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Embedding any countable semigroup in a 2-generated bisimple monoid

Published online by Cambridge University Press:  18 May 2009

Karl Byleen
Karl Byleen
Affiliation:
Department of Mathematics, Statistics, and Computer Science, Marquette University, Milwaukee, Wisconsin 53233
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G. B. Preston [10] proved that any semigroup can be embedded in a bisimple monoid. If S is a countable semigroup, his constructive proof yields a bisimple monoid which is also countable, but not necessarily finitely generated. The main result of this paper is that any countable semigroup can be embedded in a 2-generated bisimple monoid.

J. M. Howie [6] proved that any semigroup can be embedded in an idempotentgenerated semigroup. F. Pastijn [9] showed that any semigroup can be embedded in a bisimple idempotent-generated semigroup, and that any countable semigroup can be embedded in a semigroup which is generated by 3 idempotents. Easy proofs of these results using Rees matrix semigroups over a semigroup were given by the author [3]. In this paper, as a corollary to our main result, we deduce that any countable semigroup can be embedded in a bisimple semigroup which is generated by 3 idempotents.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 25 , Issue 2 , July 1984 , pp. 153 - 161
Copyright
Copyright © Glasgow Mathematical Journal Trust 1984

References

REFERENCES

1.Benzaken, C. and Mayr, H. C., Notion de demi-bande: demi-bandes de type deux, Semigroup Forum 10 (1975), 115–128.CrossRefGoogle Scholar
2.Bruck, R. H., A survey of binary systems, Ergebnisse der Math. Heft 20 (Springer, 1958).CrossRefGoogle Scholar
3.Byleen, K., Regular four-spiral semigroups, idempotent-generated semigroups, and the Rees construction, Semigroup Forum 22 (1981), 97–100.CrossRefGoogle Scholar
4.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups (Arner. Math. Soc. 1961, 1967).Google Scholar
5.Howie, J. M., An introduction to semigroup theory (Academic Press, 1976).Google Scholar
6.Howie, J. M., The subsemigroup generated by the idempotents of a full transformation semigroup, J. London Math. Soc. 41 (1966), 707–716.Google Scholar
7.McAlister, D. B., Regular Rees matrix semigroups and regular Dubreil-Jacotin semigroups, J. Austral. Math. Soc. Ser. A 31 (1981), 325–336.CrossRefGoogle Scholar
8.Munn, W. D., On simple inverse semigroups, Semigroup Forum 1 (1970), 63–74.Google Scholar
9.Pastijn, F., Embedding semigroups in semibands, Semigroup Forum 14 (1977), 247–263.CrossRefGoogle Scholar
10.Preston, G. B., Embedding any semigroup in a D-simple semigroup, Trans. Amer. Math. Soc. 93 (1959), 351–355.Google Scholar
11.Reilly, N. R., Bisimple ω-semigroups, Proc. Glasgow Math. Assoc. 7 (1966), 160–167.CrossRefGoogle Scholar