Canonical Polymorphisms of Ramsey Structures and the Unique Interpolation Property
aa r X i v : . [ m a t h . L O ] A ug CANONICAL POLYMORPHISMS OF RAMSEY STRUCTURESAND THE UNIQUE INTERPOLATION PROPERTY
BERTALAN BODOR AND MANUEL BODIRSKY,AUGUST 25, 2020
Abstract.
Let C be a model-complete core which is a reduct of a homoge-neous Ramsey structure A with finite relational signature. We present char-acterisations of when the existence of a pseudo-Siggers polymorphism of C implies the existence of a pseudo-Siggers polymorphism of C which is canoni-cal over A . This has applications for the complexity of constraint satisfaction:Barto and Pinsker showed that an ω -categorical model-complete core struc-ture C which does not have a pseudo-Siggers polymorphism has an NP-hardconstraint satisfaction problem (CSP). On the other hand, if C is a reduct ofa finitely bounded homogeneous structure B and C has a pseudo-Siggers poly-morphism which is canonical with respect to B , then the CSP for C can besolved in polynomial time, by a reduction to a finite-domain CSP of Bodirskyand Mottet and the finite-domain dichotomy theorem of Bulatov and Zhuk.Our results allow to re-derive and generalise some of the existing complexityclassifications for infinite-domain CSPs, for example for the class of all struc-tures with exponential labelled growth. We also verify the infinite-domaintractability conjecture for first-order expansions of the basic relations of thespatial reasoning formalism RCC5. Contents
1. Introduction 12. Countably Categorical Structures 53. Oligomorphic Clones 74. Independence 125. Interpolation 146. The Unique Interpolation Property 177. Binary Verification of the UIP 188. Structures with exponential labelled growth 229. First-order expansions of the basic relations of RCC5 2410. First-order expansions of the random poset 35References 361.
Introduction
Many results in universal algebra only hold for algebras over a finite domain,for instance the important theorem about the existence of cyclic terms in Taylor
The authors have received funding from the European Research Council (Grant Agreementno. 681988, CSP-Infinity). algebras [BK12]. Every cyclic term is in particular a weak near unanimity term,and the existence of a weak near unanimity term is the starting point for Zhuk’salgorithm [Zhu17], proving in 2017 the famous Feder-Vardi dichotomy conjecturefor finite-domain constraint satisfaction problems [FV99] (an independent proof,also based on universal algebra, has been given by Bulatov [Bul17]).Some of the results about finite algebras can be lifted to infinite algebras thatsatisfy a certain finiteness condition, called oligomorphicity : the requirement is thatthe permutation group of invertible unary term operations in the algebra has onlyfinitely many orbits in its componentwise action on n -tuples. Examples of suchalgebras arise systematically in model theory: every algebra whose term operationsare the polymorphisms of a (reduct of a) homogeneous structure B with finiterelational signature is of this type. An additional finiteness condition is to requirethat the class of finite substructures of B is described by finitely many forbiddensubstructures, in which case case B is called finitely bounded . The class of reductsof finitely bounded homogeneous structures is a huge generalisation of the class ofstructures with finite domain and signature.There is a generalisation of the Feder-Vardi dichotomy conjecture, which is stillopen, to reducts of finitely bounded homogeneous structures. In fact, there existsa known NP-hardness condition for the constraint satisfaction problem of sucha structure which is conjectured to be at the boarder between polynomial-timetractable and NP-complete CSPs. The infinite-domain tractability conjecture statesthat the CSP for every structure that does not satisfy the mentioned hardnesscondition is in P (this will be recalled in detail in Section 3). There are many classesof infinite-domain structures where the infinite-domain tractability conjecture hasbeen verified. Often, these classes consist of the first-order reducts of some fixedunderlying structure B . By a first-order reduct of B we mean a reduct of theexpansion of B by all relations that are first-order definable in B . The infinite-domain dichotomy conjecture has been verified for the following classes:(1) all structures preserved by all permutations [BK08], which is precisely theclass of first-order reducts of pure sets (structures with no relations);(2) all structures with a highly set-transitive automorphism group [BK09] (i.e.,for all finite setsets X, Y with | X | = | Y | there exists an automorphismwhich maps X to Y ), which is precisely the class of first-order reducts ofunbounded dense linear orders;(3) all first-order reducts of the homogeneous universal poset [KP18];(4) all first-order reducts of the binary branching C-relation [BJP17];(5) all first-order reducts for all homogeneous graphs [BMPP19].While these classifications follow similar patterns, there is so far no general resultthat would imply them in a uniform way. In some, but not in all cases above thetractable cases come from the existence of certain canonical polymorphisms (seeSection 3.4). In these cases, the algorithmic results can be obtained by reducingto polynomial-time solvable finite-domain CSPs [BM18]. We therefore ask thefollowing question. For which classes of infinite structures is it enough to look for canonicalpolymorphisms to prove polynomial-time tractability of the CSP?
For example, canonical polymorphisms suffice for the classes in (1), (3), and (5),but not for the classes in (2) and (4). In this article we present a solution to theabove question in a fairly general setting. In some situations, the condition given
HE UNIQUE INTERPOLATION PROPERTY 3 in our characterisation can be verified directly (see Sections 8, 9, and 10) in whichcase we directly obtain a complexity classification.For our result we need an additional assumption which makes the connection be-tween CSPs and universal algebra particularly strong: we require that C is a reductof a homogeneous Ramsey structure A (see Section 2.4). While the assumption tobe Ramsey is quite strong, the assumption to be a reduct of a Ramsey structureis weak: in fact, it has been asked whether every finitely bounded homogeneousstructure has a finitely bounded homogeneous Ramsey expansion (see [Bod15]).1.1. Results.
Our first result can be seen as a characterisation of when the exis-tence of a pseudo-Siggers polymorphism implies the existence of a canonical pseudo-Siggers polymorphism. Let M be a transformation monoid on a set A and let f, g : A k → A be operations. We say that f interpolates g modulo M if g ∈ { u ( f ( v , . . . , v k )) : u, v , . . . , v k ∈ M } . Definition 1.1.
Let A be a structure and let C be an operation clone that containsAut( A ). A function ζ defined on D ⊆ C has the unique interpolation property (UIP)with respect to C over A if ζ ( g ) = ζ ( h ) for all g, h ∈ D that are interpolated by thesame operation f ∈ C over Aut( A ).We write • Can( A ) for the set of all operations that are canonical with respect to A (see Section 3.4), and • Proj for the operation clone on { , } that just contains the projections.Model-complete cores are introduced in Section 2.2. Theorem 1.2.
Let A be a homogeneous Ramsey structure with finite relationalsignature, let B be a homogeneous reduct of A , and let C be a first-order reduct of B which is a model-complete core and has a pseudo-Siggers polymorphism. Thenthe following are equivalent:(1) Pol( C ) ∩ Can( B ) contains a pseudo-Siggers operation.(2) There is no uniformly continuous minor-preserving map from Pol( C ) ∩ Can( B ) to Proj .(3) If there is a uniformly continuous minor-preserving map from
Pol( C ) ∩ Can( B ) to Proj , then there is also a uniformly continuous minor-preservingmap from
Pol( C ) ∩ Can( A ) to Proj which has the UIP with respect to
Pol( C ) over A . Recall that if B is finitely bounded and C has a finite signature, then (1) impliesthat CSP( C ) is in P by the results from [BM16, BM18]. Hence, Theorem 1.2motivates the study of the UIP in the context of complexity classification of CSPs.In Sections 8, 9, and 10 we present applications where it is substantially easier toverify (3) of Theorem 1.2 in order to prove (1). Indeed, the classifications can beobtained by verifying the conditions of the following corollary (again, the proof canbe found in Section 6). Corollary 1.3.
Let C be a class of ω -categorical structures closed under takingmodel-complete cores such that each structure C ∈ C is the reduct of a finitelybounded homogeneous structure B , which is itself a reduct of a homogeneous Ram-sey structure A with finite relational signature. Suppose that whenever there is auniformly continuous minor-preserving map from Pol( C ) ∩ Can( B ) to Proj then
BERTALAN BODOR AND MANUEL BODIRSKY, AUGUST 25, 2020 there is also a minor-preserving map from
Pol( C ) ∩ Can( A ) to Proj which has theUIP with respect to
Pol( C ) over A . Then for every C ∈ C , we have that CSP( C ) isin P or NP-complete. To apply this result, the following characterisation of the UIP is helpful, since itallows us to focus on binary polymorphism of C . But first let us mention that thereexists a uniformly continuous minor-preserving map from Pol( C ) to Proj if and onlyif there is an minor-preserving map ζ : Pol( C ) → Proj which is Aut( B ) -invariant (see Section 3.7). Theorem 1.4 (Binary UIP verification) . Let A be a homogeneous Ramsey structurewith finite relational signature and let C be a reduct of A . Let C be the clone of allpolymorphisms of C that are canonical with respect to A and let ζ : C → Proj be an
Aut( A ) -invariant minor-preserving map. Then the following are equivalent.(1) ζ has the UIP with respect to Pol( C ) over A .(2) for all f ∈ Pol( C ) (2) and u , u ∈ Aut( A ) , if f (id , u ) and f (id , u ) arecanonical with respect to A then ζ ( f (id , u )) = ζ ( f (id , u )) . Theorem 1.4 can be strengthened further. To formulate the strengthening, weintroduce a new notion and prove a fact that might be of independent interest inmodel theory; this fact will also we used in the proof of Theorem 1.2. Let D be astructure and let A, B ⊆ D . We say that A is independent from B in D if for all¯ a, ¯ b ∈ A , if typ D (¯ a ) = typ D (¯ b ), then typ D (¯ a/B ) = typ D (¯ b/B ). If A is independentfrom B in D and B is independent from A in D , then we say that A and B areindependent in D . In Section 4 we prove that every homogeneous ω -categoricalRamsey structure contains two independent elementary substructures. The ideaof the following Theorem is that it suffices to verify the UIP on such a pair ofsubstructures. Theorem 1.5.
Let A be a homogeneous Ramsey structure with finite relationalsignature and let C be a reduct of A . Let C be the clone of all polymorphisms of C that are canonical with respect to A and let ζ : C → Proj be an
Aut( A ) -invariantminor-preserving map. Then the following are equivalent.(1) ζ has the UIP with respect to Pol( C ) over A .(2) there is a pair ( A , A ) of independent elementary substructures of A suchthat for all f ∈ Pol( C ) (2) , u , u ∈ Aut( A ) with u ( A ) ⊆ A , u ( A ) ⊆ A ,if f (id , u ) and f (id , u ) are canonical with respect to A then ζ ( f (id , u )) = ζ ( f (id , u )) . We present three applications of these results to prove complexity dichotomies: • in the first application (Section 8) we prove that every structure with finiterelational signature and at most exponential labelled growth has a CSPwhich is in P or NP-complete. In forthcoming work we will prove such adichotomy even for the class of structures of labelled growth bounded by cn dn for constants c, d with d <
1; it can be shown that this is preciselyLachlan’s class of cellular structures ; see [Lac92, Bra20, BB19]. • In our second application (Section 9) we prove a new complexity dichotomyresult about the spatial reasoning formalism RCC-5, solving the essential
HE UNIQUE INTERPOLATION PROPERTY 5 part of Problem 4 in [BJ17]. In particular, we show that the CSP for first-order expansions of the basic relations in RCC-5 is in P if and only if it isin Datalog. • In our third application (Section 10) we re-derive the central step in a recentcomplexity classification for poset constraint satisfaction problems [KP18]using our general results.1.2.
Outline.
In Section 2 we introduce some fundamental notions and resultsfrom model theory, and in Section 3 some fundamental notions and results fromuniversal algebra. Section 4 contains new material about a consequence of theRamsey property of an ω -categorical structure A for the existence of independentelementary substructures of A ; this result might be of independent interest in modeltheory. In Section 5 we develop some theory of interpolation in clones modulo unaryoperations which is necessary to prove our main results. The first main result,Theorem 1.2, is proved in Section 6, and the second main result, Theorem 1.4, inSection 7. We close with applications to obtain complexity dichotomies for the classof structures with exponential labelled growth (Section 8), for first-order expansionsof the random poset (Section 9), and for first-order expansions of the basic relationsof RCC-5 (Section 10).2. Countably Categorical Structures
All the material recalled in this paragraph and in Section 2.1 is standard andcan for instance be found in [Hod93]. A structure A is called ω -categorical if theset of all first-order sentences that hold in A has exactly one countable model upto isomorphism. This concept has an equivalent characterisation based on theautomorphism group Aut( A ) of A . For l ∈ N , the l -orbit of a = ( a , . . . , a l ) ∈ A l in Aut( A ) is the set { α ( a ) : α ∈ Aut( A ) } where α ( a ) := ( αa , . . . , αa l ) . We write O l ( A ) for the set of l -orbits of Aut( A ). Engeler, Svenonius, and Ryll-Nardzewski proved that a countable structure is ω -categorical if and only if Aut( A )is oligomorphic , i.e., if O l ( A ) is finite for every l ∈ N . In fact, a, b ∈ A l have thesame l -orbit if and only if they satisfy the same first-order formulas over A ; we writetyp A ( a ) for the set of all first-order formulas satisfied by a over A , called the typeof a over A . Similarly, for B ⊆ A , we write typ A ( a/B ) for the set of all first-orderformulas with parameters from B satisfied by a over A .2.1. Homogeneous Structures.
Many examples of ω -categorical structures arisefrom structures A that are homogeneous . A relational structure is homogeneous ifevery isomorphism between finite substructures of A can be extended to an auto-morphism of A . It follows from the above that if a homogeneous structure A has afinite relational signature, then A is ω -categorical. If a ∈ A n , we write qf-typ A ( a )for the set of all quantifier-free formulas that hold on a over A . Clearly, in homo-geneous structures A the quantifier-free type of a determines the type of a . Everyhomogeneous structure A is uniquely given (up to isomorphism) by its age , i.e., bythe class Age( A ) of all finite structures that embeds into A . Conversely, every class C of structures with finite relational signature which is an amalgamation class , i.e., BERTALAN BODOR AND MANUEL BODIRSKY, AUGUST 25, 2020 is closed under isomorphism, substructures, and which has the amalgamation prop-erty , is the age of a homogeneous structure [Hod93], which we call the
Fra¨ıss´e-limitof C . The Fra¨ıss´e-limit of C embeds all countable structures whose age equals C .2.2. Model-complete cores. An ω -categorical structure C is model-complete ifevery self-embedding e : C ֒ → C preserves all first-order formulas; an ω -categoricalstructure C is called a core if every endomorphism of C is a self-embedding of C . Ifthere is a homomorphism from a structure A to a structure B and vice versa, then A and B are called homomorphically equivalent . Every ω -categorical structureis homomorphically equivalent to a model-complete core, which is unique up toisomorphism, and again ω -categorical, and which will be called the model-completecore of C [Bod07, BHM12].2.3. Powers.
Three notions of powers of relational τ -structures A play a role inthis article. The first is the notion of the k -th direct power of A (also called the categorical power ), denoted by A k . It is the τ -structure with domain A k suchthat (( a , , . . . , a ,k ) , . . . , ( a n, , . . . , a n,k )) ∈ R A k if and only if ( a i, , . . . , a i,n ) ∈ R A for every i ∈ { , . . . , k } . The k -th algebraic power of A , denoted by A ( k ) ,is the expansion of A k by the relations E , . . . , E k where E i denotes the relation { (( a , . . . , a k ) , ( b , . . . , b k )) : a i = b i } . The k -th full power of A , denoted by A [ k ] , isthe expansion of A k by the relations E i,j , for i, j ∈ { , . . . , k } , where E i,j denotesthe relation { (( a , . . . , a k ) , ( b , . . . , b k )) : a i = b j } . Note that a map α : A k → A k is • an automorphism of A ( k ) if and only if there are α , . . . , α k ∈ Aut( A )such that α ( a , . . . , a k ) = (cid:0) α ( a ) , . . . , α k ( a k ) (cid:1) for all a , . . . , a k ∈ A , i.e.,Aut( A ( k ) ) = Aut( A ) k . • an automorphism of A [ k ] if and only if there exists α ′ ∈ Aut( A ) such that α ( a , . . . , a k ) = (cid:0) α ′ ( a ) , . . . , α ′ ( a k ) (cid:1) for all a , . . . , a k ∈ A .2.4. Ramsey structures.
For τ -structures A , B we write (cid:0) BA (cid:1) for the set of allembeddings of A into B . If A , B , C are τ -structures and r ∈ N then we write C → ( B ) A r if for every function χ : (cid:0) CA (cid:1) → { , . . . , r } (also referred to as a colouring , where { , . . . , r } are the different colours) there exists e ∈ (cid:0) CB (cid:1) such that χ is constant on (cid:26) e ◦ f : f ∈ (cid:18) BA (cid:19)(cid:27) ⊆ (cid:18) CA (cid:19) . Definition 2.1.
A class of finite τ -structures has the Ramsey property if for all A , B and r ∈ N there exists C such that C → ( B ) A r . A homogeneous structure hasthe Ramsey property if its age has the Ramsey property.The Ramsey property appears to be quite strong; however, note that everyhomogeneous structure with a finite relational signature known to the authors hasa homogeneous expansion with a finite relational signature which is additionallyRamsey [Bod15]. The Ramsey property has the following well-known consequence(see, e.g., Proposition 2.21 in [Bod15]). Let r ∈ N . We write C → ( B ) r if for every χ : C | B | → { , . . . , r } there exists e : B ֒ → C such that if qf-typ B ( s ) = qf-typ B ( t )for r, s ∈ B | B | , then χ ( e ( s )) = χ ( e ( t )). Lemma 2.2.
Let C be a Ramsey class. Then for every r ∈ N and B ∈ C thereexists C ∈ C such that C → ( B ) r . HE UNIQUE INTERPOLATION PROPERTY 7
An important consequence of the Ramsey property will be presented in Sec-tion 3.5; another consequence, which seems to be new, can be found in Section 4.3.
Oligomorphic Clones
In this section we introduce fundamental concepts and results about polymor-phism clones of finite and countably infinite ω -categorical structures. The resultsin Sections 3.1, 3.2, 3.3, and 3.4 are standard. The concepts and results from Sec-tion 3.6 (Rich Subsets) and Section 3.7 (Invariance) are mostly easy or follow fromthe literature; but they have not been presented in this form in the literature before.A polymorphism of a structure A is a homomorphism from a finite direct power A k to A . The set of all polymorphisms of A , denoted by Pol( A ), forms a clone(over A ) , i.e., it is closed under composition and contains for every k all the k -aryprojections, which will always be denoted by π k , . . . , π kk . The clone of all projectionson the set { , } is denoted by Proj. A function ζ : C → C between two clones iscalled minor-preserving if ζ (cid:0) f ( π k , . . . , π kn ) (cid:1) = ζ ( f )( π k , . . . , π kn )for all n -ary functions f ∈ C and k -ary projections π k , . . . , π kk . The set of alloperations on A , O A := [ k ∈ N ( A k → A ) , can be equipped with a complete metric such that the closed clones with respectto this metric are precisely the polymorphism clones of relational structures withdomain A ; when we write that a map between clones is uniformly continuous thenthis is meant with respect to this metric.3.1. Siggers polymorphisms.
An operation s : A → A is called a Siggers oper-ation [Sig10] if it satisfies s ( x, x, y, y, z, z ) = s ( y, z, x, z, x, y )for all x, y, z ∈ A . An operation f : A k → A is called • idempotent if f ( x, . . . , x ) = x for every x ∈ A . • cyclic if f ( x , . . . , x k ) = f ( x , . . . , x k , x ) for all x , . . . , x k ∈ A .If f : A k → A and H , . . . , H k ⊆ A , then we use the set-notation f ( H , . . . , H k ) := { f ( a , . . . , a k ) : a ∈ H , . . . , a k ∈ H k } . Theorem 3.1 (Combining [Sig10, BK12, BOP18, BJ01]) . Let B be a structurewith a finite domain. Then the following are equivalent. • B has no Siggers polymorphism. • B has no cyclic polymorphism. • Pol( B ) has a minor-preserving map to Proj .If furthermore all polymorphisms of B are idempotent, then the two statements arealso equivalent to • there are non-empty disjoint H , H ⊆ B such that for every f ∈ Pol( B ) ( k ) there exists j ∈ { , . . . , k } such that for all i , . . . , i k ∈ { , } f ( H i , . . . , H i k ) ⊆ H i j . BERTALAN BODOR AND MANUEL BODIRSKY, AUGUST 25, 2020
Now suppose that B is a finite structure with a finite relational signature. If theconditions given in Theorem 3.1 apply, then CSP( B ) is NP-hard [BKJ05, Sig10].On the other hand, if the conditions in Theorem 3.1 do not apply, then CSP( B ) isin P, providing a positive answer to the Feder-Vardi dichotomy conjecture [FV99]. Theorem 3.2 (Bulatov [Bul17], Zhuk [Zhu17]) . If B has a Siggers polymorphism,then CSP( B ) is in P. Pseudo-Siggers polymorphisms. A pseudo-Siggers polymorphism of astructure B is a polymorphism of B of arity 6 such that there exist endomorphisms e , e of B such that e (cid:0) s ( x, x, y, y, z, z ) (cid:1) = e (cid:0) s ( y, z, x, z, x, y ) (cid:1) for all x, y, z ∈ B . Theorem 3.1 has a generalisation to ω -categorical struc-tures [BP16a]. If we even assume that the growth is less than doubly exponential,this generalisation has a stronger formulation, which will be important in this arti-cle. The (orbit) growth is the function l
7→ |O l ( A ) | ; we say that the growth is lessthan doubly exponential if it is smaller than 2 n −
1. The following theorem canreadily be deduced from the literature; see the references given in the proof belowfor details.
Theorem 3.3.
Suppose that B is an ω -categorical model-complete core such that Aut( B ) has less than doubly exponential growth. Then exactly one of the followingtwo cases applies. • B has a pseudo-Siggers polymorphism. • Pol( B ) has a uniformly continuous minor-preserving map to Proj .Proof.
The main result from [BP16a] states that either B has a pseudo-Siggerspolymorphism, or there are finitely many constants c , . . . , c n ∈ B such thatPol( B , c , . . . , c n ) has a continuous clone homomorphism to Proj. This in turnis equivalent to the existence of a uniformly continuous minor-preserving map fromPol( B ) to Proj: this follows from one of the main results from [BKO +
17] becauseof the assumption that Aut( B ) has less than doubly exponential growth. (cid:3) As in the finite, it is known that if B is an ω -categorical structure such thatPol( B ) does not have a uniformly continuous minor-preserving map to Proj, thenCSP( B ) is NP-hard. The present paper is motivated by the following conjec-ture from [BPP19], in a reformulation from [BKO + Conjecture 3.4.
Let B be a reduct of a finitely bounded homogeneous structuresuch that Pol( B ) has no uniformly continuous minor-preserving map to Proj . Then
CSP( B ) is in P. Behaviours.
Let B , C be structures. In this section we discuss the conceptof a behaviour of a function f : B → C with respect to B and C , mostly follow-ing [BP11, BP16b]. In order to have some flexibility when using the terminology,we work in the setting of partial functions from B → C , i.e., functions to C thatare only defined on some subset of B . Definition 3.5 (Behaviours) . Let n ∈ N . Then an n -behaviour over ( A , B ) is apartial function from O n ( A ) to O n ( B ). A behaviour over ( A , B ) is a partial function B from S ∞ i = n O n ( A ) to S ∞ n =1 O n ( B ) so that for every n we have B ( O n ( A )) ⊆ O n ( B ). HE UNIQUE INTERPOLATION PROPERTY 9 An n -behaviour (behaviour) is called complete if it is defined on all of O n ( A ) (onall of S ∞ i = n O n ( A )).Note that every n -behaviour is in particular a behaviour. Definition 3.6 (Behaviours of functions) . Let B and C be two structures. A partialfunction f : B → C realises a behaviour B over ( B , C ) if for all b , . . . , b n ∈ B , if O is the orbit of ( b , . . . , b n ) then B ( O ) is the orbit of ( f ( b ) , . . . , f ( b n )).If A ⊆ B and B is a behaviour over ( B , C ), then we say that f : B → C realises B on A if the restriction f | A realises B . The following proposition can be shownby a standard compactness argument. Proposition 3.7.
Let A be an ω -categorical structure and k, n ∈ N . Then an n -behaviour over A is realised by some f ∈ Pol( A ) ( k ) if and only if it is realised bysome f ∈ Pol( A ) ( k ) on X k for every finite subset X ⊆ A . We leave the proof of this fact to the reader; a similar compactness argumentis presented in full detail for Proposition 4.2 below. Also the following propositioncan again be shown by a compactness argument.
Lemma 3.8.
Let B be a first-order reduct of an ω -categorical structure A . Let k, n, m ∈ N and let B , . . . , B m be n -behaviours over ( A ( k ) , A ) . Suppose that forevery finite F ⊆ A and i ∈ { , . . . , m } there exist α i, , . . . , α i,k and f ∈ Pol( B ) suchthat f ( α i, , . . . , α i,k ) realises B i on F k . Then there exist e , , . . . , e m,k ∈ Aut( A ) and f ∈ Pol( B ) such that for every i ∈ { , . . . , m } the operation f ( e i, , . . . , e i,k ) realises B i on all of A k . Moreover, if I ⊆ { , . . . , k } and α i,j = id for all j ∈ I andevery finite X ⊆ A then we can guarantee e i,j = id for all j ∈ I . Canonicity.
A partial function f : B → C is called n -canonical over ( B , C )if it realises some complete n -behaviour over ( B , C ), and canonical over ( B , C ) if itrealises some complete behaviour over ( B , C ). If A ⊆ B , then we say that f : B → C is ( n -) canonical on A if the restriction f | A is ( n -) canonical.Suppose that B is a homogeneous relational structure with maximal arity m . If f : B → C realises some complete m ′ -behaviour over ( B , C ) for m ′ = max( m, f is canonical over ( B , C ) (in other words, in this situation a complete m -behaviour uniquely determines a complete behaviour; the requirement m ′ ≥ m ′ -behaviour allows, for instance, to distinguish constantfunctions from injective functions). Hence, if B is homogeneous with finite rela-tional signature, then there are only finitely many complete behaviours.We can apply the concept of canonical functions to polymorphisms of a structure A by choosing B := A ( k ) for k ∈ N and C := A . The set of all operations f : A k → A that are canonical over ( A ( k ) , A ) is denoted by Can( A ). Note that Can( A ) is a cloneand closed in O A . If f ∈ Can( A ) then we also say that f is canonical over A andcorrespondingly we also use the terminology of n -behaviour over A and behaviourover A . Note that for every n ∈ N , the binary relation { ( u, α ( u )) : u ∈ A n , α ∈ Aut( A ) } is a congruence of Can( A ), and so there is a uniformly continuous clone homo-morphism ξ A n from Can( A ) to a clone whose domain O n ( A ) is the (finite) set ofcongruence classes, and which we denote by ξ A n (Can( A )). If C ⊆ Can( A ), then wewrite ξ A n ( C ) for the subclone of ξ (Can( A )) induced by the image of C under ξ A n . Note that if f ∈ Can( A ), then g : A k → A has the same behaviour as f over A ifand only if f interpolates g and g interpolates f modulo A .Another compactness argument can be used to show the following. Lemma 3.9 (Proposition 6.6 in [BPP19]) . Let A be a homogeneous structure witha finite relational signature. Let m be the maximal arity of A and suppose that f, g ∈ Can( A ) are such that ξ A m ( f ) = ξ A m ( g ) . Then there are e , e ∈ Aut( A ) suchthat e ◦ f = e ◦ g . Canonisation.
For a proof of the following lemma, see [BP16c]; it can alsobe shown using Lemma 2.2 and a compactness argument.
Lemma 3.10 (Canonisation lemma) . Let A be a homogeneous Ramsey structure,let B be ω -categorical, and let f : A → B . Then { α ◦ f ◦ β : α ∈ Aut( A ) , β ∈ Aut( B ) } contains a function which is canonical over ( A , B ) . Some of the results where we assume that A is Ramsey in fact only use that A hasthe so-called canonisation property , which states that every f : A k → A interpolatesmodulo Aut( A ) an operation g ∈ Can( A ). Corollary 3.11.
Let C be a homogeneous ω -categorical Ramsey structure. Then C has the canonisation property.Proof. Apply Lemma 3.10 to the algebraic power B := C ( k ) (Section 2.3). (cid:3) In some situations it seems most practical to use a finite version of the canoni-sation lemma (e.g. in the proof of Lemma 5.2 or Lemma 7.1), which can be derivedeasily from Corollary 3.11 by a compactness argument, or directly from a proof ofLemma 3.10 based on Lemma 2.2.
Lemma 3.12.
Let A be a homogeneous ω -categorical Ramsey structure and B ⊆ A finite. Then there exists a finite C ⊆ A such that for every f : A k → A there are α , . . . , α k ∈ Aut( A ) such that α ( B ) ⊆ C, . . . , α k ( B ) ⊆ C and f ( α , . . . , α k ) iscanonical on B k . Rich subsets.
To prove the existence of canonical functions it is useful tointroduce a notion of rich substructures of a structure.
Definition 3.13.
Let A be a countable homogeneous structure with finite relationalsignature and let m be the maximal arity of A . Let C ⊆ O A be a closed clonecontaining Aut( A ) and let k ∈ N . Then we say that X ⊆ A is k -rich with respectto C if • every m -orbit of A ( k ) contains a tuple from ( X k ) m ; • every behaviour over A which is realised on X k by some operation in C isalso realised on A k by some operation in C .Note that if X is k -rich then every behaviour which is realisable on X k is com-plete. It is clear from the definition that if X ⊆ A is k -rich, and X ⊆ Y , then Y is also k -rich. We present an example that satisfies the first, but not the secondcondition in Definition 3.13. HE UNIQUE INTERPOLATION PROPERTY 11
Example 3.14.
Let A be the countable homogeneous undirected graph whose ageconsists of all finite graphs that do not embed K (the homogeneous triangle-freegraph, also called Henson graph ). Let X ⊆ A be a set of cardinality three thatcontains precisely one edge. Note that every 3-element X ⊆ A that contains anedge and a non-edge is such that for every k ∈ N every orbit of pairs in A ( k ) intersects a tuple from ( X k ) . However, X is not even 2-rich. To show this, wespecify a 2-behaviour which is realised on X by some operation in C , but notrealised on A by some operation in C . Let E be the orbit of some (equivalently,any) pair of adjacent vertices, and let N be the orbit of some pair of non-adjacentvertices in Aut( A ). Finally, we write ≡ for the orbit of all pairs of the form ( a, a )for a ∈ A . We order all the orbits of pairs in Aut( A ) as ≡ < N < E . Let B bethe 2-behaviour which maps ( O , O ), for O , O ∈ {≡ , N, E } , to max( O , O ) withrespect to the above order. Since A embeds all countable graphs that do not embed K , it is easy to see that B is realised on X by some operation from C . On the otherhand, B is not realised on every set X = { a, b, c } which induces precisely two edges (cid:8) { a, b } , { b, c } (cid:9) , because B would force a copy of K on { f ( a, c ) , f ( b, a ) , f ( c, b ) } . △ Lemma 3.15.
Let A be a countable homogeneous structure with a finite relationalsignature. Let C be a closed clone containing Aut( A ) and let k ∈ N . Then thereexists a finite X ⊆ A which is k -rich.Proof. Let m be the maximal arity of the relations of A . The structure A ( k ) is ω -categorical and hence has finitely many m -orbits. Therefore, we can choose a finitesubset X of A such that every orbit of m -tuples in Aut( A ( k ) ) intersects ( X k ) m .Let X ⊂ X ⊂ · · · be finite subsets of A such that S ∞ i =1 X i = A . Let B i denotethe (finite) set of all m -behaviours which are realised on X ki by some operation of C . Then B ⊇ B ⊇ · · · and hence there exists an N such that B N = B N +1 = · · · .Then Proposition 3.7 implies that every behaviour in B N is realised on A k by someoperation in C . This proves that X N is k -rich: the first item is satisfied because X ⊆ X N , and the second item because every behaviour that is realised in X iN isalso realised on A k . (cid:3) Definition 3.16.
Let A and C be as in Definition 3.13, let η be a function definedon C ∩ Can( A ), let F be k -rich for some k ≥
1, and let f ∈ C ( k ) . Suppose that f has on F k the same complete behaviour as g ∈ Can( A ). Then we will also write η ( f | F k ) for η ( g ).3.7. Invariance.
Let C be an operation clone and M ⊆ C (1) a transformationmonoid. We say that a function ζ defined on C is M -invariant if for all f ∈ C ( k ) and u, v , . . . , v k ∈ M we have ζ ( f ) = ζ ( u ( f ( v , . . . , v k ))) . Lemma 3.17.
Let A be a homogeneous in a finite relational signature and let C bea subclone of Can( A ) . Let ζ : C → Proj be Aut( A ) -invariant. Then ζ is uniformlycontinuous.Proof. Let m be the maximal arity of A , and let X ⊆ A be finite so that everyorbit of m -tuples contains a witness in X m . Let k ∈ N and f, g ∈ C ( k ) be suchthat f | X k = g | X k . Then ζ ( f ) = ζ ( g ) by the choice of X . By Lemma 3.9, there are e , e ∈ Aut( A ) such that e ◦ f = e ◦ g . The Aut( A )-invariance of ζ implies that ζ ( f ) = ζ ( e ◦ f ) = ζ ( e ◦ g ) = ζ ( g ), proving the uniform continuity of ζ . (cid:3) The following can be obtained by combining known results in the literature.
Proposition 3.18.
Let B be homogeneous in a finite relational signature withmaximal arity m ∈ N . Let C be a first-order reduct of B which is a model-completecore. Suppose that C := Pol( C ) ⊆ Can( B ) . Then the following are equivalent.(1) There is no uniformly continuous minor-preserving map from C → Proj .(2) There is no
Aut( B ) -invariant minor-preserving map C → Proj .(3) ξ B m ( C ) has no minor-preserving map to Proj .(4) ξ B m ( C ) has a cyclic operation.(5) ξ B m ( C ) has a Siggers operation.(6) C has a pseudo-Siggers operation.Proof. (1) ⇒ (2): An immediate consequence of Lemma 3.17.(2) ⇒ (3): If ξ A m ( C ) has a minor-preserving map to Proj, then we can composeit with ξ A m and obtain a Aut( B )-invariant minor-preserving map from C to Proj,and (2) does not hold.(3) ⇒ (4) and (4) ⇒ (5) follow from Theorem 3.1.(5) ⇒ (6) is a consequence of Lemma 3.9.(6) ⇒ (1). Since A is homogeneous in a finite relational signature, it has less thandoubly exponential growth, and the same applies to B . Therefore, the statementfollows from Theorem 3.3. (cid:3) Independence
We now present a consequence of the Ramsey property for A which concerns theexistence of large independent sets in A and is relevant for verifying the UIP. Definition 4.1.
Let D be a structure and let A, B ⊆ D . We say that A is independent from B in D if for all ¯ a, ¯ b ∈ A , if typ D (¯ a ) = typ D (¯ b ), thentyp D (¯ a/B ) = typ D (¯ b/B ). If the reference to D is clear we might drop it. If A is independent from B and B is independent from A , then we say that A and B are independent . Two substructures of D are called independent if their domainsare independent.A substructure of a τ -structure A is called elementary if the identity mappingpreserves all first-order τ -formulas. We will be interested in the existence of inde-pendent elementary substructures of homogeneous ω -categorical structures. Notethat every ω -categorical structure can be turned into a homogeneous ω -categoricalstructure by expanding it by all first-order definable relations. The following propo-sition can be shown by an easy compactness argument which we present for com-pleteness. Proposition 4.2.
Let A be a countable ω -categorical homogeneous structure. Thenthe following are equivalent.(1) A contains two independent elementary substructures.(2) For all B , C ∈ Age( A ) there exists D ∈ Age( A ) and embeddings e : B ֒ → D and f : C ֒ → D such that e ( B ) and f ( C ) are independent in A .Proof. The forward implication is immediate from the definitions. For the converseimplication, let a , a , . . . be an enumeration of A and let B = { b , b , . . . } and C = { c , c , . . . } be sets of new constant symbols. Let Φ be the set of all sentencesthat express that for all n ∈ N , ¯ b, ¯ b ′ ∈ B n , and ¯ c, ¯ c ′ ∈ C n HE UNIQUE INTERPOLATION PROPERTY 13 • if typ A (¯ b ) = typ A (¯ b ′ ) then typ A (¯ b, ¯ c ) = typ A (¯ b ′ , ¯ c ). • if typ A (¯ c ) = typ A (¯ c ′ ) then typ A (¯ c, ¯ b ) = typ A (¯ c ′ , ¯ b ).For n ∈ N let φ n ( x , . . . , x n ) be a formula that expresses that typ A ( x , . . . , x n ) =typ B ( a , . . . , a n ). By assumption, all finite subsets of T := Th( A ) ∪ Φ ∪ { φ n ( b , . . . , b n ) ∧ φ n ( c , . . . , c n ) : n ∈ N } are satisfiable. By compactness of first-order logic, it follows that T has a model A ′ .By the downward L¨owenheim-Skolem theorem, we may assume that A ′ is countablyinfinite. The reduct of A ′ with the same signature as A satisfies Th( A ), and since A is ω -categorical this reduct is isomorphic to A , so that we may assume that A ′ isan expansion of A . Let B be the substructure of A induced by the constants from B . Since A ′ | = { φ n ( b , . . . , b n ) : n ∈ N } and by the homogeneity of A we have that B is an elementary substructure of A . Likewise, the substructure C of A inducedby the constants from C is an elementary substructure of A . Since A ′ | = Φ the twosubstructures B and C are independent. (cid:3) To find independent sets, we use the Ramsey property via the following lemma.
Lemma 4.3.
Let A be a countable homogeneous ω -categorial Ramsey structure, C ⊆ A finite, and B ∈ Age( A ) . Then there exists D ∈ Age( A ) such that for every e : D ֒ → A there exists f : B ֒ → A such that f ( B ) ⊆ e ( D ) and for all a, b ∈ B | B | , if typ A ( f ( a )) = typ A ( f ( b )) then typ A ( f ( a ) /C ) = typ A ( f ( b ) /C ) .Proof. Let C = { c , . . . , c n } . Let r be the number of types of | B | + n -tuples in A . Since A is homogeneous Ramsey there exists D ∈ Age( A ) such that D → ( C ) r .We claim that D satisfies the statement of the lemma. Let e : D ֒ → A be anembedding. We color e ( D ) | B | as follows: the color of t ∈ e ( D ) | B | is the orbit of( t , . . . , t | B | , c , . . . , c n ). There are at most r such orbits. Since D → ( C ) r we getthat there exists an f : B ֒ → A such that f ( B ) ⊆ e ( D ) and if u, v ∈ B | B | are suchthat typ A ( f ( u )) = typ A ( f ( v )), then typ A ( f ( u ) /C ) = typ A ( f ( v ) /C ). (cid:3) We are not aware of a reference for the following theorem and believe that it ofindependent interest in the theory of ω -categorical structures. Theorem 4.4.
Let A be a countable homogeneous ω -categorical Ramsey structure.Then A contains two independent elementary substructures.Proof. We use Proposition 4.2. Let B , B ∈ Age( A ); we may assume that B and B are substructures of A . Then Lemma 4.3 implies that there exists D ∈ Age( A )such that for every embedding e : D → A there exists f : B → A such that f ( B ) ⊆ e ( D ) and for all n ∈ N , b, b ′ ∈ ( B ) n if typ A ( f ( b )) = typ A ( f ( b ′ )) then typ A ( f ( b ) /B ) = typ A ( f ( b ′ ) /B ) . (1)We may assume that D is a substructure of A . Another application of Lemma 4.3gives us an embedding f : B ֒ → A such that for all n ∈ N , b, b ′ ∈ ( B ) n , iftyp A ( f ( b )) = typ A ( f ( b ′ )) then typ A ( f ( b ) /D ) = typ A ( f ( b ′ ) /D ). By the homo-geneity of A , there exists α ∈ Aut( A ) extending f . The property of D impliesthat there exists f : B → A such that f ( B ) ⊆ α − ( D ) such that (1) holds forall b, b ′ ∈ ( B ) n .We claim that f ( B ) and B are independent. Clearly, f ( B ) is independentfrom B . To show that B is independent from f ( B ), let n ∈ N and b, b ′ ∈ ( B ) n be such that typ A ( b ) = typ A ( b ′ ). Then typ A ( α ( b )) = typ A ( α ( b ′ )), and hence that typ A ( α ( b ) /D ) = typ A ( α ( b ′ ) /D ) by the choice of f . This in turn implies thattyp A ( b/α − ( D )) = typ A ( b ′ /α − ( D )) which implies the statement since f ( B ) ⊆ α − ( D ). (cid:3) Interpolation
Interpolation modulo a transformation monoid (such as the endomorphismmonoid or the automorphism group of a given structure), defined in the beginningof Section 1.1, already appeared in the canonisation lemma 3.10. In this section wepresent some new general results in the connection with proving uniform continuityof maps defined on polymorphism clones (Section 5.1), and about the variant of diagonal interpolation that will play an important role in the context of the uniqueinterpolation property (Definition 1.1).5.1.
Interpolation invariance.
It has been shown in [BP15] (see Theorem 6.4in [BOP18]) that if C ⊆ O B is a closed clone and G an oligomorphic permutationgroup over the same base set B , then every G -invariant continuous map ζ definedon C is uniformly continuous. In Lemma 5.2 we present a sufficient condition foruniform continuity which does not require that the map ζ we start from is contin-uous, for the special case that G = Aut( C ) for a homogeneous Ramsey structure C with finite relational signature. Definition 5.1.
Let C be a closed operation clone and M be a transformationmonoid over the same base set B . A function ζ defined on C is called interpolationinvariant modulo M if ζ ( f ) = ζ ( g ) whenever f ∈ C interpolates g ∈ C modulo M .Clearly, interpolation invariance modulo M implies M -invariance from Sec-tion 3.7. Lemma 5.2.
Let B be a first-order reduct of a homogeneous Ramsey structure A with finite relational signature. Let ζ be a function defined on Pol( B ) which isinterpolation invariant over A . Then ζ is uniformly continuous.Proof. Let m be the maximal arity of the relations in A . Suppose for contradictionthat ζ is not uniformly continuous. Then there exists a k ∈ N so that for everyfinite X ⊆ A there exist f X , g X ∈ Pol( B ) such that ( f X ) | X k = ( g X ) | X k , but ζ ( f X ) = ζ ( g X ). By Lemma 3.15, there exists a finite A ⊆ A which is k -rich withrespect to Pol( B ). Claim.
For every finite F ⊆ A that contains A there exists an h ∈ Pol( B ) ( k ) and h ′ ∈ Pol( B ) ( k ) ∩ Can( A ) such that h and h ′ have the same complete behaviouron F k and ζ ( h ) = ζ ( h ′ ). By Lemma 3.12 there exists a finite C ⊆ A such that forevery h : A k → A there are α , . . . , α k ∈ Aut( A ) such that α ( F ) ⊆ C, . . . , α k ( F ) ⊆ C and h ( α , . . . , α k ) is canonical on F k . In particular, this holds for the operations f C , g C introduced above, so that f ′ := f C ( α , . . . , α k ) and g ′ := g C ( α , . . . , α k ) arecanonical on F k . We have ( f C ) | C k = ( g C ) | C k and ζ ( f C ) = ζ ( g C ), and ζ ( f C ) = ζ ( f ′ )and ζ ( g C ) = ζ ( g ′ ) because f C interpolates f ′ and g C interpolates g ′ modulo Aut( A ).Since A ⊆ F is k -rich there exist h ′ ∈ Pol ( k ) ( B ) ∩ Can( A ) such that h ′ has the samecomplete behaviour as f ′ on F k , and therefore also the same complete behaviouras g ′ on F k . Hence, we must have ζ ( h ′ ) = ζ ( f ′ ) or ζ ( h ′ ) = ζ ( g ′ ), and hence either( f ′ , h ′ ) or ( g ′ , h ′ ) provide us witnesses for the claim.Let A ⊆ A ⊆ · · · be finite such that S l ∈ N A l = A . By the claim above, for each l ∈ N we can find a function h ∈ Pol( B ) and h ′ ∈ Pol( B ) ( k ) ∩ Can( A ) such that HE UNIQUE INTERPOLATION PROPERTY 15 h and h ′ have the same m -behaviour H l on F k and ζ ( h ) = ζ ( h ′ ). By Lemma 3.10the operation h interpolates an operation g with a complete behaviour B l over A ,and ζ ( g ) = ζ ( h ) = ζ ( h ′ ). Since there are finitely many m -behaviours over A , thereexists ( H , B ) such that ( H l , B l ) = ( H , B ) for infinitely many l ∈ N .Lemma 3.8 shows that there exists f ∈ Pol( B ) and e , e ′ , . . . , e k , e ′ k ∈ Aut( A )such that f ◦ ( e i , . . . , e k ) has behaviour H and f ◦ ( e ′ i , . . . , e ′ k ) has behaviour B .Hence, f interpolates two functions that take different values under ζ , contradictinginterpolation invariance of ζ . (cid:3) Diagonal interpolation.
In our proofs we need the concept of diagonal in-terpolation , which is a more restricted form of interpolation.
Definition 5.3.
Let M be a transformation monoid over the base set B . We saythat f : B k → B diagonally interpolates g : B k → B modulo M if g ∈ { u ( f ( v, . . . , v )) : u, v ∈ M } . We mention that diagonal interpolation preserves pseudo identities (e.g., if f is apseudo-Siggers polymorphism of B and diagonally interpolates g modulo End( B ),then g is a pseudo-Siggers polymorphism of B , too). This fact will not be neededhere; instead, we show here that in a certain sense diagonal interpolation is alsowell-behaved with respect to minor identities, too. Lemma 5.4.
Let M be a transformation monoid over the base set B , let f : B k → B be an operation, and let π = ( π ni , . . . , π ni k ) be a vector of k projections, each ofarity n . Suppose that f diagonally interpolates g modulo M . Then f ◦ π diagonallyinterpolates g ◦ π modulo M .Proof. By assumption we have that g ∈ { u ( f ( v, . . . , v )) : u, v ∈ M } . Then g ◦ π ∈ { u ( f ( v, . . . , v )) ◦ π : u, v ∈ M } (composition with π is continuous)= { u ( f ◦ π )( v, . . . , v ) : u, v ∈ M } which means that f ◦ π diagonally interpolates g ◦ π modulo M . (cid:3) We want to stress that the innocent-looking Lemma 5.4 fails for interpolationinstead of canonical interpolation, as the following example shows. An operation f : A k → A is called diagonally canonical over A if it is canonical over ( A [ k ] , A ) (i.e.,we use a full power of A instead of an algebraic power, see Section 2.3). Example 5.5.
Let A be a structure with countable domain and two disjoint infiniteunary relations U and U . Let f : A → A be an injective function such that forevery i ∈ { , }• f ( a, b, c ) ∈ U i if a ∈ U i and b = c , • f ( a, b, c ) ∈ U i if b ∈ U i and b = c .Note that f is diagonally canonical but not canonical. Let e , e be two self-embeddings of A with disjoint images and let g : A → A be given by g ( x, y, z ) := f ( x, e ( y ) , e ( z )) . Then g is injective and for each i ∈ { , } we have g ( U i , A, A ) ⊆ U i . Hence, g iscanonical and ξ ( g ) = π . We claim that f ′ := f ◦ ( π , π , π ) does not interpolate g ′ := g ◦ ( π , π , π ). For each i ∈ { , } we have f ′ ( A, U i ) ⊆ U i , so f ′ is canonicaland ξ ( f ′ ) = π . On the other hand ξ ( g ′ ) = ξ ( g )( π , π , π ) = π ( π , π , π ) = π which proves the claim. △ Now let A be a homogeneous ω -categorical Ramsey structure. The canonisationlemma (Lemma 3.10) applied to the structure A [ k ] shows that every operation f : A k → A diagonally interpolates an operation that is diagonally canonical over A . Of course, we can in general not assume that such a map is canonical over A .However, we have the following amazing lemma about diagonal interpolation andcanonicity. Lemma 5.6.
Let B be a reduct of a countable homogeneous Ramsey structure A with finite relational signature. Let f ∈ Pol( B ) . Then there exists g ∈ Pol( B ) thatdiagonally interpolates both f and an operation in Can( A ) .Proof. Let k be the arity of f . We first show a local version of the statement, andthen derive the statement by a compactness argument. The local version is that forevery finite X ⊆ A there exists g ∈ Pol( B ) and e ∈ Aut( A ) such that g ( e, . . . , e ) iscanonical on X k over B and g agrees with f on X k .Let c = ( c , . . . , c n ) be such that { c , . . . , c n } = X . By Theorem 4.4 thereare two independent elementary substructures C and C of A . Since A is ω -categorical, for i ∈ { , } there exists an isomorphism e i from A to C i ; by thehomogeneity of A we have e i ∈ Aut( A ). The restriction of e to X can be extendedto an automorphism δ ∈ Aut( A ). Then δ − ( C ) and δ − ( C ) induce independentelementary substructures of A , and X ⊆ δ − ( C ). So we may assume without lossof generality that X ⊆ C .Since A is Ramsey, f ( e, . . . , e ) interpolates a canonical function modulo Aut( A );so there are δ , . . . , δ k ∈ Aut( A ) such that f ( e ◦ δ , . . . , e ◦ δ k ) is canonical on X k .In other words, for Y i := e ◦ δ ( X ), the map f is canonical on Y × · · · × Y k over A .Since X ⊆ C and Y i ⊆ C , the tuples e ◦ δ ( c ) , . . . , e ◦ δ k ( c ) all have the same typeover c . Hence, there exist α i ∈ Aut( A ) such that α i ( Y ) = Y i and α ( c ) = c . Define g := f ( α , . . . , α k ). Then f and g agree on X k . We claim that h := g ( eδ , . . . , eδ )is canonical on X k over A . This follows from f being canonical on Y × · · · × Y k = α ( Y ) × · · · × α k ( Y ) = α eδ ( X ) × · · · α k eδ k ( X )since h equals f ( α eδ , . . . , α k eδ k ) on X k .To show how the local version implies the statement of the lemma, let σ be thesignature of B and let C be an expansion of B by countably many constants suchthat every element of C is named by a constant symbol.Let ρ be the signature of C together with a new a k -ary function symbol g .Consider the ρ -theory T consisting of the union of the following first-order sentences:(1) Th( C );(2) A first-order sentence which asserts that g preserves all relations from σ ;(3) For all l ∈ N and constant symbols c , c , . . . , c k ∈ ρ such that f ( c C , . . . , c C k ) = c C the sentence g ( c , . . . , c k ) = c .(4) for all constant symbols c , . . . , c l ∈ σ the sentence ψ c ,...,c l which expressesthat there exist y , . . . , y l such that typ A ( y , . . . , y l ) = typ A ( c , . . . , c l ) and g is canonical on { y , . . . , y l } k . HE UNIQUE INTERPOLATION PROPERTY 17
It follows from the basic facts about ω -categorical structures from Section 2 thatsuch sentences exist. Claim.
Every finite subset S of T has a model. Let X be the (finite) set ofall constants c such that the respective constant symbol appears in a sentence of S or in { c , . . . , c l } for some ψ c ,...,c l ∈ S . Let d , . . . , d n be some enumeration of X .Then ψ d ,...,d n implies every sentence in S ′ from item (4). By the local version ofthe statement there exists g ∈ Pol( B ) and e ∈ Aut( A ) such that • g ( e, . . . , e ) is canonical on X k over A , and • g agrees with f on X k .Let D be the ρ -expansion of C where g D := g . Then D satisfies S . This is clear forthe sentences from item (1), (2), and (3). Finally, since g ( e, . . . , e ) is canonical on X k over A the sentences of S ′ from item (4) can be satisfied by setting y i := e ( c i ).By the compactness theorem of first-order logic, T has a model D . By thedownward L¨owenheim-Skolem theorem we may also assume that D is countable.Then the σ -reduct of D satisfies Th( B ) and hence is isomorphic to B by the ω -categoricity of B . We may therefore identify the element of D with the elementsof B along this isomorphism and henceforth assume that D is an expansion of A .Because of the sentences under (2) we have that g D is a polymorphism of B . Themap e : B → B given by c C c D is an elementary self-embedding. The sentencesunder (3) imply that g ( e, . . . , e ) equals e ( f ), so g diagonally interpolates f . Finally,the sentences under (4) imply that g diagonally interpolates some function fromCan( B ). (cid:3) The Unique Interpolation Property
The main result of this section, Lemma 6.1, gives a sufficient condition for theexistence of an extension of a minor-preserving map defined on the canonical poly-morphisms of a structure B to a uniformly continuous minor-preserving map definedon all of Pol( B ). The proof uses some ideas from [BM18]; however, we do not needthe so-called ‘mashup property’ which was the basis of the approach there. Lemma 6.1 (The Extension Lemma) . Let B be a reduct of a homogeneous Ramseystructure with finite relational signature A and let D be an operation clone. Supposethat ζ : Pol( B ) ∩ Can( A ) → D is minor-preserving and has the UIP with respect to Pol( B ) over A . Then ζ can be extended to a uniformly continuous minor-preservingmap ˜ ζ : Pol( B ) → D .Proof. Let f ∈ Pol( B ) ( k ) . Since A is a homogeneous Ramsey structure, we canapply the canonisation lemma (Lemma 3.10) and obtain the existence of g ∈ Can( A ) ∩ { u ( f ( v , . . . , v k )) : u, v , . . . , v k ∈ Aut( A ) } . Define ¯ ζ ( f ) := ζ ( g ); this is well-defined because ζ has the UIP with respect toPol( B ). Note that if f ∈ Can( A ), then ¯ ζ ( f ) = ζ ( f ), so ¯ ζ extends ζ . Claim 1. ¯ ζ is Aut( A )-invariant. Suppose that f, g ∈ Pol( B ) are such that f interpolates g modulo Aut( A ). Since A is Ramsey, Lemma 3.10 implies that g interpolates modulo Aut( A ) an operation h ∈ Can( A ). Then f interpolates h modulo Aut( A ), too, and we have ¯ ζ ( f ) = ζ ( h ) = ¯ ζ ( g ). Claim 2. ¯ ζ is uniformly continuous. This follows from Claim 1 by Lemma 5.2. Claim 3. ¯ ζ is minor-preserving. Arbitrarily choose f ∈ Pol( B ) ( k ) and a vector π = ( π ni , . . . , π ni k ) of projections of arity n ∈ N . We have to show that ¯ ζ ( f ◦ π ) = ¯ ζ ( f ) ◦ π . By Lemma 5.6, there exists g ∈ Pol( B ) ( k ) that diagonally interpolates f and diagonally interpolates h ∈ Pol( B ) ( k ) ∩ Can( A ). We obtain¯ ζ ( f ◦ π ) = ¯ ζ ( g ◦ π ) (Lemma 5.4 and interpolation-invariance of ¯ ζ )= ¯ ζ ( h ◦ π ) (Lemma 5.4 and interpolation-invariance of ¯ ζ )= ζ ( h ◦ π ) ( h ◦ π ∈ Pol( B ) ∩ Can( A ))= ζ ( h ) ◦ π ( ζ is minor preserving)= ¯ ζ ( h ) ◦ π = ¯ ζ ( g ) ◦ π = ¯ ζ ( f ) ◦ π. Claim 2 and 3 imply the statement of the theorem. (cid:3)
With Lemma 6.1 we can prove the most interesting implication between thethree items of Theorem 1.2, (3) ⇒ (2). Proof of Theorem 1.2.
For the equivalence of (1) and (2), let D be a structuresuch that Pol( D ) = Pol( C ) ∩ Can( B ). Proposition 3.18 implies that D has apseudo-Siggers polymorphism if and only if there is no uniformly continuous minor-preserving map from Pol( D ) = Pol( C ) ∩ Can( B ) to Proj.The implication from (2) to (3) is trivial. For the implication from (3) to (2),suppose for contradiction that Pol( C ) ∩ Can( B ) has a uniformly continuous minor-preserving map to Proj. By assumption, there is also a uniformly continuous minor-preserving map ζ : Pol( C ) ∩ Can( A ) → Proj that has the UIP with respect to Pol( C )over A . Lemma 6.1 shows that ζ can be extended to a uniformly continuous minor-preserving map ¯ ζ : Pol( C ) → Proj. Note that A , and hence also B and C , haveless than doubly exponential growth since A is homogeneous in a finite relationallanguage. Therefore, Theorem 3.3 implies that C cannot have a pseudo-Siggerspolymorphism, in contradiction to the assumptions. (cid:3) Proof of Corollary 1.3.
Let C ∈ C ; then C has the same CSP as its model-completecore, which is again in C , so we may assume without loss of generality that C isa model-complete core. If C has no pseudo-Siggers polymorphism, then CSP( C )is NP-complete by the main results of [BP16a, BP15]. Otherwise, we may applyTheorem 1.2. The assumption and the implication (3) ⇒ (1) of Theorem 1.2imply that Pol( C ) ∩ Can( B ) contains a pseudo-Siggers operation. Since B is afinitely bounded homogeneous structure, the results from [BM16, BM18] implythat CSP( C ) is in P. (cid:3) Two examples of classes C as in the statement of Corollary 1.3 will be presentedin Sections 8 and 9. However, we first give a characterisation of the UIP that iseasier to verify in Section 7.7. Binary Verification of the UIP
In this section we show that if a minor-preserving map to the clone of projectionsdoes not have the UIP, then this is witnessed by binary operations of a very specialform, proving Theorem 1.4. In many cases the non-existence of such witnesses canbe verified easily; this will be illustrated in Sections 8 and 9.We need the following ‘higher-dimensional checker board’ canonisation lemma.
HE UNIQUE INTERPOLATION PROPERTY 19
Lemma 7.1.
Let A be a countable homogeneous Ramsey structure with finiterelational signature and f : A k → A . Suppose that f interpolates over A theoperations h , . . . , h m ∈ Can( A ) . Then for every finite X ⊆ A there exist α , , . . . , α m,k ∈ Aut( A ) such that • for every i ∈ { , . . . , m } the operation f ( α i, , . . . , α i,k ) has the same be-haviour as h i on X k , and • for all u , . . . , u k ∈ { , . . . , m } the operation f ( α u , , . . . , α u k ,k ) is canonicalon X k .Proof. Let u , . . . , u p be an enumeration of { , . . . , m } k . We show by induction on q ∈ { , . . . , p } that for every finite X ⊆ A there exist α , , . . . , α m,k ∈ Aut( A ) suchthat for every i ∈ { , . . . , m } the operation f ( α i, , . . . , α i,k ) has the same behaviouras h i on X k , and for all l ∈ { , . . . , q } the operation f ( α u l , , . . . , α u lk ,k ) is canonicalon X k . For q = p we obtain the statement of the lemma.If q = 0 the statement of the claim follows from the assumptions of the lemma.For the inductive step, we assume that the statement holds for q − q . Let X ⊆ A be finite. Lemma 3.12 asserts the existence ofa finite C ⊆ A such that for every f : A k → A there are β , . . . , β k ∈ Aut( A )such that β ( X ) ⊆ C, . . . , β k ( X ) ⊆ C and f ( β , . . . , β k ) is canonical on X k . Weapply the induction hypothesis to C , and obtain γ , , . . . , γ m,k such that for every i ∈ { , . . . , m } the operation f ( γ i, , . . . , γ i,k ) has the same behaviour as h i on C k and f ( γ u l , , . . . , γ u lk ,k ) is canonical on C k for every l ∈ { , . . . , q − } . Let f ′ := f ( γ u q , , . . . , γ u qk ,k ). The property of C implies that there are β , . . . , β k ∈ Aut( A )such that β ( X ) ⊆ C, . . . , β k ( X ) ⊆ C and f ′ ( β , . . . , β k ) is canonical on X k . For i ∈ { , . . . , m } and j ∈ { , . . . , k } define α i,j := γ i,j ◦ β j . Observe α i,j ( X ) ⊆ γ i,j ( C )and hence • f ( α u l , , . . . , α u lk ,k ) is canonical on X k for l ∈ { , . . . , q − } : this followsfrom the inductive assumption that f ( γ u l , , . . . , γ u lk ,k ) is canonical on C k ; • f ( α u q , , . . . , α u qk ,k ) is canonical on X k : this follows from the property of β , . . . , β k that f ′ ( β , . . . , β k ) = f ( α u q , , . . . , α u qk ,k ) is canonical on X k ; • for every i ∈ { , . . . , m } , the operation f ( α i, , . . . , α i,k ) has the same be-haviour as h i on X k because for every j ∈ { , . . . , k } and f ( γ i, , . . . , γ i,k )has the same behaviour as h i on C k by the inductive assumption.This concludes the proof that α , , . . . , α m,k satisfy the inductive statement. (cid:3) We introduce some useful notation for the proofs of Theorem 1.4 and Theo-rem 1.5. If f : B k → B , u : B → B , and ℓ ∈ { , . . . , k } , we write f uℓ for the k -aryoperation defined by( x , . . . , x k ) f ( x , . . . , x ℓ − , u ( x ℓ ) , x ℓ +1 , . . . , x k ) . The central step of the proof is the following proposition.
Proposition 7.2.
Let B be a reduct of a homogeneous Ramsey structure A withfinite relational signature, let C := Pol( B ) ∩ Can( A ) , and let ζ : C → Proj be an
Aut( A ) -invariant minor-preserving map. Then the following are equivalent.(1) ζ has the UIP with respect to Pol( B ) over A .(2) For all f ∈ C (2) , u, v ∈ Aut( A ) , if f u , f v ∈ Can( A ) then ζ ( f u ) = ζ ( f v ) . Proof of Proposition 7.2.
The forward implication is trivial. For the converse im-plication, suppose that ζ does not have the UIP with respect to Pol( C ) over A .That is, there are operations g, g , g ∈ C ( k ) such that g interpolates both g and g modulo Aut( A ) and ζ ( g ) = ζ ( g ). By Lemma 3.15 there exists a finite X ⊆ A which is k -rich with respect to Pol( B ). By Lemma 7.1 applied to g and X thereexist α , , . . . , α ,k ∈ Aut( A ) such that • for all u , . . . , u k ∈ { , } the operation g ( α u , , . . . , α u k ,k ) is canonical on X k over A , and • for i ∈ { , } the operation g ( α i, , . . . , α i,k ) has the same behaviour as g i on X k .Let i ∈ { , . . . , k } be the smallest index such that ζ (cid:0) g ( α , , . . . , α ,i − , α ,i , α ,i +1 , . . . , α ,k ) (cid:1) = ζ (cid:0) g ( α , , . . . , α ,i − , α ,i , α ,i +1 , . . . , α ,k ) (cid:1) . We know that such an index exists since ζ (cid:0) g ( α , , . . . , α ,k ) (cid:1) = ζ ( g ) = ζ ( g ) = ζ (cid:0) g ( α , , . . . , α ,k ) (cid:1) . Let h := g ( α , , . . . , α ,i − , id , α ,i +1 , . . . , α ,k ). So we have shown the following. Claim.
For every finite X ⊆ A that contains X there exist i ∈ { , . . . , k } , h ∈ Pol( B ) ( k ) , α, β ∈ Aut( A ), and h, h ′ ∈ Pol( B ) ( k ) ∩ Can( A ) such that • f αℓ has the same behaviour as h on X k , • f βℓ has the same behaviour as h ′ on X k , and • ζ ( h ) = ζ ( h ′ ).Let X ⊂ X ⊂ · · · be finite subsets of A such that S ∞ i =0 X i = A . Then theclaim applied to X = X i , for i ∈ N , asserts the existence of f i , α i , β i , and ℓ i suchthat ( f i ) α i ℓ i and ( f i ) β i ℓ i are canonical on X ki and ζ (( f i ) α i ℓ i ) = ζ (( f i ) β i ℓ i ). By thinningout the sequences ( f i ) i ∈ N , ( α i ) i ∈ N , ( β i ) i ∈ N , and ( ℓ i ) i ∈ N we can assume that all ℓ i are equal, say ℓ , and that the complete behaviour B of f α i i on X and B of f β i i on X does not depend on i . Note that we must have B = B . By Lemma 3.8(and in particular the statement at the end starting with “moreover”) there exist g ∈ Pol( B ) and u, v ∈ Aut( A ) such that g uℓ has behaviour B on all of A , and g vℓ has behaviour B on all of A . So they are canonical over A and ζ ( g uℓ ) = ζ ( g vℓ ).Suppose that ζ ( g uℓ ) = π kr and ζ ( g vℓ ) = π ks . Let f ( x, y ) := f ( y, . . . , y | {z } r − , x, y, . . . , y ) . HE UNIQUE INTERPOLATION PROPERTY 21
Then note that ζ (cid:0) f ( π , u ( π )) (cid:1) = ζ (cid:0) g ( u ( π ) , . . . , u ( π ) | {z } r − , π , u ( π ) , . . . , u ( π )) (cid:1) = ζ (cid:0) g uℓ ( π , . . . , π | {z } r − , π , π , . . . , π ) (cid:1) ( ζ is Aut( A )-invariant)= π kr ( π , . . . , π | {z } r − , π , π , . . . , π ) ( ζ is minor-preserving)= π Similarly, ζ (cid:0) f ( π , v ( π )) (cid:1) = π ks ( π , . . . , π | {z } r − , π , π , . . . , π )= π since r = s .Therefore, f, u, v show that item (2) from the statement does not hold, concludingthe proof that (2) ⇒ (1). (cid:3) Proof of Theorem 1.4.
Note that if ζ has the UIP, it is in particular Aut( A )-invariant, and hence uniformly continuous by Proposition 3.18. Conversely, if thereexists a uniformly continuous minor-preserving map from C to Proj, then therealso exists an Aut( A )-invariant minor-preserving map from C to Proj by Proposi-tion 3.18. The statement now follows from Proposition 7.2. (cid:3) We finally prove a strengthening of Theorem 1.4 which allows the verification ofthe UIP to take place in two independent elementary substructures of the underlyingRamsey structure.
Proof of Theorem 1.5.
There is a pair ( A , A ) of independent elementary substruc-tures of A by Theorem 4.4, so the forward implication holds trivially.We now show the converse, (2) ⇒ (1). So let ( A , A ) be the pair of independentelementary substructures of A from the statement of (2). It suffices to verify (2) inTheorem 1.4. Let f ∈ Pol( C ) (2) , u , u ∈ Aut( A ) be such that f u , f u ∈ Can( A ).Let F ⊆ A be 2-rich with respect to Pol( C ). We have to verify that f u and f u have the same complete behaviour on F .For j ∈ { , } let e j be an embedding of A into A j . Since A is homogeneous thereexists ǫ ∈ Aut( A ) such that e and ǫ agree on u ( F ) ∪ u ( F ). By Lemma 3.10 appliedto f (id , ǫ − e ) there are β , β ∈ Aut( A ) such that f ( β , ǫ − e β ) is canonical on F . Let v i := ǫu i and w := e β . Note that v i ( F ) ⊆ A and w ( F ) ⊆ A . Let h := f ( β , ǫ − ) ∈ Pol( C ), and note that h v i = f ( β , ǫ − ǫu i ) = f ( β , u i )is canonical since f u i = f (id , u i ) is canonical, and that h w = f ( β , ǫ − e β )is canonical on F as we have seen above. We thus apply (2) two times to obtainthat ζ ( h v ) = ζ ( h w ) = ζ ( h v ). Finally, note that ζ ( f (id , u i )) = ζ ( f ( β , u i )) = ζ ( f ( β , ǫ − ǫu i )) = ζ ( h v i ) . We conclude that ζ ( f (id , u )) = ζ ( f (id , u )). (cid:3) Structures with exponential labelled growth
Our first application of the general results from Section 6 and Section 7 concernsstructures of bounded labelled growth. The labelled growth function is the function ℓ : N → N which maps n ∈ N to the number of orbits of n -tuples with pairwisedistinct entries in Aut( A ). For example, the labelled growth of a finite structureis eventually 0, while it is constant 1 for the structure ( N ; =) and all its first-order reducts. Structures with exponential labelled growth, i.e., labelled growthwhich is bounded by c n for some constant c , have been classified in [BB19] upto first-order interdefinability: they are precisely the first-order reducts of unarystructures [BM18]; by a unary structure we mean a countable structure that justcarry finitely many unary predicates. The complexity of constraint satisfactionproblems for first-order reducts of unary structures has been classified in [BM18];in fact, Conjecture 3.4 has been verified for this class. However, the original proofcontains a mistake (in the proof the mashup theorem, Theorem 5.5 in [BM18], morespecifically in the claim that φ is a minor-preserving map) and fixing this with ageneral approach is one of the contributions of the present article.Our original motivation to the present work was the generalisation of the clas-sification result in [BM18] to the class of all structures with exponential growthbounded by a function of the form cn dn for some constants c, d with d <
1. Again,this class of structures has alternative descriptions: e.g., these structures are pre-cisely the finite covers of first-order reducts of unary structures [BB19]. The proofthat this much larger class satisfy a CSP complexity dichotomy is long and will bepublished elsewhere; it is based on the machinery developed here.In this section we focus on illustrating our general results about the uniqueinterpolation property to confirm Conjecture 3.4 for first-order reducts of unarystructures. Some of the proof steps are taken literally from [BM18] and not provedagain here. Our method differs at the central part of the classification, wheremashups are used in [BM18], while we use our general results instead.Note that the property to have exponential labelled growth is preserved by tak-ing model-complete cores, because for each n , the number of orbits of n -tuples ofthe model-complete core of B is at most the number of orbits of n -tuples of B (see [Bod20]). Thus, we have already verified the first of the assumptions fromCorollary 1.3. The next assumption we need to verify is that every first-orderreduct C of a unary structure can be expanded to a finitely bounded homogeneousstructure B , which itself is the reduct of a homogeneous Ramsey structure A withfinite relational signature. This is easy in our case: • every unary structure B = ( B ; U , . . . , U n ) is clearly homogeneous andfinitely bounded, and by definition C can be expanded to such a structure; • a homogeneous Ramsey expansion of B can be obtained by adding a linearorder ≺ such that if u ∈ U i , v ∈ U j , and i < j , then u ≺ v , and such that ≺ is dense and without endpoints on U i whenever U i is infinite (see the proofof Proposition 6.5 in [BM18]).We use another observation from [BM18], namely that by suitably expanding C with constants, the classification task can be reduced to the situation where C and B are even first-order interdefinable (Proposition 6.8 in [BM18]). The central HE UNIQUE INTERPOLATION PROPERTY 23 step of the classification is the verification of the final remaining assumption fromCorollary 1.3 which concerns the UIP.
Proposition 8.1.
Let C be a model-complete core which is first-order interdefinablewith a unary structure B . Let A be the homogeneous Ramsey expansion of B introduced above. Suppose that there is a uniformly continuous minor-preservingmap from Pol( C ) ∩ Can( B ) to Proj . Then there is also a minor-preserving mapfrom
Pol( C ) ∩ Can( A ) to Proj which has the UIP with respect to
Pol( C ) over A .Proof. Suppose that O , . . . , O n are the orbits of the unary structure that is first-order interdefinable with C . Then for every i ∈ { , . . . , n } , the set O i is primitivepositive definable in C because C is a model-complete core. Hence, the map η i thatsends each operation in Pol( C ) to its restriction to O i is uniformly continuous andminor-preserving. If the image under this map has a uniformly continuous minor-preserving map to Proj, then Pol( C ) has a uniformly continuous minor-preservingmap to Proj, too. In this case the restriction of this map to Pol( C ) ∩ Can( A ) hasthe UIP and we are done.So let us suppose that for all i ∈ { , . . . , n } there is no uniformly continuousminor-preserving map from η i (Pol( C )) to Proj. If ξ A (Pol( C ) ∩ Can( A )) also doesnot have a minor-preserving map to Proj, then Proposition 6.6 in [BM18] impliesthat Pol( C ) ∩ Can( B ) does not have a minor-preserving map to the projections,a contradiction to our assumption. So suppose that ξ A (Pol( C ) ∩ Can( A )) doeshave a minor-preserving map to Proj. Let D be a (finite) structure such thatPol( D ) = ξ A (Pol( C ) ∩ Can( A )); we identify the elements of the domain D of D with { O , . . . , O n } . Then Theorem 3.1 implies that there are subsets H , H ⊆ D such that for every f ∈ Pol( C ) ∩ Can( A ) ( k ) there exists j ∈ { , . . . , k } such that forall i , . . . , i k ∈ { , } ξ A ( f )( H i , . . . , H i k ) ⊆ H i j . The map ζ that sends f ∈ Pol( C ) ∩ Can( A ) ( k ) to π kj is a minor-preserving map; weverify that it has the UIP. Suppose for contradiction that there are f ∈ Pol( C ) (2) and u , u ∈ Aut( A ) such that f u , f u ∈ Can( A ) and ζ ( f u ) = ζ ( f u ). Let e : A ֒ → A be an embedding such that for all i ∈ { , . . . , n } and j ∈ { , } we have e ( O i ) ⊆ u j ( O i ) ⊆ O i if O i ∈ H j ; the structure A clearly has such self-embeddings.Define g := f (id , e ). Since A is Ramsey g interpolates a canonical function h byLemma 3.10. Let O r ∈ H and O s ∈ H . Then g ( O r , O s ) = f ( O r , e ( O s )) ⊆ f ( O r , u ( O s )) = f u ( O r , O s )and g ( O s , O r ) = f ( O s , e ( O r )) ⊆ f ( O s , u ( O r )) = f u ( O s , O r ) . Hence, ζ ( g )( O r , O s ) = ζ ( f u )( O r , O s ) = ζ ( f u )( O r , O s ) = ζ ( g )( O r , O s ) , a contradiction. Therefore, Theorem 1.4 implies that ζ has the UIP which concludesthe proof. (cid:3) Corollary 8.2.
Let B be a countable structure with exponential labelled growth andfinite relational signature. Then CSP( B ) is in P or NP-complete.Proof. It follows from Proposition 8.1 and the discussion above that the assump-tions of Corollary 1.3 are satisfied, which immediately implies the statement. (cid:3) First-order expansions of the basic relations of RCC5
Our second application of the general results from Sections 6 and 7 is a newcomplexity dichotomy for spatial constraint satisfaction problems. RCC5 is a rela-tion algebra studied in qualitative spatial reasoning [Ben94]; it can be viewed as aset of binary relations defined on some general set of regions ; a formal definitioncan be found below. Renz and Nebel [RN01] showed that the CSP for RCC5 isNP-complete, and Nebel [Neb95] showed that the CSP for the basic relations ofRCC5 is in P (via a reduction to 2SAT ). Renz and Nebel [RN01] have extendedthis polynomial-time tractability result to a superclass of the basic relations, andthey showed that their expansion is maximal in the sense that every larger subsetof the RCC5 relations has an NP-hard CSP. Drakengren and Jonsson [JD97] classi-fied the computational complexity of the CSP for all subsets of the RCC5 relations.In this section, we classify the complexity of the CSP for expansions of the basicrelations of RCC5 by first-order definable relations of arbitrary arity .9.1. Introducing RCC5.
There are many equivalent ways of formally introducingRCC5; we follow the presentation in [BC09] and then provide references for otherdefinitions and their equivalence. Let S be the structure with domain S := 2 N \{∅} ,i.e., the set of all non-empty subsets of the natural numbers N . The signature of S consists of the five binary relation symbols EQ , PP , PPI , DR , PO and for x, y ⊆ N wehave ( x, y ) ∈ EQ iff x = y, “ x and y are equal”(2) ( x, y ) ∈ PP iff x ⊂ y, “ x is strictly contained in y ”(3) ( x, y ) ∈ PPI iff x ⊃ y, “ x strictly contains y ”(4) ( x, y ) ∈ DR iff x ∩ y = ∅ , “ x and y are disjoint”(5) ( x, y ) ∈ PO iff x y ∧ y x ∧ x ∩ y = ∅ , “ x and y are properly overlap”.(6)Note that by definition every pair ( x, y ) ∈ S is contained in exactly one of therelations DR S , PO S , PP S , PPI S , EQ S . Note that the structure S is not ω -categorical;however, Age( S ) is a strong amalgamation class (Theorem 30 in [BC09]), and hencethere exists a countable homogeneous structure R with the same age as S . We referto the relations EQ R , PP R , PPI R , DR R , PO R as the basic relations of RCC5 .The composition of two binary relations R and R is the binary relation R ◦ R := (cid:8) ( x, y ) : ∃ z (cid:0) R ( x, z ) ∧ R ( z, y ) (cid:1)(cid:9) . The converse (sometimes also called inverse ) of a relation R is the relation { ( y, x ) : ( x, y ) ∈ R } , and denoted by R ⌣ . The converse of PP is PPI , and EQ R , DR R , and PO R are their own converse. The full binary relation containing all pairsof elements of R is denoted by . It is straightforward to verify that the relations EQ R , PP R , PPI R , DR R , PO R compose as shown in Table 1. As a relation algebra ,RCC5 is given by the composition table and the data about the converses, and ourstructure R introduced above is a representation of RCC5. We do not introducerelation algebras formally, because they will not be needed in the following, andrather refer to [BJ17]. The proof of Nebel was formulated for RCC8 instead of RCC5, but the result for RCC5 canbe shown analogously.
HE UNIQUE INTERPOLATION PROPERTY 25 ◦ DR PO PP PPI EQDR PP ∪ DR ∪ PO PP ∪ DR ∪ PO DR DRPO PPI ∪ DR ∪ PO PP ∪ PO PPI ∪ PO POPP DR PP ∪ DR ∪ PO PP PPPPI PPI ∪ DR ∪ PO PPI ∪ PO \ DR PPI PPIEQ DR PO PP PPI EQ
Table 1.
The composition table for the relations of R .9.2. Classifying first-order expansions of R . A first-order expansion of a struc-ture A is an expansion of a A by relations that are first-order definable over A . Themain result of this section is the following. Theorem 9.1.
Let C be a first-order expansion of R . Then either • Pol( C ) has a pseudo-Siggers operation which is canonical with respect to R ,in which case CSP( C ) is in P, or • Pol( C ) does not have a pseudo-Siggers operation, in which case CSP( C ) isNP-complete. We prove Theorem 9.1 using Theorem 1.2 and Theorem 1.5, and hence first needto introduce a Ramsey expansion ( R ; ≺ ) of R (Section 9.3). Suppose that C is asin Theorem 9.1 and has a pseudo-Siggers polymorphism. We verify item (3) inTheorem 9.1 in two steps. First we prove that if there is a uniformly continuousminor-preserving map from Pol( C ) ∩ Can( R ) to Proj, then there exists • a specific minor-preserving map η : Pol( C ) ∩ Can( R , ≺ ) → Proj which arisesfrom the action of the canonical polymorphisms on the two relations PP and( DR ∪ PO ) ∩ ≺ , or • a specific minor-preserving map ρ : Pol( C ) ∩ Can( R , ≺ ) → Proj which arisesfrom the action of the canonical polymorphisms on the two relations DR ∩ ≺ and PO ∩ ≺ (Section 9.6).In the second step, we prove that if such a map η, ρ : Pol( C ) ∩ Can( R , ≺ ) → Projexists, then it has the UIP with respect to Pol( C ) (Section 9.7). The statementthen follows from Theorem 1.2.9.3. A Ramsey expansion of R . The structure R is not Ramsey, but it hasa homogeneous expansion by a linear order. Let C be the class of all expansionsof structures from Age( S ) with the signature { EQ , DR , PO , PP , PPI , ≺} such that ≺ denotes a linear extension of PP . Proposition 9.2.
The class C defined above is a strong amalgamation class.Proof. It is clear from the definition that C is closed under isomorphisms and sub-structures. It is well-known that in order to prove the amalgamation property, itsuffices to verify the 1-point amalgamation property (see, e.g., [Bod20]): for allstructures A , B , B ∈ C such that A = B ∩ B and B i = A ∪ { b i } , for i ∈ { , } and b = b , there exists C ∈ C with C = B ∪ B = A ∪ { b , b } such that B and B are substructures of C . It is easy to see that such a structure C can bedetermined by specifying which relations of C contain the pair ( b , b ). Bodirsky and Chen [BC09] (also see [Bod20]) proved that there exists R ∈{ DR , PO , PP , PPI } such that • if we add the pair ( b , b ) to R and the pair ( b , b ) to R ⌣ , then the { EQ , DR , PO , PP , PPI } -reduct of the resulting structure C is in Age( S ), and • ( b , b ) ∈ PP C only if there exists a ∈ A such that ( b , a ) ∈ PP B and( a, b ) ∈ PP B .If R equals PP we add ( b , b ) to ≺ ; if R equals PPI we add ( b , b ) to ≺ . If R isfrom { DR , PO } , then we add ( b , b ) or ( b , b ) to ≺ according to an order-amalgamof the {≺} -reducts of B and B over A . In each of these cases, ≺ C is a linearorder that extends PP C , and hence C is in C . (cid:3) One can check by an easy back-and-forth argument that the { EQ , DR , PO , PP , PPI } -reduct of the Fra¨ıss´e-limit of C is isomorphic to R ; hence, we denote this Fra¨ıss´e-limit by ( R , ≺ ). To show that ( R , ≺ ) has the Ramsey property, by the followingtheorem it suffices to prove that ( R , ≺ ) is the model-complete core of a Ramseystructure. Theorem 9.3 (Theorem 3.18 in [Bod15]) . The model-complete core of an ω -categorical Ramsey structure is again a Ramsey structure. The countable atomless Boolean algebra A = ( A ; ∩ , ∪ , · , ,
1) has an expansionby a linear order ≺ so that the expansion ( A , ≺ ) is a homogeneous ω -categoricalRamsey structure [KPT05]. The age of ( A , ≺ ) can be described as follows. If F is a finite substructure of A then the {∩ , ∪ , · , , } -reduct of F is a finite Booleanalgebra; then there exists an enumeration a , . . . , a n of the atoms of F such that forall u, v ∈ F with u = S ni =1 ( δ i ∩ a i ) and v = S ni =1 ( ǫ i ∩ a i ) for δ i , ǫ i ∈ { , } we have u ≺ v if and only if there exists some j ∈ { , . . . , n } such that δ j < ǫ j and δ i = ǫ j for all i > j .Let A ′ be the { EQ , DR , PO , PP , PPI } -structure with domain A \ { } whose relationsare defined by the expressions in (2), (3) (here, x ⊂ y stands for ‘( x ∩ y ) = x and x = y ’), (4), (5), (6) interpreted over A and restricted to A ′ . Let ≺ ′ be therestriction of ≺ to A ′ . Note that ( A ′ , ≺ ′ ) and ( A , ≺ ) have automorphism groupsthat are topologically isomorphic. It follows that ( A ′ , ≺ ′ ) is Ramsey, too (see, e.g.,Proposition 2.28 in [Bod15]). Proposition 9.4.
The structure ( R , ≺ ) is the model-complete core of ( A ′ , ≺ ) andis a Ramsey structure.Proof. Since ( R , ≺ ) is homogeneous, it is model-complete. It is easy to see that( R , ≺ ) is a core, because the negation of every relation of R can be defined exis-tentially positively as a union of the other relations of R , and the complement of ≺ has the positive quantifier-free definition x = y ∨ y ≺ x . We claim that ( A ′ , ≺ )and ( R , ≺ ) have the same age. We have already seen that A ′ has the same age as S and hence the same age as R . Also, ≺ is a linear extension of PP A ′ , and henceevery finite substructure of ( A ′ , ≺ ) is also a substructure of ( R , ≺ ).Conversely, let ( F , ≺ ) be a finite substructure of ( R , ≺ ). Let e be an embeddingof F into A ′ . Let u , . . . , u k be an enumeration of F such that u ≺ · · · ≺ u k . Let v , . . . , v k ∈ A be such that ( v i , e ( u j )) ∈ DR A ′ for all i, j ∈ { , . . . , k } and such that( v i , v j ) ∈ DR A ′ for i = j . Then we define f : F → A by b ( u ) := e ( u ) ∪ [ ( u i ,u ) ∈ PP F v i . HE UNIQUE INTERPOLATION PROPERTY 27
Note that v , . . . , v k are atoms in the Boolean algebra generated by e ( F ) ∪{ v , . . . , v n } in A , and let w , . . . , w ℓ be the other atoms. We may assume that ≺ isdefined on this Boolean algebra according the enumeration w , . . . , w ℓ , v , . . . , v k ofthe atoms. We prove that f is an embedding of ( F , ≺ ) into ( A ′ , ≺ ). Let u, u ′ ∈ F .If ( u, u ′ ) ∈ PP F , then e ( u ) ⊂ e ( v ) and { i : ( u i , u ) ∈ PP F } ⊂ { i : ( u i , u ′ ) ∈ PP F } by the transitivity of PP F , so ( b ( u ) , b ( u ′ )) ∈ PP A ′ . It is also clear that b pre-serves EQ , PPI , DR . To see that b preserves PO , note that if ( u, u ′ ) ∈ PO F ,then ( e ( u ) , e ( u ′ )) ∈ PO A ′ , so e ( u ) ∩ e ( u ′ ), e ( u ) ∩ e ( u ′ ), and e ( u ) ∩ e ( u ′ ) are non-empty. Note that e ( u ) ∩ e ( u ′ ) ⊆ b ( u ) ∩ b ( u ′ ), e ( u ) ∩ e ( u ′ ) ⊆ b ( u ) ∩ b ( u ′ ), and e ( u ) ∩ e ( u ′ ) ⊆ b ( u ) ∩ b ( u ′ ), so ( b ( u ) , b ( u ′ )) ∈ PO A ′ .To prove that b preserves ≺ , let i, j ∈ { , . . . , k } be such that i < j and u i ≺ u j .Then PP A ′ ( v j , b ( u j )), DR A ′ ( v j , b ( u i )), and for every m ∈ { j + 1 , . . . , k } we have DR A ′ ( v m , b ( u i )) and DR A ′ ( v m , b ( u j )). Indeed, if DR A ′ ( v m , b ( u i )) does not hold, then PP A ′ ( v m , b ( u i )) and hence PP F ( u m , u i ) by the definition of b . This in turn impliesthat u m ≺ u i and hence m < i , a contradiction. By the definition of ≺ on A thisimplies that b ( u i ) ≺ b ( u j ) and finishes the proof of the claim.The claim implies via a compactness argument the existence of homomorphismsin both directions between the two countable structures (see, e.g., Lemma 4.1.7in [Bod20]). So ( R , ≺ ) is indeed the model-complete core of ( A ′ , ≺ ). The statementnow follows from the comments above and Theorem 9.3. (cid:3) From now on, we identify the symbols EQ , PP , PPI , DR , PO with the respectiverelations of R , and we write ≻ for the converse of ≺ . Note that PP ∪ DR ∪ PO and PPI ∪ DR ∪ PO are primitively positively definable in R , since they are entries inTable 1. Hence, their intersection DR ∪ PO is primitively positively definable in R ,too. We also write ⊥ instead of DR ∪ PO , and ⊥≺ for the relation ( DR ∪ PO ) ∩ ≺ .The composition table for the binary relations with a first-order definition over( R ; ≺ ) can be derived conveniently from the composition table of R (Table 1) usingthe following lemma. Lemma 9.5.
Let R , R ∈ { EQ , PP , PPI , DR , PO } and let O , O be two orbits of pairsof ( R ; ≺ ) such that O i ⊆ R i . Then O ◦ O = ( R ◦ R ) ∩ ≺ if O , O ⊆ ≺ ( R ◦ R ) ∩ ≻ if O , O ⊆ ≻ R ◦ R otherwise.Proof. It is clear that O ◦ O ⊆ R ◦ R and that if O , O ⊆ ≺ then O ◦ O ⊆ ≺ ◦ ≺ = ≺ . So the ⊆ -containment in the statement of the lemma holds in the first case, and bysimilar reasoning also in the other two cases. The reverse containment in the firsttwo cases is also clear since the homogeneity of ( R , ≺ ) implies that the expressionin the statement on the right describes an orbit of pairs in Aut( R , ≺ ) (we havealready seen that it is non-empty). To show the equality in the third case, let( x, z ) ∈ R ◦ R . If R ◦ R equals PP (or PPI ) then x ≺ z (or z ≻ x ), andagain R ◦ R is an orbit of pairs in Aut( R , ≺ ) and the equality holds. If R ◦ R equals EQ then the statement is clear, too. Otherwise, let x ′ , y ′ , z ′ ∈ R be suchthat ( x ′ , y ′ ) ∈ O and ( y ′ , z ′ ) ∈ O . If ( x ′ , z ′ ) lies in the same orbit as ( x, z )in Aut( R , ≺ ) then ( x, z ) ∈ O ◦ O . Otherwise, consider the structure induced by ( R , ≺ ) on { x ′ , y ′ , z ′ } ; we claim that if we replace the tuple ( x ′ , z ′ ) in ≺ bythe tuple ( z ′ , x ′ ), the resulting structure still embeds into ( R ; ≺ ). By assumption, O ⊆ ≺ and O ⊆ ≻ , or O ⊆ ≻ and O ⊆ ≺ , so the modified relation ≺ is stillacyclic. Moreover, since R ◦ R ∈ { DR , PO } , the modified relation ≺ is still a linearextension of PP and hence in Age( R ; ≺ ). The homogeneity of ( R ; ≺ ) implies that( x ′ , z ′ ) lies in the same orbit as ( x, z ), and hence ( x, z ) ∈ O ◦ O . This concludesthe proof that R ◦ R ⊆ O ◦ O . (cid:3) Corollary 9.6.
Let O ⊆ ≺ be an orbit of pairs in Aut( R ; ≺ ) . Then PP ⊆ O ◦ O .Proof. If O = PP , then O ◦ O = O = PP . If O = ( DR ∩ ≺ ) then O ◦ O = ( DR ◦ DR ) ∩ ≺ (Lemma 9.5)= R ∩ ≺ (Table 1)= ≺ . Similarly we may compute that PP is contained in ( PO ∩ ≺ ) ◦ ( PO ∩ ≺ ). (cid:3) Independent substructures.
To verify the UIP property, we will use The-orem 1.5 and therefore need certain pairs ( A , A ) of independent elementary sub-structures of ( R , ≺ ). Lemma 9.7.
There are elementary substructures A and A of ( R , ≺ ) that thatfor all a ∈ A and a ∈ A we have ( a , a ) ∈ DR and a ≺ a ; in particular, A and A are independent.Proof. By the homogeneity of ( R , ≺ ) it suffices to show that for every structure( B , ≺ ) ∈ Age( R , ≺ ) there are embeddings e , e : B → ( R , ≺ ) such that for all b , b ∈ B we have ( e ( a ) , e ( a )) ∈ DR and e ( b ) ≺ e ( b ). Choose an embedding f of B into the structure S from the definition of R ; so f ( b ) ⊆ N for each b ∈ B .Then f : b
7→ { n : n ∈ f ( b ) } and f : b
7→ { n + 1 : n ∈ f ( b ) } are two embeddingsof B into S such that ( f ( b ) , f ( b )) ∈ DR for all b , b ∈ B . Let B ′ be thesubstructure of S with domain B ′ := f ( B ) ∪ f ( B ). For all b , b ∈ B with b ≺ b , define the linear order ≺ on B ′ by f ( b ) ≺ f ( b ), f ( b ) ≺ f ( b ), f ( b ) ≺ f ( b ). Then it is straightforward to check that ≺ extends PP , and hence( B ′ , ≺ ) ∈ Age( R , ≺ ), which concludes the proof. (cid:3) Independent substructures are used in the proof of the following lemma thatplays an important role when verifying the UIP later.
Lemma 9.8.
Let C be a first-order expansion of R and let ζ : Pol( C ) ∩ Can( R , ≺ ) → Proj be a clone homomorphism which does not have the UIP with respect to
Pol( C ) .Then for every finite 2-rich finite subset F of the domain of R there exist f ∈ Pol( C ) (2) and α , α ∈ Aut( R ; ≺ ) such that • f is canonical on F × α ( F ) and on F × α ( F ) with respect to ( R , ≺ ) , • ζ ( f (id , α ) | F ) = ζ ( f (id , α ) | F ) (recall Definition 3.16), • for all a, b ∈ F we have ( α ( a ) , α ( b )) ∈ PP if ( a, b ) ∈ PP ∪ EQ , and • for all a, b ∈ F we have ( α ( a ) , α ( b )) ∈ ⊥ if ( a, b ) ∈ ⊥ .Proof. Let A , A be the two independent elementary substructures of ( R , ≺ ) fromLemma 9.7. Since ζ does not have the UIP with respect to Pol( C ), by Theorem 1.5there exists f ′ ∈ Pol( C ) (2) and u , u ∈ Aut( A ) such that for i ∈ { , } the image ℑ ( u i ) of u i is contained in A i , the operation f ′ (id , u i ) is canonical with respect to HE UNIQUE INTERPOLATION PROPERTY 29 ( R , ≺ ), and ζ ( f ′ (id , u )) = ζ ( f ′ (id , u )). By Lemma 3.12 there exists a finite subset X of the domain of R such that for all g ∈ Pol( R ) (2) there exist β , β ∈ Aut( R , ≺ )with β ( F ) , β ( F ) ⊆ X such that g is canonical on β ( F ) × β ( F ). For i ∈ { , } ,let ǫ i ∈ Aut( R , ≺ ) be such that ǫ i ( X ) ⊆ ℑ ( u i ). We may view the substructure of R induced by ǫ ( X ) ∪ ǫ ( X ) as a substructure of S . By the homogeneity of R we mayalso assume that the substructure of S on ǫ ( X ) ∪ ǫ ( X ) ∪ { ǫ ( x ) ∪ ǫ ( x ) : x ∈ X } has an embedding e into R such that e ( ǫ i ( x )) = ǫ i ( x ) for all x ∈ X and i ∈ { , } .Define ǫ : X → R by ǫ ( x ) := e (cid:0) ǫ ( x ) ∪ ǫ ( x ) (cid:1) . Let a, b ∈ X . Note that(1) if ( a, b ) ∈ PP ∪ EQ then ǫ i ( a ) = e ( ǫ i ( a )) ⊂ e (cid:0) ǫ ( a ) ∪ ǫ ( a ) (cid:1) ⊆ e (cid:0) ǫ ( b ) ∪ ǫ ( b ) (cid:1) = ǫ ( b )and hence ( ǫ i ( a ) , ǫ ( b )) ∈ PP ;(2) if ( a, b ) ∈ ⊥ then ( ǫ i ( a ) , ǫ i ( b )) ∈ ⊥ . Since ǫ i ( a ) ∈ A i it follows that ǫ i ( a )is disjoint from ǫ j ( b ) for j = i . This implies that ǫ i ( a ) \ ǫ ( b ) = ǫ i ( a ) \ (cid:0) ǫ ( b ) ∪ ǫ ( b ) (cid:1) = ǫ i ( a ) \ ǫ i ( b ) = ∅ and that for j = i we have ǫ j ( b ) = ǫ j ( b ) \ ǫ i ( a ) ⊆ ǫ ( b ) \ ǫ i ( a ) . In particular, ǫ ( b ) \ ǫ i ( a ) = ∅ . We obtained that the sets ǫ i ( a ) \ ǫ ( b ) and ǫ ( b ) \ ǫ i ( a ) are both non-empty. Therefore ( ǫ i ( a ) , ǫ ( b )) ∈ ⊥ .We define an order ≺ ′ on ǫ ( X ) ∪ ǫ ( X ) ∪ ǫ ( X ) by setting a ≺ ′ b if one of thefollowing holds. • a, b ∈ ǫ ( X ) ∪ ǫ ( X ) and a ≺ b ; • a ∈ ǫ ( X ) ∪ ǫ ( X ) and b ∈ ǫ ( X ); • a, b ∈ ǫ ( X ) and ǫ − ( a ) ≺ ǫ − ( b ).Then it is easy to see that ≺ ′ defines a partial order that extends PP ; let ≺ ′′ be a linear order that extends ≺ ′ . By the definition of ( R ; ≺ ) there exists anautomorphism γ of R that maps ≺ ′′ to ≺ . This shows that we may assume that ǫ preserves ≺ (otherwise, replace ǫ by γ ◦ ǫ ).By the definition of X there are β , β ∈ Aut( R , ≺ ) such that β ( F ) ⊆ X , β ( F ) ⊆ X , and f ′ (id , ǫ ) is canonical on β ( F ) × β ( F ) over ( R , ≺ ). Since ǫ i β ( F ) ⊆ ǫ i ( X ) ⊆ ℑ ( u i )for i ∈ { , } we have that ζ ( f ′ | β ( F ) × ǫ β ( F ) ) = ζ ( f ′ | β ( F ) × ǫ β ( F ) ). Then for some i ∈ { , } we have that ζ ( f ′ | β ( F ) × ǫ i β ( F ) ) = ζ ( f ′ | β ( F ) × ǫβ ( F ) ). We claim that f := f ′ ( β , id), α := ǫ i ◦ β and α := ǫ ◦ β satisfy the conclusion of the lemma: • f is canonical on F × α i ( F ), for i ∈ { , } , because f ′ (id , ǫ ) is canonical on β ( F ) × β ( F ). • for all ( a, b ) ∈ F ∩ ( PP ∪ EQ ) we have that ( β ( a ) , β ( b )) ∈ X ∩ ( PP ∪ EQ )and by (1) we obtain ( α ( a ) , α ( b )) = ( ǫ i ( β ( a )) , ǫ ( β ( b ))) ∈ PP . • for all ( a, b ) ∈ F ∩ ⊥ we have that ( β ( a ) , β ( b )) ∈ X ∩ ⊥ and by (2) weobtain ( α ( a ) , α ( b )) = ( ǫ i ( β ( a )) , ǫ ( β ( b ))) ∈ ⊥ . (cid:3) Polymorphisms of ( R , ≺ ) . The following general observations will be usefulwhen working with operations that preserve binary relations over some set B . If f : B k → B is an operation that preserves R , . . . , R k , R ′ , . . . , R ′ k ⊆ B then f ( R ◦ R , . . . , R k ◦ R ′ k ) ⊆ f ( R , . . . , R k ) ◦ f ( R ′ , . . . , R ′ k ) . (7)Also note that f ( R ⌣ , . . . , R ⌣ ) = f ( R , . . . , R ) ⌣ . (8) Lemma 9.9.
Every polymorphism of R which is canonical with respect to ( R , ≺ ) preserves ≺ and ⊥≺ .Proof. let a , . . . , a k , b , . . . , b k be elements of R such that a i ≺ b i for all i ∈{ , . . . , k } and f ( b , . . . , b k ) (cid:22) f ( a , . . . , a k ). For i ∈ { , . . . , k } , let O i be theorbit of ( a i , b i ) in Aut( R , ≺ ). We then have f ( PP , . . . , PP ) ⊆ f ( O ◦ O , . . . , O k ◦ O k ) (Corollary 9.6) ⊆ f ( O , . . . , O k ) ◦ f ( O , . . . , O k ) (7) ⊆ ( (cid:23) ◦ (cid:23) ) (canonicity)= (cid:23) which is a contradiction to f being a polymorphism of R . Since ⊥ is primitivelypositively definable in R , it also follows that every polymorphism of R which iscanonical with respect to ( R , ≺ ) preserves ⊥≺ . (cid:3) Uniformly continuous minor-preserving maps.
Let B be a first-orderexpansion of R . In this section we show that if there exists a uniformly continu-ous minor-preserving map from Pol( B ) ∩ Can( R ) to Proj, then a specific minor-preserving map η (introduced in Section 9.6.1) or ρ (introduced in Section 9.6.2) isa uniformly continuous minor-preserving map from Pol( B ) ∩ Can( R , ≺ ) → Proj.9.6.1.
Factoring ≺ . In the following, 0 stands for { DR ∩ ≺ , PO ∩ ≺} and 1 stands for { PP } . Note that ≺ = ( DR ∩ ≺ ) ∪ ( PO ∩ ≺ ) ∪ PP . Let F be the equivalence relationon { DR ∩ ≺ , PO ∩ ≺ , PP } with the equivalence classes 0 and 1. Lemma 9.10.
For every f ∈ Pol( R ) ∩ Can( R , ≺ ) , the operation ξ ( f ) preserves F .Proof. Let f ∈ Pol( R ) ( k ) be canonical with respect to ( R , ≺ ) and suppose that O , . . . , O k are orbits of pairs contained in ≺ such that f ( O , . . . , O k ) ⊆ PP . Wehave to show that then f ( O ′ , . . . , O ′ k ) ⊆ PP for all orbits of pairs O ′ i such that E ( O ′ i , O i ). We claim that O ′ i ⊆ O i ◦ O i . If O i = PP then O ′ i = PP and the statementis clear since PP ◦ PP = PP . If O i = PO then O i ◦ O i = ≺ contains DR ∩ ≺ and PO ∩ ≺ .Similarly, if O i = DR then O i ◦ O i = ≺ , and again the claim follows. By (7), wehave f ( O ′ , . . . , O ′ k ) ⊆ f ( O ◦ O , . . . , O k ◦ O k ) ⊆ f ( O , . . . , O k ) ◦ f ( O , . . . , O k ) = PP . (cid:3) Proposition 9.11.
Let C ⊆ Pol( R ) ∩ Can( R , ≺ ) be a clone and let η : C → O { , } be given by η ( f ) := ( ξ ( f ) | { DR ∩ ≺ , PO ∩ ≺ , PP } ) /F. Then either every operation in η ( C ) is a projection, or C contains an operation f such that η ( f ) = ∧ . HE UNIQUE INTERPOLATION PROPERTY 31
Proof.
First observe that η ( C ) is an idempotent clone because every f ∈ C pre-serves PP , and preserves ⊥≺ = ( DR ∩ ≺ ) ∪ ( PO ∩ ≺ ) by Lemma 9.9. The well-knownclassification of clones over a two-element set by Post [Pos41] implies that if η ( C )contains an operation that is not a projection, then it must contain the Booleanminimum, maximum, minority, or majority operation. We will prove that the lastthree cases are impossible. Note that in these three cases C contains a ternaryoperation g such that η ( g ) is cyclic and η ( g )(0 , ,
1) = 1 or g (0 , ,
1) = 1. Hence, g ( ⊥≺ , ⊥≺ , PP ) , g ( ⊥≺ , PP , ⊥≺ ) , g ( PP , ⊥≺ , ⊥≺ ) ⊆ PP or g ( ⊥≺ , PP , PP ) , g ( PP , ⊥≺ , PP ) , g ( PP , PP , ⊥≺ ) ⊆ PP . Note that ⊥≺ ◦ ⊥≺ ◦ PP = ⊥≺ ◦ PP ◦ ⊥≺ = PP ◦ ⊥≺ ◦ ⊥≺ = ≺ and ⊥≺ ◦ PP ◦ PP = PP ◦ ⊥≺ ◦ PP = PP ◦ PP ◦ ⊥≺ = ≺ and we obtain g ( ≺ , ≺ , ≺ ) ⊆ PP from applying (7) twice. In particular, we have g ( DR ∩ ≺ , DR ∩ ≺ , DR ∩ ≺ ) ⊆ PP , which contradicts the fact that g preserves ⊥≺ . (cid:3) If C contains an operation f such that η ( f ) = ∧ , then C also contains such anoperation which is not only canonical with respect to ( R , ≺ ), but also with respectto R . To prove this, we first show the following lemma. Lemma 9.12.
Let f ∈ Pol( R ) be an operation of arity k which is canonicalwith respect to ( R , ≺ ) . Let O , . . . , O k be orbits of pairs of distinct elements in Aut( R ) such that f ( O , . . . , O k ) ⊆ ⊥ . Then there exists O ∈ { DR , PO } such that f ( O , . . . , O k ) ⊆ O .Proof. If O i ∈ { DR , PO } , then let O ′ i := ( O i ∩ ≺ ); otherwise, let O ′ i := O i . Then O ′ , . . . , O ′ i are orbits of pairs in Aut( R , ≺ ). Let O := ξ ( f )( O ′ , . . . , O ′ k ). By as-sumption, O ∈ { DR ∩ ≺ , DR ∩ ≺} . We have to show that for all orbits of pairs O ′′ , . . . , O ′′ k in Aut( R , ≺ ) such that O ′′ i ⊆ O i for all i ∈ { , . . . , k } we have that ξ ( f )( O ′′ , . . . , O ′′ k ) ∈ { O, O ⌣ } . Note that PP is contained in O ′ i ◦ ( O ′′ i ) ⌣ , and (7) im-plies that PP = ξ ( f )( PP , . . . , PP ) is contained in ξ ( f )( O ′ , . . . , O ′ k ) ◦ ξ ( f )( O ′′ , . . . , O ′′ k )and in ξ ( f )( O ′′ , . . . , O ′′ k ) ◦ ξ ( f )( O ′ , . . . , O ′ k ). On the other hand, we know that PP isnot contained in PO ◦ DR , which implies that if O ⊆ PO then O ⌣ ⊆ PO and if O ⊆ DR then O ⌣ ⊆ DR . (cid:3) Lemma 9.13.
Let f ∈ Pol( R ) (2) ∩ Can( R , ≺ ) be such that η ( f ) = ∧ . Then • f is injective, and • if O , O ∈ { PP , PPI , DR , PO } are distinct then f ( O , O ) ⊆ ⊥ .Proof. Let O and O be orbits of pairs of Aut( R , ≺ ). If both O and O aredistinct from EQ , then O ◦ O and O ◦ O contain PP or PPI by Corollary 9.6.Suppose for contradiction that f ( O , O ) = EQ . Then f ( O ◦ O , O ◦ O ) ⊆ f ( O , O ) ◦ f ( O , O ) ⊆ EQ ◦ EQ = EQ and if follows that f ( O ′ , O ′ ) ⊆ EQ for some O ′ , O ′ ∈ { PP , PPI } . Since f ( PP , PP ) ⊆ PP and f ( PPI , PPI ) ⊆ PPI ) we must have f ( PP , PPI ) ⊆ EQ or f ( PPI , PP ) ⊆ EQ .Note that f ( PP , PPI ) ⊆ EQ and f ( PPI , PP ) ⊆ EQ are equivalent, by (8). Since PP ⊆ PP ◦ PPI and PP ⊆ PPI ◦ PP we obtain that f ( PP , PP ) ⊆ EQ ◦ EQ = EQ , acontradiction to f ( PP , PP ) ⊆ PP . No suppose that one of O and O , say O , equals EQ . If O = EQ , there is nothingto be shown; otherwise, either PP or PPI is contained in O ◦ O by Corollary 9.6.In the first case we have f ( EQ , PP ) = f ( EQ ◦ EQ , O ◦ O ) ⊆ EQ ◦ EQ = EQ and in the second case, analogously, f ( EQ , PPI ) ⊆ EQ . Note that f ( EQ , PPI ) ⊆ EQ and f ( EQ , PP ) ⊆ EQ are equivalent by (8). So we have that f ( EQ , PP ) ⊆ EQ . Alsonote that f ( PP , ⊥ ) ⊆ ⊥ because η ( f ) = ∧ . Then f ( PP , PP ) ⊆ f ( EQ ◦ PP , PP ◦ ⊥ ) ⊆ EQ ◦ ⊥ = ⊥ , which contradicts the assumption that f preserves PP .To prove the second statement, first observe that if O , O ⊆ ≺ then ξ ( f )( O , O ) ∈ { DR ∩ ≺ , PO ∩ ≺} since η ( f ) = ∧ . If O , O ⊆ ≻ then we applythe same argument to O ⌣ instead of O and O ⌣ instead of O . Otherwise, bythe injectivity of f we know that ξ ( f )( O , O ) = EQ . Assume for contradictionthat ξ ( f )( O , O ) = PP . For i ∈ { , } , let O ′ i := O i if O i ⊆ ≺ and otherwise let O ′ i := PO ∩ ≺ . Then one can check using the composition table (Table 1) andLemma 9.5 that O ′ i ⊆ O i ◦ PP . Therefore, f ( O ′ , O ′ ) ⊆ f ( O ◦ PP , O ◦ PP ) ⊆ f ( O , O ) ◦ f ( PP , PP ) = PP ◦ PP = PP . On the other hand, O i ⊆ ≻ for some i ∈ { , } , and hence O ′ i = PO ∩ ≺ . Since η ( f ) = ∧ we have f ( O ′ , O ′ ) ⊆ ⊥ , a contradiction. (cid:3) Lemma 9.14.
Let C ⊆ Pol( R ) ∩ Can( R , ≺ ) be a clone that contains a binaryoperation f such that η ( f ) = ∧ . Then C contains an operation g such that η ( g ) = ∧ and g ∈ Can( R ) .Proof. We claim that the operation g ( x, y ) := f (cid:0) f ( x, y ) , f ( y, x ) (cid:1) satisfies the requirements. It is clear that η ( f ) = ∧ . To show that g is canonicalwith respect to R , let O , O be two orbits of pairs in Aut( R ). We have to showthat g ( O , O ) is contained in one orbit of pairs in Aut( R , ≺ ). If O = O then g ( O , O ) ⊆ O , and we are done, so suppose that O = O . Case 1. O = EQ and O = EQ . Then f ( O , O ) ⊆ ⊥ by item (2) of Lemma 9.13.Hence, the statement follows from Lemma 9.12. Case 2. O = EQ and O ∈ { PP , PPI } . Then the statement follows directly fromthe assumption that f is canonical with respect to ( R , ≺ ). Case 3. O = EQ and O ∈ { PO , DR } . Define O ′ := ( O ∩ ≺ ).Note that O = O ′ ∪ ( O ′ ) ⌣ . Since f is injective (Lemma 9.13), we have ξ ( f )( EQ , O ′ ) ∈ { ( PO ∩ ≺ ) , ( DR ∩ ≺ ) , ( PO ∩ ≻ ) , ( DR ∩ ≻ ) } . Using observation (8) weget that f ( EQ , O ) = f ( EQ , O ′ ) ∪ f ( EQ , O ′ ) ⌣ ∈ { DR , PO } . Case 4. O = EQ and O = EQ . The statement can be shown analogously tothe cases above. This concludes the proof. (cid:3) Restricting to ⊥≺ . Recall that ⊥≺ = ( DR ∪ PO ) ∩ ≺ is preserved by everypolymorphism of R (Lemma 9.9). In the following, 0 stands for DR ∩ ≺ and 1stands for PO ∩ ≺ . HE UNIQUE INTERPOLATION PROPERTY 33
Proposition 9.15.
Let C ⊆ Pol( R ) ∩ Can( R , ≺ ) be a clone and let ρ : C → O { , } be given by ρ ( f ) := ξ ( f ) | { , } . Then either every operation in ρ ( C ) is a projection, or ρ ( C ) contains a ternarycyclic operation.Proof. The statement follows from Post’s result [Pos41], because each of the clonesgenerated by min, max, majority, and minority contains a ternary cyclic operation. (cid:3)
Canonical pseudo-cyclic polymorphisms.
In this section we prove the goalstatement from the beginning of Section 9.6 (Corollary 9.17).
Theorem 9.16.
Let C ⊆ Pol( R ) ∩ Can( R , ≺ ) be a clone that contains • a binary operation g such that η ( f ) = ∧ , and • a ternary operation f such that ρ ( f ) is cyclic.Then ξ (cid:0) C ∩ Can( R ) (cid:1) contains a cyclic operation.Proof. Let h ( x, y, z ) := g ( g ( x, y ) , z ) h ′ ( x, y, z ) := h (cid:0) h ( x, y, z ) , h ( y, z, x ) , h ( z, x, y ) (cid:1) h ′′ ( x, y, z ) := f (cid:0) h ′ ( x, y, z ) , h ′ ( y, z, x ) , h ′ ( z, x, y ) (cid:1) We verify that h ′′ is canonical with respect to Pol( B ) and that ξ ( h ′′ ) is cyclic.Let O , O , O be orbits of pairs in Aut( R ). Note that h ′′ ∈ Pol( R ); hence, if O = O = O then h ′′ ( O , O , O ) , h ′′ ( O , O , O ) , h ′′ ( O , O , O ) ⊆ O and weare done in this case. Suppose that O , O , and O are not pairwise distinct. Since g is injective (Lemma 9.13), so is h . By Proposition 9.14, we have that g, h ∈ Can( R )and O ′ := ξ ( h )( O , O , O ) = EQ ,O ′ := ξ ( h )( O , O , O ) = EQ ,O ′ := ξ ( h )( O , O , O ) = EQ . We claim that O ′′ := h ′ ( O , O , O ) ⊆ h ( O ′ , O ′ , O ′ ) ⊆ ⊥ . If O ′ = O ′ then g ( O ′ , O ′ ) ⊆ ⊥ by Lemma 9.13. If O ′ , O ′ ⊆ ⊥ then g ( O ′ , O ′ ) ⊆ ⊥ since g ∈ Pol( R ) preserves ⊥ . In both cases, g ( g ( O ′ , O ′ ) , O ′ ) ⊆ g ( ⊥ , O ′ ) ⊆ ⊥ . Oth-erwise, O ′ = O ′ ∈ { PP , PPI } and O ′ = O ′ in which case g ( g ( O ′ , O ′ ) , O ′ ) ⊆ g ( O ′ , O ′ ) ⊆ ⊥ . Similarly, O ′′ := h ′ ( O ′ , O ′ , O ′ ) ⊆ ⊥ and O ′′ := h ′ ( O ′ , O ′ , O ′ ) ⊆⊥ . Since f ∈ Pol( R ) preserves ⊥ we have that f ( O ′′ , O ′′ , O ′′ ) ⊆ ⊥ .Lemma 9.12 implies that there exists O ′′′ ∈ { DR , PO } such that f ( O ′′ , O ′′ , O ′′ ) ⊆ O ′′′ . Similarly, there are O ′′′ , O ′′′ ∈ { DR , PO } such that f ( O ′′ , O ′′ , O ′′ ) ⊆ O ′′′ and f ( O ′′ , O ′′ , O ′′ ) ⊆ O ′′′ . For i ∈ { , , } , let O ∗ i := O ′′ i ∩ ≺ . Then f ( O ∗ , O ∗ , O ∗ ) , f ( O ∗ , O ∗ , O ∗ ) , f ( O ∗ , O ∗ , O ∗ ) ⊆ ≺ by Lemma 9.9. Since f is cyclicon { DR ∩ ≺ , PO ∩ ≺} it follows that f ( O ∗ , O ∗ , O ∗ ) ⊆ ( O ′′′ ∩ ≺ ), which impliesthat O ′′′ = O ′′′ . Similarly, we obtain O ′′′ = O ′′′ ; this concludes the proof that h ′′ ∈ Can( R ) and ξ ( h ′′ ) is cyclic. (cid:3) Corollary 9.17.
Let B be a first-order expansion of R . If there exists a uniformlycontinuous minor-preserving map from Pol( B ) ∩ Can( R ) to Proj , then η or ρ is auniformly continuous minor-preserving map from Pol( B ) ∩ Can( R , ≺ ) → Proj . Proof.
We prove the contraposition, and suppose that neither η nor ρ is not a uni-formly continuous minor-preserving map from Pol( B ) ∩ Can( R , ≺ ) to Proj. ByProposition 9.11, Pol( B ) contains a binary operation f such that η ( f ) = ∧ . ByProposition 9.15, C contains a ternary operation g such that ρ ( g ) is cyclic. ByTheorem 9.16, C contains a ternary operation c such that ξ ( c ) is cyclic, and hence C does not have a uniformly continuous minor-preserving map to Proj by Propo-sition 3.18. (cid:3) Verifying the UIP.
In this section we show that if η , or ρ , is a uniformlycontinuous map from Pol( C ) ∩ Can( R , ≺ ) to Proj, then is has the UIP. Theorem 9.18.
Let C be a first-order expansion of R such that the clone η (cid:0) Pol( C ) ∩ Can( R , ≺ ) (cid:1) is isomorphic to Proj . Then η has the UIP with respect to Pol( C ) over ( R , ≺ ) .Proof. We use Theorem 1.5. Let A , A be the two independent elementary sub-structures of ( R , ≺ ) from Lemma 9.7. Note that ⊥ ⊆ ⊥≺ ◦ PP ◦ PP and ⊥ ⊆ PP ◦ PP ◦ ⊥≺ . so there are elements a j,k of R , for j ∈ { , } and k ∈ { , , , } , such that( a , , a , ) ∈ PP , ( a , , a , ) ∈ PP , ( a , , a , ) ∈ ⊥≺ , ( a , , a , ) ∈ ⊥ , and ( a , , a , ) ∈ ⊥≺ , ( a , , a , ) ∈ PP , ( a , , a , ) ∈ PP , ( a , , a , ) ∈ ⊥ . Let F be a finite subset of the domain of R and 2-rich with respect to Pol( B ) (seeLemma 3.15); we may assume without loss of generality that F contains a j,k for all j ∈ { , } and k ∈ { , , , } .If η does not have the UIP, then we may apply Lemma 9.8 to η and F ; let f ∈ Pol( C ) (2) and α , α ∈ Aut( R ; ≺ ) be the operations as in the statement ofLemma 9.8. Let b j,k := α ( a j,k ) for k ∈ { , } and let b j,k := α ( a j,k ) for k ∈ { , } .Then ( b , , b , ) , ( b , , b , ) ∈ ⊥≺ and ( b , , b , ) , ( b , , b , ) ∈ PP , because α i pre-serves ⊥≺ and PP . Moreover, for j ∈ { , } we have ( b j, , b j, ) ∈ PP and ( b j, , b j, ) ∈ ⊥≺ by the properties of α i . This implies that ( f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , )) ∈ ⊥ .On the other hand, we have that • (cid:0) f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , ) (cid:1) ∈ PP , because η ( f (id , α ) | F ) = π ℓ ,( a , , a , ) ∈ PP , and ( b , , b , ) ∈ PP . • (cid:0) f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , ) (cid:1) ∈ PP , because ( a ℓ, , a ℓ, ) ∈ PP and( b ℓ +1 , , b ℓ +1 , ) ∈ PP . • (cid:0) f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , ) (cid:1) ∈ PP , because η ( f (id , α ) | F ) = π ℓ +1 ,( b , , b , ) ∈ PP , and ( a , , a , ) ∈ PP .Since PP ◦ PP ◦ PP = PP , the pair (cid:0) f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , ) (cid:1) is also in PP , acontradiction. The statement now follows from Theorem 1.5. (cid:3) Theorem 9.19.
Let C be a first-order expansion of R such that the clone η (Pol( C ) ∩ Can( R , ≺ )) is isomorphic to Proj . Then ρ has the UIP with respect to Pol( C ) over ( R , ≺ ) .Proof. We again use Theorem 1.5; the proof is very similar to the proof of Theo-rem 9.18. Let A , A be the two independent elementary substructures of ( R , ≺ )from Lemma 9.7. Let f ∈ Pol( C ) (2) and u , u ∈ Aut( R , ≺ ) be such that for HE UNIQUE INTERPOLATION PROPERTY 35 i ∈ { , } we have ℑ ( u i ) ⊆ A i and f (id , u i ) is canonical with respect to ( R , ≺ ).Suppose for contradiction that ρ ( f (id , u )) = ρ ( f (id , u )). Note that PP ⊆ ( DR ∩ ≺ ) ◦ PP ◦ ( DR ∩ ≺ ) PP ⊆ ( PO ∩ ≺ ) ◦ PP ◦ ( PO ∩ ≺ )so there are elements a j,k of R , for j ∈ { , } and k ∈ { , , , } , such that( a , , a , ) ∈ DR ∩ ≺ , ( a , , a , ) ∈ PP , ( a , , a , ) ∈ DR ∩ ≺ , ( a , , a , ) ∈ PP , ( a , , a , ) ∈ PO ∩ ≺ , ( a , , a , ) ∈ PP , ( a , , a , ) ∈ PO ∩ ≺ , ( a , , a , ) ∈ PP . Let F be a finite subset of the domain of R which is 2-rich with respect to Pol( B )(see Lemma 3.15); we may assume without loss of generality that F contains a j,k for all j ∈ { , } and k ∈ { , , , } .Let α and α be the automorphisms of ( R ; ≺ ) obtained from applyingLemma 9.8 to ρ and F so that ρ ( f (id , α ) | F ) = π ℓ and ρ ( f (id , α ) | F ) = π ℓ +1 where indices are considered modulo two.Let b j,k := α ( a j,k ) for k ∈ { , } and let b j,k := α ( a j,k ) for k ∈ { , } . Then( b , , b , ) , ( b , , b , ) ∈ DR ∩ ≺ and ( b , , b , ) , ( b , , b , ) ∈ PO ∩ ≺ , because α i pre-serves DR ∩ ≺ and PO ∩ ≺ . Moreover, for j ∈ { , } we have ( b j, , b j, ) , ( b j, , b j, ) ∈ PP by the properties of α i . This implies that ( f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , )) ∈ PP .On the other hand, we have that • ( f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , )) ∈ DR ∩ ≺ , because η ( f (id , α ) | F ) = π ℓ ,( a , , a , ) ∈ DR ∩ ≺ , and ( b , , b , ) ∈ DR ∩ ≺ . • ( f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , )) ∈ PP , because ( a ℓ, , a ℓ, ) ∈ PP and( b ℓ +1 , , b ℓ +1 , ) ∈ PP . • ( f ( a ℓ, , b ℓ +1 , ) , f ( a ℓ, , b ℓ +1 , )) ∈ PO ∩ ≺ , because η ( f (id , α ) | F ) = π ℓ +1 ,( b , , b , ) ∈ PO ∩ ≺ , and ( a , , a , ) ∈ PO ∩ ≺ .Since (cid:0) ( DR ∩ ≺ ) ◦ PP ◦ ( PO ∩ ≺ ) (cid:1) ∩ PP = ∅ , we reached a contradiction. The statementnow follows from Theorem 1.5. (cid:3) Proof of Theorem 9.1.
To prove Theorem 9.1, we verify item (3) of Theo-rem 1.2; the complexity dichotomy then follows as in the proof of Corollary 1.3). IfPol( B ) ∩ Can( R ) has a uniformly continuous minor-preserving map to Proj, thenby Corollary 9.17 either η or ρ is a uniformly continuous minor-preserving mapfrom Pol( B ) ∩ Can( R , ≺ ) to Proj. Item (3) of Theorem 1.2 then follows fromTheorem 9.18 and from Theorem 9.19, respectively.10. First-order expansions of the random poset
Our third application concerns poset constraint satisfaction problems, re-derivingresults of Kompatscher and Van Pham [KP18]. Let ( P ; ≤ ) be the countable homoge-neous poset whose age consists of all finite posets. We write x < y for x ≤ y ∧ x = y ,and x ⊥ y if neither x ≤ y nor y ≤ x holds. Theorem 10.1.
Let C be a first-order expansion of ( P ; <, ⊥ ) . Then CSP( C ) is inP or NP-complete. In fact, Kompatscher and Van Pham classify the complexity of the CSP for theeven larger class of first-order reducts of ( P ; <, ⊥ ). However, they reduce the moregeneral case to the classification for first-order expansions of ( P ; <, ⊥ ). The structure ( P ; <, ⊥ ) is not Ramsey (because it has no first-order definablelinear order; see [Bod15]); however, it is the reduct of a homogeneous Ramseystructure with a finite relational signature. To define this structure, consider theclass P of all finite structures of the form ( P ; <, ⊥ , ≺ ) where ( P ; <, ⊥ ) embeds into( P ; <, ⊥ ) and ≺ is a linear extension of < . We write y (cid:23) x instead of ¬ ( x ≺ y ). Itis easy to show that P is an amalgamation class and that the { <, ⊥} -reduct of itsFra¨ıss´e-limit is isomorphic to ( P ; <, ⊥ ); so we denote the limit by ( P ; <, ⊥ , ≺ ). For arecent and transparent proof that ( P ; <, ⊥ , ≺ ) has the Ramsey property, see [NR18].Note that the structure that we obtain from R by replacing the two relations DR and PO by the relation ⊥ := DR ∪ PO , identifying PP with < , is isomorphic to( P ; <, ⊥ ), and that an isomorphism can be chosen so that it also preserves ≺ . Withthis observation it is straightforward to specialise the proof of the dichotomy forfirst-order expansions of R from Section 9 to first-order expansions of ( P ; <, ⊥ ). References [BB19] Manuel Bodirsky and Bertalan Bodor. Structures with small orbit growth, 2019.Preprint available at ArXiv:1810.05657.[BC09] Manuel Bodirsky and Hubie Chen. Qualitative temporal and spatial reasoning revis-ited.
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