Absolute flatness of the full transformation semigroups (Q1112966)

\({\mathcal T}_ X\) denotes the full transformation semigroup on a set X. A monoid S is defined to be left absolutely flat if every left S-set is flat. This paper is devoted to a proof of the result that \({\mathcal T}_ X\) is left absolutely flat if and only if X is finite. This, together with previous results by the author and others, yield the fact that \({\mathcal T}_ X\) has the strong representation property if and only if X is finite and also that \({\mathcal T}_ X\) is an amalgamation base in the class of all semigroups whenever X is finite.