## On the Burnside problem for semigroups.(English)Zbl 0545.20051

A finitely generated semigroup S is proved to be finite if and only if S is torsion and possesses the following permutation property: There exists an integer $$n\geq 2$$ such that to each $$(s_ 1,...,s_ n)$$ in $$S^ n$$ there corresponds a permutation $$\sigma$$ of 1,...,n, $$\sigma \neq id$$, so that $$s_ 1s_ 2...s_ n=s_{\sigma(1)}s_{\sigma(2)}...s_{\sigma(n)}.$$
Reviewer: T.J.Harju

### MSC:

 20M10 General structure theory for semigroups 20M05 Free semigroups, generators and relations, word problems
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### References:

 [1] Herstein, I.N, Noncommutative rings, (), Chap. 8 · Zbl 0874.16001 [2] Lothaire, M, Combinatorics on words, (1983), Addison-Wesley Reading, Mass · Zbl 0514.20045 [3] Morse, M; Hedlund, G, Unending chess, symbolic dynamics and a problem in semigroups, Duke math. J., 11, 1-7, (1944) · Zbl 0063.04115 [4] Rowen, L.H, Polynomial identities in ring theory, (1980), Academic Press New York/London · Zbl 0461.16001
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