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Register a new userMeasuring the complexity of reductions between equivalence relations
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Computable reducibility is a well-established notion that allows to compare the complexity of various equivalence relations over the natural numbers. We generalize computable reducibility by introducing degree spectra of reducibility and bi-reducibility. These spectra provide a natural way of measuring the complexity of reductions between equivalence relations. We prove that any upward closed collection of Turing degrees with a countable basis can be realised as a reducibility spectrum or as a bi-reducibility spectrum. We show also that there is a reducibility spectrum of computably enumerable equivalence relations with no countable basis and a reducibility spectrum of computably enumerable equivalence relations which is downward dense, thus has no basis.
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The complexity of equivalence relations has received much attention in the recent literature. The main tool for such endeavour is the following reducibility: given equivalence relations $R$ and $S$ on natural numbers, $R$ is computably reducible to $S$ if there is a computable function $f colon omega to omega$ that induces an injective map from $R$-equivalence classes to $S$-equivalence classes. In order to compare the complexity of equivalence relations which are computable, researchers considered also feasible variants of computable reducibility, such as the polynomial-time reducibility. In this work, we explore $mathbf{Peq}$, the degree structure generated by primitive recursive reducibility on punctual equivalence relations (i.e., primitive recursive equivalence relations with domain $omega$). In contrast with all other known degree structures on equivalence relations, we show that $mathbf{Peq}$ has much more structure: e.g., we show that it is a dense distributive lattice. On the other hand, we also offer evidence of the intricacy of $mathbf{Peq}$, proving, e.g., that the structure is neither rigid nor homogeneous.
We study the class of Borel equivalence relations under continuous reducibility. In particular , we characterize when a Borel equivalence relation with countable equivalence classes is $Sigma$ 0 $xi$ (or $Pi$ 0 $xi$). We characterize when all the equivalence classes of such a relation are $Sigma$ 0 $xi$ (or $Pi$ 0 $xi$). We prove analogous results for the Borel equivalence relations with countably many equivalence classes. We also completely solve these two problems for the first two ranks. In order to do this, we prove some extensions of the Louveau-Saint Raymond theorem which itself generalized the Hurewicz theorem characterizing when a Borel subset of a Polish space is G $delta$ .
We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are locally quotients of compact Polish groups. It follows that they are smooth if and only if they are type-definable, and that a quotient of a type-definable group by an analytic subgroup is either finite or of cardinality at least continuum.
We generalise the main theorems from the paper The Borel cardinality of Lascar strong types by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to get the conclusion.
Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $leq_c$. This gives rise to a rich degree-structure. In this paper, we lift the study of $c$-degrees to the $Delta^0_2$ case. In doing so, we rely on the Ershov hierarchy. For any notation $a$ for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by $leq_c$ on the $Sigma^{-1}_{a}smallsetminus Pi^{-1}_a$ equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of $c$-degrees.
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