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Relevant bibliographies by topics / Borel complexity of equivalence relations

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Journal articles

Academic literature on the topic 'Borel complexity of equivalence relations'

Author: Grafiati

Published: 13 April 2024

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Journal articles on the topic "Borel complexity of equivalence relations"

1

Gao, Su, and Michael Ray Oliver. "Borel complexity of isomorphism between quotient Boolean algebras." Journal of Symbolic Logic 73, no. 4 (2008): 1328–40. http://dx.doi.org/10.2178/jsl/1230396922.

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In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

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2

MARKS, ANDREW. "The universality of polynomial time Turing equivalence." Mathematical Structures in Computer Science 28, no. 3 (2016): 448–56. http://dx.doi.org/10.1017/s0960129516000232.

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We show that polynomial time Turing equivalence and a large class of other equivalence relations from computational complexity theory are universal countable Borel equivalence relations. We then discuss ultrafilters on the invariant Borel sets of these equivalence relations which are related to Martin's ultrafilter on the Turing degrees.

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3

Ding, Longyun, and Su Gao. "Diagonal actions and Borel equivalence relations." Journal of Symbolic Logic 71, no. 4 (2006): 1081–96. http://dx.doi.org/10.2178/jsl/1164060445.

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AbstractWe investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in the Borel reducbility hierarchy determined.

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4

KRUPIŃSKI, KRZYSZTOF, ANAND PILLAY, and SŁAWOMIR SOLECKI. "BOREL EQUIVALENCE RELATIONS AND LASCAR STRONG TYPES." Journal of Mathematical Logic 13, no. 02 (2013): 1350008. http://dx.doi.org/10.1142/s0219061313500086.

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The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three (modest) aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L(T) of T, have well-defined Borel cardinalities (in the sense of the theory of complexity of Borel equivalence relations). The second is to compute the Borel cardinalities of the known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 (2001) 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 (2012) 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint (2012), arXiv:1201.5221v1].

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5

KECHRIS, ALEXANDER S., ANDRÉ NIES, and KATRIN TENT. "THE COMPLEXITY OF TOPOLOGICAL GROUP ISOMORPHISM." Journal of Symbolic Logic 83, no. 3 (2018): 1190–203. http://dx.doi.org/10.1017/jsl.2018.25.

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AbstractWe study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.

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6

Lecomte, Dominique. "On the complexity of Borel equivalence relations with some countability property." Transactions of the American Mathematical Society 373, no. 3 (2019): 1845–83. http://dx.doi.org/10.1090/tran/7942.

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7

Calderoni, Filippo, Heike Mildenberger, and Luca Motto Ros. "Uncountable structures are not classifiable up to bi-embeddability." Journal of Mathematical Logic 20, no. 01 (2019): 2050001. http://dx.doi.org/10.1142/s0219061320500014.

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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].

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8

HJORTH, GREG. "TREEABLE EQUIVALENCE RELATIONS." Journal of Mathematical Logic 12, no. 01 (2012): 1250003. http://dx.doi.org/10.1142/s0219061312500031.

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There are continuum many ≤B-incomparable equivalence relations induced by a free, Borel action of a countable non-abelian free group — and hence, there are 2α0 many treeable countable Borel equivalence relations which are incomparable in the ordering of Borel reducibility.

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9

JACKSON, S., A. S. KECHRIS, and A. LOUVEAU. "COUNTABLE BOREL EQUIVALENCE RELATIONS." Journal of Mathematical Logic 02, no. 01 (2002): 1–80. http://dx.doi.org/10.1142/s0219061302000138.

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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.

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10

Rosendal, Christian. "Cofinal families of Borel equivalence relations and quasiorders." Journal of Symbolic Logic 70, no. 4 (2005): 1325–40. http://dx.doi.org/10.2178/jsl/1129642127.

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AbstractFamilies of Borel equivalence relations and quasiorders that are cofinal with respect to the Borel reducibility ordering. ≤B, are constructed. There is an analytic ideal on ω generating a complete analytic equivalence relation and any Borel equivalence relation reduces to one generated by a Borel ideal. Several Borel equivalence relations, among them Lipschitz isomorphism of compact metric spaces, are shown to be Kσ complete.

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