Measurable dynamical systems have been extensively studied in the last decades, due to the several relations with other areas of Mathematics (in particular Operator Algebras).

In this work we consider a discrete measure-preserving equivalence relation $R$ on a standard measure space $(X,\mu)$. The triple $(X,\mu,R)$ naturally gives rise to the \emph{full group} $[R]$ and \emph{semigroup} $[\![R]\!]$ of $R$, and a well-known result of Dye (1963) states that, in the ergodic case, $[R]$ completely determines $R$ (up to isomorphism). In the first part of this work, we present a generalization of Dye's result to the non-ergodic case.

The second part of this work is concerned with the \emph{sofic} property, which is a weak notion of approximability by finite structures and has been of great interest recently. It was introduced by Gromov (1999) for groups in the context of symbolic dynamics, and later given by Elek and Lippner (2010) for equivalence relations. We present a natural description of soficity in terms of full groups and semigroups.

The present work was supported by CAPES, Coordenação de Aperfeiçoamento de Pessoal de N\ível Superior - Brasil.