Hostname: page-component-77c89778f8-5wvtr Total loading time: 0 Render date: 2024-07-20T02:10:55.207Z Has data issue: false hasContentIssue false
  • English
  • Français

GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS

Published online by Cambridge University Press:  10 September 2003

P. M. HIGGINS ,
J. D. MITCHELL  and
N. RUšKUC
P. M. HIGGINS
Affiliation:
Department of Mathematics, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom e-mail: peteh@essex.ac.uk
J. D. MITCHELL
Affiliation:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom
N. RUšKUC
Affiliation:
Mathematical Institute, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom e-mail: nr1@st-and.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

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.

For a linearly ordered set $X$ we consider the relative rank of the semigroup of all order preserving mappings $\mathcal{O}_{X}$ on $X$ modulo the full transformation semigroup $\mathcal{T}_{X}$. In other words, we ask what is the smallest cardinality of a set $A$ of mappings such that $\genset{\mathcal{O}_{X}\cup A}=\mathcal{T}_{X}$. When $X$ is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when $X=\mathbb{R}$ (the set of real numbers) it is uncountable.

Keywords

20M2006A05
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 45 , Issue 3 , September 2003 , pp. 557 - 566
Copyright
© 2003 Glasgow Mathematical Journal Trust