Reconstructing the Topology on Monoids and Polymorphism Clones of the Rationals

Behrisch, Mike; Truss, John K.; Vargas-García, Edith

doi:10.1007/s11225-016-9682-z

Cited by 14 publications

(53 citation statements)

References 5 publications

(23 reference statements)

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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%

“…In fact it will usually be isomorphic to the superhomogeneous substructure. To see that we are not talking about an empty concept, let us have a look onto some examples: Another example are the maximally spread out substructures of (Q, ≤) introduced in [2] for showing that Emb(Q, ≤) has automatic homeomorphicity: Consider the 2-colored rationals (Q 2 , ≤). They are obtained from (Q, ≤) by assigning one of two colors (say, red and blue) to every rational in such a way that the red rationals and the blue rationals both form dense unbounded chains that lie dense in each other in the sense that between any two blue rationals there is a red one and between any two red rationals there is a blue one.…”

Section: Superhomogeneous Substructuresmentioning

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

Section: Resultsmentioning

confidence: 99%

Section: Superhomogeneous Substructuresmentioning

confidence: 99%

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

Pech¹,

Pech

2017

Stud Logica

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“…In this setting, as a by-product, we prove a new characterization of automatic homeomorphicity for transformation monoids on arbitrary carrier sets. It involves a weakening of the technical assumption that the only injective monoid endomorphism fixing every group member be the identical one, which has featured in several earlier reconstruction results [10,25,30,3]. We believe that our weakened version, so, by our characterization, automatic homeomorphicity, would also have been sufficient in any of these previous cases.…”

Section: Introductionmentioning

confidence: 89%

“…This is because the endomorphisms ψ ∈ E G are not necessarily closed maps, and therefore their images do not necessarily belong to the class K of closed transformation monoids over sets equipotent with the carrier of M . This is the same type of complication that has made additional arguments necessary in proving automatic homeomorphicity of End(Q, ≤), see [3,Lemma 4.1,p. 79] and the discussion next to it.…”

Section: And Assumementioning

confidence: 97%

On a stronger reconstruction notion for monoids and clones

Behrisch¹,

Vargas-Garc\'\ia²

2019

Preprint

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Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.

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“…(1) show that Aut(U) has automatic homeomorphicity with respect to K, (2) show that Aut(U) has automatic homeomorphicity with respect to K, (3) show that every isomorphism from End(U) to the the endomorphism monoid of a member of K is continuous, (4) show that every isomorphism from Pol(U) to the the polymorphism clone of a member of K is continuous, (5) show that every continuous isomorphism from Pol(U) to the polymorphism clone of a member of K is a homeomorphism.…”

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confidence: 99%

Polymorphism clones of homogeneous structures: gate coverings and automatic homeomorphicity

Pech¹,

Pech

2018

Algebra Univers.

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A. Every clone of functions comes naturally equipped with a topology-the topology of pointwise convergence. A clone C is said to have automatic homeomorphicity with respect to a class C of clones, if every clone-isomorphism of C to a member of C is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we study automatic homeomorphicity-properties for polymorphism clones of countable homogeneous relational structures. To this end we introduce and utilize universal homogeneous polymorphisms. Next to two generic criteria for the automatic homeomorphicity of the polymorphism clones of free homogeneous structures we show that the polymorphism clone of the generic poset with reflexive ordering has automatic homeomorphicity and that the polymorphism clone of the generic poset with strict ordering has automatic homeomorphicity with respect to countable ω-categorical structures. Our results extend and generalize previous results by Bodirsky, Pinsker, and Pongrácz.

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Reconstructing the Topology on Monoids and Polymorphism Clones of the Rationals

Cited by 14 publications

References 5 publications

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

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

On a stronger reconstruction notion for monoids and clones

Polymorphism clones of homogeneous structures: gate coverings and automatic homeomorphicity

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