Generating the full transformation semigroup using order preserving mappings
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For a linearly ordered set X we consider the relative rank of the semigroup of all order preserving mappings O-X on X modulo the full transformation semigroup Ex. In other words, we ask what is the smallest cardinality of a set A of mappings such that <O-X boolean OR A> = T-X. When X is countably infinite or well-ordered (of arbitrary cardinality) we show that this number is one, while when X = R (the set of real numbers) it is uncountable.
Higgins , P M , Mitchell , J D & Ruskuc , N 2003 , ' Generating the full transformation semigroup using order preserving mappings ' Glasgow Mathematical Journal , vol 45 , no. 3 , pp. 557-566 . DOI: 10.1017/S0017089503001460
Glasgow Mathematical Journal
(c)2003 Glasgow Mathematical Journal Trust
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