aa r X i v : . [ m a t h . L O ] M a r FROM KRUSKAL’S THEOREMTO FRIEDMAN’S GAP CONDITION
ANTON FREUND
Abstract.
Harvey Friedman’s gap condition on embeddings of finite labelledtrees plays an important role in combinatorics (proof of the graph minor the-orem) and mathematical logic (strong independence results). In the presentpaper we show that the gap condition can be reconstructed from a small num-ber of well-motivated building blocks: it arises via iterated applications of auniform Kruskal theorem. Introduction
In this paper, a tree is a finite partial order T = ( T, ≤ T ) such that • the order T has a unique minimal element hi , called the root of T , and • for each t ∈ T , the set { s ∈ T | s ≤ T t } is linearly ordered by ≤ T .For each pair of elements s, t ∈ T there is a ≤ T -maximal element s ∧ t ∈ T with s ∧ t ≤ T s and s ∧ t ≤ T t . An embedding of trees is given by a function f : S → T that satisfies f ( s ∧ t ) = f ( s ) ∧ f ( t )for all s, t ∈ S . Since s ≤ S t is equivalent to s ∧ t = s , this entails that f isan embedding of partial orders and in particular injective. Kruskal’s theorem [11]asserts the following: For any infinite sequence T , T , . . . of finite trees, there areindices i < j such that T i can be embedded into T j .Let us point out that Kruskal’s theorem has important implications for theoret-ical computer science (cf. the work of N. Dershowitz [1]) and mathematical logic.Concerning the latter, a classical result of D. Schmidt [18] and H. Friedman [19]shows that Kruskal’s theorem cannot be proved in ATR , a relatively strong ax-iom system that is associated with the predicative foundations of mathematics(see [8, 20] for detailed explanations). The precise logical strength of Kruskal’stheorem has been determined by M. Rathjen and A. Weiermann [17].By an n -tree we mean a tree T together with a function l : T → { , . . . , n − } .An embedding between n -trees ( S, l ) and (
T, l ′ ) is given by an embedding f : S → T of trees that satisfies the following conditions:(i) We have l ′ ( f ( s )) = l ( s ) for any s ∈ S .(ii) If t is an immediate successor of r ∈ S (i. e. if t is ≤ S -minimal with r < S t )and we have f ( r ) < T s < T f ( t ), then we have l ′ ( s ) ≥ l ′ ( f ( t )) = l ( t ).(iii) If we have s < T f ( hi ), then we have l ′ ( s ) ≥ l ′ ( f ( hi )) = l ( hi ).Part (ii) and (iii) constitute the famous gap condition due to H. Friedman [19].More precisely, part (ii) on its own is known as the weak gap condition. In thepresent paper we are only concerned with the strong gap condition, which is theconjunction of (ii) and (iii). The following result is known as Friedman’s theorem: For each number n and any infinite sequence T , T , . . . of finite n -trees, there is anembedding T i → T j of n -trees for some indices i < j .Friedman’s theorem plays a role in N. Robertson and P. Seymour’s proof of theirfamous graph minor theorem. In fact, Friedman, Robertson and Seymour [7] haveshown that Friedman’s theorem is equivalent to the graph minor theorem for graphsof bounded tree-width, over the weak base theory RCA . From the viewpointof mathematical logic it is very significant that Friedman’s theorem is unprovablein Π -CA , which is even stronger than the axiom system ATR mentioned above.The present paper shows that Friedman’s gap condition results from iteratedapplications of a uniform Kruskal theorem. This provides a systematic and trans-parent reconstruction of the gap condition, which may otherwise feel ad hoc. Fur-thermore, our reconstruction prepares the computation of maximal order types, asbegun by J. van der Meeren, M. Rathjen and A. Weiermann [12, 13, 14, 15].Let us explain the uniform Kruskal theorem that was mentioned in the previousparagraph. Given a partial order X , we write W ( X ) for the set of finite multisetswith elements from X . Such a multiset can be written as [ x , . . . , x n − ], wherethe multiplicity of the entries is relevant but the order is not. To define a partialorder on W ( X ), we declare that [ x , . . . , x m − ] ≤ W ( X ) [ y , . . . , y n − ] holds if, andonly if, there is an injection h : { , . . . , m − } → { , . . . , n − } such that we have x i ≤ X y h ( i ) for all i < m . Write T W for the set of trees, where isomorphic treesare identified. We get a bijection κ : W ( T W ) → T W if we define κ ([ T , . . . , T n − ]) as the tree in which the root has immediate sub-trees T , . . . , T n − . Indeed, the set T W can be charaterized as the initial fixedpoint of the transformation W . For S, T ∈ T W we write S ≤ T W T if there is anembedding S → T of trees. This relation can also be reconstructed in terms of theorder on multisets: Writing [ X ] <ω for the set of finite subsets of X , we define afamily of functions supp WX : W ( X ) → [ X ] <ω by settingsupp WX ([ x , . . . , x n − ]) = { x , . . . , x n − } . For multisets σ and τ in W ( T W ) one can verify( ⋆ ) κ ( σ ) ≤ T W κ ( τ ) ⇔ ( σ ≤ W ( T W ) τ or κ ( σ ) ≤ T W T for some T ∈ supp W T W ( τ )) . Indeed, the first disjunct on the right corresponds to an embedding κ ( σ ) → κ ( τ )that maps the root to the root, and immediate subtrees to immediate subtrees.The second disjunct corresponds to an embedding that maps all of κ ( σ ) into oneimmediate subtree of κ ( τ ).A PO-dilator is a particularly uniform transformation W of partial orders thatcomes with a family of functions supp WX : W ( X ) → [ X ] <ω . In Section 2 we willrecall the precise definition, as well as a normality condition for PO-dilators. Forany normal PO-dilator W one can construct a “Kruskal fixed point” T W that ispartially ordered according to ( ⋆ ). Recall that a partial order X is a well partialorder if any infinite sequences x , x , . . . in X admits indices i < j with x i ≤ X x j . APO-dilator W is called a WPO-dilator if W ( X ) is a well partial order whenever thesame holds for X . The uniform Kruskal theorem asserts that T W is a well partialorder for any normal WPO-dilator W . In the previous paragraph we have see thatthe usual Kruskal theorem arises as a special case. It is instructive to check thatHigman’s lemma is another special case (take W ( X ) = 1 + Z × X to generate finite ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 3 lists with entries in Z ). As shown by A. Freund, M. Rathjen and A. Weiermann [6],the uniform Kruskal theorem is equivalent to Π -comprehension (the main axiomof Π -CA ), over RCA together with the chain antichain principle. This resultbuilds on a corresponding equivalence in the context of linear orders, which is dueto the present author [2, 3, 4].In this paper we show how the construction of T W can be relativized to a givenpartial order X . The result is a partial order T W ( X ) with a bijection X ⊔ W ( T W ( X )) → T W ( X ) . The point of the relativization is that T W becomes a transformation of partialorders. We will show that T W can itself be equipped with the structure of anormal PO-dilator, which we call the Kruskal derivative of W . The axiom of Π -comprehension is still equivalent to the principle that T W is a normal WPO-dilatorwhenever the same holds for W . This principle will also be referred to as the uniformKruskal theorem.Our main aim is to reconstruct Friedman’s gap condition by taking iteratedKruskal derivatives. In the following we write M ( X ) for the set of multisets withelements from X . One can equip M with the structure of a normal WPO-dilator.Our reconstruction of Friedman’s gap condition proceeds via the following steps:(1) Start with the normal WPO-dilator T given by T ( X ) = X .(2) Assuming that the normal WPO-dilator T n is already constructed, definethe normal WPO-dilator T − n +1 as the Kruskal derivative of M ◦ T n .(3) Define the normal WPO-dilator T n +1 as the composition T n ◦ T − n +1 .(4) Verify that T n ( ∅ ) is isomorphic to the set of n -trees, ordered according toFriedman’s strong gap condition.In particular, the statement that T n is a WPO-dilator follows from n applicationsof the uniform Kruskal theorem. If Π -induction is available, then one can concludethat the statement holds for all n ∈ N . This helps to explain why Π -CA doesnot prove that T n ( ∅ ) is a well partial order for every n ∈ N , even though it provesthe statement for each fixed number.To conclude this introduction, we discuss related results from the literature.The original proof of Friedman’s theorem [19] involves iterated applications of theminimal bad sequence argument, which broadly resemble steps (2) and (3) above.It does not, however, translate these iterations into a recursive definition of thegap condition. Instead, it seems that the latter was originally motivated by certainordinal notation systems. Our transformation of W into T W is very similar toa construction by R. Hasegawa [10]. Without giving a detailed proof, Hasegawaeven states that iterations of the construction lead to a variant of Friedman’s gapcondition for trees with edge labels. Van der Meeren, Rathjen and Weiermann havereconstructed suborders of the trees with gap condition, with certain restrictionson the distribution of labels (see e. g. [14, Definition 16] and [15, Definition 12]).As far as we know, the present paper is the first to give a detailed reconstructionof the gap condition in its original form. Acknowledgements.
I am very grateful to Jeroen van der Meeren, MichaelRathjen and Andreas Weiermann. I owe them many of the ideas that were funda-mental for the present paper.
ANTON FREUND Relativized Kruskal fixed points
In this section we recall the definition of normal PO-dilator. We then constructthe relativized Kruskal fixed points T W ( X ) that were mentioned in the introduc-tion. We will introduce these fixed points in terms of notation systems. A moresemantic characterization will follow in the next section.Jean-Yves Girard [9] has introduced dilators as particularly uniform transform-ations of linear orders. A corresponding definition for partial orders has been givenby Freund, Rathjen and Weiermann [6]. In order to recall the precise definition,we need some terminology: A function f : X → Y between partial orders is calleda quasi embedding if f ( x ) ≤ Y f ( y ) implies x ≤ X y . If the converse implicationholds as well, then we have an embedding. The category PO consists of the partialorders as objects and the quasi embeddings as morphisms. We say that a functor W : PO → PO preserves embeddings if W ( f ) : W ( X ) → W ( Y ) is an embeddingwhenever the same holds for f : X → Y . As in the introduction, we write [ X ] <ω for the set of finite subsets of a given set X . To turn [ · ] <ω into a functor, we define[ f ] <ω ( a ) = { f ( x ) | x ∈ a } ∈ [ Y ] <ω for f : X → Y and a ∈ [ X ] <ω . We also apply [ · ] <ω to partial orders, omitting the forgetful functor to the under-lying set. Conversely, subsets of partial orders are often considered as suborders. Definition 2.1.
A PO-dilator consists of(i) a functor W : PO → PO that preserves embeddings and(ii) a natural transformation supp W : W ⇒ [ · ] <ω that satisfies the followingsupport condition: Given any embedding f : X → Y of partial orders, theembedding W ( f ) : W ( X ) → W ( Y ) has rangerng( W ( f )) = { σ ∈ W ( Y ) | supp WY ( σ ) ⊆ rng( f ) } . If W ( X ) is a well partial order (wpo) for any wpo X , then W is a WPO-dilator.The reader may have observed that the previous definition focuses on embeddingsrather than quasi embeddings. The latter are important for applications to thetheory of well partial orders (see e. g. [6]). Also, the inclusion ⊆ in part (ii) of thedefinition is automatic, since the naturality of supports yieldssupp WY ( W ( f )( σ )) = [ f ] <ω (supp WX ( σ )) ⊆ rng( f ) . When the partial order X is clear from the context, then ι a : a ֒ → X denotes theinclusion of a suborder a ⊆ X . For σ ∈ W ( X ) we write σ = NF W ( ι a )( σ ) with a ∈ [ X ] <ω and σ ∈ W ( a )if the equality holds and we have supp Wa ( σ ) = a . The latter is a uniquenesscondition, which is required for the following result: Lemma 2.2.
Consider a PO-dilator W and a partial order X . Any σ ∈ W ( X ) hasa unique normal form σ = NF W ( ι a )( σ ) . For the latter we have a = supp WX ( σ ) .Proof. Let us first show that the representation is unique. Since supp W is natural,we can observe that σ = NF W ( ι a )( σ ) entails supp Wa ( σ ) = a and hencesupp WX ( σ ) = supp WX ( W ( ι a )( σ )) = [ ι a ] <ω (supp Wa ( σ )) = [ ι a ] <ω ( a ) = a. This means that a is determined by σ . Just as any embedding, the function W ( ι a )is injective. Hence σ is uniquely determined as well. In order to prove existence, ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 5 we put a = supp WX ( σ ). Then we have supp WX ( σ ) ⊆ a = rng( ι a ), so that the supportcondition yields σ = W ( ι a )( σ ) for some σ ∈ W ( a ). We also have a = supp WX ( σ ) = [ ι a ] <ω (supp Wa ( σ )) . This implies supp Wa ( σ ) = a and hence σ = NF W ( ι a )( σ ). (cid:3) The normal forms from the previous lemma can be used to represent PO-dilatorsin second order arithmetic, as worked out in [6]. In the present paper we do notwork within a particular meta theory. Given a partial order X , we define a quasiorder ≤ fin X on the set [ X ] <ω by stipulating a ≤ fin X b ⇔ for any x ∈ a there is a y ∈ b with x ≤ X y. We will write a ≤ fin X y rather than a ≤ fin X { y } in the case of a singleton. Thefollowing normality condition turns out to be crucial: Definition 2.3.
A PO-dilator W is called normal if we have σ ≤ W ( X ) τ ⇒ supp WX ( σ ) ≤ fin X supp WX ( τ ) , for any partial order X and arbitrary elements σ, τ ∈ W ( X ).In many applications, the elements σ, τ ∈ W ( X ) are finite structures with labelsin X . Then the inequality supp WX ( σ ) ≤ fin X supp WX ( τ ) corresponds to the conditionthat each label is mapped to a bigger one. In [6], the Kruskal fixed point T W of anormal PO-dilator has been generated by the following inductive clause: • Assuming that we have already generated a finite suborder a ⊆ T W , weadd a term ◦ ( a, σ ) ∈ T W for each element σ ∈ W ( a ) with supp Wa ( σ ) = a .The point is that one can now define a bijection κ : W ( T W ) → T W by stipulating κ ( σ ) = ◦ ( a, σ ) for σ = NF W ( ι a )( σ ). We will relativize the construction by in-cluding constant symbols x ∈ T W ( X ) for elements x ∈ X of a given partial order.At various places in the following definition, we require that ≤ T W ( X ) is a partialorder on certain subsets of T W ( X ). We will later show that all of T W ( X ) is par-tially ordered by ≤ T W ( X ) , so that these requirements become redundant. A moredetailed justification of the following recursion can be found below. Definition 2.4.
Consider a normal PO-dilator W . For each partial order X wedefine a set T W ( X ) of terms and a binary relation ≤ T W ( X ) on this set by simul-taneous recursion. The set T W ( X ) is generated by the following clauses:(i) For each element x ∈ X we have a term x ∈ T W ( X ).(ii) Given a finite set a ⊆ T W ( X ) that is partially ordered by ≤ T W ( X ) , we adda term ◦ ( a, σ ) ∈ T W ( X ) for each σ ∈ W ( a ) with supp Wa ( σ ) = a .For s, t ∈ T W ( X ) we stipulate that s ≤ T W ( X ) t holds if, and only if, one of thefollowing clauses applies:(i’) We have s = x and t = y with x ≤ X y .(ii’) We have t = ◦ ( b, τ ) and s ≤ T W ( X ) t ′ for some t ′ ∈ b (where s can be of theform x or ◦ ( a, σ )).(iii’) We have s = ◦ ( a, σ ) and t = ◦ ( b, τ ), the restriction of ≤ T W ( X ) to a ∪ b is apartial order, and we have W ( ι a )( σ ) ≤ W ( a ∪ b ) W ( ι b )( τ ) , where ι a : a ֒ → a ∪ b and ι b : b ֒ → a ∪ b are the inclusions. ANTON FREUND
To justify the recursion in detail, one can argue as follows: First generate a set T W ( X ) ⊇ T W ( X ) by including all terms ◦ ( a, σ ) for finite a ⊆ T W ( X ), where a is not assumed to be ordered and σ ∈ W ( a ) holds with respect to some partial orderon a . Then define a length function l X : T W ( X ) → N by the recursive clauses l X ( x ) = 0 , l X ( ◦ ( a, σ )) = 1 + P r ∈ a · l X ( r ) . One can now decide r ∈ T W ( X ) and s ≤ T W ( X ) t by simultaneous recursionon l X ( r ) and l X ( s ) + l X ( t ). As an example, we consider the case of r = ◦ ( a, σ ). For s, t ∈ a we have l X ( s ) + l X ( t ) < l X ( r ), even when s and t are the same term (due tothe factor 2 above). Recursively, we can thus determine the restriction of ≤ T W ( X ) to a . If the latter is a partial order, we check whether σ ∈ W ( a ) and supp Wa ( σ ) = a hold with respect to this order. When this is the case, we have r ∈ T W ( X ). Inaddition to the length functions, we need the height functions h X : T W ( X ) → N given by h X ( x ) = 0 , h X ( ◦ ( a, σ )) = max( { } ∪ { h X ( r ) + 1 | r ∈ a } ) . When there us no danger of confusion, we sometimes omit the index X . Thefollowing important observation relies on the assumption that W is normal. Itconfirms the intuition that T W ( X ) can be seen as a tree-like structure. Lemma 2.5.
Consider a normal PO-dilator W and a partial order X . For anyelements s, t ∈ T W ( X ) , the inequality s ≤ T W ( X ) t implies h X ( s ) ≤ h X ( t ) .Proof. One argues by induction on l ( s ) + l ( t ). The case of s = x and t = y isimmediate. The remaining cases are similar to the proof of [6, Lemma 3.5]. Firstassume that s ≤ T W ( X ) ◦ ( b, τ ) = t holds because we have s ≤ T W ( X ) t ′ for some t ′ ∈ b . In view of l ( t ′ ) < l ( t ) theinduction hypothesis yields h ( s ) ≤ h ( t ′ ) < h ( t ). Now assume that s = ◦ ( a, σ ) ≤ T W ( X ) ◦ ( b, τ ) = t holds because of W ( ι a )( σ ) ≤ W ( a ∪ b ) W ( ι b )( τ ). Since W is normal, we get a = [ ι a ] <ω (supp Wa ( σ )) = supp Wa ∪ b ( W ( ι a )( σ )) ≤ fin T W ( X ) supp Wa ∪ b ( W ( ι b )( τ )) = b. Given any s ′ ∈ a we thus have s ′ ≤ T W ( X ) t ′ for some t ′ ∈ b . By induction hypothesiswe obtain h ( s ′ ) ≤ h ( t ′ ) < h ( t ). As s ′ ∈ a was arbitrary, this yields h ( s ) ≤ h ( t ). (cid:3) The proof of the following result is similar to the one of [6, Proposition 3.6].Since the present notation is somewhat different, we reproduce the proof for thereader’s convenience.
Proposition 2.6.
The relation ≤ T W ( X ) is a partial order on T W ( X ) , for anynormal PO-dilator W and any partial order X .Proof. One uses simultaneous induction on n to establish r ≤ T W ( X ) r for l ( r ) ≤ n, ( s ≤ T W ( X ) t ∧ t ≤ T W ( X ) s ) ⇒ s = t for l ( s ) + l ( t ) ≤ n, ( r ≤ T W ( X ) s ∧ s ≤ T W ( X ) t ) ⇒ r ≤ T W ( X ) t for l ( r ) + l ( s ) + l ( t ) ≤ n. Reflexivity is readily verified. Concerning antisymmetry, we consider the case where s ≤ T W ( X ) ◦ ( b, τ ) = t holds because we have s ≤ T W ( X ) t ′ for some t ′ ∈ b . By the ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 7 previous lemma we get h ( s ) ≤ h ( t ′ ) < h ( t ), which makes t ≤ T W ( X ) s impossible.Still for antisymmetry, we also consider the case where s ≤ T W ( X ) t ≤ T W ( X ) s with s = ◦ ( a, σ ) and t = ◦ ( b, τ ) holds because of W ( ι a )( σ ) = W ( ι b )( τ ). Similarly to theproof of Lemma 2.2, we get a = [ ι a ] <ω (supp Wa ( σ )) = supp Wa ∪ b ( W ( ι a )( σ )) = supp Wa ∪ b ( W ( ι b )( τ )) = b. Since W ( ι a ) = W ( ι b ) is injective, it follows that W ( ι a )( σ ) = W ( ι b )( τ ) implies σ = τ and hence s = t . For transitivity we consider t = ◦ ( c, ρ ). If r ≤ T W ( X ) s ≤ T W ( X ) t holds because we have s ≤ T W ( X ) t ′ for some t ′ ∈ c , then the induction hypothesisyields r ≤ T W ( X ) t ′ and hence r ≤ T W ( X ) t . Now assume that s = ◦ ( b, τ ) ≤ T W ( X ) t holds due to W ( ι b )( τ ) ≤ W ( b ∪ c ) W ( ι c )( ρ ) . Since W is normal this implies b ≤ fin T W ( X ) c , as in the proof of the previous lemma.So if r ≤ T W ( X ) s holds because we have r ≤ T W ( X ) s ′ for some s ′ ∈ b , then we get r ≤ T W ( X ) s ′ ≤ T W ( X ) t ′ for some t ′ ∈ c. By induction hypothesis this yields r ≤ T W ( X ) t ′ , which implies r ≤ T W ( X ) t . Fi-nally, assume that r = ◦ ( a, σ ) ≤ T W ( X ) s holds because we have W ( ι a )( σ ) ≤ W ( a ∪ b ) W ( ι b )( τ ) . To conclude r ≤ T W ( X ) t it suffices to consider the inclusions into the set a ∪ b ∪ c ,which is partially ordered due to the simultaneous induction hypothesis. (cid:3) For a suitable formalization of PO-dilators in second order arithmetic, the fol-lowing has been shown by Freund, Rathjen and Weiermann [6]: The principle that T W ( ∅ ) is a well partial order for any normal WPO-dilator W is equivalent toΠ -comprehension, over RCA together with the chain antichain principle. A for-tiori, Π -comprehension does also follow from the principle that T W ( X ) is a wellpartial order whenever the same holds for X . The following result shows that theconverse implication remains true as well, since the minimal bad sequence argumentin its proof can be justified by Π -comprehension. Even though the proof is similarto the one of [6, Theorem 3.10], we provide it for the reader’s convenience. Proposition 2.7.
Consider a normal WPO-dilator W . If X is a well partial order,then so is T W ( X ) .Proof. We use Nash-Williams’ [16] minimal bad sequence argument. Given a partialorder Y , an infinite sequence y , y , . . . ⊆ Y is called good if there are indices i < j with y i ≤ Y y j ; otherwise it is called bad. Hence Y is a well partial order if, andonly if, it contains no bad sequence. Aiming at a contradiction, we assume thatthere is a bad sequence t , t , . . . ⊆ T W ( X ) while X a well partial order. Wemay assume that t , t , . . . is minimal, in the sense that t , . . . , t i − , t ′ i , t ′ i +1 , . . . isgood whenever we have h X ( t ′ i ) < h X ( t i ). This step requires Π -comprehension;a detailed justification can, for example, be found in the proof of [6, Theorem 3.10].For i ∈ N we now define a i ⊆ T W ( X ) by a i = ( a if t i = ◦ ( a, σ ) for some σ ∈ W ( a ) , ∅ if t i is of the form x. Let us show that Z := S { a i | i ∈ N } ⊆ T W ( X ) is a well partial order. Assumingthe contrary, we get a bad sequence s , s , . . . in Z . Since each set a i is finite, there ANTON FREUND are strictly increasing functions i, j : N → N with s i ( k ) ∈ a j ( k ) for all k ∈ N . Inparticular we get h X ( s i (0) ) < h X ( t j (0) ). Since the sequence t , t , . . . was assumedto be minimal, this means that t , t , . . . , t j (0) − , s i (0) , s i (1) , s i (2) , . . . ⊆ T W ( X )must be good. As t , t , . . . and s , s , . . . are bad, this is only possible if we have t k ≤ T W ( X ) s i ( l ) for some k < j (0) and l ∈ N . In view of s i ( l ) ∈ a j ( l ) we can write t j ( l ) = ◦ ( a j ( l ) , σ ) and conclude t k ≤ T W ( X ) t j ( l ) . This inequality contradicts theassumption that t , t , . . . is bad, so that Z must be a well partial order after all.Since X is a well partial order, the bad sequence t , t , . . . can only have finitelymany entries of the form x . Passing to a subsequence, we may assume that allentries have the form t i = ◦ ( a i , σ i ). Note that this subsequence is bad but notnecessarily minimal; we still have a i ⊆ Z for any entry of the subsequence. Write ι i : a i ֒ → Z for the inclusions and consider the sequence W ( ι )( σ ) , W ( ι )( σ ) , . . . ⊆ W ( Z ) . Since W is a WPO-dilator and Z is a well partial order, we obtain indices i < j with W ( ι i )( σ i ) ≤ W ( Z ) W ( ι j )( σ j ). By factoring ι i = ι ◦ ι ′ i into ι ′ i : a i ֒ → a i ∪ a j and ι : a i ∪ a j ֒ → Z , one readily deduces W ( ι ′ i )( a i ) ≤ W ( a i ∪ a j ) W ( ι ′ j )( a j ). Due toclause (iii’) of Definition 2.4 we get t i = ◦ ( a i , σ i ) ≤ T W ( X ) ◦ ( a j , σ j ) = t j . So t , t , . . . cannot be bad after all. (cid:3) A categorical characterization
The term systems T W ( X ) from the previous section can be hard to handle, bothin general arguments and in concrete examples. To resolve this issue, the presentsection provides a more semantic approach. We begin with a general notion: Definition 3.1.
Consider a normal PO-dilator W and a partial order X . A Kruskalfixed point of W over X consists of a partial order Z and functions ι : X → Z and κ : W ( Z ) → Z that satisfy rng( ι ) ∩ rng( κ ) = ∅ and ι ( x ) ≤ Z ι ( y ) ⇒ x ≤ X y (for x, y ∈ X ) ,ι ( x ) ≤ Z κ ( τ ) ⇔ ι ( x ) ≤ fin Z supp WZ ( τ ) (for x ∈ X and τ ∈ W ( Z )) ,κ ( σ ) Z ι ( y ) for all σ ∈ W ( Z ) and y ∈ X,κ ( σ ) ≤ Z κ ( τ ) ⇔ σ ≤ W ( Z ) τ or κ ( σ ) ≤ fin Z supp WZ ( τ ) (for σ, τ ∈ W ( Z )) . Note that we do not demand that x ≤ X y implies ι ( x ) ≤ Z ι ( y ). This will becomeimportant in the proof of Theorem 4.2. The following is justified by Lemma 2.2. Definition 3.2.
Consider a normal PO-dilator W . For each partial order X wedefine functions ι X : X → T W ( X ) and κ X : W ( T W ( X )) → T W ( X ) by stipulating ι X ( x ) = x,κ X ( σ ) = ◦ ( a, σ ) for σ = NF W ( ι a )( σ ) . Let us verify that T W ( X ) has the desired structure: Theorem 3.3.
We consider a normal PO-dilator W and a partial order X . Theorder T W ( X ) and the functions ι X and κ X form a Kruskal fixed point of W over X . ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 9
Proof.
In view of Definition 2.4 it is immediate that we have rng( ι X ) ∩ rng( κ X ) = ∅ ,that ι X ( x ) = x ≤ T W ( X ) y = ι X ( y ) implies (and is indeed equivalent to) x ≤ X y ,and that κ X ( σ ) = ◦ ( a, σ ) ≤ T W ( X ) y = ι X ( y ) is always false. For τ = NF W ( ι b )( τ )we also get ι X ( x ) = x ≤ T W ( X ) ◦ ( b, τ ) = κ X ( τ ) ⇔ ι X ( x ) ≤ fin T W ( X ) b = supp W T W ( X ) ( τ ) . For the remaining equivalence we need to show ◦ ( a, σ ) ≤ T W ( X ) ◦ ( b, τ ) ⇔⇔ W ( ι a )( σ ) ≤ W ( T W ( X )) W ( ι b )( τ ) or ◦ ( a, σ ) ≤ fin T W ( X ) b, with ι a : a ֒ → T W ( X ) and ι b : b ֒ → T W ( X ). In view of Definition 2.4 it suffices toobserve that we have W ( ι a )( σ ) ≤ W ( T W ( X )) W ( ι b )( τ ) ⇔ W ( ι ′ a )( σ ) ≤ W ( a ∪ b ) W ( ι ′ b )( τ ) , with ι ′ a : a ֒ → a ∪ b and ι ′ b : b ֒ → a ∪ b . To establish this equivalence one considersthe inclusion ι : a ∪ b ֒ → T W ( X ) and composes the right side with W ( ι ). (cid:3) To obtain a unique characterization, we use the following categorical notion.
Definition 3.4.
Consider a normal PO-dilator W and a partial order X . A Kruskalfixed point ( Z, ι, κ ) is called initial if any Kruskal fixed point ( Z ′ , ι ′ , κ ′ ) of W over X admits a unique quasi embedding f : Z → Z ′ with f ◦ ι = ι ′ and f ◦ κ = κ ′ ◦ W ( f ).Like all initial objects, initial Kruskal fixed points are unique up to isomorphism.The following criterion will be very useful. Theorem 3.5.
For a Kruskal fixed point ( Z, ι, κ ) of a normal PO-dilator W overa partial order X , the following are equivalent:(i) We have rng( ι ) ∪ rng( κ ) = Z , and x ≤ X y implies ι ( x ) ≤ Z ι ( y ) for x, y ∈ X .Furthermore, there is a function h : Z → N such that s ∈ supp WZ ( σ ) ⇒ h ( s ) < h ( κ ( σ )) holds for any s ∈ Z and any σ ∈ W ( Z ) .(ii) The Kruskal fixed point ( Z, ι, κ ) is initial.Proof. Let us first show that condition (i) implies (ii). For s ∈ Z we define l ( s ) ∈ N by recursion on h ( s ), setting l ( ι ( x )) = 0 and l ( κ ( σ )) = 1 + P s ∈ supp WZ ( σ ) · l ( s ) . Note that each element of Z is covered by exactly one clause, since Definition 3.1 andpart (i) of the present theorem provide rng( ι ) ∩ rng( κ ) = ∅ and rng( ι ) ∪ rng( κ ) = Z .Now consider another Kruskal fixed point ( Z, ι ′ , κ ′ ). We first show that there is atmost one quasi embedding f : Z → Z ′ with f ◦ ι = ι ′ and f ◦ κ = κ ′ ◦ W ( f ). Theseequations amount to f ( ι ( x )) = ι ′ ( x ) for x ∈ X,f ( κ ( σ )) = κ ′ ( W ( f )( σ )) = κ ′ ( W ( f ↾ a )( σ )) for σ = NF W ( ι a )( σ ) ∈ W ( Z ) , where f ↾ a = f ◦ ι a : a → Z ′ is the restriction of f . Once again, each argumentof f is covered by exactly one of these clauses. From Lemma 2.2 we know that σ = NF W ( ι a )( σ ) implies supp WZ ( σ ) = a . Now a straightforward induction on l ( s ) shows that f ( s ) is uniquely determined. To establish existence we read the aboveas recursive clauses. We verify r ∈ Z ⇒ f ( r ) ∈ Z ′ ,f ( s ) ≤ Z ′ f ( t ) ⇒ s ≤ Z t by simultaneous induction on l ( r ) and l ( s ) + l ( t ). Let us verify the first claim for r = κ ( σ ) with σ = NF W ( ι a )( σ ). For s, t ∈ a we have l ( s ) + l ( t ) < l ( r ). Hence thesimultaneous induction hypothesis ensures that f ↾ a is a quasi embedding. We maythus form W ( f ↾ a ), as needed for the clause that defines the value f ( r ) ∈ Z ′ . Letus now show that f is a quasi embedding. For s = ι ( x ) and t = ι ( y ) we see that f ( s ) = f ( ι ( x )) = ι ′ ( x ) ≤ Z ′ ι ′ ( y ) = f ( ι ( y )) = f ( t )implies x ≤ X y . By the condition in (i) this implies s = ι ( x ) ≤ Z ι ( y ) = t , asrequired. For s = ι ( x ) and t = κ ( τ ) with τ = NF W ( ι b )( τ ), a glance at Definition 3.1reveals that f ( s ) = ι ′ ( x ) ≤ Z ′ κ ′ ( W ( f ↾ b )( τ )) = f ( t ) implies f ( s ) ≤ fin Z ′ supp WZ ′ ( W ( f ↾ b )( τ )) = [ f ↾ b ] <ω (supp Wb ( τ )) = [ f ] <ω ( b ) . As t ′ ∈ b = supp WZ ( τ ) implies h ( t ′ ) < h ( κ ( τ )) = h ( t ), the induction hypothesisyields s ≤ fin Z supp WZ ( τ ), which implies s = ι ( x ) ≤ Z κ ( τ ) = t . For s = κ ( σ ) and t = ι ( y ) it suffices to observe that f ( s ) ≤ Z ′ f ( t ) cannot hold. Finally, we considerthe case of s = κ ( σ ) and t = κ ( τ ) with σ = NF W ( ι a )( σ ) and τ = NF W ( ι b )( τ ). If f ( s ) = κ ′ ( W ( f ↾ a )( σ )) ≤ Z ′ κ ′ ( W ( f ↾ b )( τ )) = f ( t )holds because of f ( s ) ≤ fin Z ′ supp WZ ′ ( W ( f ↾ b )( τ )), then one argues as above. Nowassume that we have W ( f ↾ a )( σ ) ≤ W ( Z ′ ) W ( f ↾ b )( τ ) . The induction hypothesis ensures that f ↾ ( a ∪ b ) : a ∪ b → Z ′ is a quasi embedding.Here it is crucial that we argue by induction on l ( s ) + l ( t ), not on h ( s ) + h ( t ). Letus factor f ↾ a = f ↾ ( a ∪ b ) ◦ ι ′ a and f ↾ b = f ↾ ( a ∪ b ) ◦ ι ′ b , where ι ′ a : a ֒ → a ∪ b and ι ′ b : b ֒ → a ∪ b are the inclusions. Then the last inequality amounts to W ( f ↾ ( a ∪ b )) ◦ W ( ι ′ a )( σ ) ≤ W ( Z ′ ) W ( f ↾ ( a ∪ b )) ◦ W ( ι ′ b )( τ ) . Since W ( f ↾ ( a ∪ b )) is a quasi embedding, we get W ( ι ′ a )( σ ) ≤ W ( a ∪ b ) W ( ι ′ b )( τ ).Now compose both sides with the embedding W ( ι ), where ι : a ∪ b ֒ → Z is theinclusion. This yields σ = W ( ι a )( σ ) = W ( ι ◦ ι ′ a )( σ ) ≤ Z W ( ι ◦ ι ′ b )( τ ) = W ( ι b )( τ ) = τ. The latter implies s = κ ( σ ) ≤ Z κ ( τ ) = t , which completes the proof that (i)implies (ii). To show that (ii) implies (i), we first establish (i) for the Kruskalfixed point ( T W ( X ) , ι X , κ X ) from Theorem 3.3. Any ◦ ( a, σ ) ∈ T W ( X ) arises as κ X ( σ ) for σ = W ( ι a )( σ ). Here the condition supp Wa ( σ ) = a from Definition 2.4ensures that σ is in normal form. This shows rng( ι X ) ∪ rng( κ X ) = T W ( X ). Therequirement that x ≤ X y implies ι X ( x ) = x ≤ T W ( X ) y = ι X ( y ) is immediate byDefinition 2.4. A height function h X : T W ( X ) → N has been defined before thestatement of Lemma 2.5. For arbitrary elements σ = NF W ( ι a )( σ ) ∈ W ( T W ( X ))and s ∈ supp W T W ( X ) ( σ ) = a , the construction of h X entails h X ( s ) < h X ( ◦ ( a, σ )) = h X ( κ X ( σ )) , ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 11 just as needed. Since we have already shown that (i) implies (ii), we can concludethat ( T W ( X ) , ι X , κ X ) is an initial Kruskal fixed point of W over X . If ( Z, ι, κ ) isany initial Kruskal fixed point as in (ii), we get an isomorphism f : Z → T W ( X )with f ◦ ι = ι X and f ◦ κ = κ X ◦ W ( f ). As W ( f ) is an isomorphism, the ranges of κ X and κ X ◦ W ( f ) coincide. Hence we get T W ( X ) = rng( f ◦ ι ) ∪ rng( f ◦ κ ) andthen Z = rng( ι ) ∪ rng( κ ). We also learn that x ≤ X y implies ι ( x ) = f ◦ ι X ( x ) ≤ Z f ◦ ι X ( y ) = ι ( y ) . Finally, we define h : Z → N by h ( s ) = h X ( f ( s )). For s ∈ supp WZ ( σ ) we have f ( s ) ∈ [ f ] <ω (supp WZ ( σ )) = supp W T W ( X ) ( W ( f )( σ )) . We can conclude h ( s ) = h X ( f ( s )) < h X ( κ X ◦ W ( f )( σ )) = h X ( f ◦ κ ( σ )) = h ( κ ( σ )) , as required for (i). (cid:3) The following result was shown as part of the previous proof. It is important,because it establishes the existence of initial Kruskal fixed points.
Corollary 3.6.
For each normal PO-dilator W and each partial order X , theKruskal fixed point ( T W ( X ) , ι X , κ X ) is initial. For later use we also record the following result.
Lemma 3.7.
Let ( Z, ι, κ ) be an initial Kruskal fixed point of a normal PO-dilator W over a partial order X . Consider another Kruskal fixed point ( Z ′ , ι ′ , κ ′ ) and theunique quasi embedding f : Z → Z ′ with f ◦ ι = ι ′ and f ◦ κ = κ ′ ◦ W ( f ) . If x ≤ X y implies ι ′ ( x ) ≤ Z ′ ι ′ ( y ) for all x, y ∈ X , then f is an embedding.Proof. Define l : Z → N as in the proof of Theorem 3.5. In the latter we have usedinduction on l ( s ) + l ( t ) to show that f ( s ) ≤ Z ′ f ( t ) implies s ≤ Z t . Assuming that x ≤ X y implies ι ′ ( x ) ≤ Z ′ ι ′ ( y ), one can read the given argument in reverse, to showthat s ≤ Z t does also imply f ( s ) ≤ Z ′ f ( t ). (cid:3) So far, the notation T W ( X ) has been reserved for the term systems constructedin Definition 2.4. In the following sections we will also use T W ( X ) for other initialKruskal fixed points of W over X . This is harmless, since we have shown that allthese fixed points are equivalent.4. Kruskal derivatives
Consider a normal PO-dilator W . As shown in the previous section, each partialorder X gives rise to an initial Kruskal fixed point ( T W ( X ) , ι X , κ X ). In the presentsection we show that the transformation X
7→ T W ( X ) of partial orders can beextended into a normal PO-dilator T W . More precisely, we will show that there isan essentially unique extension in the sense of the following definition. Definition 4.1.
A Kruskal derivative of a normal PO-dilator W is tuple ( T W, ι, κ )that consists of a normal PO-dilator T W and two families of functions ι X : X → T W ( X ) and κ X : W ( T W ( X )) → T W ( X )indexed by the partial order X , such that the following properties are satisfied:(i) The tuple ( T W ( X ) , ι X , κ X ) is an initial Kruskal fixed point of W over X ,for each partial order X . (ii) We have ι Y ◦ f = T W ( f ) ◦ ι X and T W ( f ) ◦ κ X = κ Y ◦ W ( T W ( f )), forany quasi embedding f : X → Y between partial orders.Let us begin by proving existence: Theorem 4.2.
Each normal PO-dilator has a Kruskal derivative.Proof.
Consider a normal PO-dilator W . For each partial order X , Corollary 3.6provides an initial Kruskal fixed point ( T W ( X ) , ι X , κ X ) of W over X . Given aquasi embedding f : X → Y , it is easy to see that ( T W ( Y ) , ι Y ◦ f, κ Y ) is a Kruskalfixed point of W over X as well. Since ( T W ( X ) , ι X , κ X ) is initial, there is a uniquequasi embedding T W ( f ) : T W ( X ) → T W ( Y ) with ι Y ◦ f = T W ( f ) ◦ ι X and T W ( f ) ◦ κ X = κ Y ◦ W ( T W ( f )) . In order to obtain a Kruskal derivative ( T W, ι, κ ), it suffices to turn T W into anormal PO-dilator. To show that T W is a functor, one checks that T W ( g ) ◦T W ( f )satisfies the equations that characterize T W ( g ◦ f ). If f : X → Y is an embedding,then x ≤ X y implies ι Y ◦ f ( x ) ≤ T W ( Y ) ι Y ◦ f ( y ), since ι Y must satisfy the conditionfrom part (i) of Theorem 3.5. Hence Lemma 3.7 ensures that T W ( f ) is again anembedding, as required in part (i) of Definition 2.1. It remains to exhibit suitablesupport functions supp T WX : T W ( X ) → [ X ] <ω . In view of Theorem 3.5 we can recursively definesupp T WX ( ι X ( x )) = { x } , supp T WX ( κ X ( σ )) = [ { supp T WX ( s ) | s ∈ supp W T W ( X ) ( σ ) } . To show naturality, one verifiessupp T WY ( T W ( f )( s )) = [ f ] <ω (supp T WX ( s ))by induction on h X ( s ), where h X : T W ( X ) → N is as in part (i) of Theorem 3.5.To satisfy the support condition from part (ii) of Definition 2.1, we need to establishsupp T WY ( s ) ⊆ rng( f ) ⇒ s ∈ rng( T W ( f ))for an embedding f : X → Y (recall that the converse implication is automatic).We use induction on h Y ( s ). For s = ι Y ( y ) we see that { y } = supp T WY ( s ) ⊆ rng( f )yields y = f ( x ) for some x ∈ X . This entails s = ι Y ◦ f ( x ) = T W ( f ) ◦ ι X ( x ) ∈ rng( T W ( f )) . Now consider s = κ Y ( σ ). For any s ′ ∈ supp W T W ( Y ) ( σ ) we have h Y ( s ′ ) < h Y ( s ) andsupp T WY ( s ′ ) ⊆ supp T WY ( s ) ⊆ rng( f ) , so that the induction hypothesis yields s ′ ∈ rng( T W ( f )). Thus we getsupp W T W ( Y ) ( σ ) ⊆ rng( T W ( f )) . Now the support condition for the PO-dilator W yields σ = W ( T W ( f ))( σ ) forsome σ ∈ W ( T W ( X )). We then obtain s = κ Y ◦ W ( T W ( f ))( σ ) = T W ( f ) ◦ κ X ( σ ) ∈ rng( T W ( f )) , as required. It remains to show that the PO-dilator T W is normal. We verify s ≤ T W ( X ) t ⇒ supp T WX ( s ) ≤ fin X supp T WX ( t ) ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 13 by induction on h X ( s ) + h X ( t ). If we have s = ι X ( x ) ≤ T W ( X ) ι X ( y ) = t , then wemust have x ≤ X y and hencesupp T WX ( s ) = { x } ≤ fin X { y } = supp T WX ( t ) . Now consider the case of an inequality s ≤ T W ( X ) κ X ( τ ) = t that holds becausewe have s ≤ T W ( X ) t ′ for some t ′ ∈ supp W T W ( X ) ( τ ). In view of h X ( t ′ ) < h X ( t ), theinduction hypothesis yieldssupp T WX ( s ) ≤ fin X supp T WX ( t ′ ) ⊆ supp T WX ( t ) . Finally, assume that s = κ X ( σ ) ≤ T W ( X ) κ X ( τ ) = t holds due to σ ≤ W ( T W ( X )) τ .Since W is normal, we get supp W T W ( X ) ( σ ) ≤ fin T W ( X ) supp W T W ( X ) ( τ ). Given an arbit-rary s ′ ∈ supp W T W ( X ) ( σ ), we may then pick a t ′ ∈ supp W T W ( X ) ( τ ) with s ′ ≤ T W ( X ) t ′ .By induction hypothesis we getsupp T WX ( s ′ ) ≤ fin X supp T WX ( t ′ ) ⊆ supp T WX ( t ) . Since s ′ ∈ supp W T W ( X ) ( σ ) was arbitrary, this establishessupp T WX ( s ) = [ { supp T WX ( s ′ ) | s ′ ∈ supp W T W ( X ) ( σ ) } ≤ fin X supp T WX ( t ) , as required. (cid:3) Let us highlight some of the information that is implicit in the previous proof:
Remark 4.3.
In order to construct a Kruskal derivative of a specific PO-dilator,one can follow the proof of Theorem 4.2. The latter shows that we only need to finda family of initial Kruskal fixed points. The extension into a Kruskal derivative isthen automatic. In particular, the fact that one obtains a normal PO-dilator doesnot need to be verified in each specific case. Also observe that the functor T W was uniquely determined by the initial Kruskal fixed points T W ( X ). The choiceof support functions is also unique (as for any PO-dilator), since supp T WX ( s ) mustbe the smallest set a ⊆ X with s ∈ rng( T W ( ι a )), where ι a : a ֒ → T W ( X ) is theinclusion: In one direction, the support condition from part (ii) of Definition 2.1ensures s ∈ rng( T W ( ι a )) for a = supp T WX ( s ). In the other direction, naturalityentails that s = T W ( ι a )( s ) yields a ⊆ [ ι a ] <ω (supp T Wa ( s )) = supp T WX ( T W ( ι a )( s )) = supp T WX ( s ) . We have not included this information in the statement of Theorem 4.2, because amore general uniqueness result will be shown below.To prepare our uniqueness result, we recall that two PO-dilators ( V, supp V ) and( W, supp W ) are equivalent if there is a natural isomorphism η : V ⇒ W of functors.It may also seem reasonable to demandsupp WX ◦ η X = supp VX for any partial order X . However, the latter turns out to be automatic. Girard [9]has shown that this is the case for any natural transformation between dilators oflinear orders (cf. also [5, Lemma 2.17], which is closer to our notation). One cancheck that the proof remains valid for partial orders. In the case of an isomorphism, the argument is particularly simple: Given σ ∈ W ( X ), we invoke Lemma 2.2 towrite σ = W ( ι a )( σ ) with a = supp VX ( σ ). We then getsupp WX ◦ η X ( σ ) = supp WX ( η X ◦ V ( ι a )( σ )) = supp WX ( W ( ι a ) ◦ η a ( σ )) == [ ι a ] <ω (supp Wa ( η a ( σ ))) ⊆ rng( ι a ) = a = supp VX ( σ ) . By applying the same argument to the inverse of η , we also getsupp VX ( σ ) = supp VX ◦ η − X ( η X ( σ )) ⊆ supp WX ( η X ( σ )) = supp WX ◦ η X ( σ ) . The following result shows that Kruskal derivatives are essentially unique.
Theorem 4.4.
For any two Kruskal derivatives ( T W, ι , κ ) and ( T W, ι , κ ) ofa normal PO-dilator W , there is a natural isomorphism η : T W ⇒ T W suchthat we have η X ◦ ι X = ι X and η X ◦ κ X = κ X ◦ W ( η X ) for any partial order X .Proof. For each order X , the fact that ( T W ( X ) , ι X , κ X ) and ( T W ( X ) , ι X , κ X )are initial Kruskal fixed points of W over X implies that there is an isomorphism η X : T W ( X ) → T W ( X ) with η X ◦ ι X = ι X and η X ◦ κ X = κ X ◦ W ( η X ). Itremains to show that the resulting family η is natural. Given a quasi embedding f : X → Y between partial orders, we show η Y ◦ T W ( f )( s ) = T W ( f ) ◦ η X ( s )by induction on h X ( s ), where h X : T W ( X ) → N is as in part (i) of Theorem 3.5.To cover elements of the form s = ι X ( x ), it suffices to observe η Y ◦ T W ( f ) ◦ ι X = η Y ◦ ι Y ◦ f = ι Y ◦ f = T W ( f ) ◦ ι X = T W ( f ) ◦ η X ◦ ι X . Given an element s = κ X ( σ ), we invoke Lemma 2.2 to write σ = NF W ( ι a )( σ ),where ι a : a ֒ → T W ( X ) is the inclusion. For any element s ′ ∈ a = supp W T W ( X ) ( σ )we have h X ( s ′ ) < h X ( s ), so that the induction hypothesis yields η Y ◦ T W ( f ) ◦ ι a = T W ( f ) ◦ η X ◦ ι a . We can deduce η Y ◦ T W ( f )( s ) = η Y ◦ T W ( f ) ◦ κ X ( σ ) = η Y ◦ κ Y ◦ W ( T W ( f ))( σ ) == κ Y ◦ W ( η Y ) ◦ W ( T W ( f ) ◦ ι a )( σ ) = κ Y ◦ W ( T W ( f )) ◦ W ( η X ◦ ι a )( σ ) == T W ( f ) ◦ κ X ◦ W ( η X )( σ ) = T W ( f ) ◦ η X ◦ κ X ( σ ) = T W ( f ) ◦ η X ( s ) , as required. (cid:3) Given a normal PO-dilator W , we will write T W for “its” Kruskal derivative,even though the latter is only determined up to isomorphism. The following is animmediate consequence of Proposition 2.7. Corollary 4.5. If W is a normal WPO-dilator, then its Kruskal derivative T W is a normal WPO-dilator as well. In the following sections we will consider iterated Kruskal derivatives. To ensurethat the iterations are essentially unique, we now show that equivalent PO-dilatorshave equivalent Kruskal derivatives.
Proposition 4.6.
Consider a natural isomorphism η : V ⇒ W between normalPO-dilators. If ( T W, ι, κ ) is a Kruskal derivative of W , then ( T W, ι, κ ◦ η ) is aKruskal derivative of V , where κ ◦ η is defined by ( κ ◦ η ) X = κ X ◦ η T W ( X ) . ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 15
Proof.
It is straightforward to verify that ( T W ( X ) , ι X , ( κ ◦ η ) X ) is an initial Kruskalfixed point of V over X , for any partial order X . To provide a representative partof the verification, we show that ( κ ◦ η ) X ( σ ) ≤ T W ( X ) ( κ ◦ η ) X ( τ ) is equivalent to σ ≤ V ( T W ( X )) τ or ( κ ◦ η ) X ( σ ) ≤ fin T W ( X ) supp V T W ( X ) ( τ ) , as required by Definition 3.1. Since ( T W ( X ) , ι X , κ X ) is a Kruskal fixed point of W over X , the same definition entails that ( κ ◦ η ) X ( σ ) ≤ T W ( X ) ( κ ◦ η ) X ( τ ) is equivalentto the disjunction of η T W ( X ) ( σ ) ≤ W ( T W ( X )) η T W ( X ) ( τ ) and( κ ◦ η ) X ( σ ) ≤ fin T W ( X ) supp W T W ( X ) ( η T W ( X ) ( τ )) . The first disjunct is equivalent to σ ≤ V ( T W ( X )) τ . To relate the second disjuncts,it suffices to recall that we havesupp W T W ( X ) ( η T W ( X ) ( τ )) = supp V T W ( X ) ( τ ) . To conclude that ( T W, ι, κ ◦ η ) is a Kruskal derivative in the sense of Definition 4.1,we compute T W ( f ) ◦ ( κ ◦ η ) X = T W ( f ) ◦ κ X ◦ η T W ( X ) = κ Y ◦ W ( T W ( f )) ◦ η T W ( X ) == κ Y ◦ η T W ( Y ) ◦ V ( T W ( f )) = ( κ ◦ η ) Y ◦ V ( T W ( f ))for an arbitrary quasi embedding f : X → Y . (cid:3) The gap orders as PO-dilators
In the present section we give a recursive definition of the set of finite trees withlabels in { , . . . , n − } , ordered according to Friedman’s gap condition. We alsoshow that one obtains a normal PO-dilator if one relativizes the gap orders to agiven partial order X . This prepares the reconstruction of Friedman’s gap conditionin the following section.As a preparation, we give a more precise account of finite multisets: Let uswrite X <ω for the set of finite sequences h x , . . . , x n − i with entries x i ∈ X . Saythat two sequences h x , . . . , x m − i and h y , . . . , y n − i in X <ω are equivalent if, andonly if, there is a bijective function h : { , . . . , m − } → { , . . . , n − } such that wehave x i = y h ( i ) for all i < m = n . We write [ x , . . . , x n − ] for the equivalence classof h x , . . . , x n − i with respect to this equivalence relation. From [ x , . . . , x n − ] onecan recover the multiplicity but not the order of the entries. The quotient set M ( X ) = { [ x , . . . , x n − ] | h x , . . . , x n − i ∈ X <ω } is called the set of finite multisets with elements from X . We declare that[ x , . . . , x m − ] ≤ M ( X ) [ y , . . . , y n − ]holds if, and only if, there is an injection g : { , . . . , m − } → { , . . . , n − } suchthat we have x i ≤ X y g ( i ) for all i < m . One can check that this is well defined andyields a partial order on M ( X ) (for antisymmetry, use induction on the number ofelements). Higman’s lemma entails that M ( X ) is a well partial order if the sameholds for X . Given a (quasi) embedding f : X → Y , one can define a (quasi)embedding M ( f ) : M ( X ) → M ( Y ) by setting M ( f )([ x , . . . , x n − ]) = [ f ( x ) , . . . , f ( x n − )] . A family of functions supp MX : M ( X ) → [ X ] <ω can be given bysupp MX ([ x , . . . , x n − ]) = { x , . . . , x n − } . It is straightforward to check that this turns M into a normal WPO-dilator in thesense of Definitions 2.1 and 2.3.Given a partial order X , the underlying set of the partial order T n ( X ) consists ofthe finite trees with labels in { , . . . , n − }∪ X , where labels from X may only occurat the leafs. More formally, this set admits the following recursive description: Definition 5.1.
Given a number n ∈ N and a partial order X , we generate a set T n ( X ) by the following recursive clauses:(i) For each x ∈ X we have an element x ∈ T n ( X ).(ii) Whenever we have constructed an element σ = [ t , . . . , t m − ] ∈ M ( T n ( X )),we add an element i ⋆ σ ∈ T n ( X ) for each natural number i < n .Let us also define T − n ( X ) = { x | x ∈ X } ∪ { ⋆ σ | σ ∈ M ( T n ( X )) } ⊆ T n ( X ) , provided that we have n > h nX : T n ( X ) → N by the recursive clauses h nX ( x ) = 0 , h nX ( i ⋆ [ t , . . . , t m − ]) = max( { } ∪ { h nX ( t k ) + 1 | k < m } ) . The following definition decides s ≤ T n ( X ) t by recursion on h nX ( s ) + h nX ( t ). Definition 5.2.
To define a binary relation ≤ T n ( X ) on the set T n ( X ) we stipulate x ≤ T n ( X ) t ⇔ ( either t = y with x ≤ X y, or t = j ⋆ [ t , . . . , t m − ] and x ≤ T n ( X ) t l for some l < m,i ⋆ σ ≤ T n ( X ) t ⇔ ( t = i ⋆ τ with σ ≤ M ( T n ( X )) τ, or t = j ⋆ [ t , . . . , t m − ]with j ≥ i and i ⋆ σ ≤ T n ( X ) t l for some l < m. In the case of n >
0, we define ≤ T − n ( X ) as the restriction of ≤ T n ( X ) to T − n ( X ).A straightforward induction shows s ≤ T n ( X ) t ⇒ h nX ( s ) ≤ h nX ( t ) . Similarly to the proof of Proposition 2.6, one can deduce that ≤ T n ( X ) is a partial or-der on T n ( X ). In the introduction we have given the usual definition of Friedman’sgap condition for embeddings of n -trees. The following shows that the recurs-ive clauses from Definition 5.2 yield the same result. We assume that isomorphic n -trees are identified. Proposition 5.3.
The partial order T n ( ∅ ) is isomorphic to the set of n -trees,ordered according to Friedman’s strong gap condition.Proof. For s = i ⋆ [ s (0) , . . . , s ( k − ∈ T n ( ∅ ) we recursively define T s as the n -treewith root label i and immediate subtrees T s (0) , . . . , T s ( k − . It is clear that thisyields a bijection. By induction on h nX ( s ) + h nX ( t ) one can show that s ≤ T n ( ∅ ) t holds if, and only if, there is an embedding f : T s → T t that satisfies Friedman’sgap condition. An inequality s = i ⋆ σ = i ⋆ [ s (0) , . . . , s ( k − ≤ T n ( ∅ ) i ⋆ [ t (0) , . . . , t ( m − i ⋆ τ = t that holds because of σ ≤ M ( T n ( X )) τ corresponds to an embedding f : T s → T t thatmaps the root to the root. Indeed, the inequalities s ( j ) ≤ T n ( ∅ ) t ( l j ) that witness σ ≤ M ( T n ( X )) τ correspond to the restrictions f j = f ↾ T s ( j ) : T s ( j ) → T t ( l j ) ⊆ T t . ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 17
At this point it is crucial that we consider the strong gap condition: Writing root( T )for the root of T , the gap below f j (root( T s ( j ) )) ∈ T t ( l j ) corresponds to the gapbetween f (root( T s )) and f (root( T s ( j ) )) in T s . An inequality s = i ⋆ σ ≤ T n ( X ) j ⋆ [ t , . . . , t m − ] = t that holds because of j ≥ i and s ≤ T n ( X ) t l with l < m corresponds to an embedding f : T s → T t with range contained in T t ( l ) ⊆ T t . The condition j ≥ i accounts forthe fact that root( T t ) lies in the gap below f (root( T s )) in T t but not in T t ( l ) . (cid:3) Our next goal is to extend T n and T − n +1 into PO-dilators. Definition 5.4.
Given a quasi embedding f : X → Y between partial orders, wedefine T n ( f ) : T n ( X ) → T n ( Y ) by the recursive clauses T n ( f )( x ) = f ( x ) , T n ( f )( i ⋆ [ t , . . . , t m − ]) = i ⋆ [ T n ( f )( t ) , . . . , T n ( f )( t m − )] . For n > T n ( f ) restricts to T − n ( f ) : T − n ( X ) → T − n ( Y ). We alsodefine a family of functions supp T n X : T n ( X ) → [ X ] <ω by stipulatingsupp T n X ( x ) = { x } , supp T n X ( i ⋆ [ t , . . . , t m − ]) = [ { supp T n X ( t l ) | l < m } . We will write supp T − n X for the restriction of supp T n X to T − n ( X ).Let us verify that we obtain the desired structure: Proposition 5.5.
The previous definition yields normal PO-dilators T n and T − n +1 .Proof. Given a quasi embedding f , an easy induction on h nX ( s ) + h nX ( t ) shows T n ( f )( s ) ≤ T n ( Y ) T n ( f )( t ) ⇒ s ≤ T n ( X ) t. If f is an embedding, then the converse implication holds as well. Also by induction,one readily checks that T n is a functor and that supp T n is a natural transformation.To conclude that T n a PO-dilator, one needs to establish the support condition frompart (ii) of Definition 2.1. By induction on s , one can indeed showsupp T n Y ( s ) ⊆ rng( f ) ⇒ s ∈ rng( T n ( f ))for s ∈ T n ( Y ), where f : X → Y is an embedding (recall that the converse implic-ation is automatic). To see that T − n +1 does also satisfy the support condition, oneshould observe that T n +1 ( f )( s ) ∈ T − n +1 ( Y ) implies s ∈ T − n +1 ( X ). To establish thenormality condition from Definition 2.3, one verifies s ≤ T n ( X ) t ⇒ supp T n X ( s ) ≤ fin X supp T n X ( t )by induction on h nX ( s ) + h nX ( t ). (cid:3) Corollary 6.6 below will establish the stronger result that T n and T − n +1 arenormal WPO-dilators. In view of Proposition 5.3 this implies that the trees withFriedman’s gap condition form a well partial order. To prove this fact one needsiterated applications of Π -comprehension. Reconstructing the gap condition
In the introduction we have sketched the reconstruction of Friedman’s gap con-dition in terms of iterated Kruskal derivatives. The reader may wish to recallsteps (1) to (4) from the introduction, which describe a recursive construction ofnormal WPO-dilators T n and T − n +1 . We now show that the latter are unique upto natural isomorphism: Inductively, we may assume that this is the case for T n and hence for M ◦ T n (see below for the composition of PO-dilators). Theorem 4.4and Proposition 4.6 ensure that T − n +1 , which is the Kruskal derivative of M ◦ T n ,is unique up to natural isomorphism as well. Finally, the same holds for the com-position T n +1 = T n ◦ T − n +1 . The recursive construction via steps (1) to (4) mayseem at odds with the ad hoc definition of T n and T − n +1 in the previous section.However, this objection is easily resolved: In the following we will show that thePO-dilators T n and T − n +1 from the previous section are related as specified bysteps (1) to (4) from the introduction. Due to uniqueness, this means that our adhoc definition coincides with the result of the recursive construction.Let us first observe that the normal WPO-dilator T from the previous section isequivalent to the identity functor on the category of partial orders. Hence step (1)from the introduction is satisfied, at least up to natural isomorphism. In Proposi-tion 5.3 we have shown that T n ( ∅ ) is isomorphic to the set of n -trees with Friedman’sstrong gap condition, as claimed by step (4). Our next goal is to verify step (3) fromthe introduction, which requires that T n +1 is equivalent to T n ◦ T − n +1 . Let us firstdiscuss the composition of dilators in general: To compose PO-dilators V and W one first takes their composition as functors. In order to obtain a PO-dilator, onedefines a family of functions supp V ◦ WX : V ◦ W ( X ) → [ X ] <ω by settingsupp V ◦ WX ( σ ) = [ { supp WX ( s ) | s ∈ supp VW ( X ) ( σ ) } . If V and W are WPO-dilators, then so is V ◦ W . One readily checks that V ◦ W isnormal if the same holds for V and W . As explained in Section 4, two PO-dilatorsare equivalent if they are equivalent as functors. One can verify that V ◦ W isequivalent to V ′ ◦ W ′ if V is equivalent to V ′ and W is equivalent to W ′ . To realizestep (3), we will show that the following defines an equivalence. Definition 6.1.
For each partial order X , we define π nX : T n ◦ T − n +1 ( X ) → T n +1 ( X )by the recursive clauses π nX ( t ) = t, π nX ( i ⋆ [ s , . . . , s m − ]) = ( i + 1) ⋆ [ π nX ( s ) , . . . , π nX ( s m − )] , where the first clause relies on the inclusion T − n +1 ( X ) ⊆ T n +1 ( X ).Intuitively speaking, an element of T n ◦ T − n +1 ( X ) is a finite tree with labels from { , . . . , n − } ∪ T − n +1 ( X ), where the labels from T − n +1 ( X ) can only occur at leafs.The function π nX increases the labels from { , . . . , n − } and “unravels” the leaflabels. Hence the leafs of s ∈ T n ◦ T − n +1 ( X ) correspond to the minimal nodes of π nX ( s ) ∈ T n +1 ( X ) that have a label in { } ∪ X . It is interesting to observe that theinverse of π nX is similar to the transformation T T ∗ from [19, Section 4]. Let usverify the promised result: Proposition 6.2.
The family π n : T n ◦ T − n +1 ⇒ T n +1 is a natural isomorphism.Proof. In order to show that π nX is surjective we verify t ∈ rng( π nX ) by inductionon t ∈ T n +1 ( X ). If t is of the form x or 0 ⋆ σ , then we have t ∈ T − n +1 ( X ), which ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 19 yields t ∈ T n ◦ T − n +1 ( X ) and t = π nX ( t ) ∈ rng( π nX ). Now consider an element of theform t = ( i + 1) ⋆ [ t , . . . , t m − ], with i + 1 < n + 1 and t l ∈ T n +1 ( X ) for l < m .Inductively we get t l = π nX ( s l ), which yields i ⋆ [ s , . . . , s m − ] ∈ T n ◦ T − n +1 ( X ) and t = ( i + 1) ⋆ [ t , . . . , t m − ] = π nX ( i ⋆ [ s , . . . , s m − ]) ∈ rng( π nX ) . To conclude that π nX is an isomorphism, we show s ≤ T n ◦ T − n +1 ( X ) t ⇔ π nX ( s ) ≤ T n +1 ( X ) π nX ( t )by induction on h n T − n +1 ( X ) ( s ) + h n T − n +1 ( X ) ( t ). For s = s ′ and t = t ′ it suffices to invokeDefinition 5.2. Now consider s = s ′ and t = j ⋆ [ t , . . . , t m − ]. Inductively we get s ≤ T n ◦ T − n +1 ( X ) t ⇔ π nX ( s ) ≤ T n +1 ( X ) π nX ( t l ) for some l < m. Note that π nX ( s ) = s ′ ∈ T − n +1 ( X ) must be of the form x or 0 ⋆ σ . In view of j + 1 = 0and j + 1 ≥
0, the right side of the previous equivalence is thus equivalent to π nX ( s ) ≤ T n +1 ( X ) ( j + 1) ⋆ [ π nX ( t ) , . . . , π nX ( t m − )] = π nX ( t ) . For s = i ⋆ [ s , . . . , s k − ] and t = t ′ we cannot have s ≤ T n ◦ T − n +1 ( X ) t . We also see π nX ( s ) = ( i + 1) ⋆ [ π nX ( s ) , . . . , π nX ( s k − )] T n +1 ( X ) t ′ = π nX ( t ) , since an inequality would require t ′ = j ⋆ [ t , . . . , t m − ] with j ≥ i + 1, in contrastto t ′ ∈ T − n +1 ( X ). For s = i ⋆ [ s , . . . , s k − ] and t = j ⋆ [ t , . . . , t m − ] the claim isreadily deduced from the induction hypothesis (due to i ≥ j ⇔ i + 1 ≥ j + 1). Tocomplete the proof we verify the naturality property π nY ◦ ( T n ◦ T − n +1 )( f )( t ) = T n +1 ( f ) ◦ π nX ( t ) , arguing by induction on t ∈ T n ◦ T − n +1 ( X ). For t = s we compute π nY ◦ ( T n ◦ T − n +1 )( f )( t ) = π nY ◦ T n ( T − n +1 ( f ))( s ) = π nY ( T − n +1 ( f )( s )) == T − n +1 ( f )( s ) = T n +1 ( f )( s ) = T n +1 ( f ) ◦ π nX ( t ) . The induction step for t = j ⋆ [ t , . . . , t m − ] is straightforward. (cid:3) The following lemma will be needed below. Intuitively, the equivalence says thata tree with root label 0 can be embedded into another tree if, and only if, it can beembedded into a subtree with root label 0. This is true because the gap conditionbelow a node with label 0 is automatic.
Lemma 6.3.
We have s ≤ T n +1 ( X ) π nX ( t ) ⇔ s ≤ fin T − n +1 ( X ) supp T n T − n +1 ( X ) ( t ) for all s ∈ T − n +1 ( X ) and all t ∈ T n ◦ T − n +1 ( X ) .Proof. We establish the claim by induction on t . For t = t ′ it suffices to observe π nX ( t ) = t ′ and supp T n T − n +1 ( X ) ( t ) = { t ′ } . To prove the claim for t = j ⋆ [ t , . . . , t m − ],we recall a step from the previous proof: For s ∈ T − n +1 we have observed s ≤ T n +1 ( X ) π nX ( t ) ⇔ s ≤ T n +1 ( X ) π nX ( t l ) for some l < m. Together with supp T n T − n +1 ( X ) ( t ) = [ { supp T n T − n +1 ( X ) ( t l ) | l < m } , this reduces the claim to the induction hypothesis. (cid:3) To complete the reconstruction of the gap condition, it remains to realize step (2)from the introduction. For this purpose we show that T − n +1 is a Kruskal derivativeof M ◦ T n , where M is the finite multiset dilator from the beginning of Section 5.In view of Definition 4.1, we introduce the following objects: Definition 6.4.
For any partial order X we define a function ι nX : X → T − n +1 ( X )by setting ι nX ( x ) = x . To define κ nX : M ◦ T n ◦ T − n +1 ( X ) → T − n +1 ( X ) we stipulate κ nX ([ s , . . . , s m − ]) = 0 ⋆ [ π nX ( s ) , . . . , π nX ( s m − )] , for s , . . . , s m − ∈ T n ◦ T − n +1 ( X ). We will write ι n and κ n for the families offunctions ι nX and κ nX that are indexed by the partial order X .Let us now prove the central result of our reconstruction: Theorem 6.5.
For any number n ∈ N , the tuple ( T − n +1 , ι n , κ n ) is a Kruskal deriv-ative of the normal PO-dilator M ◦ T n .Proof. From Proposition 5.5 we know that T − n +1 is a normal PO-dilator. It remainsto verify conditions (i) and (ii) from Definition 4.1. Let us begin by showing that( T − n +1 ( X ) , ι nX , κ nX ) is an initial Kruskal fixed point of M ◦ T n over X , for each partialorder X . Invoking the fact that π nX : T n ◦ T − n +1 ( X ) → T n +1 ( X ) is surjective, wesee that T − n +1 ( X ) is the disjoint union of rng( ι nX ) and rng( κ nX ), as required forDefinition 3.1 and Theorem 3.5. In view of Definition 5.2 we also have ι nX ( x ) = x ≤ T − n +1 ( X ) y = ι nX ( y ) ⇔ x ≤ X y,κ nX ([ s , . . . , s k − ]) = 0 ⋆ [ π nX ( s ) , . . . , π nX ( s k − )] T − n +1 ( X ) y = ι nX ( y ) . To verify the remaining conditions from Definition 3.1, we observe that the supportof an element τ = [ t , . . . , t m − ] ∈ M ◦ T n ◦ T − n +1 ( X ) is given bysupp M ◦ T n T − n +1 ( X ) ( τ ) = [ { supp T n T − n +1 ( X ) ( t ) | t ∈ supp M T n ◦ T − n +1 ( X ) ( τ ) } == [ { supp T n T − n +1 ( X ) ( t l ) | l < m } . For s ∈ T − n +1 ( X ) we can thus invoke Lemma 6.3 to get s ≤ fin T − n +1 ( X ) supp M ◦ T n T − n +1 ( X ) ( τ ) ⇔ s ≤ T n +1 ( X ) π nX ( t l ) for some l < m. Writing σ = [ s , . . . , s k − ] and τ = [ t , . . . , t m − ], we now see that the secondcondition from Definition 3.1 requires that ι nX ( x ) = x ≤ T − n +1 ( X ) ⋆ [ π nX ( t ) , . . . , π nX ( t m − )] = κ nX ( τ )holds if, and only if, we have x ≤ T − n +1 ( X ) π nX ( t l ) for some l < m . This is trueaccording to Definition 5.2. The last condition from Definition 3.1 requires that κ nX ( σ ) = 0 ⋆ [ π nX ( s ) , . . . , π nX ( s k − )] ≤ T − n +1 ( X ) ⋆ [ π nX ( t ) , . . . , π nX ( t m − )] = κ nX ( τ )is equivalent to the disjunction σ ≤ M ◦ T n ◦ T − n +1 ( X ) τ or κ nX ( σ ) ≤ T n +1 ( X ) π nX ( t l ) for some l < m. To reduce this to Definition 5.2 it suffices to note that we have σ ≤ M ◦ T n ◦ T − n +1 ( X ) τ ⇔ [ π nX ( s ) , . . . , π nX ( s k − )] ≤ M ◦ T n +1 ( X ) [ π nX ( t ) , . . . , π nX ( t m − )] , ROM KRUSKAL’S THEOREM TO FRIEDMAN’S GAP CONDITION 21 since π nX is an embedding. Now recall the function h n +1 X : T n +1 ( X ) → N that wasspecified before the statement of Defintion 5.2 above. We will also write h n +1 X for therestriction of this function to T − n +1 ( X ) ⊆ T n +1 ( X ). In order to apply Theorem 3.5,we need to establish s ∈ supp M ◦ T n T − n +1 ( X ) ( τ ) ⇒ h n +1 X ( s ) < h n +1 X ( κ nX ( τ ))for s ∈ T − n +1 ( X ) and τ ∈ M ◦ T n ◦ T − n +1 ( X ). So assume we have s ∈ supp M ◦ T n T − n +1 ( X ) ( τ )with τ = [ t , . . . , t m − ]. By the above we get s ∈ supp T n T − n +1 ( X ) ( t l ) for some l < m .Then Lemma 6.3 yields s ≤ T n +1 ( X ) π nX ( t l ). As observed after Definition 5.2, thisimplies h n +1 X ( s ) ≤ h n +1 X ( π nX ( t l )) and hence h n +1 X ( s ) < h n +1 X (0 ⋆ [ π nX ( t ) , . . . , π nX ( t m − )]) = h n +1 X ( κ nX ( τ )) . We have now verified all conditions from Definition 3.1 and Theorem 3.5, whichshows that ( T − n +1 ( X ) , ι nX , κ nX ) is an initial Kruskal fixed point of M ◦ T n over X .To conclude that ( T − n +1 , ι n , κ n ) is a Kruskal derivative of M ◦ T n , it remains toestablish condition (ii) from Definition 4.1. Given a quasi embedding f : X → Y ,we first compute ι nY ◦ f ( x ) = f ( x ) = T n +1 ( f )( x ) = T − n +1 ( f ) ◦ ι nX ( x ) . For τ = [ t , . . . , t m − ] ∈ M ◦ T n ◦ T − n +1 ( X ) we also get κ nY ◦ ( M ◦ T n )( T − n +1 ( f ))( τ ) = κ nY ([( T n ◦ T − n +1 )( f )( t ) , . . . , ( T n ◦ T − n +1 )( f )( t m − )]) == 0 ⋆ [ π nY ◦ ( T n ◦ T − n +1 )( f )( t ) , . . . , π nY ◦ ( T n ◦ T − n +1 )( f )( t m − )] == 0 ⋆ [ T n +1 ( f ) ◦ π nX ( t ) , . . . , T n +1 ( f ) ◦ π nX ( t m − )] == T n +1 ( f )(0 ⋆ [ π nX ( t ) , . . . , π nX ( t m − )]) = T − n +1 ( f ) ◦ κ nX ( τ ) , just as required by Definition 4.1. (cid:3) As mentioned in the introduction, we can draw the following conclusion. In viewof Proposition 5.3, the corollary implies Friedman’s result that the gap conditioninduces a well partial order on the set of finite trees with labels from { , . . . , n − } . Corollary 6.6.
The normal PO-dilators T n and T − n +1 preserve well partial orders(which means that they are normal WPO-dilators), for each number n ∈ N .Proof. We argue by induction on n . Due to T ( X ) ∼ = X it is clear that T preserveswell partial orders. If T n is a normal WPO-dilator, then so is M ◦ T n . Since T − n +1 is the Kruskal derivative of M ◦ T n , Corollary 4.5 implies that it is also a normalWPO-dilator. In view of Proposition 6.2, the same holds for T n +1 ∼ = T n ◦ T − n +1 . (cid:3) References
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