Thread: Abstract Algebra: Semigroups

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 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.