  We consider reducibility of equivalence relations (ERs, for brevity), in a
nonstandard domain, in terms of the Borel reducibility and the countably
determined (CD, for brevity) reducibility. This reveals phenomena partially
analogous to those discovered in descriptive set theory. The Borel reducibility
structure of Borel sets and (partially) CD reducibility structure of CD sets in
*N is described. We prove that all CD ERs with countable equivalence classes
are CD-smooth, but not all are B-smooth, for instance, the ER of having finite
difference on *N. Similarly to the Silver dichotomy theorem in Polish spaces,
any CD ER on *N either has at most continuum-many classes or there is an
infinite internal set of pairwise inequivalent elements. Our study of monadic
ERs on *N, i.e., those of the form x E y iff |x-y| belongs to a given additive
Borel cut in *N, shows that these ERs split in two linearly families,
associated with countably cofinal and countably coinitial cuts, each of which
is linearly ordered by Borel reducibility. The relationship between monadic ERs
and the ER of finite symmetric difference on hyperfinite subsets of *N is
studied.
