J Austral Math Soc Ser A 43 pp16--20, 1987.

Growth Sequences of Finite Semigroups

James Wiegold

Abstract

The growth sequence of a finite semigroup S {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) ³ cⁿ for all n ³ 2.

1980 AMS Subject Classification: 20M99

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Authors

James Wiegold

Department of Pure Mathematics, University College, Cardiff CF1 1XL, Wales, United Kingdom.