"Local and nice structures of the groupoid of an equivalence relation"

(joint work with Tomasz Rybicki, Cahiers de Topologie et Geométrie Differentielle categoriques, 2003 (in print))
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A comparison between the concepts of local and nice structures of the groupoid of an equivalence relation is presented. It is shown that these concepts are closely related, and that generically they characterize the equivalence relations induced by regular foliations. The first concept was introduced by J.Pradines (1966), and studied by R.Brown and O.Mucuk (1996), while the second one was given by the first author (1987). The importance of these concepts in the nontransitive geometry is indicated.