Hostname: page-component-7dd5485656-6kn8j Total loading time: 0 Render date: 2025-10-30T09:00:07.315Z Has data issue: false hasContentIssue false
  • English
  • Français

On the semigroups of order-decreasing finite full transformations

Published online by Cambridge University Press:  14 November 2011

Abdullahi Umar
Abdullahi Umar
Affiliation:
Department of Mathematical Sciences, University of St Andrews, St Andrews KY16 9SS, Scotland, U.K.
Get access Rights & Permissions [Opens in a new window]

Synopsis

Let Singn be the subsemigroup of singular elements of the full transformation semigroup on a totally ordered finite set with n elements. Let be the subsemigroup of all decreasing maps of Singn. In this paper it is shown that is a non-regular abundant semigroup with n − 1 -classes and . Moreover, is idempotent-generated and it is generated by the n(n − 1)/2 idempotents in J*n−1. Let

and

Some recurrence relations satisfied by J*(n, r) and sh (n, r) are obtained. Further, it is shown that sh (n, r) is the complementary signless (or absolute) Stirling number of the first kind.

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 1-2 , 1992 , pp. 129 - 142
Copyright
Copyright © Royal Society of Edinburgh 1992

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.)

Article purchase

Temporarily unavailable

References

1Anderson, Ian. A first course in combinatorial mathematics (Oxford: Oxford University Press, 1974).Google Scholar
2Borwein, D., Rankin, S. and Renner, L.. Enumeration of injective partial transformations. Discrete Math. 73 (1989), 291–296.CrossRefGoogle Scholar
3Charalambides, Ch. A. and Singh, J.. A review of the Stirling numbers, their generalisations and statistical applications. Comm. Statist. A-Theory Methods 17 (8) (1988), 2533–2595.Google Scholar
4El-Qallali, A. and Fountain, J. B.. Idempotent-connected abundant semigroups. Proc. Roy. Soc. Edinburgh Sect. A 91 (1981), 79–80.CrossRefGoogle Scholar
5Fountain, J. B.. Abundant semigroups. Proc. London Math. Soc (3) 44 (1982), 103–129.CrossRefGoogle Scholar
6Gomes, G. M. S. and Howie, J. M.. On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum (to appear).Google Scholar
7Howie, J. M.. The subsemigroup generated by the idempotents of a full transformation semigroup. J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
8Howie, J. M.. Product of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. (2) 17 (1971), 223–236.CrossRefGoogle Scholar
9Howie, J. M.. An introduction to semigroup theory (London: Academic Press, 1976).Google Scholar
10Howie, J. M.. Idempotent generators in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 81 (1978) 317–323.CrossRefGoogle Scholar
11Howie, J. M.. Combinatorial and arithmetical aspects of the theory of transformation semigroups. (Lectures given in the University of Lisbon, March, 1990).Google Scholar
12Howie, J. M. and McFadden, R. B.. Idempotent rank in finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 161–167.CrossRefGoogle Scholar
13Howie, J. M., Robertson, E. F. and Schein, B. M.. A combinatorial property of finite full transformation semigroups. Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 319–328.CrossRefGoogle Scholar
14Pin, J. E.. Varieties of formal languages (Paris: Masson, 1984; English translation, translated by Howie, A. (Oxford: North Oxford Academic Publishers Ltd., 1986).Google Scholar
15Woodall, D. R.. A market problem. J. Combin. Theory 10 (1971) 275–287.CrossRefGoogle Scholar