Reconstructing the topology of clones

Bodirsky, Manuel; Pinsker, Michael; Pongrácz, András

doi:10.1090/tran/6937

Cited by 25 publications

(82 citation statements)

References 51 publications

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“…A natural, yet unresolved problem when comparing the finite with the ω-categorical setting then is whether the topological structure of the polymorphism clone is really essential in the infinite, or whether the abstract algebraic structure, i.e., the identities that hold in Pol(A), is sufficient to determine the complexity of the CSP. This problem motivates, in particular, the related concept of reconstruction of the topology of a clone from its algebraic structure introduced in [25], which has its own purely mathematical interest.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Topology Is Irrelevant (In a Dichotomy Conjecture for Infinite Domain Constraint Satisfaction Problems)

Barto¹,

Pinsker²

2020

SIAM J. Comput.

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The tractability conjecture for finite domain Constraint Satisfaction Problems (CSPs) stated that such CSPs are solvable in polynomial time whenever there is no natural reduction, in some precise technical sense, from the 3-SAT problem; otherwise, they are NP-complete. Its recent resolution draws on an algebraic characterization of the conjectured borderline: the CSP of a finite structure permits no natural reduction from 3-SAT if and only if the stabilizer of the polymorphism clone of the core of the structure satisfies some nontrivial system of identities, and such satisfaction is always witnessed by several specific nontrivial systems of identities which do not depend on the structure.The tractability conjecture has been generalized in the above formulation to a certain class of infinite domain CSPs, namely, CSPs of reducts of finitely bounded homogeneous structures. It was subsequently shown that the conjectured borderline between hardness and tractability, i.e., a natural reduction from 3-SAT, can be characterized for this class by a combination of algebraic and topological properties. However, it was not known whether the topological component is essential in this characterization.We provide a negative answer to this question by proving that the borderline is characterized by one specific algebraic identity, namely the pseudo-Siggers identity αs(x, y, x, z, y, z) ≈ βs (y, x, z, x, z, y). This accomplishes one of the steps of a proposed strategy for reducing the infinite domain CSP dichotomy conjecture to the finite case. Our main theorem is also of independent mathematical interest, characterizing a topological property of any ω-categorical core structure (the existence of a continuous homomorphism of a stabilizer of its polymorphism clone to the projections) in purely algebraic terms (the failure of an identity as above).

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Topology Is Irrelevant (In a Dichotomy Conjecture for Infinite Domain Constraint Satisfaction Problems)

Barto¹,

Pinsker²

2020

SIAM J. Comput.

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“…• all the examples from [5] and [2] (i.e., (N, =), the Rado graph, the countable universal homogeneous digraph, (Q, ≤),. .…”

Section: Resultsmentioning

confidence: 99%

“…• the endomorphism monoid of the Rado-graph ( [5]), • the self-embedding monoid of the countable universal homogeneous digraph ( [5]), • the endomorphism monoid of (Q, <)…”

Section: Automatic Homeomorphicitymentioning

confidence: 99%

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Reconstructing the Topology of the Elementary Self-embedding Monoids of Countable Saturated Structures

Pech¹,

Pech

2017

Stud Logica

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Abstract. Every transformation monoid comes equipped with a canonical topology-the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity.In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too.As a second result we strengthen a result by Lascar by showing that whenever A is a countable ℵ 0 -categorical G-finite structure whose automorphism group has a trivial center and if B is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism. Automatic homeomorphicityIf we consider a set A as a discrete topological space, then the set of self-mappings A A of A is naturally equipped with the product topology. A sub-basis of this topology is given by {Φ a,b | a, b ∈ A}, where Φ a,b is given by Φ a,b = {f ∈ A A | f (a) = b}. This topology is also known as the Tychonoff-topology or topology of pointwise convergence.The set A A , together with the composition operation is called the full transformation monoid on A. We denote it by T A . Every submonoid M of T A is naturally equipped with the subspace-topology (in particular, the composition on M is continuous with respect to this topology).When using the predicates open or closed in connection with transformation monoids we make an implicit distinction between transformation monoids in general and the special case of permutation groups. A transformation monoid on A is called closed (open) whenever it is Date: March 23, 2017.

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“…It has been asked in [BPP13] (Question 1 on page 29), and, independently, in [Tar14] ('open question' on page 215), whether this theorem can be generalised from permutation groups to transformation monoids. Here, we present such a generalisation (Theorem 2): a topological monoids M is topologically isomorphic to a closed submonoid of the full transformation monoid N N if and only if M is separable and admits a compatible complete left non-expansive ultrametric (Theorem 2).…”

Section: Introductionmentioning

confidence: 99%

A topological characterisation of endomorphism monoids of countable structures

Bodirsky

Schneider

2017

Algebra Univers.

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Abstract. A topological monoid is isomorphic to an endomorphism monoid of a countable structure if and only if it is separable and has a compatible complete ultrametric such that composition from the left is non-expansive. We also give a topological characterisation of those topological monoids that are isomorphic to endomorphism monoids of countable ω-categorical structures. Finally we present analogous characterisations for polymorphism clones of countable structures and for polymorphism clones of countable ω-categorical structures.

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Reconstructing the topology of clones

Cited by 25 publications

References 51 publications

Topology Is Irrelevant (In a Dichotomy Conjecture for Infinite Domain Constraint Satisfaction Problems)

Topology Is Irrelevant (In a Dichotomy Conjecture for Infinite Domain Constraint Satisfaction Problems)

Reconstructing the Topology of the Elementary Self-embedding Monoids of Countable Saturated Structures

A topological characterisation of endomorphism monoids of countable structures

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