Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-20T00:36:20.877Z Has data issue: false hasContentIssue false
  • English
  • Français

A combinatorial property of finite full transformation semigroups

Published online by Cambridge University Press:  14 November 2011

John M. Howie ,
Edmund F. Robertson  and
Boris M. Schein
John M. Howie
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland, U.K.
Edmund F. Robertson
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, Scotland, U.K.
Boris M. Schein
Affiliation:
Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let E be the set of idempotents in the semigroup Singn of singular self-maps of N = {1, …, n}. Let α ∊ Singn. Then α ∊ E2 if and only if for every x in im α the set xα−1 either contains x or contains an element of (im α)′.

Write rank α for |im α| and fix α for |{x ∊ N: xa = x}|. Define (x, xα, xα2) to be an admissible α-triple if x ∊ (im α)′, xα3 ≠ xα2. Let comp α (the complexity of α) be the maximum number of disjoint admissible α-triples. Then α ∊ E3 if and only if

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 3-4 , 1988 , pp. 319 - 328
Copyright
Copyright © Royal Society of Edinburgh 1988

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

1Harris, Bernard. A note on the number of idempotents in symmetric semigroups. American Math. Monthly 74 (1967), 1234–1235.Google Scholar
2Harris, Bernard and Schoenfeld, Lowell. The number of idempotent elements in symmetric semigroups. J. Combin. Theory 3 (1967), 122–135.CrossRefGoogle 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.. Products of idempotents in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 243–254.Google Scholar
5Howie, J. M.. Some subsemigroups of infinite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), 159–167.Google Scholar
6Howie, J. M.. Products of idempotents in finite full transformation semigroups: some improved bounds. Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 25–35.Google Scholar
7Iwahori, Nobuko. A length formula in a semigroup of mappings. J. Fac. Sci. Univ. Tokyo Sect 1A Math. 24 (1977), 255–260.Google Scholar
8Tainiter, M.. A characterization of idempotents in semigroups. J. Combin. Theory 5 (1968), 370–373.Google Scholar