aa r X i v : . [ m a t h . L O ] M a r A remark on strict independence relations
Gabriel ConantUniversity of Notre [email protected] 7, 2015; updated: March 2, 2016
Abstract
We prove that if T is a complete theory with weak elimination of imaginaries, then there is anexplicit bijection between strict independence relations for T and strict independence relationsfor T eq . We use this observation to show that if T is the theory of the Fra¨ıss´e limit of finitemetric spaces with integer distances, then T eq has more than one strict independence relation.This answers a question of Adler [1, Question 1.7]. Let T be a complete first-order theory with a sufficiently saturated monster model M . We use A, B, C, . . . to denote small subsets of M , where A is small (written A ⊂ M ) if M is | T ( A ) | + -saturated. We take the following definition from Adler [1]. Definition 1.1.
Let | ⌣ be a ternary relation on small subsets of M . We define the followingaxioms.( i ) ( invariance ) If A | ⌣ C B and σ ∈ Aut( M ), then σ ( A ) | ⌣ σ ( C ) σ ( B ).( ii ) ( monotonicity ) If A | ⌣ C B , A ′ ⊆ A , and B ′ ⊆ B , then A ′ | ⌣ C B ′ .( iii ) ( base monotonicity ) If A | ⌣ D B and D ⊆ C ⊆ B then A | ⌣ C B .( iv ) ( transitivity ) If D ⊆ C ⊆ B , A | ⌣ D C and A | ⌣ C B then A | ⌣ D B .( v ) ( extension ) If A | ⌣ C B and ˆ B ⊇ B then there is A ′ ≡ BC A such that A | ⌣ C ˆ B .( vi ) ( finite character ) If A | ⌣ C B for all finite subsets A ⊆ A , then A | ⌣ C B .( vii ) ( local character ) For every A , there is a cardinal κ ( A ) such that, for every B , there is C ⊆ B with | C | < κ ( A ) and A | ⌣ C B .( viii ) ( symmetry ) If A | ⌣ C B then B | ⌣ C A .( ix ) ( anti-reflexivity ) a | ⌣ C a implies a ∈ acl( C ).A ternary relation | ⌣ is a strict independence relation for T if it satisfies ( i ) through ( ix ). T is rosy if there is a strict independence relation for T eq . In [1], Adler formulates transitivity on the left, and proves symmetry as a consequence of the rest of the axioms.We are using the more standard formulation of transitivity on the right, and therefore include symmetry. Adleralso includes an axiom called normality , which is a consequence of invariance, extension, and symmetry [1, Remark1.2(1)]. T is rosy, then there is a distinguished strict independencerelation for T eq , called thorn-forking independence , which is canonical in the sense that it is theweakest such relation [1]. This fact motivates Question 1.7 of [1], which asks whether there is atheory T such that T eq has more than one strict independence relation. Rephrased in the negative,the question asks if the axioms of strict independence relations characterize thorn-forking in T eq for rosy theories (analogous to the characterizations of forking in stable and simple theories due,respectively, to Harnik-Harrington [7] and Kim-Pillay [8]).In this note, we answer Adler’s question by exhibiting two distinct strict independence relationsfor T eq , where T is the theory of the integral Urysohn space (i.e. the Fra¨ıss´e limit of metric spaceswith integer distances). To accomplish this, we first prove a general result that if T is a completetheory with weak elimination of imaginaries, then there is an explicit bijection between strictindependence relations for T and strict independence relations for T eq . We then show that if T isthe theory of the integral Urysohn space (shown in [4] to have weak elimination of imaginaries),then T has more than one strict independence relation. We conclude with further remarks on themotivation for Adler’s question, as well as a generalization of our results to a certain family oftheories (including that of the rational Urysohn space).
Let T be a complete first-order theory with monster model M . Definition 2.1.
1. Fix e ∈ M eq . A finite tuple c ∈ M is a weak canonical parameter for e if c ∈ acl eq ( e ) and e ∈ dcl eq ( c ).2. T has weak elimination of imaginaries if every e ∈ M eq has a weak canonical parameterin M .3. Suppose A ⊂ M eq . We say a subset A ∗ ⊂ M is a weak code for A if A ∗ = S e ∈ A c ( e ), where c ( e ) is a weak canonical parameter for e .Note that if T has weak elimination of imaginaries then any subset of M eq has a weak code in M . Therefore, if | ⌣ is a ternary relation on M , we can use weak codes to define a ternary relationon M eq in the following way. Definition 2.2.
Suppose T has weak elimination of imaginaries, and | ⌣ is a ternary relation on M . Define the ternary relation | ⌣ eq on M eq such that, given A, B, C ⊂ M eq , A | ⌣ eq C B ⇔ A ∗ | ⌣ C ∗ B ∗ for some weak codes A ∗ , B ∗ , C ∗ for A, B, C .The following basic exercise in forking calculus will allow us to replace “for some weak codes”in the last definition with “for all weak codes”.
Exercise 2.3. If | ⌣ is a strict independence relation for T , and A, B, C ⊂ M , then A | ⌣ C B if andonly if acl( A ) | ⌣ acl( C ) acl( B ) . emma 2.4. Assume T has weak elimination of imaginaries, and suppose | ⌣ is a strict indepen-dence relation for T . Then, for any A, B, C ⊂ M eq , A | ⌣ eq C B ⇔ A ∗ | ⌣ C ∗ B ∗ for all weak codes A ∗ , B ∗ , C ∗ for A, B, C .Proof.
First, note that if A ⊂ M eq and A ∗ is a weak code for A , then we have acl eq ( A ∗ ) = acl eq ( A ),and so acl( A ∗ ) = acl eq ( A ) ∩ M . In particular, acl( A ∗ ) does not depend on the choice of weak code.With this observation in hand, we prove the claim. Since any subset of M eq has a weak code in M , the right-to-left implication is trivial. For left-to-right, suppose A | ⌣ eq C B and A ∗ , B ∗ , C ∗ areweak codes for A, B, C . By definition, there are weak codes A ∗∗ , B ∗∗ , C ∗∗ for A, B, C such that A ∗∗ | ⌣ C ∗∗ B ∗∗ . By Exercise 2.3 and the above observation, A ∗∗ | ⌣ C ∗∗ B ∗∗ ⇒ acl( A ∗∗ ) | ⌣ acl( C ∗∗ ) acl( B ∗∗ ) ⇒ acl( A ∗ ) | ⌣ acl( C ∗ ) acl( B ∗ ) ⇒ A ∗ | ⌣ C ∗ B ∗ , as desired. Theorem 2.5.
Suppose T has weak elimination of imaginaries. Then the map | ⌣
7→ | ⌣ eq is abijection from strict independence relations for T to strict independence relations for T eq .Proof. We first show that the map is well-defined, i.e., if | ⌣ is a strict independence relationfor T then, in M eq , | ⌣ eq satisfies the axioms of Definition 1.1. Invariance, monotonicity, basemonotonicity, symmetry, transitivity, and finite character, are all straightforward, but rely onLemma 2.4. To illustrate this, we show transitivity. Transitivity : Suppose
A, B, C, D ⊂ M eq , with D ⊆ C ⊆ B , A | ⌣ eq D C , and A | ⌣ eq C B . We wantto show A | ⌣ eq D B . We may find weak codes A ∗ , B ∗ , C ∗ , D ∗ for A, B, C, D such that D ∗ ⊆ C ∗ ⊆ B ∗ .By Lemma 2.4, we have A ∗ | ⌣ D ∗ C ∗ and A ∗ | ⌣ C ∗ B ∗ . By transitivity for | ⌣ , we have A ∗ | ⌣ D ∗ B ∗ ,and so A | ⌣ eq D B .The rest of the axioms ( anti-reflexivity , local character , and extension ) require more than justLemma 2.4. Anti-reflexivity : Fix a ∈ M eq and C ⊂ M eq , with a | ⌣ eq C a . Let a ∗ and C ∗ be weak codes for a and C (in particular, a ∗ is a weak canonical parameter for a ). We have a ∗ | ⌣ C ∗ a ∗ by Lemma 2.4,and so a ∗ ∈ acl( C ∗ ) by anti-reflexivity for | ⌣ . Then a ∈ acl eq ( a ∗ ) ⊆ acl eq ( C ∗ ) = acl eq ( C ). Local character : Fix A ⊂ M eq , and let A ∗ be a weak code for A . By local character for | ⌣ ,there is a κ such that, for all B ⊂ M , there is C ⊆ B , with | C | < κ and A ∗ | ⌣ C B . Suppose B ⊂ M eq , and let B ∗ be a weak code for B . Fix C ⊆ B ∗ such that | C | < κ and A ∗ | ⌣ C B ∗ . Byassumption, we may write B ∗ = S e ∈ B c ( e ), where c ( e ) is a fixed weak canonical parameter for e .Given d ∈ C , we fix e d ∈ B such that d ∈ c ( e d ). Let C = { e d : d ∈ C } and C ∗ = S d ∈ C c ( e d ).Then | C | < κ , C ∗ is a weak code for C , and C ⊆ C ∗ ⊆ B ∗ . By base monotonicity for | ⌣ , we have A ∗ | ⌣ C ∗ B ∗ , and so A | ⌣ eq C B . Extension : Fix
A, B, C ⊂ M eq , with A | ⌣ eq C B , and suppose B ⊆ ˆ B . Let A ∗ , B ∗ , C ∗ be weakcodes for A, B, C , with A ∗ | ⌣ C ∗ B ∗ . Then we may find a weak code ˆ B ∗ for ˆ B , with B ∗ ⊆ ˆ B ∗ .By extension for | ⌣ , there is A ′∗ ≡ B ∗ C ∗ A ∗ such that A ′∗ | ⌣ C ∗ ˆ B ∗ . Fix σ ∈ Aut( M /B ∗ C ∗ ) suchthat σ ( A ∗ ) = A ′∗ . Let A ′ = σ ( A ). Then A ′∗ is a weak code for A ′ , and so we have A ′ | ⌣ eq C ˆ B . Itremains to show that A ′ ≡ BC A and, for this, it suffices to see that σ fixes BC pointwise. Let B ∗ = S e ∈ B c ( e ). Then σ ( c ( e )) = c ( e ) for all e ∈ B and so, since e ∈ dcl eq ( c ( e )), it follows that σ ( e ) = e . Similarly, σ also fixes C pointwise.Finally, we must show that | ⌣
7→ | ⌣ eq is a bijection. For injectivity, simply use the fact anysubset A ⊂ M is a weak code for itself. For surjectivity, suppose | ⌣ is a strict independence relation3or T eq . Let | ⌣ ∗ be the restriction to subsets of M . Then one easily checks that | ⌣ ∗ is a strictindependence relation for T . Using Exercise 2.3, we have | ⌣ ∗ , eq = | ⌣ .Recall that T is real rosy if there is a strict independence relation for T . The previous theoremimmediately implies the following well-known fact. Corollary 2.6. If T is real rosy with weak elimination of imaginaries, then T is rosy. This result is shown explicitly by Ealy and Goldbring in [5] (in the context of continuous logic).The proof of Theorem 2.5 is similar to their work.Define algebraic independence , denoted | ⌣ a , in M by: A | ⌣ aC B if and only if acl( AC ) ∩ acl( BC ) =acl( C ). For the rest of the paper, our focus will be on theories in which algebraic independencecoincides with thorn-forking independence, denoted | ⌣ þ . Therefore, we omit the definition ofthorn-forking and refer the reader to [1]. The next fact is a standard result. Fact 2.7.
The following are equivalent. ( i ) | ⌣ a coincides with | ⌣ þ in M (resp. in M eq ). ( ii ) | ⌣ a satisfies base monotonicity in M (resp. in M eq ). ( iii ) | ⌣ a is a strict independence relation for T (resp. for T eq ).Proof. See Lemma 4.2, Proposition 1.5, and Theorem 4.3 of [1] for, respectively, ( i ) ⇒ ( ii ), ( ii ) ⇒ ( iii ), and ( iii ) ⇒ ( i ). Corollary 2.8.
Suppose T has weak elimination of imaginaries. If A, B, C ⊂ M then A | ⌣ aC B in M if and only if A | ⌣ aC B in M eq . Moreover, if | ⌣ þ = | ⌣ a in M then | ⌣ þ = | ⌣ a in M eq .Proof. Let | ⌣ a, eq denote the ternary relation in M eq obtained by applying the eq-map to | ⌣ a in M . Recall that, for any A ⊂ M eq and weak code A ∗ ⊂ M , we have acl( A ∗ ) = acl eq ( A ) ∩ M . Usingthis, it is routine to show that | ⌣ a, eq coincides with | ⌣ a in M eq . The first claim then follows fromLemma 2.4, and the fact that any subset of M is a weak code for itself. For the second claim,combine Theorem 2.5 and Fact 2.7.It is worth noting that Theorem 2.5 becomes false if the assumption of weak elimination ofimaginaries is removed. In particular, if T is stable then forking and thorn-forking coincide in M eq [6], and so thorn-forking is the unique strict independence relation for T eq [1]. However, thereare stable theories (failing weak elimination of imaginaries) for which T has more than one strictindependence relation. In fact, for any cardinal κ , if T is the model completion of the theory of κ many equivalence relations, then T is stable and has at least 2 κ distinct strict independencerelations (see [1, Example 1.5]). Let N denote the ordered monoid ( N , + , ≤ , K N of finite metric spaces, withinteger distances, is a Fra¨ıss´e class in the relational language L N = { d n ( x, y ) : n ∈ N } , where d n ( x, y ) is interpreted as d ( x, y ) ≤ n . In particular, K N is closed under free amalgamation ofmetric spaces. Precisely, given integer distance metric spaces A, B, C , with ∅ 6 = C ⊆ A ∩ B , the free amalgamation of A and B over C is defined by setting, for a ∈ A and b ∈ B , d ( a, b ) = d max ( a, b/C ) := inf c ∈ C ( d ( a, c ) + d ( c, b )). 4et U N denote the Fra¨ıss´e limit of K N , which we refer to as the integral Urysohn space . Then U N is the unique (up to isometry) countable, universal, and ultrahomogeneous metric space withinteger distances. Let T N = Th( U N ), and let U N be a sufficiently saturated monster model of T N .Note that U N cannot be interpreted as an integer-valued metric space in a way coherent with T N . In particular, the type {¬ d n ( x, y ) : n ∈ N } is consistent with T N , and therefore realized in U N by points of “infinite distance”. However, this is the only obstruction to viewing U N as a metricspace, and we resolve the issue as follows. Let N ∗ = ( N ∪ {∞} , + , ≤ ,
0) be an ordered monoidextension of N , where ∞ + ∞ = ∞ and, for all n ∈ N , n < ∞ and n + ∞ = ∞ = ∞ + n . Then U N can be viewed as an N ∗ -valued metric space. We use d to refer to the N ∗ -metric on U N . Given C ⊂ U N and a ∈ U N , let d ( a, C ) = inf { d ( a, c ) : c ∈ C } . Then C = ∅ implies d ( a, C ) = ∞ , and if C is nonempty then there is some c ∈ C such that d ( a, C ) = d ( a, c ). In particular, d ( a, C ) = 0 if andonly if a ∈ C . For the subsequent work, we will need the following facts about T N . Fact 3.1. ( a ) T N has quantifier elimination. Consequently, acl( C ) = C for all C ⊂ U N . Moreover, U N is a κ + -universal and κ -homogeneous N ∗ -metric space, where κ is the saturation cardinal of U N . ( b ) T N has weak elimination of imaginaries. Details on these results can be found in [4]. Our goal is to define two distinct strict independencerelations on T N . By weak elimination of imaginaries and Theorem 2.5, we will then obtain twodistinct strict independence relations on T eq N . The first strict independence relation is given to usby thorn-forking. Theorem 3.2.
Thorn-forking is a strict independence relation for T eq N (i.e. T N is rosy). Inparticular, | ⌣ þ coincides with | ⌣ a in U eq N .Proof. By Fact 3.1( a ), | ⌣ a satisfies base monotonicity in U N . So the result follows from Fact 2.7,Corollary 2.8, and Fact 3.1( b ).We now define what we will show to be a second strict independence relation for T N . Definition 3.3.
Define | ⌣ ∞ on T N such that, given A, B, C ⊂ U N , A | ⌣ ∞ C B if and only if, for all a ∈ A , if d ( a, B ) = 0 then d ( a, C ) = 0, and if d ( a, C ) = ∞ then d ( a, B ) = ∞ .Since acl( C ) = C for all C ⊂ U N , and d ( a, C ) = 0 if and only if a ∈ C , we can rephrase | ⌣ ∞ as: A | ⌣ ∞ C B if and only if A | ⌣ aC B and, for all a ∈ A , if d ( a, C ) = ∞ then d ( a, B ) = ∞ . Theorem 3.4.
The relation | ⌣ ∞ is a strict independence relation for T N .Proof. Invariance and finite character are easy.
Symmetry : Suppose A | ⌣ ∞ C B . Then B | ⌣ aC A , and so it remains to fix b ∈ B , with d ( b, C ) = ∞ ,and show d ( a, b ) = ∞ for all a ∈ A . Given a ∈ A , we have ∞ = d ( b, C ) ≤ d ( a, b ) + d ( a, C ) bythe triangle inequality. Therefore, we either directly have d ( a, b ) = ∞ , or we have d ( a, C ) = ∞ , inwhich case A | ⌣ ∞ C B implies d ( a, b ) = ∞ . Transitivity , monotonicity , and base monotonicity : Since we have shown symmetry, we canverify all three axioms with the following claim: if D ⊆ C ⊆ B then A | ⌣ ∞ D B if and only if A | ⌣ ∞ D C and A | ⌣ ∞ C B . This is clearly true for | ⌣ a , so it suffices to fix a ∈ A and show that thefollowing are equivalent:( i ) d ( a, D ) = ∞ implies d ( a, B ) = ∞ ; 5 ii ) d ( a, D ) = ∞ implies d ( a, C ) = ∞ , and d ( a, C ) = ∞ implies d ( a, B ) = ∞ .The implication ( ii ) ⇒ ( i ) is trivial; and ( i ) ⇒ ( ii ) follows from the fact that, since D ⊆ C ⊆ B ,we have d ( a, D ) ≥ d ( a, C ) ≥ d ( a, B ). Extension : It follows from Fact 3.1( a ) that, for all A, B, C ⊂ U N , there is A ′ ≡ C A such that,for all a ′ ∈ A ′ and b ∈ B , d ( a ′ , b ) = d max ( a ′ , b/C ). Note also that d ( a, C ) ≤ d max ( a, b/C ) for any a, b ∈ U N and C ⊂ U N .Now assume A | ⌣ ∞ C B and ˆ B ⊇ B . Let A ′ ≡ BC A such that, for all a ′ ∈ A ′ and b ∈ ˆ B , d ( a ′ , b ) = d max ( a ′ , b/BC ). We show A ′ | ⌣ ∞ C ˆ B . Fix a ′ ∈ A ′ and suppose d ( a ′ , ˆ B ) = 0. Then a ′ ∈ ˆ B ,and so d max ( a ′ , a ′ /BC ) = d ( a ′ , a ′ ) = 0. Therefore d ( a ′ , BC ) = 0, which implies d ( a, BC ) = 0. Then d ( a, C ) = 0, and so d ( a ′ , C ) = 0. Next, fix a ′ ∈ A ′ and suppose d ( a ′ , C ) = ∞ . Then d ( a, C ) = ∞ ,and so A | ⌣ ∞ C B implies d ( a, BC ) = ∞ . For any b ∈ ˆ B , we have d ( a ′ , b ) = d max ( a ′ , b/BC ) ≥ d ( a ′ , BC ) = d ( a, BC ) = ∞ , and so d ( a ′ , ˆ B ) = ∞ . Local character : Fix
A, B ⊂ U N . We will find C ⊆ B such that | C | ≤ | A | and A | ⌣ ∞ C B .We may assume B is nonempty. Given a ∈ A , choose b a ∈ B such that d ( a, B ) = d ( a, b a ). Let C = { b a : a ∈ A } . For any a ∈ A , we have d ( a, C ) = d ( a, B ), and this clearly implies A | ⌣ ∞ C B . Anti-reflexivity : Trivial, since | ⌣ ∞ implies | ⌣ a . Corollary 3.5. T eq N has more than one strict independence relation.Proof. We have shown that | ⌣ a and | ⌣ ∞ are strict independence relations for T N . Therefore,by Theorem 2.5 and Fact 3.1( b ), it suffices to show that these two relations are not the same in U N . To see this, fix distinct a, b ∈ U N such that d ( a, b ) < ∞ . Fix any subset C ⊂ U N such that d ( a, C ) = ∞ = d ( b, C ) (e.g. C = ∅ ). Then a | ⌣ aC b by Fact 3.1( a ), and we clearly have a ⌣ ∞ C b . A major motivation for Question 1.7 of [1] comes from the following open problem in the study ofsimple theories (asked by several authors, e.g. [1], [6]).
Question 4.1.
Suppose T is a simple theory. Are forking and thorn-forking the same in T eq ?This question is known to have a positive answer if T is additionally assumed to eliminatehyperimaginaries or satisfy the stable forking conjecture (see [6]). If T is simple then forkingindependence is the strongest strict independence relation for T eq and thorn-forking independenceis the weakest [1]. So a negative answer to Question 4.1 would follow from the existence of a simple theory T , such that T eq has more than one strict independence relation. Therefore, it is worthobserving that our example, T N , is not simple. This is shown in [4] (see Fact 4.9 below), but alsofollows by adapting the proof in [3] that, for a fixed n ≥
3, the Fra¨ıss´e limit of metric spaces withdistances in { , , . . . , n } is not simple. On the other hand, when n = 2, this Fra¨ıss´e limit is preciselythe countable random graph (where the graph relation is d ( x, y ) = 1), which is well-known to havea simple theory. Therefore, in this context of “generalized” Urysohn spaces, a natural question iswhether there is a suitable choice of distance set such that, if T is the theory of the associatedUrysohn space, then T is simple and the analog of | ⌣ ∞ still yields a strict independence relationfor T eq distinct from thorn-forking. The goal of this section is to demonstrate that this is unlikely.First, we define a precise context for studying Urysohn spaces over arbitrary distance sets. Definition 4.2.
A structure R = ( R, ⊕ , ≤ ,
0) is a distance monoid if ( R, ⊕ ,
0) is a commutativemonoid and ≤ is a total, ⊕ -invariant order with least element 0.6ix a countable distance monoid R . We define an R -metric space to be a set equipped withan R -valued metric. Let K R denote the class of finite R -metric spaces. We consider K R as aclass of L R -structures, where L R = { d r ( x, y ) : r ∈ R } and d r ( x, y ) is a binary relation interpretedas d ( x, y ) ≤ r . Then K R is a Fra¨ıss´e class by a similar argument as for K N , or by adapting thefollowing fact, due to Sauer [9], which also motivates Definition 4.2. Fact 4.3.
Suppose S ⊆ R ≥ is countable, contains , and is closed under the operation u + S v :=sup { x ∈ S : x ≤ u + v } . Then the class of finite metric spaces, with distances in S , is a Fra¨ıss´eclass if and only if + S is associative on S (and so ( S, + S , ≤ , is a distance monoid). Definition 4.4.
Given a countable distance monoid R , we define the R -Urysohn space , denoted U R , to be the Fra¨ıss´e limit of K R . Let T R = Th( U R ), and let U R denote a sufficiently saturatedmonster model of T R .When working with T R , we again face the obstacle that U R cannot be interpreted as an R -metric space. As in the last section, this is resolved by constructing a distance monoid extension R ∗ = ( R ∗ , ⊕ , ≤ ,
0) of R . The construction is quite technical, and so we refer the reader to [4]. Therough idea is that the set R ∗ corresponds to the space of quantifier-free 2-types (over ∅ ) consistentwith T R , and the ordering ≤ and operation ⊕ extend to R ∗ in a canonical way. Then, given a model M | = T R , one defines an R ∗ -metric on M by setting the distance between two points a, b ∈ M tobe the element of R ∗ corresponding the the quantifier-free 2-type realized by ( a, b ).For example, define Q := ( Q ≥ , + , ≤ , Q ∗ , the set ( Q ≥ ) ∗ can be identified with R ≥ ∪ { q + : q ∈ Q ≥ } ∪ {∞} , where q + is a new symbol for an immediate successor of q and ∞ isa new symbol for an infinite element.Suppose R is a countable distance monoid. The goal of this section is to show that if | ⌣ ∞ can be defined for T R , and moreover yields a strict independence relation for T eq R distinct fromthorn-forking, then T R is not simple. The next definition lists the properties we need to define | ⌣ ∞ on T R , prove | ⌣ ∞ is a strict independence relation, and lift | ⌣ ∞ to T eq R using Theorem 2.5. Definition 4.5.
A countable distance monoid R is suitable if R has no maximal element, T R hasquantifier elimination, and T R has weak elimination of imaginaries.The following fact provides natural examples of suitable distance monoids. Fact 4.6. [4] If ( G, ⊕ , ≤ , is a nontrivial countable ordered abelian group, and R = ( G ≥ , ⊕ , ≤ , ,then R is a suitable distance monoid. Suppose R is a suitable distance monoid. By quantifier elimination, we again have acl( C ) = C for all C ⊂ U R , and that U R is a κ + -universal and κ -homogeneous R ∗ -metric space, where κ is thesaturation cardinal of U R . Moreover, since R has no maximal element, the type {¬ d r ( x, y ) : r ∈ R } is realized in U R , and therefore corresponds to a new element ∞ in R ∗ . So we may define | ⌣ ∞ on U R exactly as in Definition 3.3 (with R in place of N ). Theorem 4.7. If R is a suitable distance monoid then | ⌣ ∞ is a strict independence relation for T R , which is distinct from | ⌣ þ .Proof. First, we have | ⌣ a = | ⌣ þ in U R by Fact 2.7. Next, it follows from the construction of R ∗ in [4] that 0 always has an immediate successor in R ∗ (e.g. 0 + in Q ∗ ). Therefore, given a ∈ U R and C ⊂ U R , we have d ( a, C ) = 0 if and only if a ∈ C . In particular, | ⌣ ∞ implies | ⌣ a , and thesame argument as in the proof of Corollary 3.5 shows that | ⌣ ∞ and | ⌣ a are distinct. Finally, the7rgument that | ⌣ ∞ is a strict independence relation for T R is the same as Theorem 3.4 for T N ,except for the following issues.To show symmetry , we need the following properties of R ∗ , which follow from [4, Proposition2.11]. First, if a, b ∈ U R and C ⊂ U R then d ( b, C ) ≤ d ( a, b ) ⊕ d ( a, C ). Second, if r, s ∈ R ∗ then r ⊕ s = ∞ if and only if r = ∞ or s = ∞ . Given this, the proof of symmetry follows as in Theorem3.4.The argument for local character requires slightly more work. Fix A, B ⊂ U R . Given a ∈ A ,define X a ⊆ R ∗ such that r ∈ X a if and only if there is b ∈ B with d ( a, b ) = r . For each a ∈ A and r ∈ X a , fix some b ar ∈ B such that d ( a, b ar ) = r . Now let C = { b ar : a ∈ A, r ∈ X a } . Then | C | ≤ | A | + | R ∗ | and d ( a, B ) = d ( a, C ) for any a ∈ A , which implies A | ⌣ ∞ C B .To show extension , we must define the analog of d max . In particular, given C ⊂ U R and a, b ∈ U R , set d max ( a, b/C ) := inf c ∈ C ( d ( a, c ) ⊕ d ( c, b )). This definition is justified by the fact thatthe ordering in R ∗ is complete (again, see [4]). The proof of extension now follows as in Theorem3.4.Applying Theorem 2.5, we have: Corollary 4.8. If R is a suitable distance monoid, then T eq R has more than one strict independencerelation. To finish our goal, we quote [4] to show that if R is suitable then T R is not simple. In fact, T R is quite complicated in the sense of certain combinatorial dividing lines invented by Shelah (see [4]or [10] for definitions). Fact 4.9. [4] If R is a suitable distance monoid then T R has SOP n for all n ≥ (but does nothave the strict order property). Therefore T R is not simple.Proof. We fix n ≥ n from results in [4]. Fix r ∈ R > . We show( n − r < nr , which implies SOP n by a technical construction in [4, Section 6]. If ( n − r = nr then ( n − r = mr for all m ≥ n −
1, and so d ( x, y ) ≤ ( n − r is a nontrivial equivalence relationon U R . This is shown in [4, Section 7] to contradict weak elimination of imaginaries.A final observation is that this is not the first time nonstandard distances in saturated modelsof metric spaces have led to interesting model theoretic phenomena. In particular, consider the rational Urysohn space U Q . In the monster model U Q , we have the type-definable binary relations d + ( x, y ) := V r ∈ Q + d r ( x, y ) and d ∞ ( x, y ) := V r ∈ Q + ¬ d r ( x, y ) describing, respectively, infinitesimaldistance and infinite distance. Note that d + ( x, y ) and the complement of d ∞ ( x, y ) are both equiv-alence relations on U Q . The work in this section uses d ∞ ( x, y ) to obtain a strict independencerelation for T eq Q distinct from thorn-forking. In [3], Casanovas and Wagner used d + ( x, y ) to obtainthe first example of non-eliminable (finitary) hyperimaginaries in a theory without the strict orderproperty (to be precise, they considered the rational Urysohn sphere ; see [4, Section 7] for details).In both of these situations, the aberrant behavior can be traced to a failure of ℵ -categoricity.Specifically, as types in S ( T Q ), d + ( x, y ) and d ∞ ( x, y ) are both non-isolated. Moreover, it is a factthat finitary hyperimaginaries are always eliminated in ℵ -categorical theories [2, Theorem 18.14].So we ask the following question. Question 4.10.
Suppose T is a complete ℵ -categorical theory. Can T eq have more than one strictindependence relation? 8 cknowledgements I would like to thank the anonymous referee for several corrections and suggestions, which signifi-cantly improved the final draft.
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