Multidimensional exact classes, smooth approximation and bounded 4-types
TTo appear in the
Journal of Symbolic LogicJournal of Symbolic Logic
Submitted 2018-05-10 · Accepted 2019-07-28 · Resubmitted with revisions 2020-05-19
MULTIDIMENSIONAL EXACT CLASSES, SMOOTH APPROXIMATIONAND BOUNDED 4-TYPES
DANIEL WOLF
Dedicated to the memories of my son Arthur and my mother Valerie
Abstract.
In connection with the work of Anscombe, Macpherson, Steinhorn and thepresent author in [11] we investigate the notion of a multidimensional exact class ( R -mec),a special kind of multidimensional asymptotic class ( R -mac) with measuring functionsthat yield the exact sizes of definable sets, not just approximations. We use results aboutsmooth approximation [2424] and Lie coordinatisation [1414] to prove the following result(Theorem 4.6.4Theorem 4.6.4), as conjectured by Macpherson: For any countable language L and anypositive integer d the class C ( L , d ) of all finite L -structures with at most d L , where a polynomial exact class is a multidimensional exactclass with polynomial measuring functions. §1. Introduction. The model-theoretic notion of an asymptotic class wasintroduced by Macpherson and Steinhorn in [3737] as a generalisation of the resultin [88] of Chatzidakis, van den Dries and Macintyre regarding the size of defin-able sets in finite fields. This notion has been further generalised by Anscombe,Macpherson, Steinhorn and the present author in [11] and [4444] to that of a multi-dimensional asymptotic class, also known as an R -mac. Details of the historicaldevelopment of these notions can be found in § 1.1 of [4444].In the present work we focus on multidimensional exact classes, also known as R -mecs, which are a special kind of multidimensional asymptotic class wherethe measuring functions yield the exact sizes of definable sets, not just ap-proximations. We show that multidimensional exact classes and smooth ap-proximation (in the sense of [2424]) are intimately related by proving that ev-ery smoothly approximable structure gives rise to a muldimensional exact class(Proposition 3.2.1Proposition 3.2.1). Using the framework of Lie coordinatisation, as developedby Cherlin and Hrushovski in [1414], we then build on Proposition 3.2.1Proposition 3.2.1 to provethe main result of this paper, as conjectured by Macpherson: Main result (Theorem 4.6.4Theorem 4.6.4) . For any countable language L and any posi-tive integer d the class C ( L , d ) of all finite L -structures with at most d L , where a polynomial exact class is a multidimensionalexact class with polynomial measuring functions. Mathematics Subject Classification.
Key words and phrases.
Asymptotic class, smooth approximation, Lie coordinatisation.The author (born Daniel Wood) was funded by the Leeds School of Mathematics through aGraduate Teaching Assistantship. The present work is taken largely from his PhD thesis [4444].1 a r X i v : . [ m a t h . L O ] M a y DANIEL WOLF
We outline the structure of the present work. In § 2§ 2 we state the definition of an R -mec (and an R -mac), prove some technical lemmas and provide some examplesand non-examples. § 3§ 3 is about smooth approximation and is where we provethe aforementioned Proposition 3.2.1Proposition 3.2.1. In § 4§ 4 we move on to Lie coordinatisation,which we use to prove the main result Theorem 4.6.4Theorem 4.6.4.We make extensive use of the Ryll-Nardzewski Theorem throughout this paper.This is well covered in the literature, for example § 1.3 of [1818], Theorem 7.3.1in [2121], Theorem 5.1 in [2525] and Theorem 4.3.2 in [4242]. We refer to [3939] and [4242]for general model-theoretic notation and terminology. §2. Multidimensional exact classes. We introduce the central definitionof this paper, state and prove some handy lemmas in § 2.3§ 2.3 and then provide some(non-)examples in § 2.4§ 2.4.
Let L be a finitary, first-order language and let C bea class of finite L -structures. For m ∈ N + define C ( m ) := { ( M , ¯ a ) : M ∈ C , ¯ a ∈ M m } . We use M (roman typeface) to denote the underlying set of the structure M (calligraphic typeface), although we do not maintain this distinction throughout.The elements of C ( m ) are sometimes referred to as pointed structures . We define C (0) := C . Definition Φ be a partition of C ( m ) . Anelement π ∈ Φ is definable if there exists a parameter-free L -formula ψ (¯ y ) with l (¯ y ) = m such that for every ( M , ¯ a ) ∈ C ( m ) we have ( M , ¯ a ) ∈ π if and only if M | = ψ (¯ a ) . The partition Φ is definable if π is definable for every π ∈ Φ .The following definition is due to Anscombe, Macpherson, Steinhorn and thepresent author. Definition R -mec). Let C be a class of finite L -structures and let R be a set of functions from C to N . Then C is a multidimensional exact class for R in L , or R -mec in L for short, if for every parameter-free L -formula φ (¯ x, ¯ y ) ,where n := l (¯ x ) ≥ and m := l (¯ y ) , there exists a finite definable partition Φ of C ( m ) such that for each π ∈ Φ there exists h π ∈ R such that | φ ( M n , ¯ a ) | = h π ( M ) (2.1)for all ( M , ¯ a ) ∈ π .Before we provide the first example of an R -mec, we make some initial remarksand observations: Remark h π the measuring functions and the L -formulas thatdefine the partition Φ the defining L -formulas . We often refer to multi-dimensional exact classes simply as exact classes .(ii) In the L -formula φ (¯ x, ¯ y ) it is important to maintain a distinction betweenthe free variables ¯ x and the free variables ¯ y . (Although we use the plu-ral variables , either of ¯ x and ¯ y could denote a single variable.) The freevariables ¯ x , which we call object variables , are slots for solutions in each ULTIDIMENSIONAL EXACT CLASSES M ∈ C . The free variables ¯ y , which we call parameter variables , are slotsfor parameters from each M ∈ C . To aid clarity we sometimes demar-cate the two kinds of free variables with a semicolon, writing φ (¯ x ; ¯ y ) . TheProjection LemmaProjection Lemma (Lemma 2.3.1Lemma 2.3.1) shows that it suffices to consider formu-las with only a single object variable.(iii) We consider two edge cases where the definition holds trivially: • Suppose that φ (¯ x, ¯ y ) is inconsistent, i.e. that φ ( M n , ¯ a ) = ∅ for every ( M , ¯ a ) ∈ C ( m ) . Then the required definable partition of C ( m ) is thetrivial partition {C ( m ) } and the measuring function is M (cid:55)→ . • Suppose that m = 0 , i.e. that the only free variables in φ (¯ x, ¯ y ) are ¯ x . Then C ( m ) = C (0) = C and φ (¯ x, ¯ y ) can be written as φ (¯ x ) . Therequired definable partition of C is the trivial partition {C} and themeasuring function is M (cid:55)→ | φ ( M n ) | .We will henceforth assume formulas to be consistent and m to be positive.(iv) R must be closed under pointwise addition and multiplication: If A and B are definable sets, then their disjoint union A ⊔ B is definable and hassize | A | + | B | and their cartesian product A × B is definable and has size | A | · | B | . So R is generated under addition and multiplication by a subsetof basic functions.(v) If we drop the requirement that the partition Φ be definable, then we call C a weak R -mec . We call (2.12.1) the size clause and the requirement that thepartition be definable the definability clause . So a weak R -mec need satisfyonly the size clause. We sometimes use the term full R -mec to emphasisethat both the size and definability clauses hold and the term strictly weak R -mec to emphasise that only the size clause holds.(vi) R -mecs are closed under taking subclasses of C and supersets of R : If C is an R -mec in L , then any subclass of C is also an R ′ -mec in L for any superset R ′ ⊇ R . Equivalently: If C is not an R -mec in L , then no superclass of C isan R ′ -mec in L for any subset R ′ ⊆ R .Weak R -mecs are closed under taking reducts: Let C be a weak R -mecin L and consider some L ′ ⊆ L . For M ∈ C , let M ′ denote the reductof M to L ′ . Then {M ′ : M ∈ C} is a weak R -mec in L ′ . Equivalently:Suppose that C is not a weak R -mec in L and consider some L ′ ⊇ L . For M ∈ C , let M ′ be an expansion of M to L ′ . Then {M ′ : M ∈ C} is nota weak R -mec in L ′ . Note that we cannot remove the prefix ‘weak’ here,since taking reducts may affect the definability clause.We now provide our first class of examples. More examples are given in § 2.4§ 2.4. Definition C be a class of finite L -structures. We say that C has quantifier elimination in L if Th L ( M ) has quantifier elimination for every M ∈ C , where Th L ( M ) denotes the L -theory of M . Proposition . Let L be a finite relational language and let C be a classof finite L -structures. If C has quantifier elimination in L , then there exists R such that C is an R -mec in L . Proof.
Consider an L -formula φ (¯ x, ¯ y ) , where n := l (¯ x ) ≥ and m := l (¯ y ) .Let A be the set of all L -literals with free variables among ¯ y . A is finite because L DANIEL WOLF is finite and relational. Let B be the set of all maximally consistent conjunctionsof literals from A . B is finite because A is finite. We can thus enumerate theelements of B as ψ (¯ y ) , . . . , ψ d (¯ y ) for some d ∈ N + . Now consider some M ∈ C .Since C has quantifier elimination, Th L ( M ) has quantifier elimination. Thereforeeach complete type in the free variables ¯ y is isolated by one of the ψ i (¯ y ) . Wedefine π i := { ( M , ¯ a ) ∈ C ( m ) : M | = ψ i (¯ a ) } . Then { π , . . . , π d } is a definable partition of C ( m ) . Moreover, for each i , if ( M , ¯ a ) , ( M , ¯ b ) ∈ π i , then tp M (¯ a ) = tp M (¯ b ) and thus, since M is finite, there isan automophism σ : M → M such that σ (¯ a ) = ¯ b , which implies that | φ ( M n , ¯ a ) | = | φ ( M n , ¯ b ) | . So we may define h i ( M ) := | φ ( M n , ¯ a ) | , where ( M , ¯ a ) ∈ π i . Then h i is themeasuring function associated with π i . □ Corollary . The class of finite sets is a multidimensional exact classin the language of pure equality.
Proof.
Let C denote the class of finite sets and L = the language of pureequality. Since L = is finite and relational and C has quantifier elimination in L = , we can apply Proposition 2.1.5Proposition 2.1.5. □ Remark
We provide the definitions of N -dimensional andmultidimensional asymptotic classes, which we have already made reference to.The former is due to Macpherson and Steinhorn [3737] and Elwes [1717]. The latteris due to Anscombe, Macpherson, Steinhorn and the present author [11]. Definition N -dimensional asymptotic class). Let C be a class of finite L -structures and let N ∈ N + . Then C is an N -dimensional asymptotic class iffor every parameter-free L -formula φ (¯ x, ¯ y ) , where n := l (¯ x ) ≥ and m := l (¯ y ) ,there exists a finite definable partition Φ of C ( m ) such that for each π ∈ Φ thereexists ( d, µ ) ∈ ( { , . . . , N n } × R + ) ∪ { (0 , } such that (cid:12)(cid:12)(cid:12) | φ ( M n , ¯ a ) | − µ | M | d / N (cid:12)(cid:12)(cid:12) = o (cid:16) | M | d / N (cid:17) for all ( M , ¯ a ) ∈ Φ ( d,µ ) as | M | → ∞ , where the meaning of the little-o notationis as follows: For every ε > there exists Q ∈ N such that for all ( M , ¯ a ) ∈ π , if | M | > Q , then (cid:12)(cid:12)(cid:12) | φ ( M n , ¯ a ) | − µ | M | d / N (cid:12)(cid:12)(cid:12) ≤ ε | M | d / N . We call ( d, µ ) a dimension–measure pair . Definition R -mac). Let C be a class of finite L -structures and let R bea set of functions from C to R ≥ . Then C is a multidimensional asymptotic classfor R in L , or R -mac in L for short, if for every parameter-free L -formula φ (¯ x, ¯ y ) , ULTIDIMENSIONAL EXACT CLASSES n := l (¯ x ) ≥ and m := l (¯ y ) , there exists a finite definable partition Φ of C ( m ) such that for each π ∈ Φ there exists h π ∈ R such that (cid:12)(cid:12)(cid:12) | φ ( M n , ¯ a ) | − h π ( M ) (cid:12)(cid:12)(cid:12) = o ( h π ( M )) for all ( M , ¯ a ) ∈ π as | M | → ∞ , where the meaning of the little-o notation isas follows: For every ε > there exists Q ∈ N such that for all ( M , ¯ a ) ∈ π , if | M | > Q , then (cid:12)(cid:12)(cid:12) | φ ( M n , ¯ a ) | − h π ( M ) (cid:12)(cid:12)(cid:12) ≤ εh π ( M ) . Remark N -dimensional asymptotic class and an R -mac is the specification of the measuring functions, those of the former beingrestricted to the form µ | M | d / N while those of latter having no restrictionon form.(ii) In Definition 2.1.2Definition 2.1.2 the codomain of the functions in R is N , while in Definition 2.2.2Definition 2.2.2it is R ≥ . The reason for this difference is the change from exact to ap-proximate measuring functions. We state and prove a number of lemmas that we willuse later on. We start with the Projection Lemma, which we already used in theproof of Corollary 2.1.6Corollary 2.1.6.
Lemma . Let C be a class of L -structures. Supposethat the definition of an R -mec (Definition 2.1.2Definition 2.1.2) holds for C and for all L -formulas φ ( x, ¯ y ) with a single object variable x (as opposed to a tuple ¯ x ). Then C is an R ′ -mec in L , where R ′ is generated under addition and multiplication bythe functions in R . A proof of the equivalent result for R -macs is given in § 2.4 of [11]. It is adaptedfrom the proof of Theorem 2.1 in [3737]. Our proof of Lemma 2.3.1Lemma 2.3.1 is a simplifiedversion of the proof in [11]. Proof of Lemma 2.3.1Lemma 2.3.1.
Consider an arbitrary L -formula φ (¯ x, ¯ y ) , where n := l (¯ x ) ≥ and m := l (¯ y ) . We need to prove that it satisfies both the size anddefinability clauses. We do this by induction on the length of ¯ x . The base caseof the induction is the hypothesis of the lemma.Let ¯ x = ( x , . . . , x n ) . By the induction hypothesis we may assume that thesize and definability clauses are satisfied by φ ( x , . . . , x n − ; x n , ¯ y ) , where thesemicolon is used to indicate the division between the object variables and theparameter variables (see Remark 2.1.3(ii)Remark 2.1.3(ii)). So we have a finite partition Γ of C (1 + m ) = { ( M , a, ¯ b ) : M ∈ C , ( a, ¯ b ) ∈ M m } with measuring functions { f i : i ∈ Γ } ⊆ R and defining L -formulas { γ i ( x n , ¯ y ) : i ∈ Γ } .Consider each γ i ( x n , ¯ y ) . By the base case of the induction, each γ i ( x n , ¯ y ) satisfies the size and definability clauses, so for each i ∈ Γ we have a finitepartition Φ i := { π i , . . . , π ir i } of C ( m ) = { ( M , ¯ b ) : M ∈ C , ¯ b ∈ M m } withmeasuring functions { g ij : 1 ≤ j ≤ r i } ⊆ R and defining L -formulas { ψ ij (¯ y ) :1 ≤ j ≤ r i } . We thus have k := | Γ | finite partitions of C ( m ) . We use them to DANIEL WOLF construct a single finite partition Φ of C ( m ) . Define π ( j ,... ,j k ) := (cid:92) i ∈ Γ π ij i and J := { ( j , . . . , j k ) : 1 ≤ j i ≤ r i , ≤ i ≤ k } . Then
Φ := { π ( j ,... ,j k ) : ( j , . . . , j k ) ∈ J } forms a finite partition of C ( m ) . Wenow need to show that this partition works.We first consider the size clause. For each π ( j ,... ,j k ) we need to find a function h ( j ,... ,j k ) ∈ R such that h ( j ,... ,j k ) ( M ) = | Φ( M n , ¯ b ) | (2.2)for all ( M , ¯ b ) ∈ π ( j ,... ,j k ) . So fix some arbitrary ( j , . . . , j k ) and consider anarbitrary pair ( M , ¯ b ) ∈ π ( j ,... ,j k ) . (If π ( j ,... ,j k ) = ∅ , then any function h ∈ R would be vacuously suitable, so we can ignore this case.) Let χ i ( x , . . . , x n , ¯ y ) denote the L -formula φ ( x , . . . , x n , ¯ y ) ∧ γ i ( x n , ¯ y ) . Then, since the L -formulas γ i ( x n , ¯ a ) define the partition Γ , φ ( M n , ¯ b ) is parti-tioned by the χ i ( M n , ¯ b ) , i.e. φ ( M n , ¯ b ) = (cid:91) i ∈ Γ χ i ( M n , ¯ b ) , (2.3)where the union is disjoint. Now, for each i ∈ Γ we have (cid:12)(cid:12) χ i ( M n , ¯ b ) (cid:12)(cid:12) = (cid:88) a ∈ γ i ( M , ¯ b ) (cid:12)(cid:12) φ ( M n − , a, ¯ b ) (cid:12)(cid:12) because χ i ( M n , ¯ b ) fibres over γ i ( M , ¯ b ) . Thus (cid:12)(cid:12) χ i ( M n , ¯ b ) (cid:12)(cid:12) = f i ( M ) · (cid:12)(cid:12) γ i ( M , ¯ b ) (cid:12)(cid:12) , (2.4)since (cid:12)(cid:12) φ ( M n − , a, ¯ b ) (cid:12)(cid:12) = f i ( M ) if M | = γ i ( a, ¯ b ) . But ( M , ¯ b ) ∈ π ( j ,... ,j k ) ⊆ π ij i and so (cid:12)(cid:12) γ i ( M , ¯ b ) (cid:12)(cid:12) = g ij i ( M ) , which gives (cid:12)(cid:12) χ i ( M n , ¯ b ) (cid:12)(cid:12) = f i ( M ) · g ij i ( M ) when put into (2.42.4). Combining this with (2.32.3) yields (cid:12)(cid:12) φ ( M n , ¯ b ) (cid:12)(cid:12) = (cid:88) i ∈ Γ f i ( M ) · g ij i ( M ) . So define h ( j ,... ,j k ) ( M ) := k (cid:88) i =1 f i ( M ) · g ij i ( M ) for all M ∈ C and (2.22.2) is satisfied as required.We now come to the definability clause. Let ψ ( j ,... ,j k ) (¯ y ) denote the formula k (cid:94) i =1 ψ ij i (¯ y ) . Then ( M , ¯ b ) ∈ π ( j ,... ,j k ) if and only if M | = ψ ( j ,... ,j k ) (¯ b ) . So the definabilityclause is also satisfied and so we are done. □ ULTIDIMENSIONAL EXACT CLASSES R -mecs are closed under adding constantsymbols: Lemma . Suppose that C is an R -mec in L . Let L ′ be an extension of L by constant symbols and for M ∈ C let M ′ be an L ′ -expansion of M . Then C ′ := {M ′ : M ∈ C} is an R -mec in L ′ . Proof.
This follows straightforwardly from the definition of an R -mec. □ The following lemma shows that if we want to prove that a class C is an R -mec in L , then it suffices to show that the definition eventually holds for each L -formula: Lemma . Suppose that the definition of a multidimensional exact class(Definition 2.1.2Definition 2.1.2) holds for φ (¯ x, ¯ y ) , R and the subclass C ( m ) >Q := { ( M , ¯ a ) : ( M , ¯ a ) ∈ C ( m ) and | M | > Q } of C ( m ) , where m := l (¯ y ) , Q is some positive integer, and R contains the constantfunction M (cid:55)→ k for each positive integer k ≤ Q . Then the definition also holdsfor φ (¯ x, ¯ y ) , R and C ( m ) . Proof.
By the hypothesis of the lemma there exists a finite partition Φ of C ( m ) >Q with measuring functions { h π : π ∈ Φ } and defining L -formulas { ψ π (¯ y ) : π ∈ Φ } . Let Γ i := { ( M , ¯ a ) : M ∈ C ( m ) \ C ( m ) >Q and | φ ( M n , ¯ a ) | = i } . Then { Γ i : 0 ≤ i ≤ Q } ∪ Φ is a finite partition of C with measuring functions { g i : 0 ≤ i ≤ Q } ∪ { h π : π ∈ Φ } , where g i ( M ) := i for all M ∈ C . So the sizeclause holds for C .Let σ Q be the L -sentence ∃ x . . . ∃ x Q ∀ y (cid:87) ≤ i ≤ Q y = x i , i.e. σ Q says that thereare at most Q elements, and let φ i (¯ y ) be the L -formula ∃ ! i ¯ x φ (¯ x, ¯ y ) , i.e. φ i (¯ a ) says that | φ ( M n , ¯ a ) | = i . Then the partition in the previous paragraph is definedby the L -formulas { φ i (¯ y ) ∧ σ Q : 1 ≤ i ≤ Q } ∪ { ψ π (¯ y ) ∧ ¬ σ Q : π ∈ Φ } . □ Our last useful lemma is a compactness-like result:
Lemma . Let C be a class of finite L -structures. For L ′ ⊆ L let C L ′ denote the class of all L ′ -reducts of structures in C . If C L ′ is an R -mec in L ′ forevery finite L ′ ⊆ L , then C is an R -mec in L . Proof.
This follows from Definition 2.1.2Definition 2.1.2, whose first (second-order) quan-tifier ranges over L -formulas, and the following two facts: Firstly, L -formulasare finite and so any L -formula is an L ′ -formula for some finite L ′ ⊆ L . Sec-ondly, for every L ′ -formula χ (¯ y ) (where m := l (¯ y ), for every L ′ -reduct M ′ of an L -structure M and for every ¯ a ∈ M m , M ′ | = χ (¯ a ) if and only if M | = χ (¯ a ) . □ Following on from Corollary 2.1.6Corollary 2.1.6, weprovide a number of examples and non-examples of R-mecs. In order to explainthe first example (Example 2.4.3Example 2.4.3) we require a definition and a lemma:
Definition C , . . . , C k , where each C i is a class of L i -structures. Define the disjoint union of C , . . . , C k to be C ⊔ · · · ⊔ C k := {M ⊔ · · · ⊔ M k : M i ∈ C i } , DANIEL WOLF where we define a first-order structure on M ⊔ · · · ⊔ M k as follows: The domainis M ∪ · · · ∪ M k , which we make formally disjoint if necessary. The language is L ⊔ · · · ⊔ L k , which has a sort S i for each M i and contains all L i -symbols forevery i ∈ { , . . . , k } , with each L i -symbol being restricted to the sort S i . Lemma . Let C i be an R i -mec in L i . Then C ⊔ · · · ⊔ C k is an R -mec in L := L ⊔ · · · ⊔ L k , where R is the set generated by R ∪ · · · ∪ R k under additionand multiplication. Proof.
We restrict our attention to the case k = 2 , the general case followingby induction.Consider an L -formula φ (¯ x , ¯ x ; ¯ y , ¯ y ) , where ¯ x i and ¯ y i are of sort S i . By aninduction on the complexity of the formula, one can show that φ ( ¯ x , ¯ x ; ¯ y , ¯ y ) isequivalent to a finite disjunction of L -formulas of the form χ (¯ x , ¯ y ) ∧ θ (¯ x , ¯ y ) ,where χ is an L -formula, θ is an L -formula, and the disjuncts are pairwiseinconsistent. Since the domains of M ∈ C and M ∈ C are disjoint, we have | χ ( M ⊔ M , ¯ a ) ∧ θ ( M ⊔ M , ¯ a ) | = | χ ( M , ¯ a ) | · | θ ( M , ¯ a ) | . One then proceeds by using the facts that the disjuncts are pairwise inconsistent,thus allowing summation, and that each C i is an R -mec. □ Example C of finite cyclic groups and for arbitrary k ∈ N + define C k := { C ⊕ · · · ⊕ C k : C i ∈ C} . Let L be the language ofgroups (with or without a constant symbol for the identity element – recallLemma 2.3.2Lemma 2.3.2). Then C k is a multidimensional exact class in L ′ , where L ′ is L adjoined with a unary predicate P i for each part of the direct sum: P iC ⊕···⊕ C k := { (0 , . . . , , a ↑ i th place , , . . . ,
0) : a ∈ C i } . Proof.
Theorem 3.14 in [3737] states that C is a -dimensional asymptotic classin L (see Definition 2.2.1Definition 2.2.1). Inspection of the proof of this theorem shows that C isin fact an exact class, since the measuring functions yield exact sizes and not justapproximaitons. So by Lemma 2.4.2Lemma 2.4.2, C ⊔ · · · ⊔ C (cid:124) (cid:123)(cid:122) (cid:125) k times is an exact class in L ⊔ · · · ⊔ L (cid:124) (cid:123)(cid:122) (cid:125) k times .We now use the work in [11] and § 2.4 of [4444] regarding interpretability: Since L ′ is equipped with the predicates P i , it follows that C k and C ⊔ · · · ⊔ C are ∅ -bi-interpretable and thus that C k is an exact class. □ Remark C of finite cyclicgroups is both a multidimensional exact class and a -dimensional asymptoticclass, so one might wonder whether it could be a “ -dimensional exact class”.However, the notion of an N -dimensional exact class is inconsistent: Considertwo disjoint definable sets A, B ⊆ M with | A | = α | M | a/N and | B | = β | M | b/N ,where a > b . Then their union A ∪ B , which is definable, has size α | M | a/N + β | M | b/N , which cannot be expressed in the form µ | M | d/N for a dimension–measure pair ( d, µ ) . This is not an issue for an N -dimensional asymptotic class,since | M | a/N swamps | M | b/N as | M | → ∞ . It is also not an issue for a multi-dimensional exact class, where one is not bound to dimension–measure pairs. ULTIDIMENSIONAL EXACT CLASSES Example C := { ( Z /p n Z ) m : p is prime and n, m ∈ N + } in the language L := { + } . This class is an R -mec, where R is generated byfunctions of the form r (cid:88) i =0 rd (cid:88) j = − rd c ij p m ( in + j ) , where r is the length of the object-variable tuple of the given L -formula (seeRemark 2.1.3(ii)Remark 2.1.3(ii)); d is a positive integer that is constructively determined bythe L -formula; and the c ij are integers that depend on the L -formula, with c ij := 0 whenever in + j < . (Each group ( Z /p n Z ) m ∈ C is determined by atriple ( p, n, m ) , so by defining a function on such triples we also define a functionon C .)The following two examples are taken from [11]: Example R be a ring and let C be the class of all finite R -modules.Then there exists R ′ such that C is an R ′ -mec. Example R such that the class of finite abelian groups isan R -mec.Further examples will arise as this paper progresses. We now turn our atten-tion to non-examples, which are often just as interesting. Non-Example C of all finite linearorders in (any extension of) the language L = { < } does not form a weak R -mecfor any R . Proof.
Let φ ( x, y ) be the formula x < y and consider the finite linear order M k := { a < · · · < a k } . Then | φ ( M k , a i ) | = i . As we let k increase and let i vary we define arbitrarily many subsets of distinct sizes. Thus no finite numberof functions can yield | φ ( M k , a i ) | for all k, i ∈ N . Let’s make that argument alittle more rigorous.By way of contradiction, suppose that there exists R such that C forms aweak R -mec. So for the formula φ ( x, y ) there exists a finite partition Φ of C (1) with measuring functions { h π : π ∈ Φ } ⊆ R . Let t := | Φ | and consider thefinite linear order M t . Then t measuring functions are not enough for thisstructure, since there are t + 1 different sizes of the definable subsets, namely | φ ( M t , a ) | = 0 , . . . , | φ ( M t , a t ) | = t . A contradiction. □ The following non-example is informative, as it shows that the choice of lan-guage in Example 2.4.5Example 2.4.5 is important:
Non-Example p be prime. Then the class { Z /p n Z : n ∈ N + } ofmultiplicative monoids in (any extension of) the language L = {×} does notform a weak R -mec for any R . Proof.
Let R be any set of functions from C to N and let φ ( x, y ) be theformula ∃ z ( x = z × y ) . Then | φ ( Z /p n Z , p i ) | = p n − i . So as we let n increase0 DANIEL WOLF and let i vary we define arbitrarily many subsets of distinct sizes. Thus, by thesame argument given in the proof of Non-Example 2.4.8Non-Example 2.4.8, no finite number ofmeasuring functions can suffice for | φ ( Z /p n Z , p i ) | for all n, i ∈ N . □ Remark strict order property here. For example, the Paley graphs form anasymptotic class (Example 3.4 in [3737]), but any ultraproduct of them hasunstable theory (see Remark 2.4.13Remark 2.4.13).(ii) The issue preventing Non-Example 2.4.9Non-Example 2.4.9 from being an R -mec is the un-bounded exponent n . If the exponent is bounded, then one can have an R -mec, as shown by the work of Bello Aguirre in [44] and [55].We now cite two non-examples concerning ultraproducts, the random graphand the random tournament, which are covered extensively in the literature,for instance [33], Exercise 2.5.19 in [3939] and Exercise 1.2.4 in [4242] (ultraproducts),p. 232 of [1010], p. 17 of [1212], §§ 1–2 of [1818], p. 435 of [3131], pp. 50–52 of [3939] and Exer-cise 3.3.1 in [4242] (the random graph and the random tournament). We cite thesetwo non-examples in order to highlight a difference between multidimensionalexact classes and multidimensional asymptotic classes (Remark 2.4.13Remark 2.4.13). Non-Example
Non-Example a (cid:95) b is incorrectlydisplayed as a · b in Non-Example 2.3.14 in [4444].) Remark -dimensional asymptotic class (Example 3.4 in [3737]),and the random tournament is elementarily equivalent to any infinite ultraprod-uct of the class of Paley tournaments, which is also a -dimensional asymptoticclass (Example 3.5 in [3737]). This is an interesting phenomenon, especially inlight of Theorem 7.5.6 in [1414] and Theorem 4.6.4Theorem 4.6.4. We will discuss it further inQuestion 5.3Question 5.3. §3. Smooth approximation and exact classes. The goal of this section isto prove Proposition 3.2.1Proposition 3.2.1, which states that finite structures smoothly approxi-mating an ℵ -categorical structure form a multidimensional exact class. In § 3.1§ 3.1 Due to its different guises, the random graph goes by various names, including the ‘Radograph’ and ‘the generic (countable homogeneous) graph’. The random tournament has similaraliases.
ULTIDIMENSIONAL EXACT CLASSES
The notion of smooth approximation wasintroduced by Lachlan in the 1980s, arising as a generalisation of ℵ -categorical, ℵ -stable structures [1313], in particular Corollary 7.4 of that paper. [99], [2828], [3232],[3333] and [3434] are also relevant, but the key texts on smooth approximation itselfare [2424] by Kantor, Liebeck and Macpherson and [1414] by Cherlin and Hrushovski.A history of the development of the notion is to be found in § 1.1 of [1414] andthere is a survey article [3535], which also contains improvements and errata to [2424].Smooth approximation also arises in the context of asymptotic classes in [1616],[1717], [3737] and [3838].For L -structures M and N we use the notation N ≤ M to mean that N isan L -substructure of M . Definition M and N be L -structures. N is a homogeneous substructure of M , notationally N ≤ hom M , if N ≤ M andfor every k ∈ N + and every pair ¯ a, ¯ b ∈ N k , ¯ a and ¯ b lie in the same Aut( M ) -orbitif and only if ¯ a and ¯ b lie in the same Aut { N } ( M ) -orbit, where Aut { N } ( M ) := { σ ∈ Aut( M ) : σ ( N ) = N } . Definition L -structure M is smoothlyapproximable if M is ℵ -categorical and there exists a sequence ( M i ) i<ω offinite homogeneous substructures of M such that M i ⊂ M i +1 for all i < ω and (cid:83) i<ω M i = M . We say that M is smoothly approximated by the M i .We provide some examples of smoothly approximable structures, starting witha trivial example: Example M be a countably infinite set in the language of equality.Enumerate M as ( a i : i < ω ) and let M i = { a , . . . , a i } . Then each M i is afinite homogeneous substructure of M and M = (cid:83) i<ω M i . Example L := { I , I } , where I and I arebinary relation symbols. Let M be a countable L -structure where I M and I M are equivalence relations such that I M has infinitely many classes, I M refines I M , every I -equivalence class contains infinitely many I -equivalence classes,and every I -equivalence class is infinite; that is, M is partitioned into infinitelymany I -equivalence classes, each of which is then partitioned into infinitelymany I -equivalence classes, each of which is infinite. Note that M is uniqueup to isomorphism and hence ℵ -categorical, since the structure is first-orderexpressible in L .Enumerate the I -equivalence classes as ( a i : i < ω ) and the I -equivalenceclasses within each a i as ( a ij : j < ω ) . Finally, enumerate the elements of each a ij as ( a ijk : k < ω ) . Let M ( r,s,t ) := { a ijk : i ≤ r, j ≤ s, k ≤ t } . Then each M ( r,s,t ) is a finite homogeneous substructure of M and M = (cid:83) r<ω M ( r,r,r ) .Note that this example straightforwardly generalises to the case of n nestedequivalence relations for any n < ω . We define ‘homogeneous substructure’ as one term, not as the conjunction of two words;that is, ‘homogeneous substructure’ does not mean a substructure that is homogeneous. DANIEL WOLF
Example M be the direct sum of ω -many copies of the additivegroup Z /p Z , where p is some fixed prime. Note that M is ℵ -categorical, whichcan be seen via Szmielew invariants (see Appendix A.2 in [2121]). Let M i consistof the first i copies of Z /p Z . Then each M i is a finite homogeneous substructureof M and M = (cid:83) i<ω M i . We now come to Proposition 3.2.1Proposition 3.2.1,the central result of this section. We first give the main proof, leaving thenecessary technical lemmas until afterwards.
Proposition . Let M be an L -structure smoothly approximated by finitehomogeneous substructures ( M i ) i<ω . Then there exists R such that C := {M i : i < ω } is an R -mec in L . Proof.
Let φ (¯ x, ¯ y ) be an L -formula with n := l (¯ x ) ≥ and m := l (¯ y ) .We first cover the size clause. We use the Ryll-Nardzewski Theorem: Since M is ℵ -categorical, Aut( M ) acts oligomorphically on M and thus M m has onlyfinitely many Aut( M ) -orbits, say Θ , . . . , Θ d . We use these orbits to define afinite partition π , . . . , π d of C ( m ) = { ( M i , ¯ a ) : i < ω, ¯ a ∈ M im } : ( M i , ¯ a ) ∈ π j iff ¯ a ∈ Θ j . Define π M i j := { ¯ a ∈ M im : ( M i , ¯ a ) ∈ π j } and let ¯ a, ¯ b ∈ M im . Then ¯ a, ¯ b ∈ π M i j ⇐⇒ ¯ a, ¯ b ∈ Θ j = ⇒ ¯ a and ¯ b lie in the same Aut { M i } ( M ) -orbit(since M i ≤ hom M )= ⇒ | φ ( M in , ¯ a ) | = | φ ( M in , ¯ b ) | . (3.5)We justify the last implication: Since ¯ a and ¯ b lie in the same Aut { M i } ( M ) -orbit, there is some σ ∈ Aut { M i } ( M ) such that σ (¯ a ) = ¯ b . But σ ↾ M i is anautomorphism of M i and thus M i | = φ (¯ c, ¯ a ) if and only if M i | = φ ( σ (¯ c ) , σ (¯ a )) .Therefore σ : φ ( M in , ¯ a ) → φ ( M in , ¯ b ) is a bijection and hence | φ ( M in , ¯ a ) | = | φ ( M in , ¯ b ) | .Define h j ( M i ) := | φ ( M in , ¯ a ) | , where ¯ a is some arbitrary element of π M i j (ifno such ¯ a exists, then the value of h j at M i can be chosen to be anything, say0); this function is well-defined by (3.53.5). Then π , . . . , π d and h , . . . , h d satisfythe size clause.We now come to the definability clause. We use the Ryll-Nardzewski Theoremagain: Each orbit Θ j is the solution set of an isolated m -type and so the L -formula isolating this type defines Θ j in M ; let ψ j (¯ y ) be the isolating formulafor Θ j . So M | = ψ j (¯ a ) if and only if ¯ a ∈ Θ j . We claim that the following iseventually true, i.e. there exists Q ∈ N such that for each ψ j , if i > Q , then M i | = ψ j (¯ a ) ⇐⇒ ¯ a ∈ π M i j (3.6)for every ¯ a ∈ M im . By Lemma 2.3.3Lemma 2.3.3 this suffices to prove the definability clause.We prove this claim: Apply Lemma 3.2.7Lemma 3.2.7 to ψ j to obtain Q j ∈ N such that if i > Q j and ¯ a ∈ M im , then M | = ψ j (¯ a ) ⇐⇒ M i | = ψ j (¯ a ) . (3.7) ULTIDIMENSIONAL EXACT CLASSES Q := max { Q j : 1 ≤ j ≤ k } . Consider ¯ a ∈ M im with i > Q . Then M i | = ψ j (¯ a ) (3.73.7) ⇐⇒ M | = ψ j (¯ a ) ⇐⇒ ¯ a ∈ Θ j ⇐⇒ ¯ a ∈ π M i j and so (3.63.6) holds. □ Remark C . In the proof of Proposition 2.1.5Proposition 2.1.5 this uniformity arisesfrom the language L directly: We found the isolating formulas ψ (¯ y ) , . . . , ψ d (¯ y ) before considering structures in C . In the proof of Proposition 3.2.1Proposition 3.2.1 this uniform-ity arises from the oligomorphicity of M , which is then passed down to thehomogeneous substructures M i . Definition canonical language of M to be L ∗ := L ∪ { P Θ : Θ is a Aut( M ) -orbit of M} , where each P Θ is a new unary predicate symbol. We expand M to an L ∗ -structure M ∗ by defining the assignment of each P Θ in M ∗ to be Θ . We expandeach M i to an L ∗ -structure M i ∗ by defining the assignment of each P Θ to be Θ ∩ M i .Lemmas 3.2.4Lemmas 3.2.4 and 3.2.53.2.5 are standard and we state them without proof: Lemma . Aut( M ) = Aut( M ∗ ) . Lemma . Th( M ∗ ) has quantifier elimination; in particular, any L ∗ -formula is equivalent in Th( M ∗ ) to a quantifier-free ( L ∗ \ L ) -formula. Lemma . M ∗ is smoothly approximated by ( M i ∗ ) i<ω . Proof.
Since M is ℵ -categorical, by Lemma 3.2.4Lemma 3.2.4 and the Ryll-NardzewskiTheorem, M ∗ is also ℵ -categorical. Also note that each M i ∗ is a finite L ∗ -substructure of M ∗ . It remains to show that M i ∗ ≤ hom M ∗ . If ¯ a, ¯ b ∈ M i ∗ lie in the same Aut( M ∗ ) { M i } -orbit, then ¯ a and ¯ b lie in the same Aut( M ∗ ) -orbit, since Aut( M ∗ ) { M i } ⊆ Aut( M ∗ ) . Now suppose that ¯ a, ¯ b ∈ M i ∗ lie in thesame Aut( M ∗ ) -orbit. By Lemma 3.2.4Lemma 3.2.4, ¯ a and ¯ b lie in the same Aut( M ) -orbit.Thus, since M i ≤ hom M , there exists σ ∈ Aut( M ) { M i } such that σ (¯ a ) = ¯ b .But σ ∈ Aut( M ∗ ) { M i } , again by Lemma 3.2.4Lemma 3.2.4, and so ¯ a and ¯ b lie in the same Aut( M ∗ ) { M i } -orbit. □ Lemma . Let χ (¯ y ) be an L -formula with m := l (¯ y ) . Then there exists Q ∈ N such that if i > Q and ¯ c ∈ M im , then M | = χ (¯ c ) ⇐⇒ M i | = χ (¯ c ) . Proof.
Consider M ∗ . By Lemma 3.2.5Lemma 3.2.5, T := Th( M ∗ ) has quantifier elimi-nation and thus there is a quantifier-free L ∗ -formula δ (¯ y ) such that ∀ ¯ y ( χ (¯ y ) ↔ δ (¯ y )) ∈ T . Thus by compactness there is an L ∗ -sentence τ ∈ T such that τ | = ∀ ¯ y ( χ (¯ y ) ↔ δ (¯ y )) . (3.8) Note that the term canonical language is sometimes used to refer to the smaller language L ∗ \ L . We avoid this usage. DANIEL WOLF
By Lemma 3.2.6Lemma 3.2.6 and the ∀∃ -axiomatisation of T (see the proof of Proposition5.4 in [2424]), there exists Q ∈ N such that M i ∗ | = τ for all i > Q . Now, considersome arbitrary ¯ c ∈ M im with i > Q . Since δ is quantifier-free and M i ∗ ≤ M ∗ , M ∗ | = δ (¯ c ) ⇐⇒ M i ∗ | = δ (¯ c ) . Hence by (3.83.8) we have M ∗ | = χ (¯ c ) ⇐⇒ M i ∗ | = χ (¯ c ) because M ∗ | = τ and M i ∗ | = τ . But χ is an L -formula and thus M | = χ (¯ c ) ⇐⇒ M i | = χ (¯ c ) , as required. □ The following example shows that the converse of Proposition 3.2.1Proposition 3.2.1 does nothold, in the following sense: An ultraproduct of an R -mec need not be elemen-tarily equivalent to a smoothly approximable structure. Example R -mec (Example 2.4.7Example 2.4.7)and thus the subclass C of all finite cyclic groups of prime order is also an R -mec. Let U be a non-principal ultraproduct of C . Then by Łos’s theorem U istorsion-free. So U has infinitely 2-types: consider pairs ( x, x k ) for k ∈ N . Thusby the Ryll-Nardzewski Theorem U cannot be elementarily equivalent to an ℵ -categorical structure. Therefore, since smoothly approximable structures are ℵ -categorical, U cannot be elementarily equivalent to a smoothly approximablestructure. §4. Lie coordinatisation. The goal of this section is to use Lie coordinati-sation to prove the main result Theorem 4.6.4Theorem 4.6.4, as conjectured by Macpherson.As such, our account of Lie coordinatisation is streamlined for this purposeand we leave some important notions from [1414] by the wayside, most notablyorientation and orthogonality. That being said, we make explicit a number ofdetails that are only implicit in [1414], especially in our proofs of Theorem 4.4.1Theorem 4.4.1and Proposition 4.5.5Proposition 4.5.5. Our presentation is based primarily on [1414], with inputfrom [1515].The history of Lie coordinatisation does not lend itself to easy synopsis andwe give only a very brief summary; see § 1 of [1111] and §§ 1.1–1.2 of [1414] for a moredetailed picture. The notion was developed by Cherlin and Hrushovski as (interalia) an attempt to find a structure theory for smoothly approximable structures,building on the work of Kantor, Liebeck and Macpherson in [2424]. Deep links be-tween other model-theoretic notions were discovered through their investigation(§ 1.2 of [1414]). In particular, it was shown that Lie coordinatisability and smoothapproximation are equivalent (Theorem 2 in [1414]). Note that the classificationof finite simple groups plays a fundamental role, albeit in the background.In contrast to its mathematical depth, Lie coordinatisation has made onlya shallow footprint in the literature, in part due to the development of simpletheories. There are significant mathematical links between the two topics (seepp. 8–10 of [1414]), but simple theories have received more attention from modeltheorists. The reasons for this are manifold and a subject for debate, but Ipresent two subjective opinions: Firstly, simple theories are quite simply easier
ULTIDIMENSIONAL EXACT CLASSES ℵ -categorical theories such as ACFA and pseudofinite fields. Note however thatLie coordinatisation and the work of Cherlin, Lachlan and Harrington had a lot ofimplicit influence on the development of simplicity theory; early versions of [2222]and [1414] significantly predate [2727]. Also note the discussion of the independencetheorem on p. 9 of [1414]. (I thank Dugald Macpherson and Sylvy Anscombe forsharing their thoughts on the topic of this paragraph.)The first publication on Lie coordinatisation was the paper [2222] by Hrushovski,in joint work with Cherlin. Some technical issues were found in this paper (seep. 7 of [1414]) and corrected results were published in [1111], which is essentiallyan abridgement of the main text [1414]. The paper [1515] by Chowdhury, Hart andSokolović makes significant contributions and Hrushovski has published somefurther work on quasifiniteness in [2323]. There are also some unpublished notes[2020] by Hill and Smart. Lie coordinatisation arises in the context of asymptoticclasses in [1616], [1717], [3737] and [3838].We now outline the structure of this section. In § 4.1§ 4.1 we go over the basic con-cepts of Lie coordinatisation and in § 4.2§ 4.2 we provide two examples of Lie coordi-natisable structures. § 4.3§ 4.3 develops the notion of an envelope, which is fundamen-tal to the rest of the section. We then move on to § 4.4§ 4.4, where we state and sketcha proof of a result (Theorem 4.4.1Theorem 4.4.1) that allows us to apply Proposition 3.2.1Proposition 3.2.1 toobtain a short version of Macpherson’s conjecture (Corollary 4.4.2Corollary 4.4.2). § 4.5§ 4.5 thenprovides us with the extra information needed to prove the full version of theconjecture in § 4.6§ 4.6. We state the definition ofLie coordinatisation. We need to go over a number of preliminaries first, startingwith Lie geometries. We refer the reader to chapter 7 of [22] for the terminologyand theory of vector spaces with forms.
Definition K be afinite field. A linear Lie geometry over K is one of the following six kinds ofstructures:1. A degenerate space.
An infinite set in the language of equality.2.
A pure vector space.
An infinite-dimensional vector space V over K withno further structure.3. A polar space.
Two infinite-dimensional vector spaces V and W over K with a non-degenerate bilinear form V × W → K .4. A symplectic space.
An infinite-dimensional vector space V over K with asymplectic bilinear form V × V → K . In contrast to Definition 2.1.4 in [1414], we use the word ‘kind’ in order to avoid overuse ofthe word ‘type’. DANIEL WOLF A unitary space.
An infinite-dimensional vector space V over K with aunitary sesquilinear form V × V → K .6. An orthogonal space.
An infinite-dimensional vector space V over K witha quadratic form V → K whose associated bilinear form is non-degenerate. Remark ( V, K ) , with asort V in the language of groups with an abelian group structure, a sort K in the language of rings with a field structure, and a function K × V → V for scalar multiplication. We call V the vector sort and K the field sort .(See pp. 5 and 12 of [4242] for a summary of multi-sorted structures andlanguages.) The elements of K are named by constant symbols. In thepolar case, the vector sort is V ∪ W in the language of groups equippedwith an equivalence relation with precisely two classes V and W , each withan abelian group structure.(ii) We have ignored quadratic Lie geometries (Definition 2.1.4 in [1414]), as wedo not need to consider them, save only to rule them out in the proof ofProposition 4.5.5Proposition 4.5.5. They arise from the fact that in characteristic 2 everysymplectic bilinear form has many associated quadratic forms. Lemma . Every linear Lie geometryhas quantifier elimination and is ℵ -categorical. Definition L be a linear Lie geometry and let acl denote the usual model-theoretic algebraicclosure in L . We define an equivalence relation ∼ on L \ acl( ∅ ) as follows: a ∼ b iff acl( a ) = acl( b ) The projectivisation of L is defined to be the quotient structure arising from thisequivalence relation: L \ acl( ∅ ) (cid:46) ∼ A projective Lie geometry is a structure that is the projectivisation of some linearLie geometry. Remark
Definition affineLie geometry is a structure of the form ( V, A, ⊕ , − ) , where V is the vector sortof a linear Lie geometry (but not a degenerate space), A is a set, ⊕ : V × A → A is a regular group action and − : A × A → V is such that a = v ⊕ b implies a − b = v . Here ‘regular’ means that for every a, b ∈ A there exists a unique v ∈ V such that a = v ⊕ b . In the polar case the structure is ( V, W, A, ⊕ , − ) ,where ⊕ : V × A → A is a regular group action and − : A × A → V is such that a = v ⊕ b implies a − b = v . Note that this is what the prefix ‘basic’ refers to in Definition 2.1.6 in [1414]. Since we alwaysname the field elements by constant symbols, we suppress this prefix.
ULTIDIMENSIONAL EXACT CLASSES Definition
Lie geometries .The notions of canonical and stable embeddedness are fundamental to Liecoordinatisation:
Definition L -structure N and an L ′ -structure M such that the underlying set M is an L N -definable subset of N . Let c ∈ N eq be a canonical parameter for M . (See§ 8.2 of [3939] or § 8.4 of [4242] for an introduction to canonical parameters.)(i) M is canonically embedded in N if the L ′ ∅ -definable relations of M areprecisely the L c -definable relations on M ; that is, for every n ∈ N + , asubset D ⊆ M n is L ′ ∅ -definable in the structure M if and only if it is L c -definable in the structure N . (The notation L ′ ∅ isn’t strictly necessary,since L ′ = L ′ ∅ , but the subscript ∅ is added to emphasise ∅ -definability.)(ii) M is stably embedded in N if every L N -definable relation on M is L M -definable in a uniform way; that is, for every L -formula φ (¯ x, ¯ y ) , where n := l (¯ x ) ≥ and m := l (¯ y ) , if φ ( N n , ¯ a ) ⊆ M n for every ¯ a ∈ N m , thenthere exists an L -formula φ ′ (¯ x, ¯ z ) , where r := l (¯ z ) , such that for every ¯ a ∈ N m there exists ¯ a ′ ∈ M r such that φ ( N n , ¯ a ) = φ ′ ( N n , ¯ a ′ ) . (Note thatwe need not have m = r .)(iii) M is fully embedded in N if M is both canonically and stably embeddedin N .Intuitively, M is fully embedded in N if N cannot place any additional struc-ture on M .We won’t need the following definition until § 4.5§ 4.5, but it follows on from theprevious definitions. Definition P be a projec-tive Lie geometry, arising from a linear Lie geometry L . Suppose that P is fullyembedded in an L -structure M . The localisation of P over a finite set A ⊂ M is defined as follows: Let f be the bilinear/sesquilinear form on L , where for adegenerate space or a pure vector space we define f ( v, w ) := 0 for all v, w ∈ L and for an orthogonal space f is the bilinear form associated to the quadraticform on L . Define L ⊥ A := { v ∈ L : f ( v, w ) = 0 for all w ∈ acl( A ) ∩ L } or, in the polar case, L ⊥ A := { v ∈ V : f ( v, w ) = 0 for all w ∈ acl( A ) ∩ W }∪ { v ∈ W : f ( v, w ) = 0 for all w ∈ acl( A ) ∩ V } . Let L ⊥ A / ( L ⊥ A ∩ acl( A )) be the quotient space, in the usual sense of a quotient ofabelian groups. (This makes sense by Remark 4.1.5Remark 4.1.5.) Then the localisation of P over A is defined to be the projectivisation of L ⊥ A / ( L ⊥ A ∩ acl( A )) ; that is, let ∼ be as in Definition 4.1.4Definition 4.1.4 and then quotient L ⊥ A / ( L ⊥ A ∩ acl( A )) by ∼ .We denote the localisation of P over A by P/A .We are now ready to state the definition of Lie coordinatisation itself:8
DANIEL WOLF
Definition M bean L -structure. A Lie coordinatisation of M is an L ∅ -definable partial order < of M that forms a tree of finite height with an L ∅ -definable root w suchthat the following condition holds: For every a ∈ M \ { w } either the immediatepredecessor u of a has only finitely many immediate successors (which implies a ∈ acl( u ) ) or, if a ̸∈ acl( u ) , then there exist b < a and an L b -definable projectiveLie geometry J fully embedded in M such that either(i) a ∈ J or, if a ̸∈ J , then(ii) there exist c ∈ M with b < c < a and an L c -definable affine Lie geometry ( V, A ) fully embedded in M such that a ∈ A , the projectivisation of V is J , and J < V < A ,where for subsets
X, Y ⊂ M the notation X < Y means that every elementof X lies in a lower level of the tree than every element of Y . We say thatthe Lie geometries J and ( V, A ) in the tree coordinatise M and refer to themas coordinatising Lie geometries (or just coordinatising geometries ). By a Liecoordinatised structure we mean a structure equipped with a Lie coordinatisation.
Definition L -structure M is Lie coordinatisable if it is ∅ -bi-interpretable (see § 2.5 of [11] or§ 2.4 of [4444]) with a Lie coordinatised structure that has finitely many -typesover ∅ . Remark
Remark M eq to M .We quote two important results from [1414]: Lemma . If M is Lie coordinatisable, then M is ℵ -categorical. Theorem . Let M be an L -structure. Then M isLie coordinatisable if and only if M is smoothly approximable. We give two examples of Lie coordinatisable structures, re-turning to Examples 3.1.4Examples 3.1.4 and 3.1.53.1.5, which by Theorem 4.1.15Theorem 4.1.15 we know must beLie coordinatisable.
Example L := { I , I } , where I and I are binary relation symbols. Let M be a countable L -structure where I M and I M are equivalence relations such that I M has infinitelymany classes, I M refines I M , every I -equivalence class contains infinitely many ULTIDIMENSIONAL EXACT CLASSES a ⌜ a/I ⌝⌜ a/I ⌝ Figure 1.
A finite fragment of the tree from Example 4.2.1Example 4.2.1,with the branch leading to the element a in bold. The nodesare shaded according to membership: The white node is ⌜ M ⌝ ,the crossed nodes are elements of M /I , the grey nodes areelements of ( a/I ) /I , and the black nodes are elements of a/I .The small dots represent the rest of the tree. I -equivalence classes, and every I -equivalence class is infinite; that is, M ispartitioned into infinitely many I -equivalence classes, each of which is thenpartitioned into infinitely many I -equivalence classes, each of which is infinite.We claim that M is Lie coordinatisable.We first outline the tree structure. At the root we place ⌜ M ⌝ (the canonicalparameter of M in M eq , which is ∅ -definable), above which we place the I -classes, as imaginary elements of M eq . Above each I -class we then place the I -classes, again as imaginary elements of M eq , with every I -class above the I -class in which the I -class is contained. Finally, above each I -class we placethe elements of M contained in that I -class. So this tree has height 3 andinfinite width at each level.Let’s explain the notation used in Figure 1Figure 1. So consider some arbitrary a ∈ M .For j = 1 or 2, let a/I j denote the I j -class that contains a and let ⌜ a/I j ⌝ denotethe same I j -class but as a member of M eq ; so ⌜ a/I j ⌝ ∈ M eq is a canonical param-eter for the a -definable subset a/I j ⊂ M . We define ( a/I ) /I and ⌜ ( a/I ) /I ⌝ similarly.We now use this notation to check that Definition 4.1.10Definition 4.1.10 holds for the tree.The imaginary element ⌜ a/I ⌝ lies in the ⌜ M ⌝ -definable degenerate projectivegeometry M /I and ⌜ M ⌝ < ⌜ a/I ⌝ . The imaginary element ⌜ a/I ⌝ lies in the ⌜ a/I ⌝ -definable degenerate projective geometry ( a/I ) /I and ⌜ a/I ⌝ < ⌜ a/I ⌝ .0 DANIEL WOLF
Finally, the real element a lies in the ⌜ a/I ⌝ -definable degenerate projectivegeometry a/I and ⌜ a/I ⌝ < a . Adjoining a finite number of sorts from M eq (recall Remark 4.1.13Remark 4.1.13), each of these geometries is fully embedded in M . (Notethat M /I is not fully embedded, since I defines extra structure on M /I that is not definable within M /I using equality alone.) So M is indeed Liecoordinatisable. Remark n equiv-alence relations I , . . . , I n such that there are infinitely many I -classes, I j +1 refines I j and every I j -class contains infinitely many I j +1 -classes (for ≤ j ≤ n − ), and every I n -class is infinite. At the base of the tree (the 0 th level) weplace ⌜ M ⌝ . At the j th level (for ≤ j ≤ n − ) we place the I j -classes, asimaginary elements of M eq , with every I j -class above the I j − -class in whichthe I j -class is contained. Finally, at the top of the tree (the n th level) we placethe elements of M , with each a ∈ M placed above ⌜ a/I n ⌝ . ba ⌜ a/ ∼ ⌝ Figure 2.
A finite fragment of the tree from Example 4.2.3Example 4.2.3,with the branch leading to the element b ∈ M a in bold. Thenodes are shaded according to membership: The white nodeis the zero vector, the crossed nodes are elements of P ( M ) ,the grey nodes are elements of a/ ∼ , and the black nodes areelements of M a . Note that there are only finitely many (in fact p − ) nodes immediately above each crossed node. The smalldots represent the rest of the tree. Example L := { , + } and let p be a fixed prime number. (The case p = 2 is allowed.) We ULTIDIMENSIONAL EXACT CLASSES M to be the direct sum of ω -many copies of Z /p Z , i.e. M := { ( a i ) i<ω : a i ∈ Z /p Z and a i = 0 for all but finitely many i } . (We specify the direct sum because it is countable, unlike the direct product.)The set M naturally forms an L -structure M , the L -structure arising component-wise from the L -structure of the group Z /p Z . Explicitly: M := (0) i<ω and ( a i ) i<ω + ( b i ) i<ω := ( a i + b i ) i<ω . For brevity we write 0 for M . We claim that M is Lie coordinatisable.We first introduce some notation: For v ∈ M let M v := { a ∈ M : pa = v } ,where pa := a + a + · · · + a (cid:124) (cid:123)(cid:122) (cid:125) p times . Observe that M has a vector space structure over F p and thus is a linear Lie geometry over F p . Let P ( M ) be the projectivisationof M (Definition 4.1.4Definition 4.1.4). Then P ( M ) = ( M \ { } ) / ∼ , where a ∼ b if andonly if a = rb for some r ∈ F p (recall Remark 4.1.5Remark 4.1.5). So | a/ ∼| = p − for all a ∈ M . Adjoining a sort for P ( M ) (recall Remark 4.1.13Remark 4.1.13), we also have that P ( M ) is fully embedded in M .We now outline the tree structure. At the root we place , above which weplace the elements of P ( M ) , considered as imaginary elements of M eq . On thenext level we place the elements of M \ { } , with each a placed above ⌜ a/ ∼ ⌝ .Finally, the top level contains the elements of M \ M , with each b ∈ M a placedabove a . So we have a tree of height 3 and infinite width at each level, althoughthe second level comprises an infinite amount of finite branching. Note that we’reusing the fact here that if b ∈ M \ M , then b ∈ M a for some a ∈ M . Theproof of this fact is straightforward: Suppose that b ∈ M \ M . Then pb ̸ = 0 .So pb = a for some a ∈ M . Then pa = p ( pb ) = p b = 0 , since p c = 0 for all c ∈ M . So b ∈ M a and a ∈ M , as required.Let’s check that Definition 4.1.10Definition 4.1.10 holds for this tree. So consider some ar-bitrary non-zero a ∈ M and b ∈ M a . See Figure 2Figure 2 for an illustration. Theimaginary element ⌜ a/ ∼ ⌝ lies in the -definable projective geometry P ( M ) ,which is fully embedded, as noted in the previous paragraph, and < ⌜ a/ ∼ ⌝ .The real element a is algebraic over ⌜ a/ ∼ ⌝ , since a/ ∼ is ⌜ a/ ∼ ⌝ -definable andfinite, again as noted in the previous paragraph, and ⌜ a/ ∼ ⌝ < a . This leaves uswith the top level of the tree, which we deal with in the next paragraph.Firstly, observe that < a < b . The real element a defines an affine geometry ( M , M a ) , where M is the F p -vector space, M a is the M -affine space, andthe action M × M a → M a is given by ( u, v ) (cid:55)→ u + v . (This action is well-defined, since p ( u + v ) = pu + pv = 0 + a = a and so u + v ∈ M a .) As we havealready noted, the projectivisation of M is P ( M ) , which is a fully embedded, -definable projective geometry, and we have b ∈ M a by assumption. So thetree structure does indeed satisfy the definition of Lie coordinatisation. Remark ω -many copiesof Z /p n Z , for any n ∈ N + . When n = 1 , the tree structure is the same as inthe case n = 2 , except that M \ { } forms the top level, since M \ M = ∅ .When n ≥ , the first three levels ( , P ( M ) and M ) are the same, but atthe third level one places the elements of { b ∈ M : b ∈ M a for some a ∈ M } ,instead of simply M \ M , and at the ( j + 1) th level (for ≤ j ≤ n ) one2 DANIEL WOLF places { c ∈ M : c ∈ M b for some b in the j th level } . The ( n + 1) th level is theupper-most level. Remark
We develop thekey notion of an envelope of a Lie coordinatised structure. Our presentation isa simplified version of that given in [1414], streamlined for the purpose of statingand proving Proposition 4.5.5Proposition 4.5.5. We begin with the notion of a standard systemof geometries:
Definition M be a Lie coordinatised L -structure. A standard system of ge-ometries in M is a ∅ -definable function J : A → M eq whose domain A is theset of realisations of a 1-type over ∅ in M (or the canonical parameter thereof)and whose image is a set of canonical parameters of coordinatising projective Liegeometries of the same kind, i.e. J ( a ) and J ( b ) are isomorphic for every a, b ∈ A ,such that each a ∈ A parametrises its image J ( a ) .By ‘ ∅ -definable’ we mean that there exists an L -formula φ ( x, y ) such that φ ( M , a ) = J ( a ) for every a ∈ A . We write dom( J ) for the domain A of J .We abbreviate the term ‘standard system of geometries’ as ‘SSG’ and its plural‘standard systems of geometries’ as ‘SSGs’.If two SSGs have the same image, then they are equivalent. (This is a sim-plication of the notion of orthogonality developed in [1414], which we purposefullycircumvent in the present work in order to avoid unnecessary complexity.) Definition approximation of a Liegeometry. For example, if J is a degenerate space, then an approximationof J is a finite set in the language of equality, or if J is the projectivisationof an infinite-dimensional pure vector space over a finite field K , then anapproximation of J is the projectivisation of a finite-dimensional pure vectorspace over K .(ii) Let M be a Lie coordinatised structure. A dimension function is a func-tion µ on a finite set S of non-equivalent SSGs in M that assigns anapproximation to each J ∈ S , i.e. µ ( J ) is an approximation of J ( a ) forsome a ∈ dom( J ) ; note that this is independent of the choice of a , since J ( a ) is by definition the same kind of projective Lie geometry for every a ∈ dom( J ) . We call S the domain of µ , which we denote by dom( µ ) . Foreach J ∈ dom( µ ) we define dim µ ( J ) to be the dimension of µ ( J ) , exceptin the degenerate case, where we instead define dim µ ( J ) := | µ ( J ) | . ULTIDIMENSIONAL EXACT CLASSES Definition µ -Envelope, Definition 3.1.1 in [1414]). Let M be a Lie co-ordinatised structure. Then a µ -envelope is a pair ( E, µ ) consisting of a finitesubset E ⊂ M and a dimension function µ for which the following three condi-tions holds:(i) E is algebraically closed in M . (Note that this implies that E is a sub-structure of M .)(ii) For every a ∈ M \ E there exist J ∈ dom( µ ) and b ∈ dom( J ) ∩ E such that acl( E ) ∩ J ( b ) is a proper subset of acl( E, a ) ∩ J ( b ) .(iii) For every J ∈ dom( µ ) and for any b ∈ dom( J ) ∩ E , J ( b ) ∩ E and µ ( J ) areisomorphic. Remark µ -envelope by E , rather than ( E, µ ) , leaving the dimen-sion function as implicit. We similarly often use the term ‘envelope’, ratherthan ‘ µ -envelope’.(ii) It may help the reader’s intuition to know that envelopes form homogeneoussubstructures of M (Lemma in 3.2.4 [1414]). Indeed, this is how the left-to-right direction of Theorem 4.1.15Theorem 4.1.15 is proved (pp. 61–62 of [1414]).(iii) In general one can have countably infinite approximations and envelopes,but we do not need to consider them.The following definition is fundamental to the work in § 4.5§ 4.5: Definition M be a Lie coordinatised structure and consider a µ -envelope ( E, µ ) in M , where dom( µ ) = { J , . . . , J s } . For each J i we define d E ( J i ) := dim µ ( J i ) .We further define d ∗ E ( J i ) := ( −√ q ) d E ( J i ) , where q is the size of the base fi-nite field of µ ( J i ) , or d ∗ E ( J i ) := d E ( J i ) in the degenerate case. (Taking −√ q ,rather than just q , does initially look strange. It is done solely for unitaryspaces: see the end of the proofthe end of the proof of Proposition 4.5.5Proposition 4.5.5.) Finally, we define ¯ d ∗ ( E ) :=( d ∗ E ( J ) , . . . , d ∗ E ( J s )) .We illustrate the preceding definitions by returning to Examples 4.2.1Examples 4.2.1 and 4.2.34.2.3: Example M is partitionedinto infinitely many I -equivalence classes, each of which is then partitioned intoinfinitely many I -equivalence classes, each of which is infinite.Put simply, an example of an envelope in this case is a subset E ⊆ M thatintersects a fixed number ( n ) of I -classes, a fixed number ( n ) of I -classeswithin each of these I -classes, and a fixed number ( n ) of elements within eachof these I -classes. So, up to L -isomorphism, an envelope is given by a triple ( n , n , n ) . Two examples of envelopes are E := { a ijk : 1 ≤ i ≤ , ≤ j ≤ , ≤ k ≤ } and E := { a ijk : 19 ≤ i ≤ , ≤ j ≤ , ≤ k ≤ } , where we use the enumerations from Example 3.1.4Example 3.1.4. The triple for both E and E is ( n , n , n ) = (3 , , . Let’s now explain this in terms of SSGs anddimension functions.4 DANIEL WOLF
Consider the following three SSGs in M :(i) J α : { ⌜ M ⌝ } → M eq , where J α ( ⌜ M ⌝ ) := ⌜ M /I ⌝ ;(ii) J β : M → M eq , where J β ( a ) := ⌜ ( a/I ) /I ⌝ ; and(iii) J γ : M → M eq , where J γ ( a ) := ⌜ a/I ⌝ .These are in fact the only SSGs in M , since ⌜ M /I ⌝ , ⌜ ( a/I ) /I ⌝ and ⌜ a/I ⌝ are the only kinds of coordinatising projective Lie geometries in the Lie coor-dinatisation of M and because there is only one 1-type over ∅ , its realisationbeing M . A dimension function µ on { J α , J β , J γ } assigns an approximation toeach of J α ( ⌜ M ⌝ ) , J β ( a ) and J γ ( a ) , where a is arbitrary. An approximationof a given Lie geometry is determined by the dimension of the approximation,which in this case is equal to the size of the approximation, since all the projec-tive Lie geometries are degenerate. Thus µ is determined by a choice of triple ( n , n , n ) . So, if µ is given by a triple ( n , n , n ) , then a µ -envelope E is achoice of n I -classes, of n I -classes within each of the chosen I -classes andfinally of n elements within each of the chosen I -classes. Furthermore, againbecause all the projective Lie geometries in this example are degenerate, we have ¯ d ∗ ( E ) = ( n , n , n ) . Example M is a directsum of ω -many copies of Z /p Z . We first find the SSGs in M . Since there isonly one coordinatising projective Lie geometry in the Lie coordinatisation of M , namely P ( M ) , there is only one possible image for an SSG in M , namely { ⌜ P ( M ) ⌝ } . Thus there is only one SSG in M up to equivalence. An exampleis the following: J : { } → M eq , where J (0) := ⌜ P ( M ) ⌝ . We now consider dimension functions. A dimension function µ on { J } assignsan approximation to J (0) . An approximation of P ( M ) is a finite-dimensionalsubspace of P ( M ) , which is determined by its dimension n (since the base field F p is fixed). Thus, since J (0) = ⌜ P ( M ) ⌝ , µ is determined by n . A µ -envelope E is then a particular choice of an n -dimensional subspace of P ( M ) . Such asubspace is a finite power of Z /p Z ; that is, a subset { ( a i ) i<ω ∈ M : a i ̸ = 0 only if i = t j for some j } given by n distinct integers t , . . . , t n ∈ N + . Since the base field is F p , whichhas size p , we have ¯ d ∗ ( E ) = (( −√ p ) n ) . (Here ¯ d ∗ ( E ) is 1-tuple, hence the apparently superfluous brackets.) We now take a big step to-wards proving Theorem 4.6.4Theorem 4.6.4 by proving a shorter version, namely Corollary 4.4.2Corollary 4.4.2,where the existence of a multidimensional exact class is asserted but the natureof the measuring functions is not specified. We first sketch a proof of part 2 ofTheorem 6 from [1414], as this result is crucial to our proof of Corollary 4.4.2Corollary 4.4.2. Thekey ingredients needed to prove the result are contained in [1414], namely Propo-sitions 4.4.3, 4.5.1 and 8.3.2 and their proofs, but the (non-trivial) argumentputting them together is not made completely explicit. We state the result in a
ULTIDIMENSIONAL EXACT CLASSES
Theorem . Let L be a finite language and let d ∈ N + . Define C ( L , d ) tobe the class of all finite L -structures with at most d F , . . . , F k of C ( L , d ) such that the L -structures in each F i smoothlyapproximate an L -structure F ∗ i . Moreover, the F i are definably distinguishable:For each F i there exists an L -sentence χ i such that for all M ∈ C ( L , d ) abovesome minimum size, M | = χ i if and only if M ∈ F i . Sketch of proof. We first show that there cannot exist infinitely manypairwise elementarily inequivalent Lie coordinatisable L -structures with the sameskeletal type, where a skeletal type is, roughly speaking, a full description of theLie coordinatising tree structure in an extended language L sk ; see § 4.2 of [1414]for the full definition. So, for a contradiction, suppose that there are in factinfinitely many such L -structures {N i : i < ω } with the same skeletal type S .Working in L sk , by a judicious choice of ultrafilter we can take a non-principalultraproduct N ∗ of the N i such that N ∗ ̸≡ N i for all i < ω . We may assume that N ∗ is countable by moving to a countable elementary substructure. Since theskeletal type S is expressible in L sk (this is a general fact of skeletal types, notjust S ) and true in each N i , by Łos’s theorem N ∗ is Lie coordinatised and hasskeletal type S . Work in chapter 4 of [1414], especially Proposition 4.4.3 and itsproof, shows that every Lie coordinatised structure is quasifinitely axiomatised.Thus N ∗ is quasifinitely axiomatised, which means that Th( N ∗ ) is axiomatisedby a sentence σ and an axiom schema of infinity specifying that every dimensionin each coordinatising Lie geometry of N ∗ is infinite, where we consider Th( N ∗ ) as an L ′ -theory in a finite language L ′ containing L sk . This axiom schema ofinfinity holds for all the N i because they each have the same skeletal type as N ∗ . Furthermore, again by Łos’s theorem, there exists j < ω such that N j | = σ .Therefore N ∗ ≡ N j , a contradiction.We now return to the original class C := C ( L , d ) . We take an infinite ultra-product U ∗ of the structures in C . We take this ultraproduct in a non-standardmodel of set theory, working with some suitable Gödel coding of formulas, whichallows us to consider U ∗ as an L ∗ -structure, where L ∗ is the ultrapower of thelanguage L ; that is, L ∗ extends L by including infinitary formulas with nonstan-dard Gödel numbers, although the number of free variables in any given formularemains finite. We may again assume that U ∗ is countable by moving to a count-able elementary substructure. U ∗ is -quasifinite (Definition 2.1.1 in [1414]) andthus by Theorem 3 in [1414] is weakly Lie coordinatisable (see Remark 4.1.12Remark 4.1.12).So by Proposition 7.5.4 in [1414] the L -reduct U of U ∗ is also weakly Lie coor-dinatisable. The L -structure U thus has a skeletal type. By the first part ofthe proof there can be only finitely many pairwise elementarily inequivalent Liecoordinatisable L -structures with this skeletal type, say F ∗ , . . . , F ∗ k . By Propo-sition 4.4.3 in [1414], each F ∗ i has a characteristic sentence, say χ i . The χ i yield The main argument was given by Hrushovski in email correspondence and Macphersonprovided essential input by working out key details. The contribution of the present authorlay in working through further details and writing up the proof. DANIEL WOLF a partition C = F ∪ . . . ∪ F k , where each χ i is true in all M ∈ F i and false inall M ∈ F j for j ̸ = i , potentially with the exception of some small structures.Moreover, again by Proposition 4.4.3, this partition is such that each M ∈ F i is an envelope of F ∗ i and so by work in chapter 3 of [1414] the structures in F i smoothly approximate F ∗ i .Note that the work cited from chapter 4 of [1414] is written in terms of Lie coor-dinatisability, but inspection of the proofs shows that weak Lie coordinatisabilitysuffices (see Remark 4.1.12Remark 4.1.12). □ Corollary . For any count-able language L and any d ∈ N + there exists R such that the class C ( L , d ) of allfinite L -structures with at most d R -mec in L . Proof.
Let C := C ( L , d ) . The reader should recall Remark 2.1.3(vi)Remark 2.1.3(vi), as wewill use it at various points in this proof.First suppose that L is finite. By Theorem 4.4.1Theorem 4.4.1, C can be finitely partitionedinto subclasses F , . . . , F k such that the structures in each F i smoothly approx-imate an L -structure F ∗ i . Thus by Proposition 3.2.1Proposition 3.2.1 each F i is an R i -mec in L for some R i . Let R L := R ∪ · · · ∪ R k . We claim that C is an R L -mec in L .We prove this claim: Let φ (¯ x, ¯ y ) be an L -formula with n := l (¯ x ) ≥ and m := l (¯ y ) . Since each F i is an R i -mec, we have a suitable finite partition Φ i ofeach F i ( m ) . Then Φ ∪ · · · ∪ Φ k is a finite partition of C ( m ) and so C is a weak R L -mec in L . It remains to show that the definability clause holds. We againuse Theorem 4.4.1Theorem 4.4.1: For each F i there is an L -sentence χ i such that M | = χ i ifand only if M ∈ F i , for sufficiently large M . So, by conjoining χ i to the defining L -formulas of each Φ i , we satisfy the definability clause, using Lemma 2.3.3Lemma 2.3.3 todeal with the finite number of potential exceptions. So the claim is proved.Now suppose that L is infinite. Consider some arbitrary finite L ′ ⊂ L and let C L ′ denote the class of all L ′ -reducts of structures in C . Each structure in C L ′ has at most d C L ′ is an R L ′ -mec in L ′ . (It couldbe the case that C L ′ is a proper subclass of the class of all finite L ′ -structureswith at most d R -mecis also an R -mec.) Let L be the set of all finite subsets of L and define R := (cid:91) L ′ ∈ L R L ′ . Then each C L ′ is an R -mec in L ′ by Remark 2.1.3(vi)Remark 2.1.3(vi). Therefore C is an R -mecin L by Lemma 2.3.4Lemma 2.3.4. □ Remark -types. Well, firstly, if there is a bound on the number of n -types, then thereis a bound on the number of k -types for all k ≤ n . So in the statement ofTheorem 4.6.4Theorem 4.6.4 we could replace -types with n -types for any n > and the resultwould still go through. As for itself, the explanation goes deeper and we willnot go into detail. However, put very roughly, the number 4 arises because theprojective linear group preserves the cross-ratio, which is a projective invarianton -tuples of colinear points. The classification of finite simple groups also plays ULTIDIMENSIONAL EXACT CLASSES
27a role. Details can be found in § 6 of [11], [2424] and [3535]. Note that in [3535] theoriginal bound on -types, as given in [2424], is improved to one on -types. Corollary 4.4.2Corollary 4.4.2 provides no informationabout the structure of R , only its existence. In this section we use Lie geometriesto ascertain information about the nature of R . We first need to define a rank,which we name CH-rank after Cherlin and Hrushovski:
Definition M be an L -structureand let D ⊆ M eq be a parameter-definable set. We define the CH-rank of D asfollows:(i) rk( D ) = − if and only if D = ∅ .(ii) rk( D ) > if and only if D is infinite.(iii) For n ∈ N , rk( D ) ≥ n + 1 if and only if there exist parameter-definablesubsets D , D ⊆ M eq and parameter-definable functions π : D → D and f : D → D such that:(a) rk( π − ( d )) = 0 for all d ∈ D ;(b) rk( D ) > ; and(c) rk( f − ( d )) ≥ n for all d ∈ D .If rk( D ) > n for all n ∈ N , then we define rk( D ) = ∞ .Note that we will often drop the prefix ‘CH-’ and simply refer to ‘rank’. Remark n + 1 iff it can be parameter-definably partitioned intoinfinitely many subsets of rank at least n . The role of π in the definition isto preserve rank under finite parameter-definable projections; however, this willnot be necessary for the pruposes of the present work and thus we will assumethroughout that D = D and π = Id , where Id denote the identity function.We provide examples of CH-rank by returning to our running examples: Example M is partitionedinto infinitely many I -equivalence classes, each of which is then partitioned intoinfinitely many I -equivalence classes, each of which is infinite. Also recall thatfor a ∈ M and j = 1 or 2, a/I j denotes the I j -class containing a while ⌜ a/I j ⌝ denotes the same I j -class but as a member of M eq ; in other words ⌜ a/I j ⌝ ∈ M eq is a canonical parameter for the a -definable subset a/I j ⊂ M .We first calculate the rank of an I -class. So let D := a/I for some a ∈ M .Thus, since D is infinite, rk( D ) ≥ . Moreover, we see that rk( D ) ≱ ,for otherwise we would be able to parameter-definably partition an I -class intoinfinitely many infinite subsets, which is not possible in the L -structure of M .So the rank of an I -class is 1.We now calculate the rank of an I -class. So let D := a/I for some a ∈ M .We set D := { ⌜ b/I ⌝ : b ∈ D } and f ( b ) := ⌜ b/I ⌝ . Since D is infiniteand each I -class has rank 1 (as shown in the previous paragraph), we see that rk( D ) ≥ . We see that rk( D ) ≱ by the same reasoning given inthe previous paragraph: In the L -structure of M the only parameter-definableinfinite partition of an I -class into infinite subsets is the partition induced by I ;8 DANIEL WOLF there is no other parameter-definable infinite partition of an I -class and thereis no parameter-definable way to refine I . So the rank of an I -class is 2.Lastly, we show that M has rank 3. So let D := M . We set D := { ⌜ a/I ⌝ : a ∈ D } and f ( b ) = ⌜ b/I ⌝ . Since D is infinite and each I -class has rank 2 (asalready shown), we see that rk( D ) ≥ . By similar reasoning given in theprevious paragraphs, we have rk( D ) ≱ . So M has rank 3. Example M is a directsum of ω -many copies of Z /p Z . We define M v := { a ∈ M : pa = v } and P ( M ) = ( M \ { } ) / ∼ , where a ∼ b if and only if a = rb for some r ∈ F p (recall Remark 4.1.5Remark 4.1.5). So | a/ ∼| = p − for all a ∈ M .Consider some arbitrary v ∈ M . Then rk( M v ) ≥ because M v is infinite.Likewise rk( P ( M )) ≥ because P ( M ) is also infinite. We further see that rk( M v ) ≱ , since there is no parameter-definable infinite partition of M v into infinite subsets. Similarly rk( P ( M )) ≱ . So rk( M v ) = rk( P ( M )) =1 . We now show that rk( M \ M ) has rank 2. Set D := M \ M , D := D , π := Id , D := M and f ( b ) := pb . Then for each a ∈ D we have f − ( a ) = M a and so rk( f − ( a )) ≥ , since M a is infinite. Thus rk( M \ M ) ≥ . Wesee that rk( M \ M ) ≱ because there is no further parameter-definablepartitioning. So rk( M \ M ) = 2 .Lastly, we see M has rank 2, since it is a superset of the rank-2 set M \ M and because there is no further parameter-definable partitioning in M .With a rank now defined, we are in a position to prove Proposition 4.5.5Proposition 4.5.5.This result provides us with information about the sizes of definable sets inenvelopes, which we will then use in § 4.6§ 4.6 to shed light on the structure of R in Corollary 4.4.2Corollary 4.4.2. It uses Definition 4.3.5Definition 4.3.5 and is a generalisation of Proposition5.2.2 in [1414]. Proposition 5.2.2 in [1414] is essentially about the formula x = x , sinceit concerns the sizes of envelopes, rather than the sizes of definable subsets ofenvelopes, and arbitrary formulas with parameters arise only as part of the proof.In contrast, Proposition 4.5.5Proposition 4.5.5 concerns arbitrary formulas with parameters fromthe outset and so more complexity arises. We also go into considerably moredetail on certain points than in the proof given in [1414]. Proposition . Let E be an ordered familyof envelopes of a Lie coordinatised L -structure M such that dom( µ ) = dom( µ ′ ) for all ( E, µ ) , ( E ′ , µ ′ ) ∈ E and such that the parity and signature of orthogonalspaces are constant on the family, where by ‘ordered family’ we mean that forall ( E, µ ) , ( E ′ , µ ′ ) ∈ E either E ⊆ E ′ or E ′ ⊆ E . Let ¯ a ∈ M m (where m is arbitrary), let D ¯ a ⊆ M be an L ¯ a -definable set and let s be the size of thecommon domian of the dimension functions. Then there exists a polynomial ρ ∈ Q [ X , . . . , X s ] and an integer Q ∈ N such that | D ¯ a ∩ E | = ρ ( ¯ d ∗ ( E )) for all ( E, µ ) ∈ E with | E | > Q and ¯ a ∈ E m . Remark V refers to dim( V ) , distinguishing between odd and even di-mension. The signature refers to the quadratic form on V , there being only two ULTIDIMENSIONAL EXACT CLASSES
Proof of Proposition 4.5.5Proposition 4.5.5.
Let φ ( x, ¯ a ) be the L ¯ a -formula that defines D ¯ a . So D ¯ a = φ ( M , ¯ a ) . By Lemma 4.1.14Lemma 4.1.14 and the Ryll-Nardzewski Theorem wemay assume without loss of generality that φ ( x, ¯ a ) defines the set of realisationsof a 1-type r ( x ) over ¯ a in M . So D ¯ a = r ( M ) . Also note that since E is orderedby ⊆ , either | D ¯ a ∩ E | = ∅ for all ( E, µ ) ∈ E or there exists Q ∈ N such that | D ¯ a ∩ E | ̸ = ∅ for all ( E, µ ) ∈ E with | E | > Q . In the former case we can set Q := 0 and ρ := 0 . So we henceforce assume that we are in the latter case. Withthese two assumptions in hand, we are now in a position to start the main lineof argument. We proceed by induction on CH-rank.First suppose that rk( D ¯ a ) = 0 . Then D ¯ a is finite. Let k := | D ¯ a | . Since D ¯ a isboth finite and ¯ a -definable, D ¯ a ⊆ acl(¯ a ) . Thus, since envelopes are algebraicallyclosed (Definition 4.3.3Definition 4.3.3), D ¯ a ⊆ E for all E ∈ E with ¯ a ∈ E m . So | D ¯ a ∩ E | = | D ¯ a | = k for all E ∈ E with ¯ a ∈ E m . Hence the constant polynomial ρ := k suffices.Now consider the case rk( D ¯ a ) > . Then D ¯ a is infinite. Assume as theinduction hypothesis that the result holds for any parameter-definable subset of M with CH-rank strictly less than rk( D ¯ a ) . Let d ∈ D ¯ a . For a contradiction,suppose that every step in the tree below d is algebraic; that is, if c < c < · · · < c t = d is the chain leading to d , where c is the root of the tree, then each c i +1 is algebraic over its immediate predecessor c i . We claim that d ∈ acl( ∅ ) .We prove this claim. We proceed by induction on i to show that c i ∈ acl( ∅ ) for every i , and so in particular d = c t ∈ acl( ∅ ) . Since the root is ∅ -definable(Definition 4.1.10Definition 4.1.10), c ∈ dcl( ∅ ) ⊆ acl( ∅ ) . Now suppose that c i ∈ acl( ∅ ) . Then c i +1 ∈ acl(acl( ∅ )) , since c i +1 ∈ acl( c i ) by our supposition. But acl(acl( ∅ )) =acl( ∅ ) , since algebraic closure is idempotent, and hence c i +1 ∈ acl( ∅ ) . So theclaim is proved.We now use the claim to derive a contradiction. Since d ∈ acl( ∅ ) , thereexists some L -formula χ ( x ) such that M | = χ ( d ) and χ ( M ) is finite. So χ ( x ) ∈ tp( d/ ∅ ) ⊆ tp( d/ ¯ a ) = r ( x ) and hence D ¯ a = r ( M ) ⊆ χ ( M ) is finite,a contradiction.So by the contradiction there exists c ≤ d such that c is not algebraic overits immediate predecessor. Take c to be minimal, i.e. lowest in the tree. ByDefinition 4.1.10Definition 4.1.10 the non-algebraicity of c implies that c lies in a coordinatisinggeometry J , where J is b -definable for some b < c . The minimality of c impliesthat J is a projective Lie geometry, since the vector and affine parts of a coor-dinatising affine Lie geometry lie above the projectivisation of the vector part.Recalling Remark 4.1.2(ii)Remark 4.1.2(ii), the same argument applies to quadratic geometries:The affine part Q of a coordinatising quadratic geometry, namely the set of qua-dratic forms on which the vector part V acts by translation, lies above V in thetree, V being a symplectic space. So the minimality of c implies that J is theprojectivisation of V .0 DANIEL WOLF
Case 1: The element b is the root. Then b ∈ dcl( ∅ ) and so J is ∅ -definable.We define a set that is central to our argument: S := { ( c ′ , d ′ ) ∈ M : tp(( c ′ , d ′ ) / ¯ a ) = tp(( c, d ) / ¯ a ) } . Let S i be the projection of S to the i th coordinate. Then S is the set ofrealisations of tp( c/ ¯ a ) and S is the set of realisations of tp( d/ ¯ a ) , as provedin the next paragraph. Then S ⊆ J , since c ∈ J and J is ∅ -definable, and S = D ¯ a , since tp( d/ ¯ a ) = r ( x ) .We prove the claim that S is the set of realisations of tp( c/ ¯ a ) : If c ′ ∈ S ,then it is immediate from the definition of S that c ′ | = tp( c/ ¯ a ) . Now supposethat c ′ | = tp( c/ ¯ a ) . By the Ryll-Nardzewski Theorem, M is saturated and thusthere exists σ ∈ Aut( M / ¯ a ) such that σ ( c ) = c ′ ; we’ll use this trick several moretimes and henceforth won’t cite it explicitly. Thus ( c ′ , σ ( d )) = ( σ ( c ) , σ ( d )) ∈ S and hence c ′ ∈ S . So the claim is proved. The proof of the claim that S is theset of realisations of tp( d/ ¯ a ) proceeds symmetrically.Let’s now consider the intersection of D ¯ a with an envelope. So take somearbitrary ( E, µ ) ∈ E with | E | > Q and ¯ a ∈ E . Since D ¯ a ∩ E ̸ = ∅ , we mayassume without loss of generality that d ∈ E , for if d / ∈ E , then we may takesome d ′ ∈ D ¯ a ∩ E and repeat the previous arguments for this new element d ′ .Define S E := { ( c ′ , d ′ ) ∈ S : d ′ ∈ E } . We will use this set to calculate the size of D ¯ a ∩ E , but we first need to go oversome preliminaries. Let S Ei be the projection of S E to the i th coordinate. Then S E = S ∩ E = D ¯ a ∩ E . We claim that S E = S ∩ E .We prove this claim. Let c ′ ∈ S E . Then ( c ′ , d ′ ) ∈ S E for some d ′ ∈ E .Now, c ′ ≤ d ′ and so c ′ ∈ dcl( d ′ ) . Thus, since envelopes are algebraically closed(by definition), c ′ ∈ E . So c ′ ∈ S ∩ E (since S E ⊆ S ), as required. Nowlet c ′ ∈ S ∩ E . Let d ′′ ∈ D ∩ E . Since tp( d ′′ / ¯ a ) = tp( d/ ¯ a ) , there exists σ ∈ Aut( M / ¯ a ) such that σ ( d ) = d ′′ . Let c ′′ := σ ( c ) . Then ( c ′′ , d ′′ ) ∈ S E . By thesame argument used earlier in this paragraph, c ′′ ∈ E . Now, tp( c ′′ / ¯ a ) = tp( c ′ / ¯ a ) and so there exists σ ′ ∈ Aut( M / ¯ a ) such that σ ′ ( c ′′ ) = c ′ . Now, since envelopesare homogeneous substructures (Lemma 3.2.4 in [1414] and Definition 3.1.2Definition 3.1.2) and c ′ , c ′′ ∈ E , we may assume that σ ( E ) = E . Let d ′ := σ ′ ( d ′′ ) . Then d ′ ∈ E , since d ′′ ∈ E . Hence ( c ′ , d ′ ) ∈ S E and so c ′ ∈ S E , as required. So the claim is proved.We introduce some further definitions: For c ′ ∈ S let c ′ /S := { d ′ : ( c ′ , d ′ ) ∈ S } and c ′ /S E := { d ′ : ( c ′ , d ′ ) ∈ S E } , and for d ′ ∈ S let d ′ /S := { c ′ : ( c ′ , d ′ ) ∈ S } and d ′ /S E := { c ′ : ( c ′ , d ′ ) ∈ S E } . The sizes of the c ′ /S E and the d ′ /S E are in fact independent of c ′ and d ′ , as we now show.First consider some arbitrary c ′ ∈ S E . Let D ¯ ac be the set of realisations of tp( d/ ¯ ac ) . Then, by the definition of S , D ¯ ac = c/S . Let d ′ ∈ c ′ /S E . Then, since tp(( c ′ , d ′ ) / ¯ a ) = tp(( c, d ) / ¯ a ) , there exists σ ∈ Aut( M / ¯ a ) such that σ ( c ′ , d ′ ) =( c, d ) . We claim that σ : c ′ /S → c/S is a bijection. Injectivity is immediate. Itis well-defined, since if d ′′ ∈ c ′ /S , then σ ( c ′ , d ′′ ) = ( c, σ ( d ′′ )) ∈ S and so σ ( d ′ ) ∈ c/S . It is surjective, since if d ′′ ∈ c/S , then σ − ( c, d ′′ ) = ( c ′ , σ − ( d ′′ )) ∈ S and so σ − ( d ′′ ) ∈ c ′ /S . So the claim is proved. Now, as mentioned previously,envelopes are homogeneous substructures. So, since d, d ′ ∈ E , we may assume ULTIDIMENSIONAL EXACT CLASSES σ ( E ) = E . Thus | c ′ /S E | = | c ′ /S ∩ E | = | c/S ∩ E | = | D ¯ ac ∩ E | (4.9)for all c ′ ∈ S E .Now consider some arbitrary d ′ ∈ S E . Since c ≤ d , c ∈ dcl( d ) . Thus, since tp( d ′ / ¯ a ) = tp( d/ ¯ a ) , there exists a unique c ′ ∈ M such that ( c ′ , d ′ ) ∈ S . But d ′ ∈ E and so ( c ′ , d ′ ) ∈ S E . Hence | d ′ /S E | = 1 (4.10)for all d ′ ∈ S E .We are now in a position to calculate the size of S E and thereby also that of D ¯ a ∩ E . Let’s first calculate | S E | in terms of | S E | : | S E | = (cid:88) c ′ ∈ S E | c ′ /S E | = | S E | · | D ¯ ac ∩ E | (by (4.94.9)) . (4.11)And now in terms of | S E | : | S E | = (cid:88) d ′ ∈ S E | d ′ /S E | = | S E | (by (4.104.10)) . (4.12)So, since S E = D ¯ a ∩ E , (4.114.11) and (4.124.12) yield | D ¯ a ∩ E | = | S E | · | D ¯ ac ∩ E | . (4.13)First consider S E . We previously proved that S E = S ∩ E . We also showedthat S is the set of realisations of tp( c/ ¯ a ) and that S is a subset of J . By theRyll-Nardzewski Theorem, tp( c/ ¯ a ) is isolated and so S is ¯ a -definable. So S isan ¯ a -definable subset of a projective geometry. Thus, as we will show lateras we will show later (afterCase 2), there exists a polynomial ρ ∈ Q [ X , . . . , X s ] such that ρ ( ¯ d ∗ ( E )) = | S ∩ E | .Now consider D ¯ ac , which is a parameter-definable subset of M , again by theRyll-Nardzewski Theorem. We have rk( D ¯ ac ) < rk( D ¯ a ) , as proved in the follow-ing paragraph, and thus by the induction hypothesis there exists a polynomial ρ ∈ Q [ X , . . . , X s ] such that | D ¯ ac ∩ E | = ρ ( ¯ d ∗ ( E )) .We prove the claim that rk( D ¯ ac ) < rk( D ¯ a ) . Let n := rk( D ¯ ac ) . We previouslyshowed that D ¯ ac = c/S . We also showed that for every c ′ ∈ S there exists σ ∈ Aut( M / ¯ a ) such that σ ( c/S ) = c ′ /S , which thus means rk( c ′ /S ) = n for every c ′ ∈ S . Define f : D ¯ a → S by f ( d ′ ) := c ′ , where c ′ is such that ( c ′ , d ′ ) ∈ S . As we showed earliershowed earlier, for every d ∈ S there is precisely one c ′ such that ( c ′ , d ′ ) ∈ S , so f is well-defined. Then, since f − ( c ′ ) = c ′ /S , wehave rk( f − ( c ′ )) = n for every c ′ ∈ S . Also note that rk( S ) > , since S isinfinite (because c is not algebraic over its immediate predecessor). Thus, taking D := D := D ¯ a , π := Id , D := S and f := f in Definition 4.5.1Definition 4.5.1, we see that rk( D ¯ a ) ≥ n + 1 > rk( D ¯ ac ) . So the claim is proved.2 DANIEL WOLF
Define ρ := ρ · ρ . Then (4.134.13) gives us the desired result: | D ¯ a ∩ E | = | S E | · | D ¯ ac ∩ E | = ρ ( ¯ d ∗ ( E )) · ρ ( ¯ d ∗ ( E ))= ρ ( ¯ d ∗ ( E )) . End of Case 1.Case 2: The element b is not the root. Since c is minimal, b and each elementbelow b (except the root) is algebraic over its immediate predecessor. Thus, bythe same induction used earlier in the proofinduction used earlier in the proof, b ∈ acl( ∅ ) . Thus, by inspection ofDefinition 4.1.10Definition 4.1.10, we see that we may add to L a constant symbol for b withoutaffecting the Lie coordinatising tree. Adding the new constant symbol preservesthe inequality rk( D ¯ ac ) < rk( D ¯ a ) , again since b ∈ acl( ∅ ) , but it makes J ∅ -definable. We may thus simply repeat the argument given in Case 1 in theextended language L b . End of Case 2.
We now prove our earlier claimearlier claim that there is a polynomial ρ ∈ Q [ X , . . . , X s ] such that ρ ( ¯ d ∗ ( E )) = | S ∩ E | . The set S is an L ¯ a -definable subset of J and thus, since J is fully embedded in M , S is ¯ a -definable in the language of J ; we may assume that ¯ a lies in J by stable embeddedness. We now considerthe localisation J/ ¯ a of J at ¯ a (Definition 4.1.9Definition 4.1.9). J fibres over J/ ¯ a , where twoelements lie in the same fibre if and only if they have the same algebraic closureover ¯ a . These fibres all have the same finite size, where this size is determinedby tp(¯ a ) . Now, S might not respect these fibres; that is, the intersection of S with each fibre might vary in size. However, since the fibres are finite, there areonly finitely many possible sizes for these intersections and so we can ¯ a -definablypartition the set of fibres according to size. We then consider the intersectionof each part of the partition with E : We calculate the size of the base of thefibres, which is a ∅ -definable subset of J/ ¯ a , and then multiply this result by thesize of the fibre. We then sum these results to obtain | S ∩ E | . So, in short,by localising J at ¯ a , it suffices to consider ∅ -definable subsets of projective Liegeometries. It remains to do the explicit calculations in each kind of projectiveLie geometry. We use quantifier elimination (Lemma 4.1.3Lemma 4.1.3). A projectivisation of a degenerate space.
Projectivisation in this case is trivial.The only ∅ -definable set is the whole space itself. (We can rule out ∅ because D ¯ a ∩ E ̸ = ∅ .) So S = J . Thus, since J ∩ E = µ ( J ) (Definition 4.3.3Definition 4.3.3), where µ is the dimension function of E , we have | S ∩ E | = d E ( J ) , as required. A projectivisation of a pure vector space.
The only ∅ -definable set is again thewhole space itself. So S = J . Thus, going via the approximation of the linearspace, which has dimension dim µ ( J ) + 1 , we have | S ∩ E | = q dim µ ( J )+1 − q − q dim µ ( J ) + 1 = ( −√ q ) µ ( J ) + 1 = d E ( J ) + 1 , as required. A projectivisation of a polar space.
This is the same as the vector space case,except that we can define either half of the space or the whole space. If theformer, then the answer is the same as that in the vector space case. If thelatter, then we multiply this answer by . ULTIDIMENSIONAL EXACT CLASSES A projectivisation of a symplectic space.
Since there is only one -type, thiscase is the same as the pure vector space case. A projectivisation of a unitary space.
The calculations can be found in theproof of Proposition 5.2.2 in [1414]. Note that it is this case that forces us toconsider ( −√ q ) dim µ ( J ) , rather than just q dim µ ( J ) . An projectivisation of an orthogonal space.
The calculations can again befound in the proof of Proposition 5.2.2 in [1414]. Note that this is where theassumption regarding constant signature and parity is used (Remark 4.5.6Remark 4.5.6). Alsonote that there is a small typographical error in the calculations: On p. 91 of [1414]it should state n (2 i + j, α ) = q i n ( j, α ) + q j − ( q i − q i ) , the original term q i n ( i, α ) being incorrect.One final note: The calculations for unitary and orthogonal spaces in [1414]are actually done in the linear Lie geometry, rather than in the projectivisa-tion. However, by a similar fibering argument to the one used earlierused earlier with thelocalisation, this is sufficient. □ We introduce the notion ofa polynomial exact class, enabling us to state and prove Theorem 4.6.4Theorem 4.6.4, the mainresult of the present work.
Definition L be a language and C a classof finite L -structures. Then C is a polynomial exact class in L if there exist(i) R ⊆ Q [ X , . . . , X k ] for some k ∈ N + ,(ii) L -formulas δ (¯ x , ¯ y ) , . . . , δ k (¯ x k , ¯ y k ) and(iii) ¯ a ∈ M l (¯ y ) , . . . , ¯ a k ∈ M l (¯ y k ) for each M ∈ C such that C is an R -mec in L where h ( M ) = h (cid:16) | δ ( M l (¯ x ) , ¯ a ) | , . . . , | δ k ( M l (¯ x ) , ¯ a k ) | (cid:17) for every h ∈ R and for every M ∈ C . Remark R -mec’ with ‘ R -mac’ in Definition 4.6.1Definition 4.6.1, then wedefine a polynomial asymptotic class . In this case we allow polynomials withirrational coefficients.Note that any -dimensional asymptotic class is a polynomial asymptotic class,since we may take δ to be the L -formula x = x and h to be the polynomial µ X d ,where ( d, µ ) is the dimension–measure pair. Example
Theorem . For any countablelanguage L and for any d ∈ N + the class C ( L , d ) of all finite L -structures withat most d L . Proof.
By Corollary 4.4.2Corollary 4.4.2 we know that C := C ( L , d ) is a multidimensionalexact class. It remains to show that the measuring functions are polynomial inthe sense of Definition 4.6.1Definition 4.6.1.Recall our use of Theorem 4.4.1Theorem 4.4.1 in the proof of Corollary 4.4.2Corollary 4.4.2: We partitioned C into subclasses F , . . . , F k such that the L -structures in each F i smoothlyapproximate an L -structure F ∗ i . By the work in [1414] each F i is a class of envelopes4 DANIEL WOLF for F ∗ i , which is Lie coordinatisable. So Proposition 4.5.5Proposition 4.5.5 implies that C is apolynomial exact class, since each coordinatising Lie geometry is fully embeddedin and thus (by definition) also definable in F ∗ i .We address some details arising from this proof. Firstly, by the Projection LemmaProjection Lemma(Lemma 2.3.1Lemma 2.3.1) it suffices to consider L -formulas in one object variable, as we doin Proposition 4.5.5Proposition 4.5.5. Secondly, by Lemma 3.2.7Lemma 3.2.7 the intersection φ ( F ∗ i , ¯ a ) ∩ M is equal to the relativisation φ ( M , ¯ a ) for all M ∈ F i above some minimum size,so by Lemma 2.3.3Lemma 2.3.3 it suffices to consider the intersection. Thirdly, since C is anexact class, rather than just an asymptotic class, the measuring functions aredetermined by the formula and thus it is not necessary to show that the poly-nomials given by Proposition 4.5.5Proposition 4.5.5 are uniform in the parameter ¯ a ; this pointis important because the measuring functions cannot depend on the parame-ters. Lastly, the hypothesis of constant parity and signature in the statementof Proposition 4.5.5Proposition 4.5.5 can be satisfied by partitioning each F i into (up to) foursubclasses, each with constant parity and signature. □ Remark m = 4 . Then the proof shows that each structurein C has at most d C is a subclass of C ( L , d ) and thus byTheorem 4.6.4Theorem 4.6.4 is a polynomial exact class in L . §5. Open questions. We pose a number of questions arising from the presentwork. In doing so we refer to the important model-theoretic notions of stabilityand (super)simplicity, which we have so far only mentioned in passing. We donot define these notions, but instead direct the reader to the vast literature onthem, [77], [2626] and [4242] being good introductions. We also consider the notion ofhomogeneity, which is easier to define:
Definition L -structure M is homogeneous if M is countable andevery isomorphism between substructures of M extends to an automorphism of M .Note that the word ‘homogeneous’ is overused in mathematics, especially inmodel theory. What we call ‘homogeneous’ might be called ‘ultrahomogeneous’by other authors. See the comment after Definition 2.1.1 in [3636]. Fact . Let L be a finite relational language.(i) If M is a homogeneous L -structure, then M is ℵ -categorical.(ii) If M is an ℵ -categorical L -structure, then Th( M ) has quantifier elimina-tion if and only if M is homogeneous.(iii) If M is a stable homogeneous L -structure, then M is ℵ -stable. Question L is a finite relationallanguage and M is a stable homogeneous L -structure, then M is smoothly ap-proximable and thus by Proposition 3.2.1Proposition 3.2.1 is elementarily equivalent to an ultra-product of a multidimensional exact class. Does the converse hold? That is,if L is a finite relational language and M is a homogeneous L -structure that ULTIDIMENSIONAL EXACT CLASSES M necessarily stable?Recalling Remark 2.4.13Remark 2.4.13, answering this question might shed some light onthe role in Theorem 7.5.6 in [1414] of the generic bipartite graph, which is neitherstable nor smoothly approximable.The following two questions were suggested to the present author by IvanTomašić: Question R -macs and R -mecs? [3030], [4141], [2929], [4040],[4343]The notion of a generalised measurable structure, as developed in [11], alsoappears to be related, but a thorough investigation has yet to be carried out. Question
Acknowledgements.
The concepts explored in this paper were introducedto me by Dugald Macpherson, my patient and supportive PhD supervisor, whoprovided me with invaluable guidance and scrutiny throughout its composition.I had many pleasant and productive discussions with my friend and collaboratorSylvy Anscombe, while Gregory Cherlin, Immanuel Halupczok, Ehud Hrushovskiand Ivan Tomašić provided further useful feedback and insight. Several peopleon the TEX Stack Exchange helped me overcome some L A TEX issues that arosewhilst writing this paper, while my wise and wonderful wife Mandy helped meovercome some non-L A TEX issues that arose whilst writing this paper. I amindebted to the anonymous referee for their careful and thorough report; thepresent work is much improved for their suggestions.
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SCHOOL OF MATHEMATICSUNIVERSITY OF LEEDSLEEDS LS2 9JTUNITED KINGDOM