scholarly journals Uncountable structures are not classifiable up to bi-embeddability

2019 ◽  
Vol 20 (01) ◽  
pp. 2050001
Author(s):  
Filippo Calderoni ◽  
Heike Mildenberger ◽  
Luca Motto Ros
Keyword(s):  
Equivalence Relations ◽  
Symbolic Logic ◽  
Large Size ◽  
Or Groups ◽  
Descriptive Set ◽  
Natural Classes

Answering some of the main questions from [L. Motto Ros, The descriptive set-theoretical complexity of the embeddability relation on models of large size, Ann. Pure Appl. Logic 164(12) (2013) 1454–1492], we show that whenever [Formula: see text] is a cardinal satisfying [Formula: see text], then the embeddability relation between [Formula: see text]-sized structures is strongly invariantly universal, and hence complete for ([Formula: see text]-)analytic quasi-orders. We also prove that in the above result we can further restrict our attention to various natural classes of structures, including (generalized) trees, graphs, or groups. This fully generalizes to the uncountable case the main results of [A. Louveau and C. Rosendal, Complete analytic equivalence relations, Trans. Amer. Math. Soc. 357(12) (2005) 4839–4866; S.-D. Friedman and L. Motto Ros, Analytic equivalence relations and bi-embeddability, J. Symbolic Logic 76(1) (2011) 243–266; J. Williams, Universal countable Borel quasi-orders, J. Symbolic Logic 79(3) (2014) 928–954; F. Calderoni and L. Motto Ros, Universality of group embeddability, Proc. Amer. Math. Soc. 146 (2018) 1765–1780].

COUNTABLE BOREL EQUIVALENCE RELATIONS

2002 ◽  
Vol 02 (01) ◽  
pp. 1-80 ◽  
Author(s):  
S. JACKSON ◽  
A. S. KECHRIS ◽  
A. LOUVEAU
Keyword(s):  
Set Theory ◽  
Equivalence Relations ◽  
Descriptive Set Theory ◽  
Polish Spaces ◽  
Borel Equivalence Relations ◽  
Universal Equivalence ◽  
Descriptive Set

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.

scholarly journals Uniformity, universality, and computability theory

2017 ◽  
Vol 17 (01) ◽  
pp. 1750003
Author(s):  
Andrew S. Marks
Keyword(s):  
Equivalence Relation ◽  
Equivalence Relations ◽  
Technical Tool ◽  
Free Products ◽  
Borel Sets ◽  
Borel Equivalence Relations ◽  
Strengthened Form ◽  
Shift Action ◽  
Natural Classes ◽  
Main Technical Tool

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.

Borel reductions and cub games in generalised descriptive set theory

2013 ◽  
Vol 78 (2) ◽  
pp. 439-458 ◽  
Author(s):  
Vadim Kulikov
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Regular Cardinal ◽  
Equivalence Relations ◽  
Descriptive Set Theory ◽  
Borel Equivalence Relations ◽  
Power Set ◽  
Borel Reducibility ◽  
Descriptive Set

AbstractIt is shown that the power set of κ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on 2κ under Borel reducibility. Here κ is an uncountable regular cardinal with κ<κ = κ.

scholarly journals JUMP OPERATIONS FOR BOREL GRAPHS

2018 ◽  
Vol 83 (1) ◽  
pp. 13-28
Author(s):  
ADAM R. DAY ◽  
ANDREW S. MARKS
Keyword(s):  
Set Theory ◽  
Equivalence Relations ◽  
Connected Components ◽  
Descriptive Set Theory ◽  
Jump Operator ◽  
Borel Equivalence Relations ◽  
Locally Countable ◽  
Descriptive Set ◽  
Effective Descriptive Set Theory

AbstractWe investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a nonseparation result for iterated Fréchet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

scholarly journals Global aspects of measure preserving equivalence relations and graphs

10.53733/96 ◽  
2021 ◽  
Vol 52 ◽  
pp. 691-726
Author(s):  
Alexander Kechris
Keyword(s):  
Topological Space ◽  
Equivalence Relation ◽  
Point Of View ◽  
Equivalence Relations ◽  
Global Theory ◽  
Descriptive Set

This paper is an introduction and survey of a “global” theory of measure preserving equivalence relations and graphs. In this theory one views a measure preserving equivalence relation or graph as a point in an appropriate topological space and then studies the properties of this space from a topological, descriptive set theoretic and dynamical point of view.

Jack H. Silver. Counting the number of equivalence classes of Borel and coanalytic equivalence relations. Annals of mathematical logic, vol. 18 (1980), pp. 1–28. - John P. Burgess. Equivalences generated by families of Borel sets. Proceedings of the American Mathematical Society. vol. 69 (1978), pp. 323–326. - John P. Burgess. A reflection phenomenon in descriptive set theory. Fundamenta mathematicae. vol. 104 (1979), pp. 127–139. - L. Harrington and R. Sami. Equivalence relations, projective and beyond. Logic Colloquium '78, Proceedings of the Colloquium held in Mons, August 1978, edited by Maurice Boffa, Dirk van Dalen, and Kenneth McAloon, Studies in logic and the foundations of mathematics, vol. 97, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1979, pp. 247–264. - Leo Harrington and Saharon Shelah. Counting equivalence classes for co-κ-Souslin equivalence relations. Logic Colloquium '80, Papers intended for the European summer meeting of the Association for Symbolic Logic, edited by D. van Dalen, D. Lascar, and T. J. Smiley, Studies in logic and the foundations of mathematics, vol. 108, North-Holland Publishing Company, Amsterdam, New York, and Oxford, 1982, pp. 147–152. - Jacques Stern. On Lusin's restricted continuum problem. Annals of mathematics, ser. 2 vol. 120 (1984), pp. 7–37.

1987 ◽  
Vol 52 (3) ◽  
pp. 869-870
Author(s):  
Alain Louveau
Keyword(s):  
New York ◽  
American Mathematical Society ◽  
Equivalence Classes ◽  
Publishing Company ◽  
Equivalence Relations ◽  
Foundations Of Mathematics ◽  
Borel Sets ◽  
North Holland Publishing Company ◽  
North Holland Publishing ◽  
Descriptive Set

Analytic equivalence relations and bi-embeddability

2011 ◽  
Vol 76 (1) ◽  
pp. 243-266 ◽  
Author(s):  
Sy-David Friedman ◽  
Luca Motto Ros
Keyword(s):  
Banach Spaces ◽  
Equivalence Relation ◽  
Equivalence Relations ◽  
Topological Spaces ◽  
Polish Spaces ◽  
Borel Reducibility ◽  
Natural Classes ◽  
Strong Contrast

AbstractLouveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of ) is far from complete (see [5, 2]).In this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids.

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