GENERATING THE FULL TRANSFORMATION SEMIGROUP USING ORDER PRESERVING MAPPINGS

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 <(mathcal(O)X (union) 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.