We show that if A1, A2,..., An, B1, B2,...,Bt are uniform objects of an abelian category C, then A1 ⊕ A2 ⊕⋯⊕ An and B1 ⊕ B2 ⊕⋯⊕ Bt are in the same monogeny class if and only if n = t and there is a permutation σ of 1, 2,...,n} such that Ai and Bσ(i) are in the same monogeny class for every i = 1, 2,..., n. 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. © 2001 Elsevier Science B.V. All rights reserved.
Uniqueness of monogeny classes for uniform objects in abelian categories
DI RACCA, LUCA;FACCHINI, ALBERTO
2002
Abstract
We show that if A1, A2,..., An, B1, B2,...,Bt are uniform objects of an abelian category C, then A1 ⊕ A2 ⊕⋯⊕ An and B1 ⊕ B2 ⊕⋯⊕ Bt are in the same monogeny class if and only if n = t and there is a permutation σ of 1, 2,...,n} such that Ai and Bσ(i) are in the same monogeny class for every i = 1, 2,..., n. 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. © 2001 Elsevier Science B.V. All rights reserved.
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