A uniform Birkhoff theorem

Schneider, Friedrich Martin

doi:10.1007/s00012-017-0460-1

Cited by 5 publications

(8 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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 [BPP17, GP,Sch15]. Then a mapping ξ : A → B, where A and B are function clones, is continuous if and only if for all f ∈ A and all finite sets B ′ ⊆ B there exists a finite set A ′ ⊆ A such that for all g ∈ A of the same arity as f , if g agrees with f on A ′ , then ξ(g) agrees with ξ(f ) on B ′ .…”

Section: Stabilizers Whenmentioning

confidence: 99%

Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

et al. 2019

View full text Add to dashboard Cite

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 ω-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 theorem stating that every ω-categorical structure is homomorphically equivalent to a model-complete core.

show abstract

Section: Stabilizers Whenmentioning

confidence: 99%

Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

et al. 2019

View full text Add to dashboard Cite

show abstract

“…As a corollary, besides the version of topological Birkhoff by Schneider [16] and Gehrke-Pinsker [10], we get new versions of the topological Birkhoff's theorem for algebras depending on the choice of the trace b.…”

mentioning

confidence: 88%

“…The last sections of the paper are devoted to the study of topological variants of Birkhoff's theorem, as initiated by Bodirsky and Pinsker [5] for locally oligomorphic algebras, and generalised recently by Schneider [16] and Gehrke-Pinsker [10]. These authors provide a Birkhoff-type characterisation of all those members T of the variety HSPpSq generated by a given algebra S, for which the natural homomorphism from CloS onto CloT is uniformly continuous with respect to the uniformity of pointwise convergence.…”

mentioning

confidence: 99%

“…These authors provide a Birkhoff-type characterisation of all those members T of the variety HSPpSq generated by a given algebra S, for which the natural homomorphism from CloS onto CloT is uniformly continuous with respect to the uniformity of pointwise convergence. Schneider [16] and Gehrke-Pinsker [10] have independently shown that for any algebra T in the variety generated by an algebra S, the induced natural clone homomorphism is uniformly continuous if and only if every finitely generated subalgebra of T is a homomorphic image of a subalgebra of a finite power of S. The above theorem, termed Uniform Birkhoff in [10], has been successfully applied in the context of constraint satisfaction problems by Barto and Pinsker [2], and by Barto, Opršal and Pinsker [3]. We generalise this theorem to t-algebras as follows.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Universal Clone Algebra

Salibra¹

2022

Preprint

View full text Add to dashboard Cite

We develop a new general framework for algebras and clones, called Universal Clone Algebra. Algebras and clones of finitary operations are to Universal Algebra what t-algebras and clone algebras are to Universal Clone Algebra. Clone algebras have been recently introduced to found a one-sorted, purely algebraic theory of clones, while t-algebras are first introduced in this article. We present a method to codify algebras and clones into t-algebras and clone algebras, respectively. We provide concrete examples showing that general results in Universal Clone Algebra, when translated in terms of algebras and clones, give more powerful versions of known theorems in Universal Algebra. We apply this methodology to Birkhoff's HSP theorem and to the recent topological versions of Birkhoff's theorem.

show abstract

“…is uniformly continuous, and the uniformities involved in this condition are the ones previously described. For more detailed information on the general product, coproduct and subspace constructions occurring in the above we refer the reader to the excellent and concise overview given in section 2 of [29]. A more thorough treatment of uniform spaces can also be found in chapter 9 of [31, p. 238].…”

Section: Uniformities and Topologies On Powersmentioning

confidence: 99%

On a stronger reconstruction notion for monoids and clones

Behrisch¹,

Vargas-Garc\'\ia²

2019

Preprint

View full text Add to dashboard Cite

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.

show abstract

A uniform Birkhoff theorem

Cited by 5 publications

References 8 publications

Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

Equations in oligomorphic clones and the constraint satisfaction problem for ω-categorical structures

Universal Clone Algebra

On a stronger reconstruction notion for monoids and clones

Contact Info

Product

Resources

About