Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-26T13:34:23.759Z Has data issue: false hasContentIssue false
  • English
  • Français

The word problem for semigroups satisfying x3 = x

Published online by Cambridge University Press:  24 October 2008

J. A. Gerhard
J. A. Gerhard
Affiliation:
University of Manitoba, Winnipeg
Get access Rights & Permissions [Opens in a new window]

Extract

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 84 , Issue 1 , July 1978 , pp. 11 - 19
Copyright
Copyright © Cambridge Philosophical Society 1978

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Brown, T. C.On the finiteness of semigroups in which x r = x. Proc. Cambridge Philos. Soc. 60 (1964), 1028–1029.CrossRefGoogle Scholar
(2)Cohn, P. M.Universal algebra (New York, Harper and Row, 1965).Google Scholar
(3)Gerhard, J. A.The lattice of equational classes of idempotent semigroups, J. Algebra 15 (1970), 195–224.CrossRefGoogle Scholar
(4)Green, J. A. & Rees, D.On semigroups in which x r = x. Proc. Cambridge Philos. Soc. 48 (1952), 35–40.CrossRefGoogle Scholar
(5)Howie, J. M.An introduction to semigroup theory (London, Academic Press, 1976).Google Scholar
(6)Mclean, D.Idempotent semigroups. Amer. Math. Monthly 61 (1954), 110–113.CrossRefGoogle Scholar