aa r X i v : . [ m a t h . L O ] S e p TROIS COULEURS: A NEW NON-EQUATIONAL THEORY
AMADOR MARTIN-PIZARRO AND MARTIN ZIEGLER
Abstract.
A first-order theory is equational if every definable set is a Booleancombination of instances of equations, that is, of formulae such that the familyof finite intersections of instances has the descending chain condition. Equa-tionality is a strengthening of stability yet so far only two examples of non-equational stable theories are known. We construct non-equational ω -stabletheories by a suitable colouring of the free pseudospace, based on Hrushovskiand Srour’s original example. Introduction
Consider a first order complete theory T . A formula ϕ ( x ; y ) is an equation (for agiven partition of the free variables into x and y ) if, in every model of T , the familyof finite intersections of instances ϕ ( x, a ) has the descending chain condition. Thetheory T is equational if every formula ψ ( x ; y ) is equivalent modulo T to a Booleancombination of equations ϕ ( x ; y ) .Determining whether a particular stable theory is equational is not obvious. Sofar, the only known natural example of a stable non-equational theory is the freenon-abelian finitely generated group [14, 10], though the first example of a non-equational stable theory is of combinatorial nature and appeared in unpublishednotes of Hrushovski and Srour [7]. They coloured the free pseudospace [4] with twocolours in order to obtain two types r ( x, y ) = r ′ ( x, y ) which are not equationallyseparated , according to the terminology of [6, Section 2.1], that is, there are se-quences ( a i , b i ) i ∈ N and ( c i , d i ) i ∈ N , which can be assumed indiscernible over ∅ , suchthat r ( a i , b i ) and r ′ ( c i , d i ) holds for all i , but r ′ ( a i , b j ) and r ( c i , d j ) holds for i < j .In an equational theory, any two distinct types are equationally separated.All previously known examples of non-equational theories are so, due to thepresence of two distinct non-equationally separated types r ( x, y ) = r ′ ( x, y ) suchthat the length of x is . In this note, we will build on Hrushovski-Srour’s examplein order to construct new examples of non-equational theories, where all distinctreal types p = q in finitely many variables are equationally separated.2. Equations and indiscernibly closed sets
Most of the results in this section come from [11, 9].Consider a first order theory T . A formula ϕ ( x ; y ) is an equation (with respectto a given partition of the free variables into x and y ) if, in every model of T , the Date : May 20, 2019 .1991
Mathematics Subject Classification.
Key words and phrases.
Model Theory, Equationality.Research partially supported by the program MTM2017-82105-P. family of finite intersections of instances ϕ ( x, b ) has the descending chain condition.An easy compactness argument shows Lemma 2.1.
The formula ϕ ( x ; y ) is an equation if there is no sequence ( a i , b i ) i ∈ N in any model M such that M | = ϕ ( a i , b j ) and M = ϕ ( a i , b i ) for all i < j . A Ramsey argument shows that, working in a sufficiently saturated model, thesequence ( a i , b i ) can be assumed to be indiscernible of any infinite order type.Thus, if ϕ ( x ; y ) is an equation, then so are ϕ − ( x ; y ) = ϕ ( y, x ) and ϕ ( f ( x ); y ) ,whenever f is a ∅ -definable function, which maps finite tuples to finite tuples.Finite conjunctions and disjunctions of equations are again equations. Note thatequations are stable formulae.In [9], an equivalent definition of equations was obtained in terms of indiscerniblyclosed sets: an element c lies in the indiscernible closure icl( X ) of a set X if thereis an indiscernible sequence ( a i ) i ∈ N such that a i lies in X for i > and a = c .Note that X ⊂ icl( X ) . A set X is indiscernibly closed if X = icl( X ) . Lemma 2.2. [9, Theorem 3.16]
A formula ϕ ( x ; y ) is an equation if and only if theset ϕ ( M, b ) is indiscernibly closed in in every model M of T .Proof. Let us work inside a sufficiently saturated model M . If ϕ ( x ; y ) is not anequation, witnessed by the indiscernible sequence ( a i , b i ) i ∈ Z , as in Lemma 2.1, theset defined by ϕ ( x, b ) is not indiscernibly closed, for it contains all a i ’s with i < ,but does not contain a . Conversely, if some instance ϕ ( x, b ) is not indiscerniblyclosed, there is an indiscernible sequence ( a i ) i ∈ Z such that M | = ϕ ( a i , b ) for i < ,but M = ϕ ( a , b ) . For every j in Z , there is an element b j in M such that M | = ϕ ( a i , b j ) for i < j , but M = ϕ ( a j , b j ) . (cid:3) The theory T is equational if every formula ψ ( x ; y ) is equivalent modulo T toa Boolean combination of equations ϕ ( x ; y ) . Since Boolean combinations of stableformulas are stable, equational theories are stable.Typical examples of equational theories are the theory of an equivalence relationwith infinite many infinite classes, the theory of R -modules for some ring R , or thetheory of algebraically closed fields.Equationality is preserved under unnaming parameters and bi-interpretability[8]. It is unknown whether equationality holds if every formula ϕ ( x ; y ) , with x asingle variable, is a boolean combination of equations.It is easy to see that T is equational if and only if all completions of T areequational. So for the rest of this section we assume that T is complete and workin a sufficiently saturated model U .Notice that a theory T is equational if and only if every type p over A is impliedby its equational part { ϕ ( x, a ) ∈ p | ϕ ( x ; y ) is an equation } . Definition 2.3.
Given two types p ( x, b ) and q ( x, b ) , define p ( x, b ) → q ( x, b ) if q ( x, b ) ⊂ icl( p ( x, b )) , or equivalently, if there is an indiscernible sequence ( a i ) i ∈ N such that all | = p ( a i , b ) for i > and | = q ( a , b ) . If p ( x, y ) and q ( x, y ) are the thecorresponding (complete) types over ∅ , we write p ( x ; y ) → q ( x ; y ) . A standard argument as in Lemma 2.2 with p instead of ϕ and and q instead of ¬ ϕ yields the following: ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 3
Lemma 2.4.
We have p ( x ; y ) → q ( x ; y ) if an only if there is a sequence ( a i , b i ) i ∈ N such that | = p ( a i , b j ) for i < j , and | = q ( a i , b i ) for all i . Furthermore, we mayassume that the sequence is indiscernible and of any given infinite order type. The above characterisation provides an easy proof of the following remark:
Remark 2.5.
Clearly p → p . If p ( x ; y ) → q ( x ; y ) , then p − → q − , where p − ( x ; y ) = p ( y ; x ) .Furthermore, if tp( a ; b ) → tp( a ′ ; b ) , then a s tp ≡ a ′ . Thus, if p ( x ; y ) implies that x (or y ) is algebraic, then p → q only when q = p . Corollary 2.6.
Let f and g be ∅ -definable functions and a, a ′ , b, b ′ finite tuples,with tp( a ; b ) → tp( a ′ ; b ′ ) . Then tp( f ( a ); g ( b )) → tp( f ( a ′ ); g ( b ′ )) . Corollary 2.7.
A formula ϕ ( x ; y ) is an equation if and only if, whenever a type p ( x, y ) contains ϕ ( x, y ) and p ( x ; y ) → q ( x ; y ) , then ϕ ( x, y ) lies in q ( x ; y ) .Proof. One direction follows clearly from Lemma 2.2. For the converse, assumethat ϕ ( x ; y ) is not an equation and choose an indiscernible sequence ( a i , b i ) i ∈ N asin Lemma 2.1. Let p be the common type of the pairs ( a i , b j ) , with i < j and q bethe common type of the pairs ( a i , b i ) . Then p → q and ϕ belongs to p , but not to q . (cid:3) Definition 2.8. A cycle of types is a sequence p ( x ; y ) → p ( x ; y ) → · · · → p n − ( x ; y ) → p ( x ; y ) . The cycle is proper if all the p i ’s are different. The theory T is indiscernibly acyclic if there is no proper cycle of types of length n ≥ .Following the terminology of [6, Section 2.1], two distinct types p ( x ; y ) and q ( x ; y ) are not equationally separated if and only p → q → p . Remark 2.9.
Every indiscernibly acyclic theory is stable.
Proof.
If there is a formula ϕ ( x ; y ) in T with the order property, find an indiscerniblesequence ( a i ) i ∈ Z in U such that | = ϕ ( a i , a j ) if and only if i < j . Set p = tp( a ; a ) and q = tp( a − ; a ) . Then p = q , and since the sequence ( a i ) i =0 is indiscernible,we have that p → q → p , so there is a proper cycle of types of length . (cid:3) Remark 2.10.
Every equational theory is indiscernibly acyclic.
Proof.
Consider a cycle p → p → · · · → p n − → p . By Corollary 2.7, all the types p i contain the same equations, so they all agree, byequationality of T . (cid:3) Definition 2.11.
The theory T satisfies the MS-criterion if there is some formula ϕ ( x, y ) and a matrix ( a ij , b ij ) i,j ∈ N such that:(1) | = ϕ ( a ij , b il ) if and only if j = l .(2) a ij , b ij ≡ a ij , b kl , whenever i < k and j < l . Lemma 2.12.
If a theory T satisfies the MS-criterion, then there is a proper cycleof types p → q → p . In particular, the theory is not equational (cf. [10, Proposition2.6] ). AMADOR MARTIN-PIZARRO AND MARTIN ZIEGLER
Proof.
We may assume that the matrix ( a ij , b ij ) i,j ∈ N is indiscernible, that is, thetype tp( a ij , b ij ) i ∈ I,j ∈ J only depends on | I | and | J | . Set p = tp( a ; b ) , q =tp( a ; b ) and r = tp( a ; b ) . Since ( a j b j ) j ∈ N is indiscernible, we have q → p .Since ( a i b i ) i ∈ N is indiscernible, we have r → q .Now, by Definition 2.11 (1) , the formula ϕ ( x ; y ) belongs to p ( x ; y ) but not to q ( x ; y ) , so p = q . By Definition 2.11 (2) we have p = r , as desired. (cid:3) Since p and q contain the same equations, it follows that ϕ cannot be a booleancombination of equations (cf. [10, Proposition 2.6]).Let us assume for the rest of this section that T is stable. Lemma 2.13.
Let p ( x, b ) → · · · → p n − ( x, b ) → p ( x, b ) be a proper cycle oftypes and b ′ be some tuple such that p ( x, b ) has only finitely many distinct non-forking extensions to bb ′ . Then there is a proper cycle of types starting with somenonforking extension p ′ ( x ; b, b ′ ) of p ( x, b ) whose length is a multiple of n .Proof. First notice that, whenever p ( x, b ) → q ( x, b ) and q ′ ( x, b, b ′ ) is a non-forkingextension of q ( x, b ) , then p ( x, b ) has a nonforking extension p ′ ( x, b, b ′ ) with p ′ ( x ; b, b ′ ) → q ′ ( x ; b, b ′ ) . Indeed, consider an indiscernible sequence ( a i ) i ∈ N such that | = p ( a i , b ) , for i > ,and q ( a , b ) . We may assume that a realises q ′ ( x, b, b ′ ) and that the sequence ( a i ) i ∈ N is independent from b ′ over b . By a Ramsey argument, we may assume thatthe sequence ( a i ) i> is indiscernible over a bb ′ . Set now p ′ ( x, b, b ′ ) to be the typeof a over bb ′ , so p ′ ( x, b, b ′ ) → q ′ ( x, b, b ′ ) , as desired.Let k now be the number of distinct nonforking extensions of p ( x, b ) to bb ′ .Working backwards in the cycle of types, we deduce from the above that there is asequence r ( x ; b, b ′ ) → · · · → r n · k ( x ; b, b ′ ) , where r n · i + j ( x, b, b ′ ) is a non-forking ex-tension of p j ( x, b ) for each i ≤ k . Since p has only finitely many distinct nonforkingextensions to bb ′ , there are two indices i < i ′ such that r n · i ( x, b, b ′ ) = r n · i ′ ( x, b, b ′ ) .Choose i and i ′ such that < i ′ − i is least possible. Then r n · i ( x ; b, b ′ ) → · · · → r n · i ′ ( x ; b, b ′ ) is a proper cycle of types. (cid:3) Corollary 2.14. If T is totally transcendental, then it is indiscernibly acyclic ifand only if so is T eq .Proof. We need only show that T eq is indiscernibly acyclic, provided that T isindiscernibly acyclic. Assume first that the type p ( x, e ) starts a proper cycle oftypes, where e is an imaginary element. Choose a real tuple b such that π E ( b ) = e for some -definable equivalence relation E . Since T is totally transcendental, thetype p ( x, b ) has only finitely many nonforking extensions to { b, e } , so there is aproper cycle starting with some nonforking extension p ′ ( x, b, e ) , by the Lemma2.13. By the Corollary 2.6, if we restrict the types in the cycle to b , we have a cycleof types which must be proper, because e is definable from b .Since the relation → is symmetric in x and y , we can now replace x by some realtuple, so T is not indiscernibly acyclic. (cid:3) Notation.
Given two stationary types p ( x, b ) and p ( x ′ , b ) , we denote by p ( x, b ) ⊗ p ( x ′ , b ) the type of the pair ( a , a ) over b , where | = p i ( a i , b ) , for i = 1 , , and a | ⌣ b a . ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 5
Observe that p ( x, b ) ⊗ (cid:16) p ( x ′ , b ) ⊗ p ( x ′′ , b ) (cid:17) = (cid:16) p ( x, b ) ⊗ p ( x ′ , b ) (cid:17) ⊗ p ( x ′′ , b ) . Lemma 2.15.
Given stationary types p j ( x j , y j , c ) and q j ( x j , y j , c ) over a tuple c in acl eq ( ∅ ) such that p j ( x j ; y j , c ) → q j ( x j ; y j , c ) , for j = 1 , , then p ( x ; y , c ) ⊗ p ( x ; y , c ) → q ( x ; y , c ) ⊗ q ( x ; y , c ) . By the above, the lemma generalises to an arbitrary finite product of types.
Proof.
For j = 1 , , choose a tuple b j and an indiscernible sequence ( a ji ) i ∈ N suchthat | = p j ( a ji , b j , c ) , for i > , and | = q j ( a j , b j , c ) . We may assume that b ∪ { a i } i ∈ N | ⌣ c b ∪ { a i } i ∈ N . Since c is algebraic over ∅ , the sequences { a i } i ∈ N and { a i } i ∈ N are both indiscernibleover c and therefore mutually indiscernible, by stationarity of strong types, so { a i , a i } i ∈ N is indiscernible. Notice that ( a i , a i ) realises p ( x ; b , c ) ⊗ p ( x ; b , c ) for i > , and ( a , a ) realises q ( x ; b , c ) ⊗ q ( x ; b , c ) , as desired. (cid:3) Proposition 2.16. If T is totally transcendental, then it is indiscernibly acyclic ifand only if there is no proper cycle of types in T eq of length .Proof. By Corollary 2.14, we need only prove one direction, so suppose p ( x, y ) → · · · → p n − ( x, y ) → p ( x, y ) is a proper cycle of types with real variables. Since T is totally transcendental,there is a finite tuple c in acl eq ( ∅ ) such that all nonforking extensions of all p i ’s to c are stationary. Lemma 2.13 gives a proper cycle of stationary types p ( x ; y, c ) → · · · → p k − ( x ; y, c ) → p ( x ; y, c ) for some k in N .Denote by ¯ x = ( x , . . . , x k − ) and ¯ y = ( y , . . . , y k − ) and consider the types r (¯ x ; ¯ y, c ) = p ( x , y , c ) ⊗ p ( x , y , c ) ⊗ . . . ⊗ p k − ( x k − , y k − , c ) r (¯ x ; ¯ y, c ) = p ( x , y , c ) ⊗ p ( x , y , c ) ⊗ . . . ⊗ p k − ( x k − , y k − , c ) r (¯ x ; ¯ y, c ) = p ( x , y , c ) ⊗ p ( x , y , c ) ⊗ . . . ⊗ p k − ( x k − , y k − , c ) ⊗ p ( x k − , y k − , c ) The Lemma 2.15 yields the cycle of types r (¯ x ; ¯ y, c ) → r (¯ x ; ¯ y, c ) → r (¯ x ; ¯ y, c ) . Given (¯ a, ¯ b ) realising r (¯ x ; ¯ y, c ) and (¯ a ′ , ¯ b ′ ) realising r (¯ x ; ¯ y, c ) , notice that ( a , a , . . . , a k − , a , b , b , . . . , b k − , b ) realise r (¯ x ; ¯ y, c ) . If f denotes the function which maps a k − -tuple ( f , . . . , f k − ) to the imaginary coding the set { f , . . . , f k − } , Corollary 2.6 implies that tp( { a , . . . , a k − } ; { b , . . . , b k − } , c ) → tp( { a , . . . , a k − } ; { b , . . . , b k − } , c ) →→ tp( { a , . . . , a k − } ; { b , . . . , b k − } , c ) AMADOR MARTIN-PIZARRO AND MARTIN ZIEGLER
In order to conclude, we need only show that the above two imaginary types aredifferent. Otherwise, if the two types are equal, we have for each ≤ i ≤ k − , twovalues ≤ ρ ( i ) , τ ( i ) ≤ k − such that ( a ρ ( i ) , b τ ( i ) ) | = p i ( x, y, c ) . Observe that notwo elements a i and a j , with i = j , can be equal since the independence a i | ⌣ c a j would imply that a i is algebraic, and thus p i +1 ( x, y, c ) = p i ( x, y, c ) , by the Remark2.5. Likewise, no two elements b i and b j can be equal, for i = j . Thus, each of themaps i ρ ( i ) and i τ ( i ) is a bijection.If ρ ( k −
1) = τ ( k −
1) = j , then ( a j , b j ) realises both p j ( x ; y, c ) and p k − ( x ; y, c ) ,which contradicts that the cycle of types is proper. Hence, the values ρ ( k − and τ ( k − are different, so there must be some ≤ i ≤ k − such that ρ ( i ) = τ ( i ) .The independences a ρ ( i ) | ⌣ c b τ ( i ) and a ρ ( k ) | ⌣ c b τ ( k ) imply that p i ( x, y, c ) = p k − ( x, y, c ) , by the Remark 2.5 and stationarity of strongtypes, which yields the desired contradiction. (cid:3) We do not know whether Corollary 2.14 and Proposition 2.16 are true for arbi-trary stable theories.All known examples of non-equational stable theories have a proper cycle of realtypes of length . Indeed, in Hrushovski and Srour’s primordial example [7], thetype of a white point and the type of a red point in a plane indiscernibly convergeto each other, whereas the non-abelian free group satisfies the MS-criterion [10,Lemmata 3.4 & 3.6]. In this note, we will provide new examples of non-equationaltotally transcendental theories, one for each natural number k , having proper cyclesof length k but no proper cycles of real types of length strictly smaller than k . Wewill do so by suitable colouring the free pseudospace, mimicking the constructionof Hrushovski and Srour. The following question seems hence natural, though wedo not have a solid guess what the answer will be. Question.
Is there a non-equational indiscernibly acyclic theory?Related to the above, we wonder whether there is a local characterisation ofequationality in terms of cycles of types:
Question.
Is a formula ϕ ( x, y ) a Boolean combination of equations if and only ifwhenever ϕ ∈ p ( x, y ) → p ( x, y ) → . . . → p n − ( x, y ) → p ( x, y ) , then ϕ belongs to p i for every i > ?Do two types p and q contain the exact same equations if and only if p and q both occur in a (proper) cycle of types?Observe that a positive answer to the second question would positively answerthe first one. 3. Indiscernible Kernels
To our knowledge, the results in this section only appeared in print form inAdler’s Master’s Thesis [1] (in German). Therefore, we will include their proofs,even if the results are most likely well-known among the community.As before, work inside a sufficiently saturated model U of the complete theory T . ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 7
Notation.
Given two subsets I and I of a linearly ordered infinite index set withno endpoints, we write I ≪ I if i < i for all i in I and i in I . If ( a i ) i ∈ I is asequence indexed by I , set acl eq ( a I ) = acl( { a i } i ∈ I ) . Definition 3.1.
The kernel of the indiscernible sequence ( a i ) i ∈ I is defined as Ker(( a i ) i ∈ I ) = [ I ,I ⊂ I I ≪ I acl eq ( a I ) ∩ acl eq ( a I ) . Note that we may assume that both I and I are finite subsets of I . Furthermore,the set acl eq ( a I ) ∩ acl eq ( a I ) only depends on | I | and | I | (possibly after enlarging I ), since ( a i ) i ∈ I \ I is indiscernible over a I . If the sequence is indiscernible as aset (which is always the case in stable theories), then we may define the kernel byconsidering all the intersections given by pairs ( I , I ) with I ∩ I = ∅ .Observe that (if I is large enough), Ker(( a i ) i ∈ I ) = acl eq ( a I ) ∩ acl eq ( a I ) , for any I < I both infinite . Lemma 3.2.
The kernel K of an indiscernible sequence ( a i ) i ∈ I is the largest subsetof acl eq (( a i ) i ∈ I ) over which the sequence is indiscernible.Proof. We may assume that I has no endpoints. Clearly, the sequence is indis-cernible over K . Given a tuple b in acl eq ( a I ) , for I ⊂ I finite, such that thesequence is indiscernible over b , the tuple b lies in acl eq ( a I ) , whenever I < I , so b lies in K . (cid:3) Lemma 3.3. If T is stable, then the kernel K of an indiscernible sequence ( a i ) i ∈ I is the smallest algebraically closed subset (in T eq ) over which the sequence is inde-pendent.Proof. Let E be an algebraically closed subset (in T eq ) such that ( a i ) i ∈ I is E -independent. In particular, for each I < I , we have that a I | ⌣ E a I , so K ⊂ E .Let now p = Av(( a i ) i ∈ I ) be the average type, that is, p = { ϕ ( x, b ) L U -formula | ϕ ( a i , b ) for all but finitely many i ∈ I } . Since p is invariant over every infinite subsequence of ( a i ) i ∈ I , its canonical base C is contained in K . Thus, the sequence is C -indiscernible and p is a nonforkingextension of the stationary type p ↾ K .It suffices to show that a i | = p ↾ K ∪ ( a j ) j
In a stable theory T , every indiscernible sequence is a Morleysequence over its kernel. Using kernels, we can provide a different characterisation of the relation → in astable theory. AMADOR MARTIN-PIZARRO AND MARTIN ZIEGLER
Corollary 3.5.
Given types p ( x ; y ) and q ( x ; y ) in a stable theory T , we have that p → q if and only if there is a set C and tuples a , a ′ and b such that: • | = p ( a, b ) and | = q ( a ′ , b ) ; • a s tp ≡ C a ′ , and • a | ⌣ C b .In particular, given a cycle p ( x, y ) → p ( x, y ) → . . . → p n − ( x, y ) → p ( x, y ) , there are tuples b , a , . . . , a n and subsets C , . . . , C n − such that: • | = p r ( a r , b ) , for ≤ r ≤ n − , and | = p ( a n , b ) . • a r s tp ≡ C r a r +1 for ≤ r ≤ n − . • a r | ⌣ C r b for ≤ r ≤ n − .Proof. If p → q , choose some tuple b and an indiscernible sequence ( a i ) i< | T | + suchthat q ( a , b ) and p ( a i , b ) for each i > . Consider the kernel K of the sequence,which is algebraically closed in T eq , so a i s tp ≡ K a j for all i, j . In particular, thesubsequence ( a i )
Remark 3.6.
In a stable theory T , a formula ϕ ( x ; y ) is an equation if and only iffor every set set C and tuples a , a ′ and b such that ϕ ( a, b ) holds with a | ⌣ C b , thenso does ϕ ( a ′ , b ) hold, whenever a ′ s tp ≡ C a . Proof.
Given C , a , a ′ and b as in the statement, Corollary 3.5 yields that tp( a, b ) → tp( a ′ , b ) . As ϕ belongs to tp( a, b ) , it must lie in tp( a ′ , b ) , by Corollary 2.7.For the other direction, it suffices to show that ϕ lies in q , whenever ϕ belongsto p and p → q , by Corollary 2.7. By Corollary 3.5, there are C , a , a ′ and b suchthat p ( a, b ) , q ( a ′ , b ) , a | ⌣ C b and a ′ s tp ≡ C a . Since ϕ ( a, b ) holds, we conclude that sodoes ϕ ( a ′ , b ) , that is, the formula ϕ belongs to q , as desired. (cid:3) A blank pseudospace
Hrushovski and Srour produced the first example [7] of a non-equational stabletheory by adding two colours to an underlying ( -dimensional) free pseudospace,a structure later studied by Baudisch and Pillay [4]. Subsequently, the free ( n -dimensional) pseudospace has been considered from different perspectives, eitheras a lattice [12, 13] or as a right-angled building [2, 3], in order to show that theample hierarchy is strict. In this section, we will recall the basic properties of thefree -dimensional pseudospace.A geometry is a graph whose vertices have levels , and . Vertices of level are called points (usually denoted by the letter c ), whereas vertices of level are lines (denoted by b ) and vertices of level are planes (denoted by a ). By an abuse ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 9 of notation, we say that the point c lies in the plane a if there is a line b containedin a passing through c , though there are no edges between points and planes. Werefer to a subgraph of the form a − b − c as a flag .A letter s is a non-empty subinterval of [0 , . Given a flag F in a geometry A , anew geometry B is obtained from A, F via the operation s by freely adding a newflag G which coincides with F on the levels in [0 , \ s : abcF c ′ G [0] Operation [0] abcF b ′ G [1] Operation [1] abcF a ′ G [2] Operation [2] cbaF b ′ a ′ G [1 , Operation [1 , abcF b ′ c ′ G [0 , Operation [0 , abcF a ′ b ′ c ′ G [0 , Operation [0 , The free pseudospace M ∞ is obtained by successively applying countably manytimes all of the above operations starting from a flag. The geometry M ∞ is inde-pendent, up to isomorphism, of the order in which the operations are applied. Itis denoted by M ∞ in [2, Definition 4.6]. Observe that the geometry obtained byonly considering the operations , and is an elementary substructure of M ∞ (namely, the prime model).We will now exhibit the axioms for the theory PS of M ∞ . Let us first fix somenotation. A word is a sequence of letters. A permutation of the word u is obtainedby successively replacing an occurrence of the subword · by the subword · ;similarly the subword · is permuted to · . The word u is reduced if it doesnot contain, up to permutation, a subword of the form s · t , where s ⊂ t or t ⊂ s (please note that our notation s ⊂ t does not imply s ( t ).A flag path F −→ s F · · · F n − −→ s n F n with word u = s · · · s n is a sequence of flags such that, for each ≤ i ≤ n , the flag F i differs from F i − exactly in the levels in [0 , \ s i . The above flag path is reduced if its word is reduced and for each i , the flags F i − and F i cannot be connectedby a splitting , that is, a flag subpath whose word consists of proper subletters of s i . It is not hard to show that every two flags are connected by a reduced path [3,Corollary 3.13]. Fact 4.1. [3, Theorem 4.12] The theory PS is axiomatised by the following prop-erties: (1) The universe is a geometry such that every vertex lies in a flag.(2) For every level i in [0 , and every flag F , there are infinitely many flags G with F −→ i G .(3) Every closed reduced flag path F −→ s F · · · F n − −→ s n F has length n = 0 .It was proved in [3, Theorem 3.26] that property (3) can be expressed by a set ofelementary sentences.We will now describe types and the geometry of forking in the pseudospace. Werefer the reader to [3, Sections 3–7] for the corresponding proofs. Since there areno non-trivial reduced closed paths of flags, the word u connecting two flags F and G by a reduced path F −→ u G is unique, up to permutation, and will be denoted by d( F, G ) . The flags F and G agree modulo a subset S of [0 , , that is, they have thesame vertices in all levels off S , if and only if the letters in d( F, G ) are all containedin S . In particular, the collection of points and lines, resp. lines and planes, form apseudoplane, so every two lines intersect in at most one point, resp. lie in at mostone plane. Furthermore, the intersection of two distinct planes is either empty, aunique point or a unique line [4]. Actually, the geometry forms a lattice, once asmallest element and a largest element are added [12].If u = d( F, G ) = u · u , given two reduced flag paths HF G,H u u u u and a vertex p in H of level i which does not wobble , that is, such that u · [ i ] or [ i ] · u is reduced, then p is also a vertex of H . In particular, the vertex p isdefinable over F, G .A non-empty subset A of M ∞ is nice if: • every vertex in A lies in a flag fully contained in A ; and • every two flags in A are connected by a reduced path of flags in A .algebraic closure and the definable closure of a set X agree [13, Corollary 5.4] andcoincide with the intersection of all nice sets A ⊃ X . If X is finite, then so isthe algebraic closure. The quantifier-free type of a nice subset determines its type.More generally: Fact 4.2. [13, Corollary 3.12] The quantifier-free type of an algebraically closedsubset determines its type in PS .Observe that if we apply one of the operations [0] , [1] or [2] to a flag in a niceset A , the resulting geometry is again nice.Given a flag F and a nice subset A , there is a flag G in A (called a base-point of F over A ) such that, for any flag G ′ in A , the word d( F, G ′ ) is the non-splittingreduction of d( F, G ) · d( F, G ) , that is, whenever a subword s · t or t · s occurs ina permutation of the product d( F, G ) · d( F, G ) , with s ⊂ t , we cancel s . If weconsider a reduced flag path P connecting F to some base-point G over A withword d( F, G ) , the set A ∪ P is again nice. Any flag occurring in the nice set P appears in a permutation of the path P . ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 11
The theory PS of M ∞ is ω -stable of rank ω , equational with perfectly trivialforking and has weak elimination of imaginaries. Forking can be easily described:Given nice sets A and B containing a common algebraically closed subset C , wehave that A | ⌣ C B if and only if for every nice set D ⊃ C and flags F in A and H in B we have that d( F, G ) is the non-splitting reduction of d( F, G ) · d( G, H ) , where G is a base-point of F over D . In particular, F | ⌣ G D. Remark 4.3. [13, Proposition 4.3 & Theorem 4.13] Assume that A , B and C = A ∩ B are algebraically closed and A | ⌣ C B . Then(1) A ∪ B is algebraically closed,(2) if a vertex x in A is directly connected to a vertex y in B , then x or y mustlie in C ,(3) if a point in A lies in a plane of B , then there is a line in C connectingthem,(4) a point c , which belongs to both a line in A \ C and to a line in B \ C , liesin C .Before introducing the k -colored pseudospace in section 5, we will prove severalauxiliary results about the free pseudospace. We hope that this will allow the readerto become more familiar with the theory PS . Lemma 4.4.
Let X and Y be algebraically closed sets independent over their com-mon intersection Z . Given a point c not contained in Y \ Z lying in the line b of X , then X ∪ { c } | ⌣ Z Y. Proof.
By the transitivity of non-forking, we may assume that Z = X . If c belongsto X , then there is nothing to prove. Otherwise, the type of c over X has Morleyrank (it is actually strongly minimal), by [4, Remark 6.2] (cf. [2, Corollary 7.13]).Since the extension tp( c/Y ) is not algebraic, it does not fork over X . (cid:3) Lemma 4.5.
The type of a set X is determined by the collection of types tp( x, x ′ ) ,with x and x ′ in X .In particular, if X ≡ Z X ′ and X ≡ Y X ′ , then X ≡ Y Z X ′ .Proof. Choose an enumeration of X = { x α } α<κ and flags F α containing x α , for α < κ , such that F α | ⌣ x α X ∪ { F β } β<α . In particular, for α = β , we have that F α | ⌣ x α x β and F α | ⌣ x α ,x β F β . Since the type of F α over x α is stationary, the type of the pair ( x α , x β ) determinesthe type of F α , F β . By [3, Theorem 7.24], the type of ( F α ) α<κ , hence the type of X , is uniquely determined by the collection of types tp( x α , x β ) , for α, β < κ . (cid:3) A colored pseudospace
Work inside a sufficiently saturated model U of the theory PS of the free pseu-dospace and consider a natural number k ≥ . For ≤ i < k , we use the notation i + 1 instead of i + 1 mod k , and likewise i − for i − mod k . We colour the lines in U , as well as the pairs ( a, c ) , where the point c lies in theplane a , with k many colours. Formally, we partition the set of lines into subsets C , . . . , C k − , and the set of pairs ( a, c ) , where c lies in the plane a , into I , . . . , I k .Given a plane a and an index ≤ i < k , we denote by the section I i ( a ) the collectionof points c with I i ( a, c ) .Consider the theory CPS k of k -colored pseudospaces with following axioms: • The axioms of PS . Universal Axioms • For each ≤ i < k , given a line b with colour i in a plane a , all the points c in b lie in the section I i ( a ) except at most one point, which lies in I i +1 ( a ) (if it exists, we call it the exceptional point of b in a ). Inductive Axioms • Every line b in a plane a contains an exceptional point, denoted by ep( a, b ) . • For each ≤ i < k , given a point c and a plane a with I i ( a, c ) , there areinfinitely many lines in a passing through c with colour i . • For each ≤ i < k , given a point c and a plane a with I i ( a, c ) , there areinfinitely many lines in a passing through c with colour i − . • For every point c in a line b , there are infinitely many planes a containing b such that c is exceptional for b in a .We can construct a model of CPS k as follows: We start with a flag A = { a − b − c } with any colouring, eg. b ∈ C and I ( a, c ) and construct an ascending sequence A ⊂ A ⊂ · · · of coloured geometries by applying one the operations [0] , [1] and [2] to a flag a − b − c in A j obtain A j +1 , extending the colouring to A j +1 in anarbitrary way whilst preserving the Universal Axioms. For example, do as follows: • Operation [0] adds a new point c ′ to b . If b has colour i , then for all a ′′ in A j containing b , paint the pair ( a ′′ , c ′ ) with the colour i , if ep( a ′′ , b ) alreadyexists in A j . Otherwise, paint ( a ′′ , c ′ ) with the colour i + 1 otherwise. • Operation [1] adds a new line b ′ between a and c . If ( a, c ) has colour i , thenpaint b ′ with the colour i or the colour i − , and see to it that each choiceoccurs infinitely often in the sequence. • Operation [2] adds a new plane a ′ which contains b . If b has colour i , thenfor all c ′′ in A j which lie in b , we give the pair ( a ′ , c ′′ ) one of the colours i or i − . Each choice should occur infinitely often.It is easy to see that the structure obtained in this fashion satisfies all axioms of CPS k , so the theory CPS k is consistent. Notation.
Given a subset X of a model of CPS k , we will denote by h X i thealgebraic closure of X in the reduct PS , and by EP( X ) = { ep( a, b ) , ( a, b ) ∈ X × X } the exceptional points of lines and planes from X . Remark 5.1.
If the point c is directly connected to a line in X , then h X, c i = h X i ∪ { c } .In particular, if X = h X i , given c in EP( X ) , then X ∪ { c } is algebraically closedin the reduct PS . ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 13
Proof.
In order to show that h X, c i = h X i ∪ { c } , it suffices to consider the casewhen X is nice. The geometry X ∪ { c } is either X or obtained from X by applyingthe operation [0] , so it is nice again, and thus algebraically closed. (cid:3) Similar to [13, Proposition 3.10] working inside two ℵ -saturated models of CPS k ,it is easy to see that the collection of partial isomorphisms between PS -algebraicallyclosed finite sets which are closed under exceptional points is non-empty and hasthe back-and-forth property, so we deduce the following: Theorem 5.2.
The theory
CPS k is complete. Given a set X in a model of CPS k with X = h X i and EP( X ) ⊂ X , then the quantifier-free type of X determines itstype. The back-and-forth system yields an explicit description of the algebraic closure,as well as showing that the theory
CPS k is ω -stable, by a standard counting typesargument. Corollary 5.3.
The theory
CPS k is ω -stable. The algebraic closure acl( X ) of aset X is obtained by closing h X i under exceptional points: acl( X ) = h X i ∪ EP( h X i ) . We deduce the following characterisation of forking over (colored) algebraicallyclosed sets.
Corollary 5.4.
Let X and Y two supersets of an algebraically closed set Z = acl( Z ) in CPS k . We have that X CPS k | ⌣ Z Y if and only if • X | ⌣ PS Z Y , and • EP( h X i ) ∩ EP( h Y i ) ⊂ Z .Types over algebraically closed sets are stationary, that is, the theory CPS k hasweak elimination of imaginaries.Proof. Since PS has weak elimination of imaginaries, we have that non-forking in CPS k implies nonforking in the reduct PS over algebraically closed sets, by [5,Lemme 2.1]. Clearly EP( h X i ) ∩ EP( h Y i ) ⊂ Z .For the other direction, we may assume that X = h X i and Y = h Y i . Lemma 4.4yields that X ∪ EP( X ) PS | ⌣ Z Y ∪ EP( Y ) . Since acl( X ) = X ∪ EP( X ) , Remark 4.3 implies that the set acl( X ) ∪ acl( Y ) isalgebraically closed in PS . We need only show that it contains all exceptional points,so it determines a unique type in the stable theory CPS k . If c is an exceptionalpoint of a plane a and a line b in acl( X ) ∪ acl( Y ) , we may assume that a lies in X and b lies in Y . Since a and b are directly connected and X | ⌣ PS Z Y , Remark4.3 implies that a or b lies in Z . Therefore c lies in EP( X ) ∪ EP( Y ) and hence iscontained in acl( X ) ∪ acl( Y ) , as desired. (cid:3) Corollary 5.5.
Let X , Y and Z = acl( Z ) be sets such that X | ⌣ Z Y. Then h X, Y i ∩ acl(
X, Z ) = h X, Y i ∩ h
X, Z i .Proof. Let ξ be in h X, Y i ∩ acl(
X, Z ) . The independence X | ⌣ Z Y yields that ξ, X | ⌣ Z Y. It follows from Corollary 5.4 that ξ, X PS | ⌣ Z Y, and thus ξ PS | ⌣ X,Z
X, Y.
Since ξ lies in h X, Y i , the above independence implies that ξ lies in h X, Z i , asdesired. (cid:3) Proposition 5.6.
Let X = h X i and Y = h Y i be two subsets of a model of CPS k .A map F : X → Y is elementary with respect to the theory CPS k if and only if itsatisfies the following conditions:(1) The map F is a partial isomorphism with respect to the reduct PS .(2) The function F preserves colours of lines and sections.(3) For all a , a ′ and b in X , we have that ep( a, b ) = ep( a ′ , b ) if and only if ep( F ( a ) , F ( b )) = ep( F ( a ′ ) , F ( b )) .Proof. We need only show that F is elementary, if it satisfies all three conditions.By Theorem 5.2, it suffices to show that F extends to a partial isomorphism ˜ F preserving colours between acl( X ) = X ∪ EP( X ) and acl( Y ) = Y ∪ EP( Y ) .For each line b in X contained in a plane a of X , set ˜ F (ep( a, b )) = ep( F ( a ) , F ( b )) .Let us first show that ˜ F is well-defined, which analogously yields that ˜ F is a bi-jection. Suppose that ep( a, b ) = ep( a , b ) , for a line b contained in the plane a ,both in X . If b = b , then ep( a, b ) is the unique intersection of b and b , both linesin X , so ep( a, b ) lies in X and hence its image is determined by F . Otherwise, weconclude that b = b , and thus ˜ F is bijective, by Condition (3) .Similarly, the map ˜ F defined above is a partial isomorphism with respect to thereduct PS . We need only show that ˜ F preserves the colours of sections. Choose anew point ep( a, b ) not in X and an arbitrary plane a = a in X containing ep( a, b ) .Since ep( a, b ) does not lie in X , the intersection of a and a cannot solely consist ofthe point ep( a, b ) . Hence, the intersection of a and a is given by a unique line b ,which lies in X and contains ep( a, b ) . We conclude as before that b = b . The colourof ep( a, b ) in a is uniquely determined according to whether ep( a, b ) = ep( a , b ) ,and thus so is the colour of its image in F ( a ) by ˜ F , by Condition (3) . (cid:3) ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 15 Colored paths
We will now show that the theory
CPS k is not indiscernibly acyclic, and hence itis not equational, yet every proper cycle of types has length at least k (cf. Theorem6.2), so we expect the complexity of these theories to increase as k grows. However,we do not know whether two of these theories are bi-interpretable. Theorem 6.1. In CPS k there is a proper cycle of types p ( x ; y ) → p ( x ; y ) → . . . → p k − ( x ; y ) → p ( x ; y ) , where both the variables x and y have length . In particular, the theory CPS k isnot equational.Proof. For each ≤ r < k , a pair ( a, c ) with colour I r has a unique type p r =tp( c, a ) , for the set { a, c } is algebraically closed, since it is the intersection of all theflags containing a and c , and it is closed under exceptional points, for it containsno line. Clearly p r = p r +1 , for each ≤ r < k .It suffices to show that p r → p r +1 : Let ( a, c ) with colour I r , and choose a line b connecting them with colour r . Let c ′ be the exceptional point of b in a , so ( c ′ , a ) | = p r +1 . Now, the set { b } is algebraically closed in CPS k . By Corollary 5.4,the points c and c ′ have the same strong type over b , and c | ⌣ b a. Corollary 3.5 implies that p r = tp( c, a ) → tp( c ′ , a ) = p r +1 , as desired. (cid:3) Theorem 6.2.
Let x and y be finite tuples of variables. In CPS k , every propercycle of types p ( x ; y ) → p ( x ; y ) → . . . → p n − ( x ; y ) → p ( x ; y ) , has length n ≥ k .Proof. A proper cycle of types p ( x ; y ) → p ( x ; y ) → . . . → p n − ( x ; y ) → p ( x ; y ) as above induces a cycle in the reduct PS , which is equational. Therefore, thecolourless reducts of p r and p s agree, for all r , s .Corollary 3.5 implies that there are tuples f , e , . . . , e n and algebraically closedsubsets Z , . . . , Z n − such that: • | = p r ( e r , f ) , for ≤ r ≤ n − , and | = p ( e n , f ) . • e r ≡ Z r e r +1 for ≤ r ≤ n − . • e r | ⌣ Z r f for ≤ r ≤ n − .Set Y = acl( f ) , X r = acl( e r ) , for ≤ r ≤ n . Since the definable and algebraicclosure coincide, and the colourless reducts of all p r agree, all the types tp PS ( X r Y ) are equal. Denote h X r Y i by P r . We find colourless isomorphisms F r : P r → P r +1 , which fix Y pointwise. Note that X r and X r +1 have the same type over Z r , for r ≤ n − . Lemma 4.5 yields that tp PS ( X r Y Z r ) = tp PS ( X r +1 Y Z r ) , for r ≤ n − .The above map F r extends to a colourless isomorphism between h X r Y Z r i and h X r +1 Y Z r i , which is the identity on h Y Z r i . We will still refer to this colourlessisomorphism as F r , keeping in mind that it is elementary in the sense of CPS k on h X r Z r i and (clearly) on h Y Z r i separately. Observe that h X r Y Z r i = h X r Z r i ∪ h Y Z r i , by the Remark 4.3 (1).If a set W is finite, so are the closures h W i and acl( W ) . Define its defect as thenatural number defect( W ) = | acl( W ) \ h W i| = | EP( h W i ) \ h W i| . Claim.
For each r ≤ n − , we have that defect( P r ) ≥ defect( P r +1 ) .Proof of Claim. Whenever ep( a, b ) = ep( a , b ) , for b and b in P r , with b = b ,then the point ep( a, b ) lies in P r by 4.3 (4). Thus, it suffices to show the following:(1) Whenever the line b in P r lies in the plane a in P r , with ep( a, b ) in P r , then ep( F r ( a ) , F r ( b )) lies in P r +1 .(2) Whenever a , a and b lie in P r and ep( a, b ) = ep( a , b ) , then ep( F r ( a ) , F r ( b )) =ep( F r ( a ) , F r ( b )) .For (1) , since X r | ⌣ Z r Y , the plane a and the line b must both lie in the same set P r ∩ h X r Z r i or in P r ∩ h Y Z r i , by the independence a PS | ⌣ Z r b and Remark 4.3 (2). For example, let a and b lie in P r ∩ h X r Z r i , so ep( a, b ) liesin P r ∩ acl( X r Z r ) = P r ∩ h X r Z r i , by Corollary 5.5. Since F r is elementary on P r ∩ h X r Z r i , we have that ep( F r ( a ) , F r ( b )) = F r (ep( a, b )) lies in P r +1 , as desired.Observe that we have actually shown that ep( a, b ) ∈ P r ⇐⇒ ep( F r ( a ) , F r ( b )) ∈ P r +1 . For (2) , we need only consider the case when a = a and the exceptional point ep( a, b ) = ep( a , b ) does not lie in P , by (1) . Again, if both a and a lie in h X r Z r i or in h Y Z r i , then so does b , and we are done by Proposition 5.6, since F r iselementary on each side. If this is not the case, and a lies in P r ∩ h X r Z r i and a in P r ∩ h Y Z r i , then the line b lies in P r ∩ Z r , by the Remark 4.3 (3). Thus the point ep( a, b ) = ep( a , b ) lies in acl( X r , Z r ) ∩ acl( Y, Z r ) = Z r , so we conclude as beforesince F r is elementary on each side separately. (cid:3) Claim As P and P n have the same type, their defect is the same, so defect( P r ) =defect( P r +1 ) , for all ≤ r ≤ n − . Hence, for all a , a ′ and b in P r , we have that ep( a, b ) = ep( a ′ , b ) if and only if ep( F r ( a ) , F r ( b )) = ep( F r ( a ′ ) , F r ( b )) . Since P r and P r +1 are closed in the reduct PS k , but tp( P r ) = tp( P r +1 ) , Proposi-tion 5.6 implies that F r restricted to P r cannot preserve colours. As F r is elementaryon each side separately, the colours of lines are preserved. Thus, there is a pair ( a, c ) in P r whose colour j , with ≤ j < k , is not preserved under F r . We will show nowthat the colour of the pair ( F r ( a ) , F r ( c )) is j + 1 .Since F r is elementary on h X r Z r i and on h Y Z r i separately, neither a nor c lie in Z r . The independence X r | ⌣ PS Z r Y and Remark 4.3 (3) yield that there is a line b in Z r connecting a and c . The characterisation of the independence in Corollary 5.4implies that c = ep( a, b ) . Hence the line b must have colour j . The map F r is theidentity on Z r , and the plane F r ( a ) is connected to the point F r ( c ) by b = F r ( b ) ,so the only possible colours for the pair ( F r ( a ) , F r ( c )) are j or j + 1 . As the colourof the pair ( a, c ) is not preserved, we deduce that ( F r ( a ) , F r ( c )) has colour j + 1 ,as desired. ROIS COULEURS: A NEW NON-EQUATIONAL THEORY 17
Let F n be the CPS k -elementary map mapping P n to P (as both ( e , f ) and ( e n , f ) realise the type p ) and write F r = F r ◦ . . . ◦ F . Notice that the map F n is the identity of P . Let ( a, c ) be one of the pairs in P whose colour j changesunder F . The colours of the pairs ( a, c ) , F ( a, c ) , . . . , F n − ( a, c ) change at each step by at most adding (modulo k ), so the colour of F n − ( a, c ) equals j + m modulo k , for some ≤ m ≤ n . Since F n preserves colours and F n ( a, c ) = ( a, c ) , we have that m is divisible by k , and thus k ≤ m ≤ n . Weconclude that the original cycle had length at least k . (cid:3) Remark 6.3.
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Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg,Germany
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