In this article we shall prove that every $(n+1)$-ary equivalence relation on a set $S$ uniquely determines one binary equivalence in the set $S^{(n)}$ of all $n$-subsets of $S$, satisfying a special property ($(i_n)$, Lemma 2). We shall also prove the converse, i.e. that there is a bijection between these two sets of relations. One of its consequences is (Corollary 11) that every lattice of $(n+1)$-ary equivalences (or of partitions of type $n$) the finite set $S$ is isomorphic to the quotient relative to one closure operation on the lattice of binary equivalences (or of partitions of type 1) on $S^{(n)}$.