Equations in oligomorphic clones and the Constraint Satisfaction Problem for -categorical structures
There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete core, equivalent to its polymorphism clone satisfying a certain non-trivial linear identity modulo outer embeddings. The second conjecture, challenging the approach via model-complete cores by reflections, states that tractability is equivalent to the linear identities (without outer embeddings) satisfied by its polymorphisms clone, together with the natural uniformity on it, being non-trivial.
We prove that the identities satisfied in the polymorphism clone of a structure allow for conclusions about the orbit growth of its automorphism group, and apply this to show that the two conjectures are equivalent. We contrast this with a counterexample showing that -categoricity alone is insufficient to imply the equivalence of the two conditions above in a model-complete core.
Taking a different approach, we then show how the Ramsey property of a homogeneous structure can be utilized for obtaining a similar equivalence under different conditions.
We then prove that any polymorphism of sufficiently large arity which is totally symmetric modulo outer embeddings of a finitely bounded structure can be turned into a non-trivial system of linear identities, and obtain non-trivial linear identities for all tractable cases of reducts of the rational order, the random graph, and the random poset.
Finally, we provide a new and short proof, in the language of monoids, of the existence and uniqueness of the model-complete core of an -categorical structure.
In order to keep the presentation of the wide topic of the present article as compact as possible, we postpone most definitions to an own preliminaries section (Section 2).
1.1. Constraint Satisfaction Problems
The Constraint Satisfaction Problem (CSP) of a structure in a finite relational language, denoted by , is the computational problem of deciding its primitive positive theory: given a sentence which is an existentially quantified conjunction of atomic formulas, decide whether or not holds in . When has a finite domain, then its CSP is in NP, and it has been conjectured that its CSP is always either NP-complete or polynomial-time solvable [FV99]. While the CSP of structures with an infinite domain can be of any complexity [BG08], and can in particular be undecidable, for a certain class of infinite-domain CSPs a similar dichotomy conjecture as for the finite case has been stated. In fact, two such conjectures have been brought up via different approaches; in the present article we first establish their equivalence, and then investigate in more detail the tractability conditions of the two conjectures.
The range of both conjectures are reducts of finitely bounded homogeneous structures, a (proper) subclass of the countable -categorical structures. It is well-known, and easy to see from the definition, that the CSP of such structures is contained in NP; both conjectures state that it is always either NP-complete or contained in P, but each conjecture gives a different delineation between the (NP-)hard and the tractable (i.e., polynomial-time solvable) cases.
1.2. The first conjecture
The first and older conjecture, formulated by Bodirsky and Pinsker (cf. [BPPa]), is based on the notion of the model-complete core of an -categorical structure, which can be viewed as the simplest representative in the class of an -categorical structure with respect to the equivalence relation of homomorphic equivalence. We have the following.
Theorem 1.1 (Bodirsky [Bod07]).
Every countable -categorical structure is homomorphically equivalent to a model-complete core. This model-complete core is unique up to isomorphism and itself -categorical.
The idea leading to the first conjecture is that the complexity of the CSP of a structure in the range of the conjecture is determined by which finite structures have a primitive positive (pp-) interpretation with parameters in its model-complete core . This approach builds on two facts: the first fact being that homomorphically equivalent structures have the same primitive positive theory, and hence and have equal CSPs; and the second fact being that if a structure has a primitive positive interpretation with parameters in an -categorical model-complete core , then reduces to in polynomial time. It is a well-known fact that the structure
pp-interprets all finite structures, and that its CSP is NP-complete.
Let be a reduct of a finitely bounded homogeneous structure, and let be its model-complete core. Then one of the following holds.
pp-interprets with parameters (and consequently, is NP-complete).
is polynomial-time solvable.
From our remarks above it follows that if condition (i) in Conjecture 1.2 holds, then is indeed NP-complete. What remains to prove is that if this condition is not satisfied, then is tractable. The following equivalent conditions have been established for this situation via the polymorphism clone of a structure ((ii) in [BP15b], and (iii), (iv) in [BP16a]). We denote the clone of projections on the set by ; then .
Let be an -categorical model-complete core. The following are equivalent.
does not pp-interpret with parameters.
No stabilizer of maps homomorphically and continuously to .
No stabilizer of maps homomorphically to .
has a Siggers term modulo outer embeddings, i.e., there exist such that the identity
holds in .
Observe that the very recent condition (iv) turns, for the first time, the supposed tractability criterion of Conjecture 1.2 into a positive statement, nourishing the hope for a positive answer to the conjecture.
1.3. The second conjecture
The second and younger conjecture was born from the observation that the usage of homomorphic equivalence and pp-interpretations might not be optimal in the order which leads to Conjecture 1.2, as the crucial structure might, for example, be homomorphically equivalent to a structure with a pp-interpretation in , but not pp-interpretable with parameters by the model-complete core of . This suggests the following weaker conjecture, which does use the reductions by homomorphic equivalence and pp-interpretations in the best possible way [BOP].
Let be a reduct of a finitely bounded homogeneous structure. Then one of the following holds.
is homomorphically equivalent to a structure with a pp-interpretation in (and consequently, is NP-complete).
is polynomial-time solvable.
It has been remarked in [BOP] that the two conjectures are equivalent for finite structures. While more likely to be true, one disadvantage of Conjecture 1.4 is that there is no unique optimal structure that can be pp-interpreted in , as opposed to the model-complete core for homomorphic equivalence. Similarly to (ii) in Theorem 1.3, the authors of [BOP] did however provide an equivalent tractability criterion using identities and topology.
Let be -categorical. The following are equivalent.
is not homomorphically equivalent to a structure with a pp-interpretation in .
does not have a uniformly continuous h1 clone homomorphism to .
1.4. Equivalence of the conjectures
The results known so far concerning identities in polymorphism clones were shown for all -categorical model-complete cores, rather than for the considerably more restricted class of structures concerned by the conjectures; it is probably fair to say that it seemed inconceivable that assumptions like finite boundedness would be useful when proving such structural results (these assumptions are, however, essential for the algorithmic aspects of the CSPs). Therefore, the most likely way of showing the equivalence of the conjectures seemed by proving that for all -categorical model-complete cores, conditions (iv) in Theorem 1.3 and (ii) in Theorem 1.5 are equivalent: that is, since the other implication is well-known and easy, that a Siggers term modulo outer embeddings prevents a uniformly continuous h1 clone homomorphism to .
We will, however, provide a counterexample, basically the atomless Boolean algebra with the right choice of relations, showing that this is not true in general.
There exists an -categorical model-complete core structure such that:
No stabilizer of has a continuous clone homomorphism to (and hence, by Theorem 1.3, has a Siggers term modulo outer embeddings).
has a uniformly continuous h1 clone homomorphism to .
Surprisingly, on the other hand, it turns out that every structure which is a counterexample as above must have at least double exponential orbit growth. This is remarkable in that it is the first instance discovered where structural higher-arity information about the polymorphism clone of an -categorical structure yields information about its automorphism group.
Let be a structure with the properties stated in Theorem 1.6. Then its automorphism group must have at least double exponential orbit growth.
From this, it is straightforward to derive the equivalence of the CSP conjectures, answering Problem 8.3 in [BOP] to the positive, and showing that the implication from (4) to (3) in Corollary 5.3 of [BP16a] holds.
Let be a reduct of a structure which is homogeneous in a finite relational language, and let be its model-complete core. Then the following are equivalent.
Some stabilizer of has a continuous clone homomorphism to .
has a uniformly continuous h1 clone homomorphism to .
has a uniformly continuous h1 clone homomorphism to .
1.5. The Ramsey property
Via an alternative approach involving Ramsey theory, we will then show a statement similar to Corollary 1.8 under different, and incomparable, conditions. Although this might seem irrelevant for CSPs considering our results above which cover the entire range of Conjectures 1.2 and 1.4, it is interesting that various conditions of very different nature seem to imply this statement, while at the same time we know from our counterexample in Theorem 1.6 that -categoricity alone is not sufficient. Observe that in the following theorem, there is no requirement of finite language, or orbit growth, or even -categoricity; on the other hand, we require the non-trivial linear identities to be satisfied modulo embeddings of an ordered Ramsey structure.
Let be a reduct of an ordered homogeneous Ramsey structure . If satisfies a non-trivial set of linear identities modulo outer embeddings of , then does not have a uniformly continuous h1 clone homomorphism to .
Note that Theorem 1.9 corresponds to the contrapositive of the non-trival implication from (iii) to (i) in Corollary 1.8, via the fact that (i) there is equivalent to the existence of a Siggers term modulo outer embeddings.
We would also like to remark that the situation of Theorem 1.9 is particularly interesting for the approach to Conjectures 1.2 and 1.4 via canonical functions, as surveyed in [BP11] (cf. also the recent [BP16b]); indeed, many of the successful CSP classifications via that approach yield tractable situations as in Theorem 1.9.
Corollary 1.8, combined with Theorem 1.3, implies that if an -categorical model-complete core has a Siggers polymorphism modulo outer embeddings, then does not have a uniformly continuous h1 clone homomorphism onto . It does not imply that in that situation, satisfies non-trivial linear identities, i.e., that does not have an h1 clone homomorphism to disregarding the uniformity on . The situation in Theorem 1.9 is similar. It is hitherto unknown under which conditions non-trivial linear identities modulo outer embeddings imply non-trivial linear identities in a polymorphism clone; as of today, we cannot even refute the possibility that the existence of an h1 homomorphism to implies the existence of a uniformly continuous such homomorphism in general. This question, for -categorical model-complete cores, corresponds to the implication from (6) to (4) in [BP16a].
Approaching this problem, we are going to show that under the assumption of finite boundedness, and stronger identities than the Siggers identity modulo outer embeddings, we can derive the satisfaction of non-trivial linear identities in a polymorphism clone.
Let be a reduct of a finitely bounded homogeneous structure which is given by a set of forbidden substructures all of which have size at most . If contains a -ary polymorphism which is totally symmetric modulo outer embeddings of , i.e., for all permutations of satisfies an identity of the form
where , then does not have an h1 clone homomorphism to .
We are further going to show that under different conditions, trading finite boundedness and total symmetry for near unanimity and strong preservation of the relations of , non-trivial linear identities can be derived as well.
Let be a structure which is homogeneous in a finite relational language. Suppose that is a -ary operation, where , satisfying the following two conditions:
is a strong polymorphism of , i.e., preserves all relations of and their negations;
is a near unanimity (nu) function modulo outer embeddings of , i.e., satisfies identities of the form
Then no reduct of with the polymorphism has an h1 clone homomorphism to .
From Theorems 1.10 and 1.11 and the classifications in [BK08], [BK09], [BP15a], and [KP] it follows directly that most reducts of the rationals, the random graph, and the random partial order with tractable CSPs have a polymorphism clone satisfying non-trivial linear identities. Using a similar proof technique for the remaining cases we obtain the following.
Let be a reduct of one of the following structures:
the order of the rational numbers;
the random partial order;
the random graph.
Then has a uniformly continuous h1 clone homomorphism to if and only if it has an h1 clone homomorphism to . When has a finite language, then its CSP is tractable if and only if satisfies a non-trivial set of linear identities.
Theorem 1.1 above stating the existence and uniqueness of the model-complete core of an -categorical structure is of central importance for Conjecture 1.2, and calculating the model-complete core of structures has been an integral part of the major successful CSP classifications so far. While the alternative more recent Conjecture 1.4 threatened to make the notion obsolete for its context, the equivalence of the conjectures established in the present article provides further evidence of the decisive role of model-complete cores for CSPs.
Observe that the notion of a model-complete core is defined via the endomorphism monoid of a structure (density of the invertibles in the monoid), so in particular structures with isomorphic (as topological monoids, cf. [BPPb, BEKP]) endomorphism monoids are either both model-complete cores, or none of them is. Moreover, by the theorem of Ryll-Nardzewski, Engeler, and Svenonius [Hod97], the condition of -categoricity of a countable structure is equivalent to oligomorphicity of its automorphism group, again captured by its endomorphism monoid. It thus seems natural to have a proof of Theorem 1.1 in the language of transformation monoids, without reference to the particular language of a structure. The original and quite lengthy proof due to Bodirsky, however, does work with structures, and it is not obvious how to translate it into a proof via monoids.
We shall provide a new, short proof of Theorem 1.1 using topological monoids, which perhaps reflects better the combinatorial content of the theorem, and in particular connects it to the recent notion of reflections (which in turn leads to the other conjecture, Conjecture 1.4). Set naturally in the language of monoids, our proof yields simultaneously the generalization of the theorem to weakly oligomorphic structures given in [PP16].
1.8. Organization of this article
We provide definitions and notation in Section 2. The main results about the two CSP conjectures, Theorems 1.6 and 1.7, Corollary 1.8, and Theorem 1.9, are shown in Section 3. In Section 4, we investigate the relationship between linear identities modulo outer embeddings and those without outer embeddings, proving Theorems 1.10, 1.11, and 1.12. The new proof of Theorem 1.1, and new insights connecting it directly to reflections, are provided in Section 5.
We explain the notions which appeared in the introduction, and fix some notation for the rest of the article. For undefined universal algebraic concepts and more detailed presentations of the notions presented here we refer to [BS81, Ber11]. For notions from model theory we refer to [Hod97].
2.1. Polymorphism clones, automorphisms, and invertibles
We denote relational structures by , etc. When is a relational structure, we reserve the symbol for its domain. We write for its polymorphism clone, i.e., the set of all finitary operations on which preserve all relations of . The polymorphism clone is always a function clone, i.e., it is closed under composition and contains all projections. The unary functions in are precisely the endomorphisms of , denoted by . The endomorphisms which are bijections and whose inverse function is also an endomorphism are precisely the automorphisms of . We denote the set of automorphisms of by .
When is any function clone (not necessarily the polymorphism clone of a structure), then still the unary functions in form a transformation monoid, and the unary invertible functions in (i.e., those having an inverse in ) form a permutation group, the group of invertibles of . We write for the domain of the function clone .
2.2. Clone homomorphisms
A clone homomorphism from a function clone to a function clone is a mapping which
preserves arities, i.e., it sends every function in to a function of the same arity in ;
preserves each projection, i.e., it sends the -ary projection onto the -th coordinate in to the same projection in , for all ;
preserves composition, i.e., for all -ary functions and all -ary functions in .
For all we denote the -ary projection onto the -th coordinate by , in any function clone and irrespectively of the domain of that clone. This slight abuse of notation allows us, for example, to express the second item above by writing .
A mapping is called an h1 clone homomorphism if it preserves arities and composition with projections, i.e., for all -ary functions in and all -ary projections . If, in addition, preserves each projection, then it is called a strong h1 clone homomorphism. Note that an h1 clone homomorphism to is automatically strong.
2.3. Identities / Equations
The clone homomorphisms are those mappings between clones preserving identities, i.e., universally quantified equations between terms built from the functions in clones (with an appropriate language providing a symbol for every element of the clones). The h1 clone homomorphisms are those mappings between clones preserving identities of height one, and strong h1 clone homomorphisms preserve all identities of height at most one, also known as linear identities, i.e., identities where no nesting of functions is allowed; cf. [BOP]. A linear identity modulo outer unary functions is a universally quantified equation of the form , where are unary and are terms of height at most one.
When is a relational structure, then a linear identity of modulo outside endomorphisms (automorphisms, embeddings) is an identity of the form which holds in , where are endomorphisms (automorphisms, embeddings) of , and terms over of height at most one. Similarly, we speak of linear identities modulo outside embeddings (automorphisms, embeddings) of , where is some other structure, with the obvious meaning.
A set of identities is non-trivial if it is unsatisfiable in the clone . Therefore, a function clone satisfies a non-trivial set of identities if and only if it does not have a clone homomorphism to ; it satisfies a non-trivial set of linear identities if and only if it does not have an h1 clone homomorphism to . It follows from the compactness theorem of first-order logic that these non-trivial sets of identities can be chosen to be finite.
When is a function clone, and is a finite subset of its domain, then the (pointwise) stabilizer of in , denoted by , is the function clone of all satisfying for all . We emphasize that we always understand stabilizers to be pointwise, and always of a finite set.
We remark that when is a relational structure and is finite, then the stabilizer of in is the polymorphism clone of the structure obtained by enriching by a unary singleton relation for every .
Every function clone is naturally equipped with the topology of pointwise convergence: in this topology, a sequence of -ary functions converges to an -ary function on the same domain if and only if for all -tuples of the domain the functions agree with on for all but finitely many . Therefore, every function clone gives rise to an abstract topological clone which reflects this topology as well as the composition structure of the clone [BPPb].
We always imagine function clones to carry the pointwise convergence topology, which is, in the case of a countable domain, in fact induced by a metric, and in general by a uniformity [BPPb, GP16, Sch15]. Then a mapping , where and are function clones, is continuous if and only if for all and all finite sets there exists a finite set such that for all of the same arity as , if agrees with on , then agrees with on . It is uniformly continuous if and only if for all and all finite there exists a finite such that whenever two -ary functions agree on , then their images agree on . Note that in the case of mappings , uniform continuity means that for every there exists a finite such that only depends on the restriction of to , for all -ary . When is an h1 clone homomorphism, then can be chosen independently of .
We remark that the polymorphism clones of relational structures are precisely the function clones which are complete with respect to this uniformity (or, put differently, closed in the function clone of all functions of the domain). Function clones on a finite domain are discrete.
2.6. Oligomorphicity, -categoricity and orbit growth
Recall that by the theorem of Ryll-Nardzewski, Engeler, and Svenonius, a countable relational structure is -categorical if and only if its automorphism group is oligomorphic, i.e., for every , the natural componentwise action of on has only finitely many orbits. In particular finite structures are always -categorical. Every countable -categorical structure thus induces a monotone function on the positive natural numbers which assigns to every the number of orbits of -tuples with respect to ; we call this function the orbit growth of (or of ). There exist -categorical structures of arbitrarily fast orbit growth.
Similarly, we say that a function clone is oligomorphic if its group of unary invertibles is, and we can hence naturally speak of the orbit growth of an oligomorphic function clone.
2.7. Homogeneity, finite boundedness, and the Ramsey property
The -categorical structures concerned by the conjectures above are reducts of finitely bounded homogeneous structures. Here, following [Tho91] and numerous subsequent authors, we define a reduct of a relational structure to be a relational structure on the same domain all of whose relations have a first-order definition in without parameters.
A relational structure is homogeneous if every partial isomorphism between finite substructures extends to an automorphism of the entire structure. A countable relational structure is finitely bounded if it has a finite signature, and there exists a finite set of finite structures in its signature such that contains precisely those structures as induced substructures which embed no member of . We are going to call every such a set of forbidden substructures (with respect to ).
A relational structure is Ramsey if for all finite induced substructures of and all functions from the isomorphic copies of in to there exists an isomorphic copy of in on which is constant. It is ordered if it first-order defines (without parameters) a linear order on its domain. For more details about Ramsey structures in this context, we refer to the surveys [BP11], [Bod12].
2.8. Homomorphic equivalence and model-complete cores
When relational structures and have the same signature, then we say that and are homomorphically equivalent if there exists a homomorphism and a homomorphism . A relational structure is called a model-complete core if is dense in , i.e., for every endomorphism of and every finite subset of there exists an automorphism of which agrees with on . When is finite, then this means that every endomorphism is an automorphism, and is simply called a core.
Similarly, we call a function clone or a transformation monoid a model-complete core if the group of its invertible functions is dense in its unary functions.
For a finite relational signature and a -structure , the constraint satisfaction problem of , or for short, is the membership problem for the class
An alternative definition of is via primitive positive (pp-) sentences. Recall that a pp-formula over is a first order formula which only uses predicates from , conjunction, equality, and existential quantification. can equivalently be phrased as the membership problem of the set of pp-sentences which are true in .
3. Equivalence of the Conjectures, and the Ramsey Property
This section is divided into three parts: we first prove Theorem 1.7 and Corollary 1.8 in Section 3.1, and then provide the counterexample of Theorem 1.6 in Section 3.2. Finally, we turn to applications of the Ramsey property in Section 3.3, proving Theorem 1.9.
3.1. Orbit Growth and Equivalence of the conjectures
Let be a function clone, and let be a subset of its domain with . Then a function is a retractional witness for with respect to if the restriction of to is a projection on for all .
For an -ary , we call an index fundamental for with respect to if there exists a retractional witness for with respect to such that is the -th -ary projection on .
The ambiguity degree of with respect to is the number of its fundamental indices with respect to . The ambiguity degree of is the supremum of the ambiguity degrees of its members with respect to sets of at least two elements:
Let be a function clone of infinite ambiguity degree. Then the componentwise action of the group of unary invertible functions in on has at least orbits, for all .
Given , pick of ambiguity degree at least with respect to of at least two elements; by taking a subset, we may assume . By identifying some variables of with variables corresponding to a fundamental index of , we may assume that all indices of are fundamental, and that . For any non-empty subset of , pick an -tuple of the form (applied componentwise), where every , and all tuples in appear as some .
We claim that when , then and lie in distinct orbits. To see this, suppose without loss of generality that . Thus there exists some . Let be the retractional witness such that projects to the -th coordinate. If for an invertible , then we would have . Observe that is a projection since and since is a retractional witness. Hence, , a contradiction. ∎
Let be a function clone which is a model-complete core and which satisfies some non-trivial linear identity modulo outer unary functions. If has a retractional witness, then it has infinite ambiguity degree.
For any of ambiguity degree , we find of ambiguity degree . So let be given, and let be a 2-element set such that has fundamental indices with respect to , witnessed by functions . By identifying variables we may assume that is -ary. Renaming the variables, we may further assume that witnesses the index , for . Set . Because is a model-complete core, in the stabilizer a nontrivial identity which is linear modulo outer unary functions is satisfied, since linear identities modulo outer functions which hold in a model-complete core also hold in all of its stabilizers (this is easy to see and well-known, but we refer to [BP16a]). Let witness this, i.e., satisfies the nontrivial identity , for variables which are not necessarily distinct. We claim that the -ary term
has the desired property.
To see this, we first observe that is a projection, and in fact a projection onto a variable of the form : inserting variables into , we obtain
with the second equation holding since is stabilized by . In particular, has ambiguity degree at least , witnessed by . Note furthermore that for the same reason, the functions and have ambiguity degree at least , projecting to a variable of the form when composed with from the left. This can be restated by saying that has ambiguity degree at least , with fundamental indices corresponding to variables of the form witnessed by two witnesses and ; we now argue that the fundamental indices witnessed by and are distinct.
To this end, fix any , and say that projects onto , where . Since the equation is non-trivial, we must have . On the other hand, we must have that , since obviously depends on its -th variable. Thus for some . This means that is a retractional witness such that projects onto the variable with index , proving our claim.
Summarizing, each fundamental index of , witnessed by some , has a corresponding fundamental index of , also witnessed by , and this assignment is injective; and moreover, each fundamental index of , witnessed by , yields two fundamental indices of , witnessed by and , respectively. ∎
We thus obtain the following theorem, which shows, in particular, how equational properties of the polymorphism clone of a structure can have implications about its automorphism group. We first formulate it in terms of function clones, and then restate it in terms of structures.
Let be an oligomorphic function clone which is a model-complete core. Suppose that
satisfies a non-trivial linear identity modulo outer unary functions, and
has a uniformly continuous h1 clone homomorphism onto .
Then has at least double exponential orbit growth.
By the results from [BOP], (ii) together with oligomorphicity implies that there exists such that the componentwise action of on , which we denote by , has a retractional witness (in the terminology of [BOP], which we avoid to fully define here, the clone is an expansion of a reflection of a finite power of , which implies our formulation – see also Section 5). By Lemma 3.3, has infinite ambiguity degree, and so it has at least double exponential orbit growth by Lemma 3.2. Hence, also has at least double exponential orbit growth. ∎
Let be an -categorical model-complete core, and suppose that satisfies (i) and (ii) of Theorem 3.4. Then has at least double exponential orbit growth.
Note that this implies, in particular, Theorem 1.7. We obtain the following result in the language of clone homomorphisms, for reducts of homogeneous structures in a finite language.
Let be a reduct of a structure which is homogeneous in a finite relational language, and suppose is a model-complete core. Then the following are equivalent.
Some stabilizer of has a continuous clone homomorphism to .
has a uniformly continuous h1 clone homomorphism to .
The implication from (i) to (ii) is a direct consequence of the results in [BOP]. For the other direction, assume that (ii) holds. By Theorem 1.3, (i) holds if and only if has no Siggers term modulo outer unary functions. If that was not the case, then Corollary 3.5 would imply that has at least double exponential orbit growth, contradicting that is a reduct of a structure which is homogeneous in a finite relational language (see [Mac11]). ∎
Finally, we obtain the equivalence of the two CSP conjectures.
Proof of Corollary 1.8.
By [BOP], has a uniformly continuous h1 clone homomorphism onto if and only if does, so (ii) and (iii) are equivalent. Applying Corollary 3.6 to the model-complete core , and taking onto account that does not have faster orbit growth than (for the latter, refer to the proof of the existence of the model-complete core in Section 5), the equivalence with (i) follows. ∎
3.2. The counterexample
We now prove Theorem 1.6. That is, we show that in Corollary 3.6, it would not be sufficient to only require the structure to be an -categorical model-complete core: the assumption of being a reduct of a homogeneous structure in finite language (or more precisely, as we can see from the proof, the assumption of less than double exponential orbit growth) is indeed needed.
Our counterexample is based on the countable atomless Boolean algebra, i.e., the (up to isomorphism) unique countable Boolean algebra without atoms (see e.g. [Hod97]). This Boolean algebra can be described, more explicitly, as the Boolean algebra that is freely generated by a countable set of generators. Among other interesting model-theoretical properties, it is -categorical and has double exponential orbit growth. In the following we occasionally view this structure as a relational structure , where the relations are the graphs of the fundamental operations of the Boolean algebra (although we will sometimes use the same symbols for the operations of the Boolean algebra, without danger of confusion). The following two statements about are essential for the construction of our counterexample.
Let be the countable atomless Boolean algebra. Then:
For every finite set there is a binary injective which stabilizes all elements of and which is symmetric modulo outer embeddings of , i.e., the identity holds for some self-embeddings of .
has a uniformly continuous h1 clone homomorphism onto .
Let be a countable set that freely generates . Since every element of can be expressed as a term over using finitely many generators, we can restrict ourselves in (i) to the stabilizers of sets of the form . The product algebra is also a countable atomless Boolean algebra and thus isomorphic to . Moreover, it is freely generated by the pairs and for all : let , let be terms over such that and in . Then we can represent the pair by
We now define to be the unique homomorphism that extends the following map between the generating sets:
By definition is a polymorphism of that stabilizes all the elements for . Furthermore, since is induced by a bijection between the generating sets of free Boolean algebras, is an isomorphism between and . It satisfies the equation , where denotes the unique automorphism of that maps to and fixes all other generating elements, which concludes the proof of (i).
In order to show (ii), let be an ultrafilter of ; we remark that some version of the axiom of choice is needed for its existence. Then for every exactly one of the elements , , …, is an element of : this follows from the fact that, since stabilizes , the disjunction is equal to ; but on the other hand is equal to whenever , since stabilizes . Let be the unique index such that . Then defines an h1 clone homomorphism from to . Furthermore is uniformly continuous, since for every the image of an -ary polymorphism only depends on the restriction of to the finite set . ∎
Note that the countable atomless Boolean algebra is not a model-complete core, since it can be homomorphically mapped to the two-element Boolean algebra. However, a slight change of language yields a model-complete core which satisfies all conditions of Theorem 1.6:
Proof of Theorem 1.6.
Let be the expansion of the countable Boolean algebra by the inequality relation. Then clearly and have the same automorphism group. Using the fact that contains the inequality relation, it can be easily verified that is dense in the endomorphisms of , and so is an -categorical model-complete core.
By Lemma 3.7 (i), every stabilizer of contains an injective binary function which is symmetric modulo outer embeddings. Since those functions are injective, they preserve in particular the inequality relation and are thus elements of . Therefore no stabilizer of has a clone homomorphism to . But, by Lemma 3.7 (ii) there is a uniformly continuous h1 clone homomorphism of to , and its restriction to shows that also has such a clone homomorphism. ∎
3.3. The Ramsey property
We now prove Theorem 1.9, which states that also under different, Ramsey-theoretic conditions, the satisfaction of a non-trivial set of linear identities modulo outer embeddings in a polymorphism clone implies that this clone has no uniformly continuous h1 clone homomorphism to . Although the cases covered by this result are not congruent with the range of Conjecture 1.2, they appear in many known classifications of CSPs over homogeneous structures; in fact such CSP classifications are often based on the fact that the underlying structures can be expanded to Ramsey structures (cf. [BP11, BP16b] for numerous examples and further references).
Let be a reduct of an ordered homogeneous Ramsey structure and let satisfy a non-trivial set of linear identities modulo outer embeddings of ; by homogeneity, those embeddings are elements of . Then Theorem 1.9 claims that there is no uniformly continuous h1 clone homomorphism from to . We provide two proofs, a combinatorial one applying the Ramsey property directly, and a more algebraic one using dynamical systems.
First proof of Theorem 1.9.
Let satisfy the non-trivial set of identities
where , , and are not necessarily distinct variables, for and . The finiteness of this set follows from the compactness theorem of first-order logic, and we can assume to have equal arity by adding dummy variables. Moreover assume, for technical reasons, that every right side of an identity also appears as a left side, simply by repeating identities. For contradiction, let us assume that there is a uniformly continuous h1 clone homomorphism .
By the uniform continuity of there is a finite such that whenever two functions of arity agree on , then . We are going to color the copies of the structures induced in by .
By the homogeneity of all such copies are have domains of the form , where . Since is totally ordered, every other that maps to has to coincide with on . Hence, since only depends on the restriction of to , the coloring on the copies of which sends every copy induced by to is well defined.
Now set to be structure induced by the union of all sets and , where . By the Ramsey property, there is an isomorphic copy of in on which the colorings are monochromatic. This implies that for any that maps to we have for all . Hence, because preserves the linear identities above, , contradicting the fact that the system of identities of the form is not satisfiable in . ∎
Second proof of Theorem 1.9.
We will use the fact due to [KPT05] that is, as the automorphism group of an ordered Ramsey structure, extremely amenable: whenever it acts continuously on a compact Hausdorff space, then this action has a fixed point.
Fix , and let be the set of all mappings from the -ary functions in to the -ary functions in . Bearing the product topology, is a compact Hausdorff space. We define an action of on by setting, for and , the mapping to be given by
For contradiction, suppose that there is a uniformly continuous h1 clone homomorphism from to , and let be its restriction to -ary functions, where is fixed. Consider the restriction of the above action of to the closure of the orbit of in , i.e., let act on
Clearly, is compact. Moreover, this restriction of the action to is continuous: to illustrate this, let us first observe that there exists a finite set such that for all and all -ary we have that implies . Now consider a basic open neighborhood
of some , where the -ary is fixed. Then by our remark above, the set
is a basic open neighborhood of that is mapped into under the action.
Since is extremely amenable, there is a fixed point of its action on , i.e., for all . This means that preserves composition with elements of from the outside, and by continuity even with elements of , i.e., with self-embeddings of . Moreover, preserves linear identities, since any mapping does, and so does any mapping in the closure of the functions of the latter form.
It follows that cannot satisfy any finite non-trivial set of identities which are linear modulo embeddings of from the outside, as otherwise they would be satisfied in by virtue of , if we choose larger than all arities of the functions in that set. ∎
We would like to remark that the atomless Boolean algebra that was used to provide the counterexample of Theorem 1.6 is the reduct of a homogeneous Ramsey structure, namely of its expansion by a linear order which extends the natural partial order on (see for instance [KPT05]). We proved in Lemma 3.7 that there are polymorphisms of satisfying the non-trivial equation . However this non-trivial equation does not satisfy the condition of Theorem 1.9, since the embedding does not preserve any linear order on the domain of .
4. Linearization of Non-trivial Identities
We are going to show that, under stronger conditions than the existence of a Siggers term modulo outer embeddings, we can derive the satisfaction of non-trivial linear identities in polymorphism clones. In Section 4.1 we prove a strengthening of Theorem 1.10. A proof of Theorem 1.11 is given in Section 4.2. Finally, in Section 4.3 we show how to apply these results to the polymorphism clones of all reducts of equality, the rational order, the random graph and the random partial order, for which complete complexity classifications of the corresponding CSPs have been obtained [BK08, BK09, BP15a, KP].
4.1. Totally symmetric polymorphisms modulo embeddings
Let be a -ary operation on a set . We define the nu-minors of as the binary functions , where and the only is located on the -th coordinate. When is a relational structure on , we say that the nu-minors of are totally symmetric modulo outer embeddings of if for all permutations of there are embeddings of such that
Clearly, whenever is totally symmetric modulo outer embeddings of , then also its nu-minors are totally symmetric modulo outer embeddings of . On the other hand, we remark that if is a weak near unanimity function modulo outer embeddings of , i.e., satisfies the identities
for embeddings of , then this does not imply in an obvious way that its nu-minors are symmetric modulo outer embeddings. We will show the following.
Let be a reduct of a finitely bounded homogeneous structure whose age is given by a finite set of forbidden substructures all of which have size at most .
If contains a -ary function whose nu-minors are totally symmetric modulo outer embeddings of , then does not have an h1 clone homomorphism to .
The proof of Proposition 4.2 is based on the following easy observation, which relies on the pigeonhole principle.
Let be a function clone, let , and assume that there are binary functions for all such that for every injective there exists whose nu-minors equal the functions . Then there is no h1 clone homomorphism from to .
If there was an h1 clone homomorphism from onto , then by the pigeonhole-principle there would be such that all are sent to the same projection. But this contradicts the fact that are the nu-minors of . ∎
It is further enough to find polymorphisms that satisfy the equations in Lemma 4.3 locally, by a simple compactness argument which yields the following lemma.
Lemma 4.4 (Lemma 3 in [BP16b]).
Let be an -categorical structure. Let be an index set, and for every , let and be functions on of the same arity such that for every finite there is with . Then there are such that for all . Moreover, if for we have for every finite set , then .
We are going to construct the functions needed for Lemma 4.3 as suitable compositions of the nu-minors of with embeddings of .
Let , , and be as in Proposition 4.2, and let be finite. Then there are binary such that for every injective there exists such that for all .
Whenever is injective, we define a mapping
Writing for the kernel of , we then naturally obtain a structure in the language of on the set of kernel classes of , in which we choose the relations to be so that the mapping from to induced by is an embedding.
The main point of our construction is the observation that because the nu-minors of are totally symmetric modulo outer embeddings of , any two structures are isomorphic via the mapping that sends any kernel class to . For example, to see that this mapping is well-defined, note that by definition if and only if ; but this is the case, by definition, if and only if . Similarly, one checks that the mapping is an isomorphism.
We define a binary relation on by setting if and only if there is a such that , and claim that it is transitive, and thus an equivalence relation. To see transitivity, note that by the total symmetry of nu-minors, is equivalent to the statement that for every containing and in its image holds. Now let with and . Since , there is an injection such that , and hence .
Since the relations of the structures agree on their intersections, we obtain a structure on the equivalence classes of , defined as the “union” of the structures . This structure does not contain any forbidden substructures of , since any -element substructure of is already contained in some structure , which in turn embeds into . Hence, embeds into via an embedding . For and , we now set .
Given as in the lemma, it is clear from the definition of that the tuples and satisfy the same relations in . By the homogeneity of , the latter can be sent to the first via an automorphism of , which is what we had to show.
We have now all the tools ready to prove Proposition 4.2.