Growth of generating sets for direct powers of classical algebraic structures
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For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.
Quick , M & Ruskuc , N 2010 , ' Growth of generating sets for direct powers of classical algebraic structures ' Journal of the Australian Mathematical Society , vol 89 , no. 1 , pp. 105-126 . DOI: 10.1017/S1446788710001473
Journal of the Australian Mathematical Society
(c) 2010 Australian Mathematical Publishing Association Inc.
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