Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-27T21:46:25.450Z Has data issue: false hasContentIssue false
  • English
  • Français

The structure of elements in finite full transformation semigroups

Published online by Cambridge University Press:  17 April 2009

Gonca Ayik ,
Hayrullah Ayik ,
Yusuf Ünlü  and
John M. Howie
Gonca Ayik
Affiliation:
Department of Mathematics, Çukurova University, Adana, Turkey, e-mail: [email protected]
Hayrullah Ayik
Affiliation:
Department of Mathematics, Çukurova University, Adana, Turkey, e-mail: [email protected]
Yusuf Ünlü
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom, e-mail: [email protected]
John M. Howie
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St Andrews, Fife KY16 9SS, United Kingdom, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with index m and period 1 is called an m-potent element. For an element α of a finite full transformation semigroup with index m and period r, a unique factorisation α = σβ such that Shift(σ) ∩ Shift(β) = ∅ is obtained, where σ is a permutation of order r and β is an m-potent. Some applications of this factorisation are given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 69 - 74
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Ayik, G., Ayik, H. and Howie, J.M., ‘On factorisations and generators in transformation semigroups’, Semigroup Forum (to appear).Google Scholar
[2]Harris, B. and Schoenfeld, L., ‘The number of idempotents in symmetric semigroup’, J. Combinatorial Theory 3 (1967), 122–135.CrossRefGoogle Scholar
[3]Howie, J.M., ‘The subsemigroup generated by the idempotents of a full transformation semigroup’, J. London Math. Soc. 41 (1966), 707–716.CrossRefGoogle Scholar
[4]Howie, J.M., Fundamentals of semigroup theory, London Mathematical Society Monographs 12 (Oxford University Press, New York 1995).CrossRefGoogle Scholar
[5]Howie, J.M. and McFadden, R.B., ‘Idempotent rank in finite full transformation semigroups’, Proc. Royal Soc. Edinburgh Sect. A 114 (1990), 161–167.Google Scholar
[6]Lipscomb, S., Symmetric inverse semigroups, Math. Surveys and Monographs 46 (American Math. Soc.,Providence, R.I., 1996).CrossRefGoogle Scholar
[7]Riordan, J., ‘Forests of labeled trees’, J. Combinatorial Theory 5 (1968), 90–103.CrossRefGoogle Scholar
[8]Sachkov, V.N., Combinatorial methods in discrete mathematics (Cambridge University Press, Cambridge, 1996).CrossRefGoogle Scholar
[9]Sioson, F.M., ‘Counting elements in semigroups’, J. Combinatorial Theory 12 (1972), 339–345.CrossRefGoogle Scholar
[10]Vorob'ev, N.N., ‘On symmetric associative systems’, (Russian), Leningrad. Gos. Ped. Inst. Uc. Zap. 89 (1953), 161–166.Google Scholar