Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T05:10:25.942Z Has data issue: false hasContentIssue false
  • English
  • Français

Products of idempotents in finite full transformation semigroups

Published online by Cambridge University Press:  14 November 2011

J. M. Howie
J. M. Howie
Affiliation:
Mathematical Institute, University of St Andrews†
Get access Rights & Permissions [Opens in a new window]

Synopsis

If |X| = n and α is a singular mapping in J(X), define c(α) to be the number of cyclic orbits of α and f(α) to be the number of fixed points. Then α is expressible as a product of n + c(α)−f(α) idempotents of rank n − 1, and no smaller number of idempotents of rank n − 1 will suffice. The maximum possible value of n + c(α)–f(α) is [3/2(n − 1)], which is thus a best possible global lower bound for the number of idempotents required to generate a singular element of J(X).

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 3-4 , 1980 , pp. 243 - 254
Copyright
Copyright © Royal Society of Edinburgh 1980

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

1Hall, T. E.. On regular semigroups. J. Algebra 24 (1973), 1–24.Google Scholar
2Howie, J. M.. An introduction to semigroup theory (London: Academic Press, 1976).Google Scholar
3Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–716.Google Scholar
4Howie, J. M.. Idempotent generators in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 81 (1978), 317–323.Google Scholar