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Growth sequences of finite semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

James Wiegold  and
H. Lausch
James Wiegold
Affiliation:
Department of Pure Mathematics, University College, Cardiff CF1 1XL Wales, United Kingdom
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Abstract

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The growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.

MSC classification

Secondary: 20M99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 43 , Issue 1 , August 1987 , pp. 16 - 20
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Wiegold, James, ‘Growth sequences of finite groups’, J. Austral. Math. Soc. 17 (1974), 133–141.CrossRefGoogle Scholar
[2]Wiegold, James, ‘Growth sequences of finite groups II’, J. Austral. Math. Soc. 20 (1975), 225–229.CrossRefGoogle Scholar
[3]Wiegold, James, ‘Growth sequences of finite groups III’, J. Austral. Math. Soc. 25 (1978), 142–144.CrossRefGoogle Scholar
[4]Wiegold, James, ‘Growth sequences of finite groups IV’, J. Austral. Math. Soc. 29 (1980), 14–16.CrossRefGoogle Scholar
[5]Meier, D. and Wiegold, James, ‘Growth sequences of finite groups V’, J. Austral. Math. Soc. 31 (1981), 374–375.CrossRefGoogle Scholar
[6]Wiegold, James and Wilson, John, ‘Growth sequences of finitely generated groups’, Arch. Math. 30 (1978), 337–343.CrossRefGoogle Scholar