Uniqueness of monogeny classes for uniform objects in abelian categories

We show that if <f>A1,A2,…,An,B1,B2,…,Bt</f> are uniform objects of an abelian category <f>C</f>, then <f>A1⊕A2⊕⋯⊕An</f> and <f>B1⊕B2⊕⋯⊕Bt</f> are in the same monogeny class if and only if <f>n=t</f> and there is a permutation <f>σ</f> of <f>{1,2,…,n}</f> such that <f>Ai</f> and <f>Bσ(i)</f> are in the same monogeny class for every <f>i=1,2,…,n</f>. This is proved by showing that strong components of bipartite digraphs with enough edges intersect the two independent sets of vertices of a bipartition of the digraph in sets of the same cardinality. [Copyright &y& Elsevier]