In this paper we discuss the relationship between two of the fundamental examples of restricted orbit equivalence: even Kakutani equivalence and α equivalence. Both equivalence relations arise in the context of representations of ergodic and measure preserving Rd actions. The first is related to the Ambrose-Kakutani Theorem [1]: every free, measure preserving and ergodic R action can be represe...

ly, we assume there is some function that extracts the angles, size, color, or whatever other property of elements we’re interested in. Two elements would be considered equivalent iff the function extracts the same value for each. � Course Notes, Week 4: Binary Relations 3 For example, if fc is the function mapping a triangle to the lengths its sides, then fc determines the congruence relation....

Families of types are fundamental objects in Martin-Löf type theory. When extending the notion of setoid (type with an equivalence relation) to families of setoids, a choice between proof-relevant or proof-irrelevant indexing appears. It is shown that a family of types may be canonically extended to a proof-relevant family of setoids via the identity types, but that such a family is in general ...

For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We study in this paper the global structure of the classes of K-structurable equivalence relations for various K. We show that K-structurability interacts well with several kinds of Borel homomorphism...

Two types of equivalence relation are used to classify functions between finite groups into classes which preserve combinatorial and algebraic properties important for a wide range of applications. However, it is very difficult to tell when functions equivalent under the coarser (“graph”) equivalence are inequivalent under the finer (“bundle”) equivalence. Here we relate graphs to transversals ...