The structure of Borel reducibility for equivalence relations on $\omega _1$ is determined.

Given a Polish group G, let $E(G)$ be the right coset equivalence relation $G^{\omega }/c(G)$, where $c(G)$ is the group of all convergent sequences in G. The connected component of the identity of a Polish group G is denoted by $G_0$.

Let $G,H$ be locally compact abelian Polish groups. If $E(G)\leq _B E(H)$, then there is a continuous homomorphism $S:G_0\rightarrow H_0$ such that $\ker (S)$ is non-archimedean. The converse is also true when G is connected and compact.

For $n\in {\mathbb {N}}^+$, the partially ordered set $P(\omega )/\mbox {Fin}$ can be embedded into Borel equivalence relations between $E({\mathbb {R}}^n)$ and $E({\mathbb {T}}^n)$.

We study the bi-embeddability and elementary bi-embeddability relation on graphs under Borel reducibility and investigate the degree spectra realized by these relations. We first give a Borel reduction from embeddability on graphs to elementary embeddability on graphs. As a consequence we obtain that elementary bi-embeddability on graphs is a $\boldsymbol {\Sigma }^1_1$ complete equivalence relation. We then investigate the algorithmic properties of this reduction. We obtain that elementary bi-embeddability on the class of computable graphs is $\Sigma ^1_1$ complete with respect to computable reducibility and show that the elementary bi-embeddability and bi-embeddability spectra realized by graphs are related.

In this article, a notion of Schauder equivalence relation ℝℕ/L is introduced, where L is a linear subspace of ℝℕ and the unit vectors form a Schauder basis of L. The main theorem is to show that the following conditions are equivalent:

(1) the unit vector basis is boundedly complete;

(2) L is a Fσ in ℝℕ;

(3) ℝℕ/L is Borel reducible to ℓ∞.

We show that any Schauder equivalence relation generalized by a basis of ℓ2 is Borel bireducible to ℝℕ/ℓ2 itself, but it is not true for bases of c0 or ℓ1. Furthermore, among all Schauder equivalence relations generated by sequences in c0, we find the minimum and the maximum elements with respect to Borel reducibility.

We prove that, for 1 ≤ p < q < ∞, the partially ordered set P(ω)/Fin can be embedded into Borel equivalence relations between ℝω/ℓp and ℝω/ℓq. Since there is an antichain of size continuum in P(ω)/Fin, there are continuum many pairwise incomparable Borel equivalence relations between ℝω/ℓp and ℝω/ℓq.

We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a ∅-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, rings, categories, groupoids and preorders which are ∅-type-definable in a small structure. For an A-type-definable group GA(where the set A may be infinite) in a small and simple structure, we deduce that

(1) if GA is included in some definable set X such that boundedly many translates of GA cover X, then GA is the conjunction of definable groups.

(2) for any finite tuple ḡ in GA, there is a definable group containing ḡ.

This note answers a questions from [2] by showing that considered up to Borel reducibility, there are more essentially countable Borel equivalence relations than countable Borel equivalence relations. Namely:

Theorem 0.1. There is an essentially countable Borel equivalence relation E such that for no countable Borel equivalence relation F (on a standard Borel space) do we have

The proof of the result is short. It does however require an extensive rear guard campaign to extract from the techniques of [1] the following

Messy Fact 0.2. There are countable Borel equivalence relationssuch that:

(i) eachExis defined on a standard Borel probability space (Xx, μx); each Ex is μx-invariant and μx-ergodic;

(ii) forx1 ≠ x2 and A μxι -conull, we haveExι/Anot Borel reducible toEx2;

(iii) if f: Xx → Xxis a measurable reduction ofExto itself then

(iv)

is a standard Borel space on which the projection function

is Borel and the equivalence relation Ê given by

if and only ifx = x′ andzExz′ is Borel;

(V)

is Borel.

We first prove the theorem granted this messy fact. We then prove the fact.

(iv) and (v) are messy and unpleasant to state precisely, but are intended to express the idea that we have an effective parameterization of countable Borel equivalence relations by points in a standard Borel space. Examples along these lines appear already in the Adams-Kechris constructions; the new feature is (iii).

Simon Thomas has pointed out to me that in light of theorem 4.4 [5] the Gefter-Golodets examples of section 5 [5] also satisfy the conclusion of 0.2.