for all j'b' =A jb. Taking the union over all b' -A b, we get cIE(i,b/A) D

clE (A) x tp(b/A). But since the fibres of ClE (a, b/A) in the other direction have to

be E-closed, too, we get CIE(j, b/A) D CIE(i /A) X CIE(b/A). If the topology is not fibrous, the lemma is obviously false. Consider the infinite

set without structure, E = all equations, and a :& b. Then U2 = clE (a/0) x cIE(b/0) + cIE(a, b/1/) = {(x, y) x 78 y}.

PROPOSITION 3.13. Suppose the E-topology isfibrous and

CIE (a/b, A)= CIE (j, b/A)b




for all a, b and A. Then a J Lb C clE(a, b/A) = CIE(a/A) X CIE(b/A) hence Jf A

is symmetric and T is almost E-equational. PROOF. If JIf is not symmetric then the above condition is not satisfied by

Lemma 3.12. -i

The condition in the proposition holds for example for the Zariski topology in algebraically closed fields and more generally for any perfect and strict E-topology.

In a dense open ordering or in an ordered vector space, a formula 0 (x; 5) is a semi-perfect equations if it is of the form yj(J, j) A 8(j) for some perfect equation qi. Hence Ef-closed sets are EP-closed, whence these structures are also almost Et-equational.

It is not known whether an equational theory has to be (almost) Ef-equational. All the standard examples of equational theories are Ef-equational. We may also ask analogous questions to 3.9:

QUESTION 3.14. Is a theory T almost equational iff T is almost Ef -equational? Is a theory T almost Ef -equational if T is almost EP-equational?

REFERENCES

[1] S. BUECHLER, A. PILLAY, and E WAGNER, Elimination of hyperimaginaries for supersimple theories, Journal of the American Mathematical Society, vol. 14 (2001), no. 1, pp. 109-124.

[2] V. HARNIK and L. HARRINGTON, Fundamentals of forking, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 245-286.

[3] E. HRUSHOVSKI, A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147-166.

[4] E. HRUSHOVSKI and G. SROUR, On stable non-equational theories, manuscript.

[5] M. JUNKER, A note on equational theories, this JOURNAL, vol. 65 (2000), no. 4, pp. 1705-1712. [6] M. JUNKER and D. LASCAR, The indiscernible topology: A mock Zariski topology, Journal of

Mathematical Logic, vol. 1 (2001), no. 1, pp. 99-124. [7] B. KIM, Simplefirst order theories, Ph.D. thesis, University of Notre Dame, 1996. [8] , Simplicity, and stability in there, this JOURNAL, vol. 66 (2001), no. 2, pp. 822-836. [9] B. KIM and A. PILLAY, Simple theories, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2-3,

pp. 149-164. [10] I. KRAus, Lokale Stabilitdt in generischen Modellen, Ph.D. thesis, Universitdt Freiburg, 2001. [11] D. LASCAR, On the category of models of a complete theory, this JOURNAL, vol. 47 (1984), no. 2,

pp. 249-266. [12] , StabilitW en theorie des modules, 1986, Cabay, Louvain-la-Neuve. [131 A. PILLAY, Geometric Stability Theory, Clarendon Press, Oxford, 1996. [14] A. PILLAY and B. POIZAT, Pas d'imaginaires dans l'infini!, this JOURNAL, vol. 52 (1987), no. 2,

pp. 400-403. [15] A. PILLAY and G. SROUR, Closed sets and chain conditions in stable theories, this JOURNAL, vol. 49

(1984), no. 4, pp. 1350-1362. [16] H. SCHEUERMANN, Generalized independence and canonical bases, in preparation. [17] , Unabhiingigkeitsrelationen, Diploma Thesis, Universitat Freiburg, 1996. [18] G. SROUR, The independence relation in separably closedfields, this JOURNAL, vol. 51 (1986), no. 3,

pp. 715-725.

UNIVERSITAT FREIBURG

INSTITUT FUR MATHEMATISCHE LOGIK

ECKERSTRASSE 1, 79104 FREIBURG, GERMANY

E-mail: junker~mathematik.uni-freiburg.de URL: http://sunpool.mathematik.uni-freiburg.de/home/junker E-mail: kraus~logik.mathematik.uni-freiburg.de



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