Abelian Groups and Representations of Finite Partially Ordered Sets

Quasi-e-locally cyclic torsion-free abelian groups

“…In the remarks prior to the theorem we have established that (1) implies (2). Since dim Q (D) = dim Q (QA), D and QA are isomorphic Q-spaces and since both are D-spaces, we get that dim D (QA) = 1.…”

Indecomposable (1,3)-Groups and a matrix problem

“…Since G has no direct summand of rank 2 the coordinate matrix has no 0-column. Moreover, the part [β 1 | β 2 ] has no 0-row, since otherwise, by Lemma 14, an S-reduced form of [β 1 | β 2 ] would have a 0-row, and G would have a direct summand of rank 2 by Corollary 26 (2). Thus [β 1 | β 2 ] has no 0-lines.…”

“…Almost completely decomposable groups are a notoriously complicated class of torsion-free abelian groups of finite rank ( [15], [2], [17]), the source of many pathological decompositions ( [13]) and have been generalized to infinite rank ( [18]). …”

The School of Kiev in Colombia; the legacy of Alexander Zavadskij

“…Finally, we would like to point that Arnold described some relationships between Butler groups and representations of finite posets over discrete valuation rings. According to Arnold [1] computations of representation type of representations of posets over discrete valuation rings lead to explicit procedures for constructing indecomposable almost completely decomposable groups of arbitrarily large finite rank with types in a fixed finite set of types. Such constructions are quite difficult without the assistance of techniques from representation theory.…”