Do Rational Equivalence Relations have Regular Cross-Sections?

The following classes of rational equivalence relations are shown to have regular cross-sections: deterministic rational equivalence relations, rational equivalence relations over a one letter alphabet, and rational equivalence relations with bounded separability. Although the general case remains open, it is shown to be reducible to that of locally-finite rational equivalence relations over a two letter alphabet. Two particular cross-sections are shown not to be regular: the set of minimum length words and the set of lexicographically minimal words.