Sokhatsky F.
On the associativity of multiplace operations

In previous paper the study of (i,j)-associative (n+1)-ary groupoids is reduced to the study of groupoids, which are (k,m)-associative for all k, m satisfying k=m=0(mod r), k=m(mod s), where r|s|n. The last groupoids are called associates of the sort (r,s,n) and of the sort (s,n), when r=1. Here, first order balanced identities are described in the associates. In other words, a bracketing rule is given.