If X is a semigroup, and Tx is the set of all transformations in X, how does one show that Tx has non-commuting idempotents?
I'm not sure you've framed your question correctly. By $\displaystyle T_X$ do you mean the full transformation semigroup? This is regular and idempotents commute iff it's inverse. Your statement implies that full transformation semigroups can't be inverse which is not correct.