Classification of ℵ 0 -categorical C -minimal pure C -sets
aa r X i v : . [ m a t h . L O ] F e b Classification of ℵ -categorical C -minimal pure C -sets Fran¸coise Delon, Marie-H´el`ene MourguesFebruary 11, 2020 C -sets are sets equipped with a C -relation. They can be understood as a slightweakening of ultrametric structures. They generalize in particular linear ordersand allow a rich combinatoric. They are therefore not classifiable, except to restricttheir class. It is what we do here: we consider ℵ -categorical and C -minimal C -sets. C -minimality is the minimality notion fitting in this context: any definablesubset in one variable is quantifier free definable using the C -relation alone. In thecase of ultrametric structures this corresponds to finite Boolean combinations ofclosed or open balls. We classify here all ℵ -categorical and C -minimal C -sets upto elementary equivalence (in other words we classify all finite or countable suchstructures).To state our result let us introduce some material. A C -set M has a canonicaltree, T ( M ), in which M appears as the set of leaves, with the C -relation definedas follows : for α ∈ M , call br ( α ) := { x ∈ T ( M ); x ≤ α } the branch α definesin T ( M ) ; then for α, β and γ in M , M | = C ( α, β, γ ) iff in T ( M ), br ( β ) ∩ br ( γ )strictly contains br ( α ) ∩ br ( β ) (which then must be equal to br ( α ) ∩ br ( γ )). Letus give a very simple example: call trivial a C -relation satisfying C ( α, β, γ ) iff α = β = γ and suppose M is not a singleton; then C is trivial on M iff T ( M )consists of a root, say r , and the elements of M as leaves, all having r as a prede-cessor. The C -set ( M, C ) and the tree ( T ( M ) , < ) are uniformly biinterpretable.As usual the Ryll-Nardzewski Theorem makes the classification of indiscernible ℵ -categorical C -minimal sets as a first step in our work. Recall that a structureis said to be indiscernible iff all its elements have the same complete type. If M is indiscernible leaves are indiscernible in T ( M ) but nodes never are (exceptfor the trivial C -relation): given a node n , T ( M ) may have leaves at infinite ordifferent finite distances from n . We prove that a pure C -set M is indiscernible, ℵ -categorical and C -minimal iff its canonical tree T ( M ) is colored, where coloredis defined by induction as follows. Consider on leaves above a node n the relation“ br ( α ) ∩ br ( β ) contains only nodes strictly bigger than n ”. Call cone above n anequivalence class. Now a 1-colored good tree either is a singleton, or consists of aunique node with m leaves, where m is an integer m ≥ ∞ , or there exists µ ,an integer ≥ ∞ , such that for any leaf α of T ( M ), ] − ∞ ; α [, the branch of α in T ( M ) deprived of its leaf α , is densely ordered and at each node of T ( M ) there are1xactly µ infinite cones, or there exist m and µ , each an integer or ∞ , such that forany leaf α of T ( M ), α has a predecessor in T ( M ), say the node α − , ] − ∞ ; α − ] isdensely ordered and at each node of T ( M ) there are exactly m leaves and µ infinitecones. An ( n + 1)-colored good tree is an n -colored good tree in which each leaf issubstituted with a copy of a 1-colored good tree, the same at each leaf, with someconstraints on the parameters m and µ occurring on both sides of the construction.The reduction of the general classification to that of indiscernible structuresuses a very precise description of definable subsets in one variable. ℵ -categoricityis combined with the classical description coming from C -minimality to producea “canonical partition” of the structure in finitely many definable subsets, eachof them maximal indiscernible. The characterization of ℵ -categorical and C -minimal C -sets is done via finite trees with labeled vertices and arrows, where thelabels may be integers or/and complete theories of indiscernible, ℵ -categorical C -minimal C -structures; these C -structures are in fact pure C -sets or very slightenrichments. The reconstruction of the structure from such a finite labeled treeuses again an induction on the depth of the tree.Chapter 2 lists some preliminaries. In Chapter 3 we draw a certain amount ofconsequences of indiscernibility, ℵ -categoricity and C -minimality of a C -structure,which leads to the notion of precolored good tree. Chapters 4 to 6 are dedicatedto colored good trees. Chapter 4 presents 1-colored good trees, which in fact arethe same thing as precolored good trees of depth 1. In Chapter 5 we define theextension of a colored good tree by a 1-colored good tree, construction which isthe core of the inductive definition of ( n + 1)-colored good trees from n -coloredgood trees. General colored good trees are defined and completely axiomatizedin Chapter 6. In Chapter 7 we show that the classes of precolored good trees, ofcolored good trees as well as of canonical trees of indiscernible, ℵ -categorical and C -minimal C -structures do in fact coincide. Chapter 8 gives a complete classifi-cation of ℵ -categorical and C -minimal C -sets. C -sets and good trees Definition 2.1 A C -relation is a ternary relation, usually called C , satisfying thefour axioms:1: C ( x, y, z ) → C ( x, z, y ) C ( x, y, z ) → ¬ C ( y, x, z ) C ( x, y, z ) → [ C ( x, y, w ) ∨ C ( w, y, z )] x = y → C ( x, y, y ) .A C - set is a set equipped with a C -relation. C -relations appear in [AN], [M–S] or [H–M], where they satisfy additional axioms.2ur present definition comes from [D]. As already mentioned in the introduction,a C -set M has a canonical tree, which is in fact bi-interpretable with M , as weexplain now. Definition 2.2
We call tree an order in which for any element x the set { y ; y ≤ x } is linearly ordered.Call a tree good if :- it is a meet semi-lattice (i.e. any two elements x and y have an infimum, or meet , x ∧ y , which means: x ∧ y ≤ x, y and ( z ≤ x, y ) → z ≤ x ∧ y ),- it has maximal elements, or leaves , everywhere (i.e. ∀ x, ∃ y ( y ≥ x ∧ ¬∃ z > y )) - and any of its elements is a leaf or a node (i.e. of form x ∧ y for some distinct x and y ). Let T be a good tree. It is convenient to consider T in the language { <, ∧ , L } where ∧ is the function T × T → T defined above and L a unary predicate for theset of leaves (cf. Definition 2.2). Proposition 2.3 C -sets and good trees are bi-interpretable classes. Let us explain these two interpretations in a few words. More details can be foundin [D].Call branch of a tree any maximal subchain. The set of branches of T carriesa canonical C -relation: C ( α, β, γ ) iff α ∩ β = α ∩ γ ( β ∩ γ . Now, leaves of T may be identified to branches via the map α br ( α ) := { β ∈ T ; β ≤ α } . Thus,if Br l ( T ) denotes the set of branches with a leaf of T , the two-sorted structure( T, <, Br l ( T ) , ∈ ) is definable in ( T, < ), and the canonical C -relation on Br l ( T )also. We denote this C -set M ( T ). This gives the definition of a C -set in agood tree. The canonical tree of a C -set provides the reverse construction. Itis (almost) the representation theorem of Adeleke and Neumann ([AN], 12.4),slightly modified according to [D]. Given a C -set ( M, C ), define on M binaryrelations ( α, β ) ( γ, δ ) : ⇔ ¬ C ( γ, α, β )& ¬ C ( δ, α, β )( α, β ) R ( γ, δ ) : ⇔ ¬ C ( α, γ, δ )& ¬ C ( β, γ, δ )& ¬ C ( γ, α, β )& ¬ C ( δ, α, β ) . Then the relation is a pre-order, R is the corresponding equivalence relationand the quotient T := M /R is a good tree. Proposition Proposition 2.4 summarizes these facts in a more precise way thanProposition (2.3) did. Adeleke and Neumann work in fact with the set of pairs of distinct elements of M , insteadof M as we do (and reverse order). It is the reason why we get maximal elements everywhere inthe tree, meanwhile they did not get any. In the other direction also, Br l ( T ) is interpretable in T meanwhile the “covering set of branches” considered by Adeleke and Neumann is not determinedby T . roposition 2.4 Given a C -set M , there is a unique good tree such that M isisomorphic to its set of branches with leaf, equipped with the canonical C -relation.This tree is called the canonical tree of M and is denoted T ( M ) .Let L be the set of leaves of T ( M ) . Then h M, C i and h T ( M ) , <, ∧ , L i are first-order bi-interpretable, quantifier free and without parameters, and M and L ( T ( M )) are definably isomorphic. Therefore an embedding M ⊆ N induces an embedding T ( M ) ⊆ T ( N ) . Moreover, given a good tree T , T ( M ( T )) and T are definablyisomorphic. C -structures and C -minimality Definition 2.5 A C - structure is a C -set possibly equipped with additional struc-ture.A C -structure M is called C - minimal iff for any structure N ≡ M any definablesubset of N is definable by a quantifier free formula in the pure language { C } . C -minimality has been introduced by Deirdre Haskell, Dugald Macpherson andCharlie Steinhorn as is the minimality notion suitable to C -relations ([H–M],[M–S]). We define now some particular definable subsets of M which, due to C -minimality, generate by Boolean combination all definable subsets of M . Ifwe want to distinguish between nodes and leaves of the tree T ( M ), we will useLatin letters x, y, etc... to denote nodes and Greek letters α, β, etc... for leaves (cf.Definition 2.2). According to the representation theorem, elements of M are alsorepresented by Greek letters. Definition 2.6 • For α and β two distinct elements of M , the subset of M : C ( α ∧ β, β ) := { γ ∈ M ; C ( α, γ, β ) } is called the cone of β at α ∧ β .We also use the notation, for elements y > x from T ( M ) , C ( x, y ) := C ( x, α ) for any (or some) α ∈ M such that br ( α ) contains y , and we say that C ( x, y ) is the cone of y at x . • For α and β in M , the subset of M : C ( α ∧ β ) := { γ ∈ M ; ¬ C ( γ, α, β } = { γ ; α ∧ β ∈ γ } is called the thick cone at α ∧ β . Note that, if α = β , thethick cone at α ∧ β is the disjoint union of all cones at α ∧ β . • For x < y ∈ T ( M ) the pruned cone at x of y is the cone at x of y minus thethick cone at y , in other words the set C (] x, y [) = { γ ∈ M ; x < γ ∧ y < y } .The interval ] x, y [ is called the axis of the pruned cone. Note that the word “cone” follows the terminology of Haskell, Macpherson andSteinhorn while our “thick cone” replace their “0-levelled set” (with the motivationthat we do not use here n -levelled sets for n = 0). We also replace “interval” by In the particular case of ultrametric spaces the C -relation is defined as follows: C ( x, y, z )iff d ( x, y ) = d ( x, z ) < d ( y, z ). The thick cones are the closed balls and cones are the open balls.Some balls may be open and closed. In the same way a closed ball, say of radius r , is partitionedin open balls of radius r , a thick cone at a node n is partitioned in cones at n . M definable by an atomic formula of thelanguage { C } are M , ∅ , singletons, cones and complements of thick cones. We cantherefore rephrase the above definition of C -minimality as follows: A C -structure M is C -minimal iff for any structure N ≡ M any definable subset of N is a Booleancombination of cones and thick cones.Given a general structure M and a subset A of M the question of the structureinduced by M on A is a delicate issue. Our particular situation prevents us ofany difficulty in the two following cases. Proposition 2.7
Let M be a C -minimal pure C -set and A a cone, thick cone orpruned cone with a dense axis in M . Then, considered as a pure C -set, A is still C -minimal. Proof:
The trace of a cone on a cone, say A , is a (relative) cone: this meansthat this trace can be described as { x ∈ A ; C ( α, β, x ) } for two parameters α and β from A . More generally the trace of a possibly thick cone on a possibly thickcone is a possibly thick cone. Thus the above statement is trivial for cones. For apruned cone, C -minimality is ensured by the axis density, see [D], p. 70, Exampleand Lemma 3.12 (the C -minimality considered there is in some sense “external”and stronger than the absolute one considered in the above statement). (cid:3) Definition 2.8
Let M be a structure and A a ∅ -definable subset of M .By defini-tion the language of the structure induced by M on A consists of all subsets ofsome A r which are definable in M without parameters.We say that A is stably embedded in M if for all integer r every subset of A r which is definable in M with parameters, is definable with parameters from A .In this case the subsets of some A r definable in M or in the structure induced by M on A are the same. Proposition 2.9
Let M be a C -minimal C -structure. Then any branch with leafof the canonical tree is stably embedded and o-minimal. Proof:
Haskell and Macpherson [H–M] have shown that each branch br ( α ) of Br l ( T ) is o-minimal in T , in the sense that, any subset of br ( α ) definable in T is a finite union of intervals with bounds in br ( α ) ∪ { + ∞} . This means exactlythat br ( α ) is 1-stably embedded in ( M , α ) in the sense of [P]. Now we can applyPillay’s criterion (Theorem 1.4 in [P]) as any C -minimal structure is NIP and anyo-minimal one rosy. (cid:3) We have defined (possibility thick or pruned) cones as subsets of M . But theyhave their counterparts in the canonical tree. So cones are subsets of M as well5s of T ( M ), we hope the context and the distinct notation C or Γ will make thechoice clear.As previously, Latin letters x, y, etc... denote nodes of T ( M ) which are not leavesand Greek letters α, β, etc... leaves. Definition 2.10 • For α and β two distinct elements of M , the subset of T ( M ): Γ( α ∧ β, β ) := { t ∈ T ( M ); α ∧ β < t ∧ β } is called the cone of β at α ∧ β . Note that it is the canonical tree of C ( α ∧ β, β ) .As for cones in M , we also use the notation, for elements y > x from T , Γ( x, y ) := Γ( x, α ) for any (or some) α ∈ M such that br ( α ) contains y andwe say that Γ( x, y ) is the cone of y at x . • For α and β in M , the subset of T ( M ) : Γ( α ∧ β ) = { t ∈ T ( M ); α ∧ β ≤ t } iscalled the thick cone at α ∧ β . Note that it is the canonical tree of C ( α ∧ β ) and that Γ( x ) = S αx ∈ α Γ( x, α ) ∪ { x } . • For x < y ∈ T ( M ) , the pruned cone at x of y is the set Γ(] x, y [) = { t ∈ T ( M ); x < t ∧ x < t ∧ b } := Γ( x, β ) \ Γ( y ) where β is the any branch suchthat x, y ∈ β . It is the canonical tree of C (] x, y [) . The interval ] x, y [ is calledthe axis of the pruned cone. Definition 2.11
We say that a leaf α of T is isolated if there exists a node x in T such that x < α and there is no node between x and α , in other words, α geta predecessor in T . If α is an isolated leaf, then its unique predecessor is denotedby p ( α ). Definition 2.12
Let x be a node of T . We say that a cone Γ at x is an innercone if:1. x has no successor on any branch br ( α ) s.t. α ∈ Γ . Note that, x has asuccessor (say x + ) on br ( α ) for some α ∈ Γ , iff Γ is a thick cone (the thickcone at x + ).2. There exists t ∈ Γ such that, for any t ′ ∈ T with x < t ′ < t , t ′ is of sametree-type as x .Otherwise, we say that Γ is a border cone . Note that the cone Γ( p ( α ) , α ) at the predecessor p ( α ) of an isolated leaf α is aborder cone which consists only of that leaf. Definition 2.13
The color of a node x of a tree T is the couple ( m, µ ) ∈ N ∪ {∞ ] where m is the number of border cones at x and µ the number of inner cones at x . Be aware that in [H–M] a cone of nodes always contains its basis, in other words a cone at a is the union of a and what we call here a cone. emma 2.14 Suppose the C -set M is ℵ -categorical. Then the color of a node of T ( M ) is definable in the pure order of T ( M ) , which means that there are unaryformulas ϕ r and ψ r , r ∈ ω ∪ {∞} , of the language { < } such that, for any node x of T ( M ) and r , T ( M ) | = ϕ r ( x ) iff there are exactly r border cones at x,T ( M ) | = ψ r ( x ) iff there are exactly r inner cones at x. Proof:
By the Ryll-Nardzewski Theorem, Condition 2 of Definition 2.12 is first-order. (cid:3) ℵ -categorical C -minimal C -sets We say that a structure is indiscernible if there is only one complete 1-type over ∅ . ℵ -categorical C -structurewith o -minimal branches For each leaf α the set br ( α ) is a chain of T with maximal element α . Definition 3.1 A basic interval of a linear ordered set O will mean a singletonor a dense convex subset with bounds in O ∪ {−∞} . So a basic interval may be a singleton or otherwise open, semi-closed or closedand denoted ( a, b ), where a < b and “(” and “)” are “[” or “]”. A singleton { x } isconsidered as the closed interval [ x, x ]. Definition 3.2
A basic one-typed interval of T is a basic interval (of some branchof T ) such that all of its element have same tree-type (over ∅ ). Theorem 3.3
Let M be a (finite or countable) indiscernible and ℵ -categorical C -structure. Let T be its canonical good tree. Assume that for each leaf α of T ,any subset of the chain br ( α ) definable in T is a finite union of intervals withbounds in br ( α ) ∪ {−∞} . Then there exists an integer n ≥ such that for any leaf α of T , the branch br ( α ) can be written as a disjoint union of n basic one-typedintervals br ( α ) = ∪ nj =1 I j ( α ) ∪ { α } , with I j ( α ) < I j +1 ( α ) . This decomposition isunique if we assume that the I j ( α ) are maximal one-typed, that is, I j ( α ) ∪ I j +1 ( α ) is not a one-typed basic interval. Possible forms of each I j ( α ) are { x } , ] x, y [ and ] x, y ] . The decomposition is independent of the leaf α , that is, the form (a singletonor not, open or closed on the right) of I j ( α ) for a fixed j as well as the tree-typeof its element do not depend on the leaf α . emark 3.4
1. Remember that Haskell and Macpherson have shown that, if M is C -minimal, then for each leaf α , any subset of br ( α ) definable in T is a finiteunion of intervals with bounds in br ( α ) ∪ {−∞} . Thus the conclusion of the abovetheorem remains the same if we add the hypothesis that M is C -minimal andremove the condition on Br l ( T ) .2. Pillay and Steinhorn have classified ([P–S], Theorem 5.1) all ℵ -categoricaland o-minimal linearly ordered structures, and not only linear pure orders. Proof of Theorem 3.3 . In the following, a “branch of T ” will always meana branch with a leaf, i.e. an element of Br l ( T ). By Ryll-Nardzewski Theoremthe ℵ -categoricity of M implies that for any integer p there is a finite numberof p -types over ∅ . Now T ( M ) is interpretable without parameters in M where itappears as a definable quotient of M . Since there is a finite number of 2 p -typesover ∅ in M , there is a finite number of p -types in T ( M ). Hence, T ( M ) is finiteor ℵ -categorical. Thus we can partition the tree T ( M ) into finitely many sets S such that two nodes in T have the same complete type over ∅ iff they are in thesame set S . The trace on any branch br ( α ) of such a set S is definable and thus,by o -minimality, a finite union of intervals. In fact it consists of a unique interval:if a node x belongs to the left first interval of S ∩ br ( α ), then by definition of thesets S any other element of S ∩ br ( α ) will too. For the same reason, if S ∩ br ( α )has a first element, then this interval is in fact a singleton. (We are here makinguse of the tree structure: the set { y ∈ T ; y < x } is linearly ordered.)Hence, for a given leaf α , br ( α ) is the order sum of finitely many maximal one-typed intervals. Using indiscernibility, the number of such basic intervals, theform (singleton, open or closed on the right) of each of them, and the tree-type ofits elements, depend only on its index and not on the branch. (cid:3) Lemma 3.5
Let α , β be two distinct leaves of T . Let j ⋆ be the unique index suchthat α ∧ β ∈ I j ⋆ ( α ) . Then, ∀ j < j ⋆ , I j ( α ) = I j ( β ) . Moreover, I j ⋆ ( α ) ∩ I j ⋆ ( β ) isan initial segment of both I j ⋆ ( α ) and I j ⋆ ( β ) . Proof:
By definition, br ( α ) ∩ br ( β ) = I ( α ) ∪· · ·∪ I j ⋆ − ( α ) ∪{ t ∈ I j ⋆ ( α ) , t ≤ α ∧ β } (or { t ∈ I j ⋆ ( α ) , t ≤ α ∧ β } if j ⋆ = 1). The same is true with β instead of α .Therefore, by definition and uniqueness of the partition of each branch into max-imal basic one-typed intervals, we get ∀ j < j ⋆ , I j ( α ) = I j ( β ). Moreover, { t ∈ I j ⋆ ( α ); t ≤ α ∧ β } = { t ∈ I j ⋆ ( β ); t ≤ α ∧ β } = I j ⋆ ( α ) ∩ I j ⋆ ( β ). (cid:3) In this subsection, T will be a good tree and L its set of leaves. Definition 3.6
One-colored basic intervalWe say that a basic interval I of a branch with a leaf of T is one-colored if I satisfies one of the following conditions: I is a singleton { x } and the k distinct cones at x are border cones. We saythat I is of color ( k, .(1.a) I is open on both left and right sides: I =] x, y [ . Any element of I is of color (0 , k ) , for a k ∈ N ∗ ∪ {∞} , that is, there are exactly k distinct cones at anyelement of I , and all are inner cones. We say that the basic interval I is ofcolor (0 , k ) .(1.b) I is open on the left side and closed on the right side: I =] x, y ] and anyelement of I is of color ( m, µ ) , for m, µ ∈ N ∗ ∪ {∞} , that is, there areexactly m border cones and µ inner cones at any point of I . We say thatthe basic interval I is of color ( m, µ ) . Definition 3.7
We say that T is a precolored good tree if there exists an integer n , such that for all α ∈ L :(1) the branch br ( α ) can be written as a disjoint union of n basic one-coloredintervals br ( α ) = ∪ nj =1 I j ( α ) ∪ { α } , with I j ( α ) < I j +1 ( α ) .(2) The I j ( α ) are maximal one-colored, that is, I j ( α ) ∪ I j +1 ( α ) is not a one-colored basic interval, and for all j ∈ { , · · · , n } , the color of I j ( α ) is inde-pendent of α .(3) For any α, β ∈ L and j ∈ { , · · · , n } , if α ∧ β ∈ I j ( α ) , then α ∧ β ∈ I j ( β ) , I j ( α ) ∧ I j ( β ) is an initial segment of both I j ( α ) and I j ( β ) ; and for any i < j , I i ( α ) = I i ( β ) .The integer n is called the depth of the precolored tree T . As an immediate consequence of Theorem 3.3:
Corollary 3.8
Let M be a finite or countable ℵ -categorical, indiscernible and C -minimal C -set. Then T ( M ) is a precolored good tree. Proposition 3.9
Let T be a precolored good tree, then all leaves of T are isolatedor all leaves of T are non isolated. Proof:
Let α be a leaf of T . Assume that α has a predecessor p ( α ), then the lastinterval I n ( α ) is closed on the right, that is either I n ( α ) = { p ( α ) } of color ( k, I n ( α ) =] x, p ( α )] of color ( m, µ ) with m = 0. By definition of precolored goodtree, for any leaf β , the last interval of br ( β ) is of color ( k, β ,the last interval of br ( β ) is of color ( m, µ ), with m = 0. In both cases, β has apredecessor. (cid:3) Definition 3.10
Definition of the partial functions e j .Let T be a precolored good tree of depth n . For < j < n , we denote e j − ( α ) thelower bound of I j ( α ) and E j the image of the function e j . By the above lemma, if j ( α ) , e j ( β ) ≤ α ∧ β , then e j ( α ) = e j ( β ) . Hence, we can extend the functions e j to partial functions from T to N in the following way: Dom ( e j ) = ∪ α ( { e j ( α ) } ∪ I j +1 ( α ) ∪ · · · ∪ I n ( α ) ∪ { α } , e j ( e j ( α )) = e j ( α ) and, ∀ x ∈ br ( α ) ∩ Dom ( e j ) , e j ( x ) = e j ( α ) . The e j and p are definable functions from L to N . Corollary 3.11
Let T be a precolored good tree of depth . Then, uniformly in α , I ( α ) is of the form,(0): { r } = { p ( α ) } ( r is the root), or (1.a): ] − ∞ , α [ , or(1.b): ] − ∞ , p ( α )] .Let T be a precolored good tree of depth n > . Then, uniformly in α ,- I ( α ) is of the form: { r } or ] − ∞ , e ( α )[ or ] − ∞ , e ( α )] or ] − ∞ , p ( e ( α ))] .- For ≤ j ≤ n − , I j ( α ) is of the form: { e j − ( α ) } or ] e j − ( α ) , e j ( α )[ or ] e j − ( α ) , e j ( α )] or ] e j − ( α ) , p ( e j ( α ))] .- Moreover, for j < n , if I j ( α ) is open on the right, then I j +1 ( α ) is a singleton.- And, uniformly in α , I n ( α ) is of the following form:if T has isolated leaves, (0): { e n − ( α ) } = { p ( α ) } , or (1.b): ] e n − ( α ) , p ( α )] ;if T has no isolated leaves, (1.a): ] e n − ( α ) , α [ . Proposition 3.12
Let T be a precolored good tree of depth n .If T has isolated leaves and I n ( α ) = { p ( α ) } , for any α ∈ L , then the set p ( L ) := { p ( α ); α ∈ L } is a maximal antichain of T .If T has isolated leaves and I n ( α ) =] e n − ( α ) , p ( α )] , then p ( L ) = ∪ α ∈ L I n ( α ) . Proof: If T has isolated leaves and I n ( α ) = { p ( α ) } for any α ∈ L , let α and β be two leaves such that p ( α ) ≤ p ( β ). Then α ∧ β = p ( α ). Hence, by Lemma 3.5, p ( α ) = p ( β ). This shows that p ( L ) is an antichain of T . To prove it is maximal,let t ∈ T ; either t is a leaf and t > p ( t ), or t is a node, hence there exists a leaf α such that t < α , thus t ≤ p ( α ).If T has isolated leaves and I n ( α ) =] e n − ( α ) , p ( α )], the result comes directly fromDefinition 3.6 (1 .b ). (cid:3) -colored good trees In section (6) we will introduce a very concrete class, the class of colored goodtrees, which will turn out to be the same thing as precolored good trees. Itsdefinition is inductive. The present section defines 1-colored good trees. Section(5) will present a construction which gives the induction step.
Definition 4.1
Let T be a good tree. We say that T is a -colored tree if T satisfies one of the following group of properties.(00) T consists of a unique element. T consists of a unique node with m leaves, where m is an integer m ≥ orinfinite.(1.a) There exists µ , an integer µ ≥ or infinite, such that for any leaf α of T , ] − ∞ , α [ is densely ordered and at each node of T there are exactly µ cones,all infinite.(1.b) There exists ( m, µ ) where m is an integer m ≥ or infinite, µ is an integer µ ≥ or infinite such that for any leaf α of T , α has a predecessor, the node p ( α ) , ] − ∞ , p ( α )] is densely ordered and at each node of T there are exactly m leaves and µ infinite cones.We will say that (00),(0), (1.a) or (1.b) is the type of the -colored tree, and (0 , , ( m, , (0 , µ ) , or ( m, µ ) its branching color . Remark 4.2
A precolored good tree T of depth is a -colored good tree of branch-ing color ( m, µ ) where ( m, µ ) is the color of any node of T . .2 Examples In the following pictures, a continous line means a dense order and a dashed linemeans that there is no node between its two extremities.(0) Trees of form (00) and (0) are canonical trees of C -sets equipped with thetrivial C -relations ( C ( α, β, γ ) iff α = β = γ ), in other words of pure sets.The first picture gives the tree of case (0), where m = 3. α α α T ype (0) rm = 3 , µ = 0(1.a) Example of type (0 , µ )Let Q be the set of rational numbers and µ an integer ≥ ℵ . Let M bethe set of applications with finite support from Q to µ , equipped with the C -relation C ( α, β, γ ) iff the maximal initial segment of Q where β and γ coincide (asfunctions) strictly contains the maximal initial segment where α and β co-incide.The thick cone at α ∧ β is the set { γ ∈ M ; γ coincide with α and β on themaximal initial segment where α and β coincide } . If α and β are differentand q is the first rational number where α ( q ) = β ( q ), then there are µ possible values for γ ( q ), in other words there are µ different cones at α ∧ β . T ype (1 .a ) m = 0 , µ = 2 α x,r α x,l x 121.b) Example of type ( m, µ ), m and µ ≥ T = N ∪ L (and this times µ = 1is possible) Example 1.b is given by the tree N × m (we remove the leavesof T and add m new leaves at each node). T ype (1 .b ) m = 2 , µ = 1 α x,l α x, α x, In what follows the tree of type (0 ,
0) is not considered.
Definition 4.3
For m and µ in N ∪ {∞} such that m + µ ≥ , we denote Σ ( m,µ ) the set of axioms in the language L := { L, N, ≤ , ∧} describing -colored goodtrees of branching color ( m, µ ) , and S the set of all these L -theories, S := { Σ ( m,µ ) ; ( m, µ ) ∈ ( N ∪ {∞} ) × ( N ∪ {∞} ) , with m + µ ≥ } . Proposition 4.4
Any theory in S is ℵ -categorical, hence complete. Moreover,it admits quantifier elimination in a natural language, Σ ( m, in { L, N } , Σ (0 ,µ ) in L = { L, N, ≤ , ∧} and Σ ( m,µ ) with m, µ = 0 in L +1 := { L, N, ≤ , ∧ , p } . Note thatin this last case, Dom ( p ) = L and Im ( p ) = N . Proof:
Trees of form (0) consist of one node and leaves. They are clearly ℵ -categorical and eliminate quantifiers in the language { L, N } .From now on, we assume that Σ = Σ m,µ , where µ = 0. Note that in this case, amodel of Σ has no root. We will prove simultaneously ℵ -categoricity and quan-tifier elimination using a back and forth between finite L -substructures in thecase where m = 0 (resp. L +1 -substructures in the case where m = 0) of any twocountable models of Σ, say T and T ′ .Forth construction:In what follows the set A = { x , · · · , x s , α , · · · , α l , } is a finite set of nodes andleaves of T which is a substructure in the language L if m = 0 (resp. L +1 if m = 0), id est closed under ∧ (resp. ∧ and p ), and ϕ is a partial L -isomorphism13resp. L +1 -isomorphism) from T to T ′ with domain A . We will use the followingfacts. Fact 1:
Let t be an element of T , t / ∈ A . Then there exists a unique node n t of T such that n t is less or equal to an element of A , and for any i and j , 1 ≤ i ≤ s and 1 ≤ j ≤ l , t ∧ x i = n t ∧ x i and t ∧ α j = n t ∧ α j . Proof:
The set B = { t ∧ x , · · · , t ∧ x s , t ∧ α , · · · , t ∧ α l } is a linearly orderedfinite set, of nodes since t is not in A . Let n t be its greatest element. So, thereexists y ∈ A such that n t = t ∧ y , and therefore n t < y . Moreover, it is easy to seethat, since n t is the greatest element of B , for any z ∈ A , t ∧ z = n t ∧ z . Unicityis clear. ⊣ Note that n t ≤ t and ( n t = t iff t is a node smaller than an element of A ). Fact 2:
Assume first that m = 0. Let t ∈ T \ A . Then the L -substructure h A ∪ { t }i generated by A and t is the minimal subset containing A , t , n t ( id est A ∪ { t, n t } if n t = t and A ∪ { t } if n t = t ).Assume now that m = 0. Let x be a node of T \ A . Then the L +1 -substructure h A ∪ { x }i generated by A ∪ { x } is the minimal subset containing A , x , n x . If α isa leaf of T , the L +1 -substructure h A ∪ { α }i generated by A ∪ { α } is the minimalsubset containing A , α , n α , and p ( α ). Proof:
Assume first that x is a node of T . Then for any t ∈ A , x ∧ t = n x ∧ t .By definition, there is z ∈ A such that n x ≤ z , so for any t ∈ A , n x ∧ t = n x or n x ∧ t = z ∧ t ∈ A .Assume now that α is a leaf of T . If α is non isolated the same argument applies.If α is isolated then p ( α ) ∈ h A ∪ { α }i . For any t ∈ A , p ( α ) ∧ t = α ∧ t = n α ∧ t .And as above, the minimal subset containing A , α , n α and p ( α ) is closed underthe function ∧ . ⊣ Fact 3:
Let Γ be a cone at x ∈ A , such that Γ ∩ A = ∅ . Then there exists a coneΓ ′ of T ′ at ϕ ( x ) such that Γ ′ ∩ ϕ ( A ) = ∅ . Moreover, if Γ is infinite, resp. consistsof a single leaf, then we can choose Γ ′ infinite, resp. consisting of a single leaf.Proof of fact 3: If Γ is an infinite cone and µ is infinite, resp. Γ = { α } and m isinfinite, the result is obvious since A is finite.If now Γ is infinite, and µ is finite, there are exactly µ infinite cones at both x and ϕ ( x ); one of the cones at ϕ ( x ), say Γ ′ , is such that Γ ′ ∩ ϕ ( A ) = ∅ . If Γ = { α } and m = 0 is finite, then, x = p ( α ) and there are exactly m leaves above both x and ϕ ( x ). So since α / ∈ A , there exists α ′ / ∈ ϕ ( A ) above ϕ ( x ). ⊣ Fact 4:
Let x / ∈ A such that n x = x . Then, x is a node and ϕ can be extended to apartial L -isomorphism (resp. L +1 -isomorphism) with domain h A ∪ { x }i = A ∪{ x } . Proof:
Since n x = x , x is smaller than an element, say a , of A . Thus h A ∪ { x }i is equal to A ∪ { x } . Since A is closed under ∧ we can take a to be the smallestelement of A , a > x . If m = 0, ] − ∞ , ϕ ( a )[ is dense. If m = 0, since A is closedunder p , a is not a leaf, neither is ϕ ( a ), so in this case too, ] − ∞ , ϕ ( a )[ is dense.So in both cases, there is x ′ in this interval such that A ∪ { x } and ϕ ( A ) ∪ { x ′ } are14somorphic trees. ⊣ Fact 5:
Let t ∈ T \ A . Then ϕ can be extended to a partial L -isomorphism(resp. a partial L +1 -isomorphism) with domain h A ∪ { n t }i . Proof:
By Fact 4. ⊣ Fact 6:
Let t ∈ T \ A . Then ϕ can be extended to a partial L -isomorphism(resp. a partial L +1 -isomorphism) with domain h A ∪ { t }i . Proof:
By Fact 5, we can assume that t = n t and n t ∈ A . Let Γ be the cone of t at n t , then by definition of n t , Γ ∩ A = ∅ . Assume first that m = 0. Since Γ isinfinite, there exists by Fact 3, an infinite cone Γ ′ at φ ( n t ) such that Γ ′ ∩ φ ( A ) = ∅ .Then we can extend φ to h A ∪ { t }i , by letting φ ( t ) = t ′ , where t ′ is any node ofΓ ′ if t is a node, or any leaf of Γ ′ is t is a leaf.Assume now that m = 0. If Γ consists in a leaf, id est, t is a leaf and n t = p ( t ),then, by fact 3, there exists a cone Γ ′ at φ ( n t ) which consists only in a leaf α ′ .Then, we can extend φ to h A ∪ { t }i , by letting φ ( t ) = α ′ . If Γ is infinite then, byfact 3, there exists an infinite cone Γ ′ at φ ( n x ) in T ′ such that Γ ′ ∩ φ ( A ) = ∅ . If t is a node, we can extend φ to h A ∪ { t }i , by letting φ ( t ) = t ′ , where t ′ is any nodeof Γ ′ . If t is a leaf, p ( t ) ∈ Γ and we can extend φ to h A ∪ { t }i , by letting φ ( t ) = t ′ ,and φ ( p ( t )) = p ( t ′ ), where t ′ is any leaf of Γ ′ . ⊣ This completes the forth construction. The back construction is the same.Hence, T and T ′ are isomorphic, thus Σ is ℵ -categorical and complete.Let ( t · · · , t n ) ∈ T , ( t ′ · · · , t ′ n ) ∈ T ′ , satisfying the same atomic formulas. Inthe above proof, we have constructed an L -isomorphism from T to T ′ , sending( t · · · , t n ) to ( t ′ · · · , t ′ n ). So these n -tuples have same complete type and Σ admitsquantifier elimination. (cid:3) Theorem 4.5
1. Precolored good trees of depth are exactly the -colored goodtrees. For such a tree its color is its branching color.2. If T is such a tree, M ( T ) is C -minimal, indiscernible and ℵ -categorical. Proof:
Let T be a 1-colored tree of color ( m, µ ). By quantifier elimination inthe langage { L, N } , L or L +1 (see Proposition 4.4) all nodes of a 1-colored tree T have the same tree-type. Singletons consisting of a leaf, if m = 0, are the bordercones and the infinite cones, if µ = 0, are the inner cones. Moreover all leaveshave same type. So, any branch of T is the union of its leaf and a one-coloredbasic interval of color ( m, µ ) and T is a precolored good tree.The converse is Remark 4.2.Again by quantifier elimination, any definable subset of T is clearly a booleancombination of cones and thick cones, which gives C -minimality. ℵ -categoricityis given by Proposition 4.4. (cid:3) Extension of a good tree by a -colored tree In this section T is a good tree considered in the langage L = {≤ , ∧ , N, L } . Thepartial function “predecessor” p and its domain are definable in the pure orderand we will make a free use of them. T ⋊ T We define now T ⋊ T , the “extension of T by T ”, where T is a 1-colored goodtree and T and T are not singletons. We require furthermore for this constructionthat:Condition ( ⋆ ): either all leaves of T are isolated or all leaves of T are non isolated.Condition ( ⋆⋆ ): if T has non isolated leaves, T should be of type (0), id est hasa root.We define T ⋊ T as the tree consisting of T in which each leaf is replaced by acopy of T . More formally, let L T and N T be respectively the set of leaves andnodes of T , L and N the set of leaves and nodes of T . As a set, T ⋊ T is thedisjoint union of N T and L T × T . The order on T ⋊ T is defined as follows: ∀ x, x ′ ∈ N T , T ⋊ T | = x ≤ x ′ iff T | = x ≤ x ′ ; ∀ ( α, t ) , ( α ′ , t ′ ) ∈ L T × T , T ⋊ T | = ( α, t ) ≤ ( α ′ , t ′ ) iff T | = α = α ′ and T | = t ≤ t ′ ; ∀ x ∈ N T , ( α, t ) ∈ L T × T , T ⋊ T | = x ≤ ( α, t ) iff T | = x ≤ α .This construction makes T ⋊ T a good tree (due to Conditions ( ⋆ ) and ( ⋆⋆ )), withset of leaves L T × L and set of nodes N T ∪ L T × N . Remark 5.1
The good tree T ⋊ T satisfies condition ( ⋆ ) .Leaves of T ⋊ T are isolated iff leaves of T are. We look now a bit more carefully at the connection between T and T ⋊ T . Byconstruction the set of nodes N T of T embeds in T ⋊ T as an initial subtree of N T ⋊ T . Let us call σ this embedding and, for each α ∈ L T , τ α the embedding of T in T ⋊ T , x ( α, x ).In the case where T has a root, L T also embeds in T ⋊ T by the map ρ : α ( α, r ), where r is the root of T . Via σ and ρ , T embeds as an initial subtree of T ⋊ T and τ α ( T ) is the thick cone at ρ ( α ).If T has no root, the embedding of N T does not extend naturally to an embeddingof T into T ⋊ T but T will appear as a quotient of T ⋊ T . Define in this case ρ : L T → T ⋊ T denote the (non injective) map α σ ◦ p ( α ). Note that by ( ⋆ ), T has isolated leaves hence p ( α ) ∈ N T and σ ◦ p ( α ) is well defined. In this case, τ α ( T ) is a cone at ρ ( α ).In both cases, ρ ( α ) = inf τ α ( T ). Definition 5.2
We define the equivalence relation ∼ corresponding to the con-struction of T ⋊ T :- ∼ is the equality on N T ; if T has a root, say r , the equivalence class of ( α, t ) ∈ L T × T is the thick coneat ρ ( α ) ; so cl ( α, t ) = cl ( α, r ) ;- if T has no root, for any ( α, t ) ∈ L T × T , the equivalence class cl ( α, t ) of ( α, t ) is the cone of t at ρ ( α ) . Remark 5.3
1. For any ( α, t ) ∈ L T × T , cl ( α, t ) = τ α ( T ) .2. Distinct equivalence classes a, b ∈ T × T / ∼ satisfy: ∃ u ∈ a, ∃ v ∈ b, u < v iff ∀ u ∈ a, ∀ v ∈ b, u < v . Consequently T × T / ∼ inherits the (good) tree structureof T × T and T × T / ∼ and T are isomorphic trees.3. The ∼ -class of any element of N T is a singleton. Consequently the embedding N T ⊆ T × T induces when taking ∼ -classes the embedding N T ⊆ T . T ⋊ T We require from now on an additional condition on T :conditions ( ⋆ ⋆ ⋆ ): if all leaves of T are isolated, then p ( L T ) is convex , id est ∀ x, y, z ∈ T, ( x, z ∈ p ( L T ) ∧ x < y < z ) → y ∈ p ( L T ).Let L := L ∪ { e, E, E ≥ } where e is a new symbol for a unary function and E and E ≥ are unary predicates. We interpret as follows these new symbols on T ⋊ T :- Dom ( e ) = L T × T if T has a root and Dom ( e ) = ( L T × T ) ∪ p ( L T ) if T hasno root;- ∀ ( α, t ) ∈ L T × T , e (( α, t )) = ρ ( α ), and if T has no root, for any α ∈ L T , e ( p ( α )) = p ( α );- E = ρ ( L T );- E ≥ = { x ; ∃ y ∈ E, y ≤ x } .The predicates E > := E ≥ \ E , E < the complement of E ≥ and E ≤ := E < ∪ E arequantifier free L -definable, and we will make a free use of them. Proposition 5.4
1. The tree T has a root iff the L -structure T ⋊ T satisfiesboth sentences: ∀ α ∈ L , α ∈ Dom ( p ) and ∀ α, β ∈ L, ¬ ( p ( α ) < p ( β )) .2. If T has a root, then E is an antichain and x ∼ y iff ( x = y or ( x, y ∈ Dom ( e ) and e ( x ) = e ( y ))) .3. If T has no root, then x ∼ y iff ( x = y or ( x, y ∈ Dom ( e ) and e ( x ) = e ( y ) First note that ( α, β ) ∈ L belongs to Dom ( p ) iff β has a predecessor in T . So Dom ( p ) ⊇ L iff T is of type (0) or (1 .b ). In this case p ( α, β ) = ( α, p ( β )).Assume that T has a root, say r .Let ( α, β ), ( α ′ , β ′ ) be two leaves of T ⋊ T such that p ( α, β ) ≤ p ( α ′ , β ′ ), so bydefinition of order in T ⋊ T and the remark above, α = α ′ and p ( β ) ≤ p ( β ′ ). But p ( β ) = p ( β ′ ) = r , so p ( α, β ) = p ( α ′ , β ′ ). So the second formula of (1) is satisfied.By definition, for any ( α, t ) ∈ L T × T = dom ( e ), e ( α, t ) = ρ ( α ) = ( α, r ). Hence,17 is an antichain. And, for all ( α, β ) ∈ L , E ∩ br (( α, β )) = { e ( α, β ) } .Moreover, the equivalence class of ( α, t ) is the thick cone at ( α, r ) = e ( α, t ).Therefore, ( α, t ) ∼ ( α ′ , t ′ ) iff e ( α, t ) ∼ e ( α ′ , t ′ ).Assume now that T has no root.In order to prove the if direction of the first assertion, we suppose in addition that T is of type (1 .b ). Let ( α, β ) be a leaf of T ⋊ T , then by definition of such a1-colored good tree, any element of ] − ∞ , p ( β )[ is the predecessor of a leaf in T ,say p ( β ′ ). So we have p ( α, β ) < p ( α, β ′ ).Since T has no root, T has isolated leaves and then for any ( α, t ) ∈ L T × T , e ( α, t ) = ρ ( α ) = p ( α ) = e ( p ( α )). By definition, the equivalence class of ( α, t ) isthe cone of t at ρ ( α ), so ( α, t ) ∼ ( α ′ , t ′ ) iff ρ ( α ) = ρ ( α ′ ) and ( α, t ) ∧ ( α ′ , t ′ ) > ρ ( α ).In other words, ( α, t ) ∼ ( α ′ , t ′ ) iff e (( α, t )) = e (( α ′ , t ′ )) < ( α, t ) ∧ ( α ′ , t ′ ). Thisprove the third assertion.We have seen that in this case, E = S α ∈ L T { p ( α ) } . So let ( α, β ) be a leaf of T ⋊ T and α ′ be a leaf of T , such that p ( α ′ ) ∈ br (( α, β )). Then, p ( α ′ ) ≤ α in T . So, p ( α ′ ) ≤ p ( α ) = e ( α, β ). Hence, e ( α, β ) is the greatest element of E ∩ br (( α, β )). (cid:3) α p ( α ) = p ( α ) L T N T N T e ( α , t ) = e ( α , t ) τ α ( T ) τ α ( T ) T ( isolated ) T ( isolated ) ⋊ T ( dense (1 .a ) or (1 .b ))19 α , β )( α , β )( α , β ) N T τ α ( T ) τ α ( T ) T ( isolated ) ⋊ T ( discrete ) ( α , β )( α , β )( α , β ) e (( α , β i )) e (( α , β i )) α ∧ α N T τ α ( T ) τ α ( T ) ( α , β )( α , β )( α , β ) e (( α , β i )) e (( α , β i ))( α , β ) ( α , β ) ( α , β ) T ( non isolated ) ⋊ T ( discrete ) Corollary 5.5 The equivalence relation ∼ is L -definable, uniformly in T anduniformly in T . This makes T uniformly L -interpretable in T ⋊ T . roof: The preceding proposition, 1, allows us to first order distinguish whether T has a root or not and gives the fitting definition for both cases. (cid:3) We assume now that T is equipped with some additional structure given by afinite set F of unary partial functions and the set P = { D f , F f ; f ∈ F } of unarypredicates satisfying Conditions (4 ⋆ ): for any f ∈ F , . Dom ( f ) = D f and Im ( f ) = F f , . D f = ( F f ) ≥ and F f ∩ L = ∅ , . ∀ t ∈ D f , f ( t ) ≤ t , . ∀ t ∈ F f , f ( t ) = t .We define L = L ∪ F ∪ P and L ′ = L ∪ F ∪ P . Note that Conditions (4 ⋆ ) arefirst order in L . We interpret L ′ on T ⋊ T as follows.- We have already defined the L ′ -structure.- For f a function in F : . Dom ( f ) T ⋊ T = ( Dom ( f ) T ∩ N T ) ˙ ∪ L T × T (recall that N T embeds as an initialsubtree in T ⋊ T ); . ∀ x ∈ ( Dom ( f ) T ∩ N T ), f T ⋊ T ( x ) = f T ( x ) and ∀ ( α, t ) ∈ L T × T , f T ⋊ T ( α, t ) = f T ( α ) (which belongs to N T since Im ( f T ) ∩ L T = ∅ by conditions (4 ⋆ ), hence to T ⋊ T ); . F T ⋊ T f = F Tf and D T ⋊ T f = ( D Tf ∩ N T ) ˙ ∪ L T × T .Conditions (4 ⋆ ) are true on T ⋊ T for the set of functions F ∪ { e } , D e = E ≥ and F e = E . We will generalize this construction in Lemma 5.9 and see in which senseit is canonical. T ⋊ T We will see how the construction of T ⋊ T can be retraced in its theory. Definition 5.6 Let Σ ′′ be the following theory in the language L :- ( ≤ , ∧ ) is a good tree;- E is convex: ∀ x, y, z, ( x, z ∈ E ∧ x < y < z ) → y ∈ E ;- E ≥ = { x ; ∃ y ∈ E, x ≥ y } ;- E ≥ = Dom ( e ) , E = Im ( e ) ;- L ⊆ E ≥ and E ∩ L = ∅ ;- ∀ x ∈ E ≥ , e ( x ) ≤ x ;- ∀ x, E ∩ br ( x ) has e ( x ) as a greatest element, where br ( x ) := { y ; y ≤ x } . For Λ | = Σ ′′ , we write E < for the interpretation of E < in Λ and we will do thesame with E ≤ , E ≤ and other symbols from L ′ . “ T has a root” will mean thatboth sentences of Proposition 5.4, item 1, are true in Λ. Lemma 5.7 Let Λ be a model of Σ ′′ . Consider on Λ the relation ∼ defined asfollows: either1. T has a root and x ∼ y iff ( x, y ∈ E < and x = y ) or ( x, y ∈ E ≥ and ( x ) = e ( y )) , or2. T has no root and x ∼ y iff ( x, y ∈ E < and x = y ) or ( x, y ∈ E ≥ and e ( x ) = e ( y ) < x ∧ y ) .Then ∼ is compatible with the order in the sense of remark 4.8, 2. More precisely,for x ∈ Λ such that ( x/ ∼ ) = { x } , ( x/ ∼ ) = Γ( e ( x )) in the first case and ( x/ ∼ ) = Γ( e ( x ) , x ) in the second case. Proof: Let x ∈ Λ such that ( x/ ∼ ) = { x } .Let y ∈ ( x/ ∼ ), then by definition e ( y ) = e ( x ). Since e ( y ) ≤ y , y ∈ Γ( e ( x )). If weare in the second case, e ( x ) = e ( y ) < x ∧ y ), thus y ∈ Γ( e ( x ) , x ).Conversely, let y ∈ Γ( e ( x )), then y ∈ E ≥ , and e ( x ) ≤ x ∧ y . Since e ( x ) ≤ y and e ( y ) ≤ y , e ( x ) and e ( y ) are comparable. In the first case, E is an antichain, thus e ( x ) = e ( y ). Assume now y ∈ Γ( e ( x ) , x ), so x ∧ y > e ( x ). Then, e ( x ) ∈ br ( y ) ∩ E ,hence e ( x ) ≤ e ( y ). If x ∧ y ≤ e ( y ), then by convexity of E , x ∧ y ∈ E , so x ∧ y ≤ e ( x )which gives a contradiction. Thus, e ( y ) ≤ x ∧ y , therefore, e ( y ) ≤ e ( x ). Finally, e ( x ) = e ( y ) < x ∧ y . (cid:3) Corollary 5.8 Let Λ be a model of Σ ′′ and ¯Λ := Λ / ∼ ; for x ∈ Λ , we note ¯ x := x/ ∼ .1. In case 1, Λ is the disjoint union E < ˙ ∪ ˙ S x ∈ E Γ( x ) , where E ≤ is an initialsubtree, E is an antichain and ∼ is the identity on E < . Hence ¯Λ is a treecanonically isomorphic to E ≤ with E its set of leaves. If all cones Γ( x ) , x ∈ E are isomorphic trees, say all isomorphic to Γ then Λ = ¯Λ ⋊ Γ .2. In case 2, Λ = E ≤ ˙ ∪ ˙ S x ∈ E > Γ( e ( x ); x ) with E ≤ an initial subtree and ∼ theidentity on E ≤ ; E ≤ embeds canonically in the tree of nodes of ¯Λ .3. Thus in both cases, E ≤ can be identified with ¯ E ≤ := { ¯ x ; x ∈ E ≤ } and E with ¯ E := { ¯ x ; e ∈ E } and considered as living in ¯Λ . Proof: 1. In this case E is an antichain and by definition of the relation ∼ , Λ isthe disjoint union of an initial tree with the union of disjoint final trees indexedby points from E , namely Λ = E < ˙ ∪ ˙ S x ∈ E Γ( x ) which is also E ≤ ˙ ∪ E > , with ∼ theidentity on E ≤ and ¯ x = e ( x ) for x ∈ E > . Thus the inclusion E ≤ ⊆ Λ induces theequality E ≤ = ¯Λ where more precisely of E < is identified with the set of nodes of¯Λ and E with its set of leaves.2. By definition of ∼ in case 2, Λ has the form indicated. Hence the inclusion E ≤ ⊆ Λ induces an inclusion E ≤ ⊆ ¯Λ. Since E ∩ L = ∅ , E ≤ embeds in fact in theset of nodes of ¯Λ. (cid:3) Lemma 5.9 Let Λ be a model of Σ ′′ , ¯Λ := Λ / ∼ . Suppose furthermore ¯Λ equippedwith an L -structure model of (4*); we note ¯ L , ¯ p and for f ∈ F , ¯ f the interpretationin ¯Λ of the symbols L , p and f from L . Then there is exactly one L ′ -structure on Λ defined as follows: for each function f ∈ F : . For x ∈ E ≤ , f is defined at x iff, in ¯Λ , ¯ f is defined at ¯ x and in this case f ( x ) is the unique y ∈ E ≤ such that ¯ y = ¯ f (¯ x ) in ¯Λ .2. For x ∈ E ≥ , f ( x ) = f ( e ( x )) .This L ′ -structure on Λ satisfies conditions (4 ∗ ) for the set of functions F ∪ { e } with F e = E and F f = F ¯ f (see Corollary 5.8, (3)) for f ∈ F . Proof: The uniqueness of x in 1. is ,given by Corollary 5.8 and 1 and 2 arecompatible since e is the identity on E . For f ∈ F and x ∈ E ≥ , f ( x ) ∈ E ≤ ; now e ( x ) := max ( E ∩ br ( x )) hence f ( x ) ≤ e ( x ) ≤ x ; for x ∈ E ≤ , “ f ( x ) = f ( x ) ≤ ¯ x = x ”. Other Conditions (4 ⋆ ) for f on Λ follow from E ∩ L = ∅ and Conditions (4 ⋆ )for ¯ f on ¯Λ. (cid:3) Proposition 5.10 Let Σ ′ be the theory in the language L ′ consisting of the con-junction of Σ ′′ and the following axioms and axiom schemes:- for ∼ the relation defined as in Lemma 5.7 and x ∈ E ≥ in case 1 or x ∈ E > incase 2, the ∼ -class of x is elementary equivalent to T (as a pure tree);- the quotient modulo ∼ and T are elementary equivalent L -structures;- if T has no root then by Condition ( ⋆⋆ ) leaves of the quotient modulo ∼ have apredecessor and, interpreted in the quotient modulo ∼ , ¯ E = ¯ p ( ¯ L ) ;- for any f ∈ F , conditions 1 and 2 of Lemma 5.9.Then Σ ′ is a complete axiomatization of T ⋊ T . If T is ℵ -categorical then Σ ′ is ℵ -categorical too. If T eliminates quantifiers in L ∪ { p, D, F } where p is thepredecessor function, D its domain and F its image, then Σ ′ eliminates quantifiersin L ′ ∪ { p, D, F } . Proof: Assume first that T has a root. Take Λ | = Σ ′ . Assume CH for short andΛ as well as T and T saturated of cardinality finite or ℵ . As an L -structure, Λmust be the extension T ⋊ T described in Corollary 5.8, case 1. By Lemma 5.9the rest of the L -structure on Λ as well is determined by its restriction to E ≤ idest by the L -structure T . So Σ ′ has a unique saturated model of cardinality finiteor ℵ . This shows the completeness of Σ ′ .About quantifier elimination now. Take any finite tuple from Λ. Close this tupleunder e . Write it in the form ( x, y , . . . , y m ) where x is a tuple from E ≤ , y , . . . , y m tuples from E > such that all components of each y i have same image under e , callit e ( y i ) (thus, e ( y ) , . . . , e ( y m ) are coordinates of x ), and e ( y i ) = e ( y j ) for i = j .Take ( x ′ , y ′ , . . . , y ′ m ) ∈ Λ having same quantifier free L ′ ∪ { p, D, F } -type than( x, y , . . . , y m ). Thus x ′ ∈ E ≤ and ( y ′ , . . . , y ′ m ) ∈ E > . If L ∪ { p, D, F } eliminatesquantifier of T , x and x ′ have same complete type in E ≤ = T and there is anautomorphism σ of T sending x to x ′ . Any automorphism, say f , of Λ extending σ will send for each i , e ( y i ) to σ ( e ( y i )). Hence f ( y i ) and y ′ i are in the same copyof T , say T i . Since T consists of one root and leaves and f ( y i ) as well as y ′ i consists of distinct leaves, there is an automorphism σ i of T i sending f ( y i ) to y ′ i . The disjoint union of σ , the σ i and the identity on other copies of T is theautomorphism of Λ we were looking for.23e consider now the case where T has no root and suppose as previouslythat Λ, T and T are saturated of cardinality finite or ℵ . This time Λ = E ≤ ˙ ∪ ˙ S x ∈ E > Γ( e ( x ); x ) and N T = E ≤ (recall that, by Lemma 5.8 (3), N T livesalso in Λ). By the third axiom scheme, ¯ E = ¯ p ( L T ) hence the L -structure on Λmust be the extension T ⋊ T described in Corollary 5.8, case 2. By Lemma 5.9again the rest of the L -structure on Λ is determined by the L -structure T . Thisshows the uniqueness of the saturated model of cardinality finite or ℵ and thecompleteness of Σ ′ .The proof of quantifier elimination runs very similarly too, with the small differ-ence that the existence of the σ i comes from quantifier elimination in T .- If T is of type (1 .a ), it eliminates quantifier in L which gives σ i as desired.- If T is of type (1 .b ), it eliminates quantifiers in L = L ∪ { p } . In T the do-main of p is L T . For each embedding of T in Λ as a cone Γ( e ( x ); x ) we have theinclusions L T ⊆ L Λ ⊆ Dom ( p Λ ) and for any leaf α of (this) T , p T ( α ) = p Λ ( α ),which gives the σ i . This shows that Λ eliminates quantifier in L ′ ∪ { p L , Im ( p L ) } where p L is the restriction of the predecessor function to the set of leaves and Im ( p L ) its image. Now adding p L to L ′ is quantifier free equivalent to adding p : Dom ( p ) = ( Dom (¯ p ) ∩ E ≤ ) ∪ L , p coincide with ¯ p on Dom (¯ p ) ∩ E ≤ and with p L on L .Suppose now T ℵ -categorical. By Theorem 4.5, T is ℵ -categorical too. Theabove proof of the completeness shows the uniqueness of the countable model ofΣ ′ and its ℵ -categoricity. (cid:3) Definition 5.11 If Σ is a complete axiomatization of T and Σ is a completeaxiomatization of T , we denote Σ ⋊ Σ the theory Σ ′ . Proposition 5.12 Σ ⋊ Σ is C -minimal iff Σ is. Proof: Let Λ | = Σ ⋊ Σ . For A ⊆ L ( ¯Λ), A Λ := { α ∈ L (Λ); ¯ α ∈ A } is a conein ¯Λ iff A is a cone in Λ, of same type (thick or not) except when A consists ofa non isolated leaf (in ¯Λ) and A Λ a is a cone. This proves two things. First ¯Λ is C -minimal if Λ is. Secondly if ¯Λ is C -minimal any subset of Λ of the form A Λ isa Boolean combination of cones and thick cones. The general case is processed byhand. Fact: For x a leaf of Λ, a composition of functions from F ∪ { p, e } applied to x is, up to equality, a constant or of the form x, p ( x ) (only if T is of type (1 .b ))or t ( e ( x )) where t is a composition of functions from F ∪ { p } (hence a term of L ∪ { p } ).Assume the first function right in the term is p . If T is of type (0) we replace p with e . If T is of type (1 .b ), p ( x ) Dom ( p ). Conclusion: at most one p right. Ifa term t is a composition of functions from F ∪ { e } , then t ( p ( x )) = t ( x ). Indeed, e ( x ) < x if x ∈ L hence e ( x ) = e ( p ( x )) (by definition e ( x ) = max ( E ∩ br x )),and f ( x ) = f ( e ( x )). Conclusion: in composition no p right needed. Finally, for24 ∈ F ∪ { e } , f ( x ) = f ( e ( x )). So, if a term is neither x nor p ( x ), we may assumeit begins right with the function e . ⊣ So non constant terms in x are all smaller that x , thus linearly ordered. Con-sequently, up to a definable partition of L (Λ) (namely in the two sets { x ; t ( x ) ≥ t ′ ( x ) } and { x ; t ( x ) < t ′ ( x ) } ), terms of the form t ( x ) ∧ t ′ ( x ) are not to be considered.To summarize, it is enough to consider subsets definable by formulas t ( x ) ≤ t ′ ( x ), t ( x ) = t ′ ( x ), t ( x ) ≤ a , t ( x ) = a , t ( x ) ∈ E, E ≥ , F f or D f , and ( t ( x ) ∧ a ) = b where t and t ′ are of the form described in the above fact. To ϕ a one variable formulafrom L without constant associate a formula ϕ Λ (also from L , one variable andwithout constant) such that Λ | = ϕ Λ ( x ) iff ¯Λ | = ϕ (¯ x ). Then ϕ ( e ( x )) is equivalentto:- ϕ Λ ( x ) when T has a root, and- ψ Λ ( x ) with ψ ( y ) = ϕ (¯ p ( y )) when T has no root,both already handled. Are left to be considered:- t ( e ( x )) < x and, if T is of type (1 .b ), t ( e ( x )) < p ( x ) < x are always true,- x ∈ E, F f always wrong, as p ( x ) ∈ E, F f are since p ( x ) occurs only if T is iftype (1 .b ),- x, p ( x ) ∈ E ≥ , D f always true,- t ( x ) (cid:3) b and ( t ( x ) ∧ a ) (cid:3) b with (cid:3) ∈ { <, = , > } , formulas that we treat now.For b ∈ E > , t ( e ( x )) ≥ b is always wrong and t ( e ( x )) < b is equivalent to t ( e ( x )) < e ( b ). For b ∈ E ≤ , Λ | = t ( e ( x )) (cid:3) b iff ¯Λ | = t ( e ( x )) (cid:3) ¯ b . For b ∈ E > ,( t ( e ( x )) ∧ a ) ≥ b is always wrong and ( t ( e ( x )) ∧ a ) < b iff ( t ( e ( x )) ∧ a ) < e ( b ).For a ∈ E > , ( t ( e ( x )) ∧ a ) = ( t ( e ( x )) ∧ e ( a )). Finally, for a and b in E ≤ ,Λ | = ( t ( e ( x )) ∧ a ) (cid:3) b iff ¯Λ | = ( t ( e ( x )) ∧ a ) (cid:3) ¯ b . We are left with formulas x (cid:3) b , p ( x ) (cid:3) b , ( x ∧ a ) (cid:3) b and ( p ( x ) ∧ a ) (cid:3) b which are routine. (cid:3) Proposition 5.13 L ( T ⋊ T ) is indiscernible iff L ( T ) is. Proof: The right-to-left implication follows clearly from our proof of C -minimalitytransfer from L ( T ) to L ( T ⋊ T ). The other direction is trivial since T is a definablequotient of L ( T ⋊ T ) (and leaves are sent to leaves in the quotient). (cid:3) Definition 6.1 A colored good tree is a tree of the form ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n for some integer n ≥ , where T , · · · , T n are -colored good trees such that, foreach i , ≤ i ≤ n − , if T i is of type (1 .a ) then T i +1 is of type (0) . Remark 6.2 - By Remark 5.1 and an easy induction on n , T = ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n is a well defined good tree.- Moreover, if T = ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n is a colored good tree then for any k ≤ n , ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T k and ( . . . ( T k +1 ⋊ T k +2 ) ⋊ · · · ) ⋊ T n are coloredgood trees.- Conversely, let T ′ = ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n and T ′′ = ( . . . ( T n +1 ⋊ T ) ⋊ · · · ) ⋊ n + m , m ≥ , be colored good trees such that (( T ′ ⋊ T n +1 ) ⋊ T n +2 is an coloredgood tree, then, T ′ ⋊ T ′′ =: ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n + m is a colored good tree.- T is a colored good tree iff T is a -colored good tree or ( T = T ′ ⋊ T n where T ′ is a colored good tree and T n is a -colored good tree). Convention: T ′ ⋊ T will always be T . Definition 6.3 Let T be a good tree and x a node of T . Extending the definition4.1, we call branching color of x and we note b - col T ( x ) the couple ( m T ( x ) , µ T ( x )) , m T ( x ) , µ T ( x ) ∈ N ≥ ∪ {∞} , where m T ( x ) is the number of cones at x which arealso thick cones (in other words the number of elements of T which have x as apredecessor) and µ T ( x ) is the number of cones at x which are not thick cones. Remark 6.4 - Branching color is definable in the pure order of T in the sense ofLemma 2.14 (no ℵ -categoricity presently needed).- If T is a -colored good tree then the branching color of any node of T is its colorin the sense of Definition 2.13 (so uniform on T ). We will denote ( m i , µ i ) thecolor (i.e. branching color) of any node of a -colored tree T i (in T i ). Lemma 6.5 Let T = T ′ ⋊ T n a colored good tree. Let E, E > , E < as defined insection 5.2. Then for x ∈ T ,- if x ∈ E < , then b - col T ( x ) = b - col T ′ ( x ) ,- if x ∈ E and T n has a root, then b - col T ( x ) is the branching color (in T n ) of theroot of T n (of the form ( m, ),- if x ∈ E and T n has no root, then b - col T ( x ) = (0 , µ T ′ ( x ) + m T ′ ( x )) ,- if x ∈ E > , then b - col T ( x ) is the branching color of any node of T n . Proof: Clear by construction of T ′ ⋊ T n . (cid:3) We intend to define the function e associated to the extension T ′ ⋊ T n in termsof change of branching color, which is not always possible. Take for example n = 2, and T = T ⋊ T . If T is of type (1 .b ), then for any α ∈ L T , e ( α ) = Sup ( br ( α ) ∩ { x ∈ N, x is of branching color ( m T , µ T ) } ) and T is of type (0),then e ( α ) = p ( α ). But assume now that T is of type (1 .b ) of color (1 , 1) and T isof type (1 .a ) with color (0 , T is a 1-colored good tree of color (0 , 2) and,by quantifier elimination (Proposition 4.4), e is not definable. Proposition 6.6 Let T = T ′ ⋊ T n be a colored tree. The function e is definable inthe pure order except when T n is of type (1 .a ) of color (0 , µ n ) and T ′ = T − ⋊ T n − ( T ′ = T if n = 2 ) and:Exception 1: T n − is -colored of type (1 .b ) of color ( m n − , µ n − ) and µ n = m n − + µ n − or,Exception 2: T n − is -colored of type (0) and T − = T = ⋊ T n − ( T − = T if n = 3 ) and T n − is of type (1 .a ) of color (0 , µ n − ) and µ n − = m n − = µ n . roof: Note that if the restriction to L T of e is definable in the pure order, then E ≥ = { x ∈ T ; ∃ α ∈ L T , x ≥ e ( α ) } is definable and for all x ∈ E ≥ , e ( x ) = e ( α ) forany α ∈ L T , α ≥ x , so e is definable.Assume first that T n has a root, then e ( α ) = p ( α ), for any α ∈ L T , so e is definable.Assume now that T n is of type (1 .b ), then by Lemma 6.5, the color of any elementof E > is ( m n , µ n ), with m n = 0, while, if x ∈ E , b - col T ( x ) = (0 , µ T ′ ( x ) + m T ′ ( x )).Therefore, for any α ∈ L T , e ( α ) = Sup ( br ( α ) ∩ { x ∈ N ; b - col ( x ) = (0 , µ ) , µ ∈ N ≥ ∪ {∞}} , so e is definable.From now on T n is of type (1 .a ). Again by Lemma 6.5, the branching color of anyelement of E > is (0 , µ n ), and if x ∈ E , b - col T ( x ) = (0 , µ T ′ ( x ) + m T ′ ( x )). We aregoing to apply again Lemma 6.5 to the tree T ′ and its corresponding subsets E ′ < , E ′ and E ′ > .If T n − is of type (1 .b ), then E ⊂ E ′ > , therefore for any x ∈ E , the branchingcolor of x is its branching color in T n − . The first exception of the propositionstatement insures that m n = m n − + µ n − . Hence e is definable as follows: forany α ∈ L T , e ( α ) = Sup ( br ( α ) ∩ { x ∈ N, b - col ( x ) = (0 , m n − + µ n − ) } .If T n − is of type (0), E = E ′ , hence for any x ∈ E , b - col T ′ ( x ) = (0 , m n − ), so b - col T ( x ) = (0 , m n − ). Therefore if µ n = m n − , e is definable as above. Now, if µ n = m n − , we must consider the branching colors of the nodes of E ′ < thus wemust look down at the tree T − and its corresponding subsets E − , E − < and E − > . If T n − is of type (0), or (1 .b ), by the previous discussion E − is definable in the pureorder and E ′ = E is the subset of all successors of nodes of E − , hence definable inthe pure order too. If T n − is of type (1 .a ), then the branching color of the nodesof E − > is (0 , µ n − ). By the second exception of the proposition, µ n − = m n − , soas previously, the function e is definable. (cid:3) Remark 6.7 Note that in exception 1, T n − ⋊ T n is a -colored tree, and inexception 2, T n − ⋊ T n ⋊ T n is also a -colored tree. In these cases e cannot bedefinable in the pure order. Definition 6.8 We define n - colored good trees by induction on n ∈ N ≥ :A -colored good tree has been defined in Definition 4.1.An ( n + 1) -colored tree is a colored tree which is not a k -colored tree for any k ≤ n . Corollary 6.9 Let T be a colored tree, then there exists a unique n ∈ N ≥ suchthat T is an n -colored tree. Definition 6.10 We define and interpret now by induction the language L n on n -colored good trees. The language L has already been defined. For n ≥ , let L n +1 = L n ∪ { e n , E n , E ≥ ,n } where e n is a partial functions and E n and E ≥ ,n areunary predicates. We consider T an n -colored good tree as an L n -structure withan interpretation of L n defined by induction as follows:- if n = 1 , L is interpreted naturally as in Proposition 4.4;- if T = T ′ ⋊ T n +1 symbols of function e i (resp predicates E i and E ≥ ,i ) of L n are nterpreted in T as functions and predicates of F ∪ P are in the construction ofparagraph 5.2, that is Dom ( e i ) = ( Dom ( e Ti ∩ N T ) ˙ ∪ L T × T n +1 = E ≥ ,i , ∀ x ∈ ( Dom ( e Ti ∩ N T ) , e i ( x ) = e Ti ( x ) , ∀ ( α, t ) ∈ L T × T n +1 , e i ( α, t ) = e Ti ( α ) , E i the image of e i and e n = e , E n = E = Im ( e n ) , E ≥ ,n = E ≥ = Dom ( e n ) .For ≤ i ≤ n , predicates E >,i = E ≥ ,i \ E i , E <,i , the complement in N T of E ≥ ,i and E <,i = E ≤ ,i \ E i are definable in L n . Remark 6.11 These definitions are legitimate since at each induction step, func-tions of F satisfy condition (4 ⋆ ) . Proposition 6.12 Let T be an n -colored tree. Then functions and predicates of L n \ L are definable in the pure order. Proof: Follows directly from Proposition 6.6 by induction. (cid:3) Definition 6.13 Let T = ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n be an n -colored good tree andfor any i , ≤ i ≤ n , let Σ m i ,µ i the complete theory of the -colored good tree T i .We denote Σ m ,µ ⋊ · · · ⋊ Σ m n ,µ n the L -theory ( . . . (Σ m ,µ ⋊ Σ m ,µ ) ⋊ · · · ) ⋊ Σ m n ,µ n defined by induction using Propositions 5.11 and 6.12.We denote S n the sets of all theories Σ m ,µ ⋊ · · · ⋊ Σ m n ,µ n in the language L when the conditions of Definition (6.8) are fullfilled. Theorem 6.14 For any integer n ≥ any theory in S n is complete and admitsquantifier elimination in the language L n ∪ { p, D, F } where p is the (partial) pre-decessor function, D its domain and F its image. Furthermore S n is the set ofcomplete theories of all n -colored good trees. Proof: We proceed by induction on n . Case n = 1 is given by Proposition 4.4and the induction step by Proposition 5.10. (cid:3) ℵ -categorical C -minimalpure C -sets Theorem 7.1 Let M be a countable or finite pure C -structure. Then the follow-ing assertions are equivalent:(i) M is ℵ -categorical, C -minimal and indiscernible(ii) T ( M ) is a precolored good tree.(iii) T ( M ) is a colored good tree. roof: ( i ) ⇒ ( ii )This is a direct consequence of section 3: Theorem 3.3, Lemma 3.5 and Proposi-tion 3.6.( ii ) ⇒ ( iii )The case of depth 1 is given by Remark 4.2.We will prove the result by induction on the depth of a precolored good tree.Asume that any precolored good tree of depth n is an n -colored good tree. Let T be a precolored good tree of depth n + 1. By 3.11, for any leaf α , the lat-est one-colored interval I n +1 ( α ) of the branch br ( α ) is either { p ( α ) } , case (0), or] e n ( α ) , α [, case (1 .a ) , or ] e n ( α ) , p ( α )], case (1 .b ) .In case (0), for any leaf α the thick cone T α at p ( α ) is a 1-colored good tree oftype (0), and in the case (1 .a ) or (1 .b ), for any leaf α , the cone T α of α at e n ( α )is a 1-colored good tree of type (1 .a ) or (1 .b ). Let us call ( m n +1 , µ n +1 ) the color(independent of α ) of the 1-colored good tree T α . Thus by Proposition 4.4, forany α , T α | = Σ m n +1 ,µ n +1 . Let T n +1 be the countable or finite 1-colored good treemodel of Σ m n +1 ,µ n +1 .Moreover, T is an L -structure, interpreting e by e n , E = Im ( e n ). We can easilycheck that, as an L -structure, T is model of Σ” (cf 5.6). Let us consider on T the equivalence relation associated to e n , as defined in 5.2, and T := T / ∼ .Then, T is a good tree whose set of nodes is ∪ α { x ∈ T ′ ; x < e n ( α ) } if T n +1 is oftype (0) and ∪ α { x ∈ T ′ ; x ≤ e n ( α ) } if T n +1 has no root; and whose set of leavesis ∪ α { cl ( x ) , x ≥ e n ( α ) } if T n +1 has a root, and ∪ α { cl ( x ) , x > e n ( α ) } otherwise.Therefore, by 5.8 and 5.9, T is an precolored good tree of depth n . From the uni-form decomposition of any branch into one-colored intervals described in Corollary3.11, if I n ( α ) is open on the right, then I n +1 ( α ) is a singleton, thus T and T n +1 satisfy conditions ( ⋆⋆ ). Then by Proposition 5 . T is elementary equivalent to T ⋊ T n +1 .By induction hypothesis, T = ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n , hence, T = ( . . . ( T ⋊ T ) ⋊ · · · ⋊ T n ) ⋊ T n +1 is well defined. So, its remains only to verify conditions ofDefinition 6.8.Assume for a contradiction that T n +1 is of type (1 .a ) of color (0 , µ n +1 ) and that T n is of type (1 .b ) of color ( m n , µ n ), with µ n +1 = m n + µ n . Then for any leaf α the intervals I n ( α ) and I n +1 ( α ) are basic intervals of same color (0 , µ n +1 ) whichcontradicts the fact that T is a precolored good tree of depth n + 1 (maximality ofthe intervals). The same contradiction arises if T n is of type (0) of color ( m n , T n − is of type (1 .a ) of color (0 , µ n − ) and m n = µ n − = µ n +1 . Hence T is an( n + 1)-colored good tree.( iii ) ⇒ ( i )We proceed again by iduction on n .The case n = 1 is given by Theorem 4.5.Assume that T ′ is an ( n + 1)-colored good tree. By definition, T ′ = T ⋊ T n +1 ,where T is an n -colored good tree and T n +1 a one-colored good tree. By induc-tion hypothesis, M ( T ) is ℵ -categorical, C -minimal and indiscernible, thus byPropositions 5.13, 5.12 and ?? , M ( T ′ ) has the same properties. (cid:3) emark 7.2 The proof of the above proposition shows that an n -colored good treeis a precolored good tree of depth n and that the color of any node x of such a treeis the color in the basics intervals I i ( α ) containing x . In this section we reduce the general classification of ℵ -categorical and C -minimal C -sets to the classification of indiscernible ones, previously achieved in section 7.By the Ryll-Nardzewski Theorem, any ℵ -categorical structure is a finite unionof indiscernible subsets. In a C -minimal structure M these subsets have a veryparticular form. Let us give an idea: there exists a finite subtree Θ of T := T ( M ),closed under ∧ and ∅ -algebraic with the following properties:- any a ∈ Θ, except its root, has a predecessor in Θ since Θ is finite, call it a − ;now, in T , ] a − , a [ is either empty or dense, and if it is dense, then the pruned cone C (] a − , a [) is indiscernible in M ,- for a as above and b ∈ Θ, b > a , then C (] a − , b [) is not indiscernible,and similar other properties, to deal with cones at a ∈ Θ for example. Anequivalence relation is defined over Θ which identifies points a and b such that C (] a − , a [) ∪ C (] b − , b [) is not empty and is indiscernible (other couples of elementsare also identified). We call vertices the elements of the quotient ¯Θ of Θ. They arefinite antichains of T and (oriented) arrows linking them are induced by the order(it is the classical order on antichains). Vertices and arrows of ¯Θ are labeled. Asan example, on a vertex A ,- a first label gives the (finite) cardinality of A seen as a subset of T ,- another labels says whether, for any a ∈ A , ] a − , a [ is either empty and dense ornot,- and if it is dense, a third label gives the complete theory of the indiscernible C -set C (] a − , a [).There are other labels which are also either cardinals in N ∪ {∞} or complete the-ories of indiscernible ℵ -categorical and C -minimal C -structures. Conversely, wehave isolated ten properties such that, given a labelled graph Ξ sharing theseten properties, there is an ℵ -categorical and C -minimal C -set M such that¯Θ( M ) = Ξ. In this sense, the classification of ℵ -categorical and C -minimal C -sets is reduced to that of indiscernible ones. Proposition 8.1 Let M be an ℵ -categorical structure, then there exists a uniquepartition of M into a finite number of ∅ -definable subsets which are maximal in-discernible. Proof: By ℵ -categoricity, there is a finite number of 1-types over ∅ . By com-pacity, each of these types is consequence of one of its formulas. (cid:3) efinition 8.2 We call this partition the canonical partition . Thereafter it willbe noted ( M , · · · , M r ) . By C -minimality definable subsets in one variable have a simple form. We re-formulate here for convenience the description given in [ D ] in the proof of Propo-sition 5, with a small difference: instead of working with T ( M ) we will work with T ( M ) ∗ defined as follows: T ∗ := T if T has a root and T ∗ := T ∪ {−∞} otherwise.In the last case, we say that “ −∞ exists”. Definition 8.3 By C -minimality each M i of the canonical description is a finiteboolean combination of cones and thick cones. Let C be the set of bases of conesand thick cones appearing in these combinations. We define Θ := { x ∈ T ( M ) ∗ ; forsome c ∈ C, x ≤ c } and Θ := { x ∈ Θ ; ∃ i = j, α ∈ M i , β ∈ M j ; x ∈ br ( α ) ∩ br ( β ) } .We define: U := { suprema of branches from Θ } B := { branching points of Θ } S := { c ∈ Θ \ ( U ∪ B ); the thick cone at c without the cone of the branch of Θ intersects non trivially both M i and M j for a couple ( i, j ) , i = j } I := (cid:8) infima ∈ Θ \ ( U ∪ B ∪ S ) of intervals on branches of Θ which aremaximal for being contained in { c ∈ Θ \ ( U ∪ B ∪ S ); the thick cone at c withoutthe cone of the branch of Θ is entirely contained in a same M i (cid:9) Θ := U ∪ B ∪ S ∪ I . Remark 8.4 Since C is finite, Θ and Θ are trees with finitely many branches,which implies that U and B are finite; S is finite since it is contained in C ; I isfinite by o-minimality of branches of Θ . Hence Θ is finite. Θ , U , B , S , I and Θ are all definable from the M i , hence ∅ -definable since the M i are. As Θ is finite, it is contained in the algebraic closure of the empty set. Θ is a subtree of T ( M ) ∗ closed under ∧ . Because it is finite each element of Θ hasa predecessor in Θ . Elements of Θ which are nodes (or leaves) in T ( M ) may notbe nodes (or leaves) in Θ . So, to avoid confusion we will use the words verticesand edges for the tree Θ .We have the equivalence: M is not indiscernible iff Θ is not empty iff the root of T ( M ) ∗ belongs to Θ . Proposition 8.5 Let M be a C -minimal, ℵ -categorical structure. Then the sub-sets M , · · · , M r of the canonical partition are the orbits over ∅ of acl ( ∅ ) -definablesubsets of the form: • a finite union of cones at a same basis • an almost thick cone (i.e. a cofinite union of cones at a same basis) • a pruned cone C (] a, b [) where a < b and ] a, b [ is a dense interval withoutextremities (namely the cone of b at a minus the thick cone at b ). ach of these sets, endowed with the structure induced by M , is C -minimal, ℵ -categorical and indiscernible. Call them ( M , · · · , M r ) . Proof: By definition of Θ, any M i is a finite union of pruned cones C (] a, b [),cones and thick cones at a , with a, b ∈ Θ and a the predecessor of b in Θ. By ∅ -definissability, M i is the union of the orbits over ∅ of these sets (for more details,see [D], Proposition 3.7). This gives almost the first assertion except the fact that] a, b [ is a dense interval without extremity. This result follows from ℵ -categoricityusing the following facts. Fact 8.6 Assume some subset of the canonical partition is of the form M j = S ni =1 C (] a i , b i [) with b i = b j when i = j . Let ( a, b ) be one of the couples ( a i , b j ) .Then all the elements of the pruned cone C (] a, b [) have same type over ( a, b ) in M . Proof: Assume M ω -homogeneous. Then, for x, y ∈ C (] a, b [) there exists anautomorphism of M sending x to y . Such an automorphism preserves M j hencepreserves C (] a, b [). Therefore x and y have the same type over ( a, b ). ⊣ Fact 8.7 All nodes of ] a, b [ have same type over ( a, b ) . Proof: This is a direct consequence of the preceding Fact, since any node of ] a, b [is of the form b ∧ x , where x ∈ C (] a, b [). ⊣ Now, since all the nodes of ] a, b [ have same type over ∅ , either ] a, b [ is denseor consists of a unique node, or contains an infinite discrete order which is notpossible by ℵ -categoricity.In the case where ] a, b [ consists of a single node, say c , C (] a, b [) is an almost thickcone, that is the thick cone at c without C ( c, b ). So, C (] a, b [) changes from thethird category to the second category of subsets.We may now assume that ] a, b [ is dense. Since all nodes of C (] a, b [) have sametype over ( a, b ), C (] a, b [) is indiscernible in M and thus for its induced structure. C -minimality follows from [D], Lemma 3.12, and ℵ -categoricity follows from Ryll-Nardzewski. (cid:3) In particular, Fact 8.6 gives the following fact. Fact 8.8 If a ∈ Θ has a predecessor in T ( M ) ∗ , then this predecessor belongs alsoto Θ . ⊣ The M i are a priori not pure C structures. It is true that any subset of M i definablein M is definable in the pure C -set M i (it is the content of the C -minimality of M i ). But why should it be also the case for subsets of Cartesian powers of M i ? Asan example, the function C (] a, b [) → Γ(] a, b [), α α ∧ b is a priori not definablein the pure C -set C (] a, b [) (and Lemmas 8.7 and 8.12 will take this point intoaccount). Nevertheless what we have done in Section 3 applies to the (due toProposition 8.5 last assertion) C -minimal pure C -set M i . In particular, if C (] a, b [)is as in Lemma 8.6, then branches of its canonical tree Γ(] a, b [) are uniformlydecomposed in finitely many basic intervals (cf. Theorem 3.3).32 emma 8.9 Let C (] a, b [) be a pruned cone as in Fact 8.6. Then ] a, b [ is includedin the first level of Γ(] a, b [) , the colored good tree associated to C (] a, b [) . Proof: Hence the statement follows immediately from Fact 8.7. (cid:3) Notation: Let T = ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n be an n -colored good tree and Σ itscomplete theory. We will denote Σ =1 the L -theory of the 1-colored good tree T which is the first level of T , and Σ > the L -theory of the ( n − . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n . For i ∈ { , · · · , n } , we will note ( m i , µ i ) the colorof any node of the 1-colored tree T i .By construction of the canonical partition, each M i is maximal indiscernible,i.e. if i = j , there do not exist α ∈ M i and β ∈ M j with same type. We investigatesome consequences below. Lemma 8.10 Let a ∈ Θ be maximal in Θ , a not the root of Θ . Let a − be itspredecessor in Θ . If the interval ] a − , a [ is empty, then a is not a leaf of T ( M ) andthere exist at least two cones at a with different complete theories as colored goodtrees. Proof: Since a is maximal, following the notation 8.3, a is in U , i.e. a is thesupremum of some branch from Θ . Since ] a − , a [ is empty, a is in Θ , hence a belongs to at least two branches of different types in M . In particular a is not aleaf. (cid:3) Lemma 8.11 Let M be a countable C -minimal C -structure.Let a, b ∈ T ( M ) , with b < a and such that the interval ] b, a [ is not empty and isdense. Assume that the canonical tree Γ(] b, a [) of the pruned cone C (] b, a [) is an n -colored good tree and let Σ ] b,a [ be its complete theory. Assume furthermore that ] b, a [ is contained in the first level of Γ(] b, a [) . Let C be the union of at least twocones at a , such that each of these cones is indiscernible. Then, T ( C (] b, a [) ∪ C ) isa model of Σ ] b,a [ if and only if one of the following cases appears:1. m = 0 , n ≥ , and the thick cone at a in T ( C (] b, a [) ∪C ) is an ( n − -coloredgood tree model of (Σ ] b,a [ ) > .2. m = 0 , and C is the union of exactly µ cones at a , all models of Σ ] b,a [ .3. m = 0 and µ = 0 . n = 1 : C is the union of exactly m = 0 cones which consist of a leaf, and µ cones which are all models of Σ ] b,a [ . n ≥ : C is the union of exactly m cones which are models of (Σ ] b,a [ ) > andexactly µ cones wich are models of Σ ] b,a [ . Proof: Note that C becomes the thick cone at a in the C -structure C (] b, a [) ∪ C =: N . So for any α ∈ C , the branch, br T ( N ) ( α ) of α in T ( N ) is the union of ] b, a [ andits branch in T ( C ), and for any β ∈ C (] b, a [) the branch of β in T ( N ) is egal to thebranch of β in Γ(] b, a [) and keeps the same decomposition into basic one-coloredintervals. 33e will prove first the ”if” direction.(1) Assume the first case appears. Then, in T ( N ), a is the root of an ( n − ] b,a [ ) > , so its color is ( m , ℵ -categoricity,Theorem 7.1, this ( n − b, a [) > . Let T be the first level of Γ(] b, a [) plus an additional element a which is now the leaf ofthe branch ] b, a [. Then, by eq for 1-colored good trees, T is a model of (Σ ] b,a [ ) =1 .Hence, T ( N ) = T ⋊ Γ(] b, a [) > . So, T ( N ) is model of Σ ] b,a [ .(2) and (3). We will show that in these cases, T ( N ) is a precolored good treewhose any branch has the same decomposition into one-colored basic intervals asany branch of Γ(] b, a [). The conclusion will then follows by Theorem 7.1. Sinceany cone of T ( C ) is a colored good tree and therefore a precolored good tree,conditions of 3.7 hold for α and α ′ belonging to the same cone of C . The sameis true for any β and β ′ belonging to C (] b, a [). Note that, for any α ∈ C and any β ∈ C (] b, a [), α ∧ β ∈ ] b, a [, so the condition (3) of Definition 3.7 holds as well inthat case.Let us now considerate the two situations in detail.Assume (2).Then, by hypothesis, the color of a in T ( N ) is (0 , µ ). Let α ∈ C . Since ] b, a [is included in the first level of Γ(] b, a [), the interval ] b, a ] is a basic one coloredinterval of color (0 , µ ). So br T ( N ) ( α ) admits a decomposition into n one coloredbasic intervals, whose first interval contains strictly ] b, a ]. Morever, let α and α ′ be two leaves belonging to two distincts cones of C . Then, I ( α ) ∩ I ( α ′ ) =] b, a ], sothis intersection is an initial segment of both I ( α ) and I ( α ′ ). Therefore, T ( N )is a precolored good tree, hence an n -colored good tree model of Σ ] b,a [ .Assume (3).Then a has the same color ( m , µ ) as any node of ] b, a [.If n = 1, any branch of T ( N ) is of the form br T ( N ) ( α ) =] b, p ( α )] ∪ { α } , where thecolor of ] b, p ( α )] is ( m , µ ) and p ( α ) = a iff α is in a cone at a which consists ina leaf. So, clearly, T ( N ) is a 1-colored tree.Assume n ≥ 2. Let Γ be a cone at a which is a model of Σ > b,a [ , then Γ is aborder cone at a , and for any α of Γ , br T ( N ) ( α ) =] b, a ] ∪ br T ( C ) ( α ). So, as in thecase (2), br T ( N ) ( α ) has the same decomposition into one colored basic interval asany branch of Γ(] b, a [).Let Γ be a cone at α which is model of Σ =1] b,a [ , then Γ is an inner cone at a , andfor any leaf α of Γ , the initial segment ] b, a ] has the same color as the first onecolored interval of br T ( C ) ( α ), so br T ( N ) ( α ) has the same decomposition into onecolored intervals as any branch of Γ(] b, a [), with ] b, a ] strictly included in the firstinterval. Hence, for any α ∈ Γ , α ′ ∈ Γ , I ( α ) ∩ I ( α ′ ) =] b, a ] and is an initialsegment of both I ( α ) and I ( α ′ ). So, in this case again, T ( N ) is a precoloredgood tree.Conversely, if C (] b, a [) ∪ C is indiscernible in M , T ( C (] b, a [) ∪ C ) is an n -coloredgood tree, and since ] b, a [ belongs to the first level of Γ ] b,a [ ∪ Γ, the color of a is( m , µ ) or ( m , µ ). 34ssume first that the color of a is ( m , µ ). Let Γ( a, α ) be a cone at a , then eitherΓ( a, α ) is an inner cone and its theory is Σ ] b,a [ , or Γ( a, α ) is a border cone and itstheory is (Σ ] b,a [ ) > . If m = 0, then there is only inner cones at a , all models ofΣ ] b,a [ , and we are in the second case.If m = 0, and n = 1, it’s clear.If n ≥ 2, then there are m = 0 border cones at a which are models of (Σ ] b, a [) > ,and we are in the third case.Assume now that the color of a is ( m , µ ). Then, the first level of T ( C (] b, a [) ∪ C )is of type (1 .a ). So, m = 0, and a is the root of an ( n − ] b,a [ ) > . So we are in the first case. (cid:3) Lemma 8.12 Let Σ ∈ S n be a complete theory n -colored good tree Σ = Σ ( m ,µ ) ⋊ · · · ⋊ Σ ( m n ,µ n ) with µ = 0 and V a unary predicate such that V / ∈ L n . Let usconsider the theory Σ( V ) in the langage L V := L ∪ { V, ∧ V } , which consists of Σ together with the axiom: Ax ( V ) : V is a “branch” (i.e. a maximal chain) inthe first level of any (some) model of Σ and V has no leaf. Let ∧ V : x x ∧ V .Then the theory Σ( V ) is complete, admits quantifier elimination in the language L Vn := L n ∪ { V, ∧ V } , is C -minimal, ℵ -categorical and indiscernible. Proof: Consider an n -colored good tree T = ( . . . ( T ⋊ T ) ⋊ · · · ) ⋊ T n modelof Σ with T countable or finite. Since µ = 0, T not only is infinite but has2 ℵ branches. Hence 2 ℵ many of them have no leaf, which shows Σ( V ) to beconsistent. Let Σ =1 be the L -theory of T , and L V = L ∪ { V, ∧ V } . Since V is included in T , we will first prove that the theory Σ =1 ( V ) = Σ =1 ∪ { Ax ( V ) } admits quantifier elimination in the language L V . We will use a back and forthargument between countable models T and T ′ as in 4.4. Let A be a finite L V -substructure of T , and ϕ a partial L V -isomorphism from T to T ′ with domain A . Let x ∈ T \ A . With the same notation as in the proof of 4.4 there exists anode n x such that x ∧ n x is the maximal element of the set { x ∧ y ; y ∈ A } .Assume first that x ∈ V T \ A , then x is a node and by Fact 2 of 4.4 the L V -substructure h A ∪ { x }i is the minimal subset containing A , x and n x . Since n x ≤ x , n x belongs to V T .Assume that x = n x , so h A ∪ { x }i = A ∪{ x } . As in Fact 4, there exists a ∈ A ∩ V T such that ] − ∞ , a [ ∩ A = ∅ and x ∈ ] − ∞ , a [. Moreover, ] − ∞ , ϕ ( a )[ is includedin V T ′ . So, there is x ′ in this interval such that A ∪ { x } and ϕ ( A ) ∪ { x ′ } areisomorphic L V structures.Now, we can assume that n x = x and n x ∈ A . So it is possible to find x ′ ∈ V T ′ , x ′ > ϕ ( n x ) such that h A ∪ { x }i is L V -isomorphic to h ϕ ( A ) ∪ { x ′ }i .Assume now that x ∈ T \ ( V T ∪ A ). Then, since A is closed under ∧ V , and n x ≤ x and n x smaller then an element of A , the L V -substructure h A ∪ { x }i isstill the minimal subset containing A , x and n x if x if T has non isolated leaf andif T has isolated leaves the minimal subset containing A , x , n x if x is a node andthe minimal subset containing A , x , n x and p ( x ) if x is a leaf.If x = n x we proceed as in Fact 4 of 4.4 to extend φ . If x = n x , either n x / ∈ V T and we proceed as in Fact 6 of 4.4, or n x ∈ V and we extend φ to A ∪ { n x } as35bove. Then, we may assume that n x ∈ A and the proof runs similarly. The backconstruction is the same.So the theory Σ =1 ( V ) is ℵ -categorical, hence complete.Let ( t · · · , t n ) ∈ T , ( t ′ · · · , t ′ n ) ∈ T ′ satisfying the same atomic formulas, thenusing the same arguments we can see that h t · · · , t n i is isomorphic to h t ′ · · · , t ′ n i .Therefore the theory Σ =1 ( V ) eliminates quantifiers in the language L V .To achieve the proof, we will proceed as in 5.10. Let T and T ′ be two countablemodels of the L Vn theory Σ( V ). Then, T and T ′ are n -colored good trees, whoserestriction to the langage L n are isomorphic. By Remark 6.2, T = T ⋊ T and T ′ = T ′ ⋊ T ′ , where T and T ′ are two L n − -isomorphic models of Σ > . Moreover,by definition of n -colored good trees, since T and T ′ are of type (1 .a ), T and T ′ have a root. Now take any finite tuple from T and close it under e . Write it in theform ( x, y , . . . , y m ) where x is a tuple from ( E ) ≤ , y , . . . , y m tuples from ( E ) > such that all components of each y i have same image under e , call it e ( y i ) (thus, e ( y ) , . . . , e ( y m ) are components of x ), and e ( y i ) = e ( y j ) for i = j . Recall that( E ) ≤ is equal to T or to N ( T ) (depending on the type of the first level of T ).Take ( x ′ , y ′ , . . . , y ′ m ) ∈ T ′ having same quantifier free L Vn -type than ( x, y , . . . , y m ).Since L V eliminates quantifiers, x and x ′ have same complete type in ( E ) ≤ , andthere exists an L V -isomorphism σ from T onto T ′ sending x to x ′ . Since Σ > eliminates quantifiers in L n − ∪ { p, D, F } , for any i , there exists an isomorphism ϕ i , from the copy of T above e ( y i ) onto the copy of T ′ above e ( y ′ i ) sending y i on y ′ i . Since σ ( e ( y i )) = e ( y ′ i ), there exists a L Vn ∪ { p, D, F } -isomorphism ϕ from T onto T ′ extending σ and each ϕ i . Therefore, Σ( V ) is complete and eliminatesquantifiers in the langage L Vn ∪ p, D, F . (cid:3) Lemma 8.13 Let a, b ∈ T ( M ) , with b < a and such that the interval ] b, a [ isnot empty and is dense. Assume that the canonical tree Γ ] b,a [ of the pruned cone C (] b, a [) is an n -colored good tree with colors ( m i , µ i ) for ≤ i ≤ n and that ] b, a [ is contained in its first level. Let Σ ] b,a [ ( V ) be the complete theory of Γ(] b, a [) .Assume furthermore that there is c ∈ T ( M ) , c > a , such that ] a, c [ is not emptyand (Γ(] a, c [) is a model of Σ ] b,a [ . Then (Γ(] b, c [)) is a model of Σ ] b,a [ ( V ) iff thereare at a exactly m + µ cones and among those that do not contain c , m aremodels of (Σ > b,a [ and µ − models of Σ ] b,a [ . Proof: The proof of the ”if” direction uses quite the same arguments as in thepreceding lemma. Note that the hypotheses imply that the color of a in the treeΓ(] b, c [) is ( m , µ ). Let α be an element of ( C (] b, c [ \C (] b, a [). Assume first that α belongs to the cone at a containing c . Then, br Γ(] b,c [ ( α ) =] b, a ] ∪ br Γ(] a,c [ ( α ), with] b, a ] strictly included in the first one colored interval of br Γ(] b,c [ ( α ). Therefore, br Γ(] b,c [ has the same decomposition into one colored interval as any branch of C (] b, a [).If α belongs to one of the µ − a models of Σ ] b,a [ , say Γ , then, Γ isan inner cone at a and br Γ(] b,c [ ( α ) =] b, a ] ∪ br Γ ( α ). Hence again ] b, a ] is strictly36ncluded in the first one-colored interval of br Γ(] b,c [ ( α ), which has the same decom-position into one-colored intervals as any branch of C (] b, a [).Now, let us consider α in one of the m cones at a model of (Σ ] b,a [ , say Γ . Then, Γ is an border cone at a , and br Γ(] b,c [ ( α ) =] b, a ] ∪ br Γ ( α ). Here, the first one-coloredinterval of br Γ(] b,c [ ( α ) is exactly ] b, a ]. It easy to verify that for any α, α ′ ∈ C (] b, c [, I ( α ) ∩ I ( α ′ ) is an initial segment of both I ( α ) and I ( α ′ ).Conversely, assume that (Γ(] b, c [ , ] b, c [) is a model of Σ ] b,a [ ( V ). Then the color of a is ( m , µ ) and a ∈ V . Since ] b, c [ is included in the first level of the tree Γ(] b, c [),the cone of c at a is one of the µ border cone at a and all the border cone at a aremodels of Σ ] b,a [ . And all the inner cone at a are clearly models of (Σ ] b,a [ ) > . (cid:3) ¯Θ The automorphism group of M acts on Θ. Let Θ := { A , . . . , A s } be the set oforbits of elements from Θ. Each A i is a finite antichain of T ∗ . Definition 8.14 For A and B antichains in T ∗ , let us define:- the relation A < B : ⇐⇒ ∀ a ∈ A, ∃ b ∈ B, a < b and ∀ b ∈ B, ∃ a ∈ A, a < b (given b this a is unique);- for (finite) antichains A < B in T ∗ such that for any a ∈ A , b, c ∈ B with a < b, c , either b = c or a = b ∧ c , the (definable) subset ] A, B [ of M consisting ofthe union of cones of elements from B at nodes from A , with the thick cones atnodes from B removed. We extend this notation to ] {−∞} , A [ , or stil ] − ∞ , A [ ,which will denote the complement of the union of thick cones at all a ∈ A . Fact 8.15 Let A and B be in Θ . Then- if there is a ∈ A and b ∈ B with a < b (or a = b ) then A < B (or A = B ).- (Θ , < ) is a finite meet-semi-lattice tree; its root, say A , is a singleton (either { r } if r is a root of T , or {−∞} ). I¸t allows to define the predecessor A − of anelement A = A of Θ .- If A < B there is k ∈ N ≥ such that each a ∈ A is smaller than exactly k elements from B .- If A = B − , a ∈ A, b, c ∈ B, a < b, a < c, b = c then a = b ∧ c . (cid:3) We now aim to collect on Θ and the indiscernible blocks M i enough informationto be able to reconstruct M from them. To each A ∈ Θ, associate- its cardinality n A ;- an integer s A , complete theories Σ A, , . . . , Σ A,s A in L all different and coefficients k A, , . . . , k A,s A ∈ N ≥ ∪ {∞} such that, at each a ∈ A , there are exactly k A, + · · · + k A,s A cones containing no branch from Θ, k A, of which are models of Σ A, ,...,and k A,s A models of Σ A,s A (we are here applying Ryll-Nardzewski again);- if A = A , ] A − , A [ = ∅ , b ∈ A − , a ∈ A and b < a , the complete L ( V )theoryΣ A − ,A of Γ(] b, a [) in the language L ∪ { V } where V is a unary predicate definedon Γ(] b, a [) and interpreted as: V ( x ) ↔ x < a . Since the interval ] b, a [ is a dense37inear order without endpoint, V is a branch without leaf from Γ(] b, a [) and thefollowing holds:( ∗ ): for all x in in a thick cone at a , for all y ∈ Γ(] b, a [), x ∧ y = V ∧ y .We consider the s A , Σ A,i and k A,i (the Σ A − ,A ) as labels on the vertices (edges) ofΘ or Θ, and the n A as labels on the vertices of Θ. The Σ A,i (Σ A − ,A ) may alsobe understood as indexing those cones at any/some a ∈ A (pruned cones Γ(] b, a [)pour b ∈ A − , a ∈ A , b < a ) which are models of it. Lemma 8.16 1. Assume A = A . There is no theory Σ A − ,A labeling ] A − , A [ iff ] A − , A [= ∅ .2. For A ∈ Θ and any/some a ∈ A , Θ has a unique branch at a iff there is aunique B ∈ Θ such that B − = A , and furthermore n A = n B holds.3. T ∗ = T iff s A = 0 , A has a unique successor in Θ , say B , and n B = 1 . Proof: (1) holds by definition of the labels of Θ.(2) is clear.(3): The direction only if is clear. Let us prove the if direction. The unique el-ement, say a , of A is either −∞ or the root of T . If A has a successor, a isnot a leaf, and if different from −∞ it must be a branching point of T . Now thehypotheses force Θ to have a unique branch at its root. Therefore a = −∞ . (cid:3) The next lemma gives a list of constraints. Lemma 8.17 Let A and A ∈ Θ , A the root of Θ .(1) If A = A , n A − divides n A ; n A = 1 .(2) If A is maximal in Θ , then either s A = 0 , or Σ ≤ i ≤ s A k A,i ≥ .(3) If −∞ exists and B ∈ Θ is such that B − = A , then ] A , B [ = ∅ .(4) If Θ has a unique branch in any/some a ∈ A , and A = {−∞} if −∞ exists,then s A ≥ .(5) Assume A = A and a ∈ A . If ] A − , A [ is not empty, then Σ A − ,A is atheory of colored tree with an initial branch, in the sense of lemma 8.12,with a = sup V .(6) At most one k A,i is infinite. The Σ A,i are complete theories of C -structurewith colored canonical tree.(7) Let A be maximal in Θ , A not the root of Θ . If ] A − , A [ is empty then s A ≥ .(8) Let A be maximal in Θ , A not the root of Θ and such that ] A − , A [ is notempty. Assume that models of Σ A − ,A are n -colored trees with colors ( m i , µ i ) for ≤ i ≤ n . Then, none of the following situation can’t appear: a) m = 0 , n ≥ , s A = 1 , Σ A, = (Σ A − ,A ) > and k A, = m .(b) m = 0 , s A = 1 , Σ A, = Σ A − ,A and k A, = µ .(c) m = 0 , µ = 0 , n = 1 , s A = 2 , Σ A, = Σ A − ,A k A, = µ , Σ A, = Σ (0 , (i.e. the theory of a tree consisting only of a leaf ) and k A, = m . m = 0 , µ = 0 , n ≥ , s A = 2 , Σ A, = Σ A − ,A , k A, = µ , Σ A, =(Σ A − ,A ) > , and k A, = m .(9) Let A ∈ Θ , A not the root of Θ and such that ] A − , A [ is not empty. Assumethat models of Σ A − ,A are n -colored trees with colors ( m i , µ i ) for ≤ i ≤ n .If A is not maximal teh conjonction of the following condition can’t appear:- at least one wedge of Θ at A has a label- let B be the successor of A on that wedge and the label of ] A, B [ is Σ A − ,A - either m ≥ , µ ≥ , s A = 2 , Σ A, = Σ A − ,A , k A, = µ − , Σ A, =(Σ A − ,A ) > , or m = 0 , s A = 1 , Σ A, = Σ A − ,A , k A, = µ , or µ = 1 , s A = 1 , Σ A, = Σ > A − ,A , k A, = m . Proof. (1) n A − divides n A by indiscernibility of elements from A . It has alreadybeen noticed in Fact 8.15 that A is a singleton.(2) If A is maximal in Θ, either any a ∈ A is a leaf of T ( M ) and then s A = 0,or any such a is a node in T ( M ) where no branch of Θ goes through and thenΣ ≤ i ≤ s A k A,i ≥ −∞ exists, no branch of T has a first element.(4) Indeed a must be a node in T ( M ) ∗ .(5) It is lemma 8.7.(6) At most one k A,i is infinite by strong minimality of the node a , for any a ∈ A .(7) It is a reformulation of lemma 8.10.(8) For A maximal, the situation has already been set out in Lemma 8.11, that weapply here with b ∈ A − , a ∈ A and C the thick cone at a . In this way T ( C (] b, a [) ∪C )becomes the cone Γ( b, a ) of a at b . Condition (8) prevents C ( b ∧ a, a ) from beinga model of Σ A − ,A hence indiscernible. Would it be the case, it would be as wellindiscernible in M contradicting maximal indiscernibility of (the orbit of) C (] b, a [).(9) Follows from Lemma 8.13. (cid:3) A last constraint is given by the next proposition. Proposition 8.18 (10) The tree Θ labeled with the coefficients n , k , s and thetheories Σ on its edges and vertices has no non trivial automorphism. By construction two elements from Θ having same type in M are identified inΘ. Thus, to prove the above proposition it is enough to show that, if M is thecountable model, then any automorphism of Θ lifts up to an automorphism of M .This proof requires some new tools that we introduce now.39 .3 Connection and sticking ⊔ of C -structures. Let κ > H i , i ∈ κ , C -structures. The underlying set of the connection H = F H i of the H i is the disjoint union of the H i and its canonicaltree the disjoint union of the T ( H i ) plus an additional root r with, for a, b ∈ T ( H ), a ≤ b in T ( H ) iff a = r or a, b ∈ T ( H i ) for some i , and a ≤ b in T ( H i ). Canonicity : If the H i are pure C -structures then H is the unique pure C -structurewhose canonical tree has a root, say r , and having exactly the H i as cones at r . Language : If each H i is a C -structure in the language L ( H i ), which contains onlyunary predicates and unary functions, we consider H := F i ∈ I H i in the language L ˙ ∪{ ( H i ) i ∈ I } ˙ ∪ ˙ S i ∈ I ( L ( H i ) \ L ) ˙ ∪{ r } where r is a new constant for the root of T ( H ), each H i is a unary predicate for the elements of H i and each L ( H i ) isnaturally interpreted in H i . Outside of the H i relations are never satisfied and weconsider for example functions as constant, ranging on the root of T ( H ). Lemma 8.19 If I is finite then Aut( F i ∈ I H i ) ≃ Π i Aut H i . Lemma 8.20 Assume that I is finite. If for any i ∈ I , H i eliminates quanti-fiers (respectively is C -minimal, or ℵ -categorical)), then F i ∈ I H i has the sameproperties. Proof: Preceding lemma for quantifier elimination (in a saturated model, thereis an automorphism sending an element to another one iff these two elements havesame quantifier free type). C -minimality follows, ℵ -categoricity is trivial. (cid:3) We will slightly weaken the language to allow permutation of certain terms ofthe connection and, in this way, keep control of model theoretical properties evenwhen the number of terms is infinite. Notation : For H a C -structure and k > H · k is the connection of k copies of H . Assume that H and T ( H ) are the unique sort in L ( H ) and L ( H )consists only of function or predicate symbols (no constant). We consider H · k as an L ( H )-structure as follows. The tree structure gives the interpretation ofsymbols of L . Each copy of H in H · k is a cone at r , the root of T ( H · k ). Thisuniform definition of the copies of H and the fact that they are disjoint in H · k ,as copies of T ( H ) are in T ( H · k ), allow us to interpret in H · k predicates andfunctions from L ( H ) \ L as the (disjoint) union of their interpretations in thedifferent copies of H .We define L ( H · k ) = L ( H ) ∪ { e r , E r } where e r is the unary function sending everyelement of the canonical tree to the root r of T ( H · k ) and E r is the unary predicateinterpreted as { r } .By definition H · H . 40 emma 8.21 Assume k > and H ℵ -categorical or finite. Then, H · k is ℵ -categorical or finite. If H eliminates quantifiers in L ( H ) , then H · k eliminatesquantifiers in L ( H · k ) . Proof: In H · k , copies of H are exactly cones at r . Therefore, if H is countableor finite, H · k is the unique model countable or finite if k is finite, and H · ℵ isthe unique countable model if k is infinite. Let us show that, in this model, giventwo finite tuples with same quantifier free type, there is an automorphism sendingone to this other. This will show quantifier elimination. We first enumerate thecopies of H in H · k ( k finite or ℵ ): H · k = F i ∈ k H i . A tuple in H · k may thus bewritten, up to a permutation of its components, x = ( x , . . . , x j , . . . , x m ) with m finite and x j tuple of H l j . Up to a permutation of the H i in H · k , l j = j . A tuple y ∈ H · k with same quantifier free type as x in the language {≤ , ∧ , E r } has, upto a permutation of the H i , the same decomposition y = ( y , ..., y j , ..., y m ) witheach y j a tuple of H j with same type as x j . Since H is homogeneous, there is anautomorphism σ j of H j sending x j to y j . The disjoint union of the σ j on H j for j = 1 , ..., m and of the identity on the other factors is an automorphism of H · k which sends x to y . (cid:3) Notation : Thereafter F i ∈ I H i · k i will denote the connection of k i copies of H i ,for i ∈ I (Σ k i > H i is a C -structure in a language L ( H i ), we consider F i ∈ I H i · k i in thelanguage L ˙ ∪{ H i ; i ∈ I } ˙ ∪ ˙ S i ∈ I ( L ( H i · k i ) \ L ) where each H i is a unary predicatefor the union of the k i copies of H i , and L ( H i · k i ) is interpreted in H i · k i asdescribed before Lemma 8.21. Lemma 8.22 Suppose I finite. If each H i eliminates quantifiers (respectively is ℵ -categorical), then so does F i ∈ I H i · k i . If each H i is C -minimal and at mostone k i is infinite, then F i ∈ I H i · k i is C -minimal too. Proof: The proof is not exactly the same as for Lemma 8.20. Indeed, if each H i is a cone, it is not true of H i · k i . But H i · k i is a finite union of cones if k i is finite,and if k i is the unique one to be infinite, then H i · k i is the complement of a finiteunion of cones. (cid:3) ⊳ in a pruned cone M of a C -structure C whose canonicaltree has a root Let be given two C -structures, first C , which has a root in its canonical tree, andthen ( M, V ), where V is a branch without leaf from T ( M ). We will define the C -structure M ⊳ C . The underlying set of M ⊳ C is the disjoint union M ˙ ∪C ,its canonical tree the disjoint union T ( M ) ˙ ∪ T ( C ) equipped with the unique orderextending those of T ( M ) and T ( C ), satisfying V = { t ∈ T ( M ); t < T ( C ) } andsuch that no element of T ( C ) is smaller than any element of T ( M ).41 anonicity : M ⊳ C is the unique C -structure which is the union of M and C and where C becomes a thick cone with basis the supremum of V . . We consider M ⊳ C in the language L ˙ ∪{ M, C} ˙ ∪ ( L ( M ) \ L ) ˙ ∪ ( L ( C ) \ L ) ˙ ∪{ V M , ∧ V M } where M and C are unary predicates for the elements from the eponym sets and L ( M )and L ( C ) are naturally interpreted in M and C respectively. Again, outside oftheir natural definition domain, functions can be defined as constant with valuethe root of T ( C ). Lemma 8.23 If M et C eliminates quantifiers (respectively are C -minimal, or ℵ -categorical), then M ⊳ C has the same property. The two previous constructions will enable us to prove proposition 8.18. Let Ξ be a finite semi-lattice tree. The depth of a vertex in Ξ isthe minimal function from Ξ to ω such that:if a is a maximal element of Ξ , depth ( a ) = 0 ;if x < y , depth ( x ) ≥ depth ( y ) + 1 . Lemma 8.25 Given a finite meet-semi-lattice tree Ξ , labeled with a coefficient n A to each A ∈ Ξ and satisfying (1) with A the root of Ξ , there is a unique tree Ξ which is the disjoint union of antichains U A , A ∈ Ξ , with | U A | = n A , and suchthat the set of the U A ordered by the order induced by the order of Ξ is isomorphicto Ξ . Furthermore Ξ is a meet-semi-lattice tree and any automorphism of Ξ liftsto an automorphism of Ξ . Proof. We define inductively an order on Ξ := ˙ S A ∈ Ξ U A . Let Ξ ⊆ Ξ satisfying[( A, B ∈ Ξ & A < B & B ∈ Ξ ) ⇒ A ∈ Ξ ], and assume ˙ S A ∈ Ξ U A alreadyordered in such a way that the U A are antichains. For X ∈ Ξ \ Ξ such that X − =: B ∈ Ξ , we extend the order on ˙ S A ∈ Ξ U A ˙ ∪ U X . Since X − = B , n B divides n X which allows us to partition U X = ˙ S y ∈ U B V y where each V y has n X ( n B ) − elements; for x ∈ U X and y ∈ U B we set x > y iff x ∈ V y , with no other orderrelation between elements from U B ∪ U X . We define in this way an order, whichis a meet-semi-lattice tree because Ξ is one and n A = 1. The other propertiesand the uniqueness (up to isomorphism) are clear. (cid:3) Proof of proposition 8.18. As already noticed, it is enough to prove that any au-tomorphism of the labeled tree Θ( M ) lifts up to an automorphism of M ( M is thecountable model). The previous lemma gives the tree Θ( M ) from the tree Θ( M ),and the full labels as well: if π : Θ( M ) → Θ( M ) is the canonical projection, avertex or an edge from Θ( M ) and its image under π have the same label. Any au-tomorphism of the labeled tree Θ( M ) comes from an automorphism of the labeledtree Θ( M ), which comes itself from an automorphism of M as we show now byan induction on the depth of vertices from Θ( M ). Let σ be an automorphism of42he labeled tree Θ( M ), a and σ ( a ) two maximal elements of Θ( M ). By canonicityof the connection, the thick cone at a is the connection of all the cones at a , aswell as the thick cone at σ ( a ). Since σ preserves labels, theories indexing a and σ ( a ) are the same. Because these theories are ℵ -categorical and M is countableor finite, cones at a or σ ( a ) are the same. So are thick cones at a and σ ( a ) bycanonicity of the connection. There are two different induction steps for a vertex:- one is similar to the case of maximal elements of Θ( M ): if all cones at b ∈ Θ( M )and at σ ( b ) are isomorphic, so are the thick cones;- if b ∈ B ∈ Θ( M ), if the thick cones at b and σ ( b ) are isomorphic, then the coneof b at c ∈ B − , c < b and of σ ( b ) at σ ( c ) are isomorphic (by canonicity of thesticking). (cid:3) M from Θ( M ) Consider a finite meet-semi-lattice Ξ , with its vertices and edges labeled as in8.25, A its root and A ∈ Ξ \ { A } . Lemma 8.25 produces another finite meet-semi-lattice Ξ labeled as Ξ plus the n A . Since here Ξ is not yet a subtree of agiven tree, we must reformulate conditions (1) to (9) of Lemma 8.17, and (10) ofProposition 8.18 in terms of meet-semi-lattice and labels only. Lemma 8.16, makesthat reformulation possible. For example, the condition “ −∞ exists in T ( M )” willbe replaced by “ S A = 0, A has a unique successor in Ξ , say B and n B = 1”.So we obtain the conditions (1 ′ ) to (10 ′ ) as follows: (1’), (2’), (6’), (8’) and (9’)are the same as (1), (2), (6), (8) and (9) in 8.17, and (10’) is the same as in 8.18.(3’) If S A = 0, A has a unique successor in Ξ , say B and n B = 1, then thereis a theory Σ A ,B labelling ] A , B [.(4’) If Ξ has a unique branch in any/some a ∈ A , for A = A in the case where S A = 0, A has a unique sucessor in Ξ , say B and n B = 1, then S A ≥ A = A , and a ∈ A , and b ∈ A − , b < a . If there exists a theoryΣ A − ,A labelling ] A − , A [, then Σ A − ,A is a theory of colored tree with a branchwithout leaf V and a = supV .(7’) Let A be maximal in Ξ , A not the root of Ξ and s A = 0, a ∈ A , b ∈ A − with b < a . If there is no theory Σ A − ,A labelling ] A − , A [, then s A ≥ M from Θ( M ). Theorem 8.26 Given a finite meet-semi-lattice tree Ξ labeled with coefficientsand theories satisfying (1’) to (10’), there exists a unique finite-or-countable pure C -structure M which is C -minimal, ℵ -categorical, and such that Θ( M ) = Ξ . We will build M by induction. We build Ξ from Ξ according to Lemma 8.25then we define C -structures M a and N a for a ∈ Ξ, by induction on the maximallength of chains in Ξ (or in Ξ, it is the same thing), M a for each such a , and43 a if furthermore a is not the root of Ξ. The M a are destined to become thickcones in M and the N a cones, and they are the only possible choice thanks to thecanonicity of both constructions of connection and sticking. The language of eachstructure is defined by induction too. Among the labels there are theories Σ, eachof which comes with its own language.- Let A be maximal in Ξ and a ∈ A . If s A = 0 then M a is a singleton. IfΣ ≤ i ≤ s A k A,i ≥ M a = F ≤ i ≤ s A Γ A,i · k A,i , where Γ A,i is the unique finite-or-countable model of Σ A,i . It is to be noticed that in both cases, T ( M a ) has a root(due to axiom (2) and the definition of the connection when a is not a leaf). Eachtheory Γ A,i is considered in each elimination language and is given by Lemma8.22.- If a is not maximal in Ξ, M a = F B − = A,b ∈ B,b>a N b · ( n B : n A ) ⊔ F ≤ i ≤ s A Γ A,i · k A,i ,where Γ A,i is the unique finite-or-countable model of Σ A,i (and again we use aslightly abusive notation: we take the connection of k A,i copies of Γ A,i , and ( n B : n A ) copies of N b , for 1 ≤ i ≤ s A and B − = A, b ∈ B, b > a . There again, bycondition (2), T ( M a ) has a root and the Γ A,i are considered in their eliminationlanguages.- For A different from the root A of Ξ and if ] A − , A [ is not empty, Σ A − ,A hasaccording to Lemma 8.12 a unique finite-or-countable model Γ A − ,A and we set N a = M a ⊲ Γ A − ,A with the C -relation given by ( ∗ ) and V . If ] A − , A [ is empty, weset N a = M a .In the case where S A = 0, A has a unique successor B in Ξ with n B = 1, thenwe define M = N b , where b is the unique element of B and then T ( M ) has no rootand A = {−∞} . Else, we define M = M a , where A = { a } . Lemma 8.27 The set M is the disjoint union of k A,i copies of Γ A,i with A ∈ Ξ and ≤ i ≤ s A , and of ( n A : n A − ) copies of Γ A − ,A , with A ∈ Ξ , A = A . Thelabelled graph Ξ deduced from Ξ and the n A using Lemma 8.25 embeds canonicallyin T ( M ) ∗ and T ( M ) ∗ is the disjoint union of Ξ , copies of T (Γ A,i ) , and copies of T (Γ A − ,A ) , with the natural tree structure. In particular the A are antichains in T ( M ) . Proof: By construction, for each A, B ∈ Ξ with B − = A , a ∈ A and b ∈ B , N b is the disjoint union of M b and Γ A,B , and T ( N b ) the disjoint union of T ( b ) and T (Γ A,B ); M b is the disjoint union of k A,i copies of Γ A,i and ( n B : n A ) copies of N b ; and T ( M A ) is the disjoint union of k A,i copies de T (Γ A,i ), ( n B : n A ) copies of T ( N b ) and a root (conditions (2) and (3)). Therefore, by induction on the depthof A in Ξ , M b is the disjoint union of k A,i ( n A : n B ) copies of Γ A,i and ( n B : n A )copies of Γ B − ,B , for B ∈ Ξ , B > A , and 1 ≤ i ≤ s B ; and T ( M A ) is the disjointunion of k B,i ( n B : n A ) copies of T (Γ B,i ), ( n B : n A ) copies of T (Γ B − ,B ), and ofroots of the M B . Taking A the singleton consisting in the root of Ξ, we noticethat T ( M ) contains roots of n B copies of M B . As this set of roots is the samething as B , Ξ embeds in T ( M ). (cid:3) Proposition 8.28 M is C -minimal and ℵ -categorical-or-with-a-finite-model. roof: The proof is by induction. The conditions on labels give the zero depth,one induction step is given by Lemmas 8.21 and 8.22, the one by Lemma 8.23. (cid:3) Proposition 8.29 Θ( M ) = Ξ and Θ( M ) = Ξ . Proof: In the following, unless otherwise specified, “indiscernible” means “indis-cernible as an autonomous C -structure”.Let us first show Θ( M a ) = Θ( N a ): either ] a − , a [= ∅ and M a = N a , or ] a − , a [is dense the interdictions given by conditions (8) and (9) apply, which imposes a ∈ Θ( N a ) and then that a is definable in N a , hence points in M a have same typein N a iff in M a , and finally Θ( M a ) = Θ( N a ).Let us show now, by induction on vertices depth, that Ξ ≥ a = Θ( M a ) = Θ( N a ).Assume A maximal in Ξ , A = A , a ∈ A and b ∈ A − . By condition (7), M a is not indiscernible, thus a ∈ Θ( M a ). By construction of M any cone in a estindiscernible, thus Θ( M a ) = { a } = Ξ ≥ a .The same argument (the use of (7) less) apply to any other point a of Ξ. So Ξembeds in Θ( M ). Otherwise, by construction of M , the direct product Y A ∈ Ξ A = A ( Aut L∪{ V, Γ } Γ A − ,A )( n A : n A − )! × Π s A i ( Aut L Γ A,i ) k A,i !embeds in Aut L M , and the ] A − , A [ are indiscernible in M , as are the unions of the n A k A copies of Γ A,i , hence Ξ = Θ( M ). That Θ( M ) = Ξ follows now from (10). (cid:3) References [AN] Samson A. Adeleke et Peter M. Neumann, Relations Related to Betweenness:Their Structure and Automorphisms, Mem. AMS 623, 1998.[D] Franoise Delon, C -minimal structures without density assumption, in MotivicIntegration and its Interaction with Model Theory and Non-Archimedean Ge-ometry (Ed. Cluckers, Nicaise & Sebag), LMS, LNS 384 Volume I, 2008, 51-86.[H–M] Deirdre Haskell and Dugald Macpherson, Cell decompositions of C-minimal structures, Annals of Pure and Applied Logic 66 (1994), 113-162.[M–S] Dugald Macpherson, Charles Steinhorn, On variants of o -minimality, An-nals of Pure and Applied Logic 79 (1996), 165-209.[P] Anand Pillay, Stable embeddedness and NIP, JSL 76 (2011), 665-672.[P–S] Anand Pillay and Charles Steinhorn, Definable sets in ordered structuresIII,