Reducibility of Monadic Equivalence Relations

Each additive cut in the nonstandard natural numbers $\!{}^*{\mathbb N}$ induces the equivalence relation $\operatorname M_U$ on $\!{}^*{\mathbb N}$ defined as $x\operatorname M_Uy$ if $|x-y|\in U$. Such equivalence relations are said to be monadic. Reducibility between monadic equivalence relations is studied. The main result (Theorem 3.1) is that reducibility can be defined in terms of cofinality (or coinitiality) and a special parameter of a cut, called its width. Smoothness and the existence of transversals are also considered. The results obtained are similar to theorems of modern descriptive set theory on the reducibility of Borel equivalence relations.