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Growth of generating sets for direct powers of classical algebraic structures

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QuickRuskuc2010JAustMathSoc89.pdf (224.1Kb)
Date
08/2010
Author
Quick, Martyn
Ruskuc, Nik
Keywords
Generating sets
Growth
Direct products
Algebraic structures
Universal algebra
QA Mathematics
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Abstract
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.
Citation
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 . https://doi.org/10.1017/S1446788710001473
Publication
Journal of the Australian Mathematical Society
Status
Peer reviewed
DOI
https://doi.org/10.1017/S1446788710001473
ISSN
1446-7887
Type
Journal article
Rights
(c) 2010 Australian Mathematical Publishing Association Inc.
Collections
  • University of St Andrews Research
URI
http://hdl.handle.net/10023/3058

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