Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T00:47:49.352Z Has data issue: false hasContentIssue false
  • English
  • Français

On the rank of completely 0-simple semigroups

Published online by Cambridge University Press:  24 October 2008

N. Ruškuc
N. Ruškuc
Affiliation:
Department of Mathematical and Computational Sciences, University of St Andrews, St Andrews, KY16 9SS, Scotland
Get access Rights & Permissions [Opens in a new window]

Abstract

Connected completely 0-simple semigroups are defined by a number of equivalent conditions, and a formula for the rank of these semigroups is proved. As a consequence an alternative proof of the result from [11] is given. In the case of a Rees matrix semigroup M0 [G, I, Λ, P] the rank is expressed in terms of |I|, |Λ|, G and a certain subgroup of G depending on P. At the end the minimal rank of all semigroups M0[G, I, Λ, P] is found for a given group G. Since every completely simple semigroup is connected, every result has a corollary for these semigroups.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 116 , Issue 2 , September 1994 , pp. 325 - 338
Copyright
Copyright © Cambridge Philosophical Society 1994

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]Clifford, A. H.. Semigroups admitting relative inverses. Ann. of Math. 42 (1941), 1037–1049.CrossRefGoogle Scholar
[2]Feigenbaum, R.. Regular semigroup congruences. Semigroup Forum 17 (1979), 373–377.CrossRefGoogle Scholar
[3]Fitz-Gerald, D. G.. On inverses of products of idempotents in regular semigroups. J. Australian Math. Soc. 13 (1972), 335–337.CrossRefGoogle Scholar
[4]Gomes, G. M. S. and Howie, J. M.. On the ranks of certain finite semigroups of transformations. Math. Proc. Camb. Philos. Soc. 101 (1987), 395–403.CrossRefGoogle Scholar
[5]Gomes, G. M. S. and Howie, J. M.. On the ranks of certain semigroups of order preserving transformations. Semigroup Forum 40 (1990), 1–15.Google Scholar
[6]Hall, T. E.. On regular semigroups. J. Algebra 24 (1973), 1–24.CrossRefGoogle Scholar
[7]Higgins, P. M.. Techniques of Semigroup Theory (Oxford University Press, 1992).CrossRefGoogle Scholar
[8]Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
[9]Howie, J. M.. An Introduction to Semigroup Theory (Academic Press, 1976).Google Scholar
[10]Howie, J. M.. Idempotents in completely 0-simple semigroups. Glasgow Math. J. 19 (1978), 109–113.CrossRefGoogle Scholar
[11]Howie, J. M. and McFadden, R. B.. Idempotent rank in finite full transformation semigroups. Proc. Roy. Soc. Edinb. Sect. A 114, (1990), 161–167.Google Scholar
[12]Petrich, M.. Inverse Semigroups (John Wiley, 1984).Google Scholar
[13]Rees, D.. On semi-groups. Proc. Cambridge Philos. Soc. 36 (1940), 387–400.CrossRefGoogle Scholar