Geodesic
GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS
Glasgow mathematical journal, Tome 45 (2003) no. 3, pp. 557-566

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DOI

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.
DOI : 10.1017/S0017089503001460
Mots-clés : 20M20, 06A05 
HIGGINS, P. M.; MITCHELL, J. D.; RUšKUC, N. GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS. Glasgow mathematical journal, Tome 45 (2003) no. 3, pp. 557-566. doi: 10.1017/S0017089503001460
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     title = {GENERATING {THE} {FULL} {TRANSFORMATION} {SEMIGROUP} {USING} {ORDER} {PRESERVING} {MAPPINGS}},
     journal = {Glasgow mathematical journal},
     pages = {557--566},
     year = {2003},
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