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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
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Abstract

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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