TC semigroups and inflations

An algebra A satisfies TC (the term condition) if p(a, (x) over tilde) = p(a, (y) over tilde) iff p(b, (x) over tilde) = p(b, (y) over tilde) for any a,b is an element of A, (x) over tilde, (y) over tilde is an element of A(n) and any n + 1-ary term p. TG algebras have been extensively studied. We previously determined the structure of all TC semigroups. We use this result to show that if S is a TC semigroup then S-E = {alpha is an element of S\alpha x is an idempotent for some x is an element of S} is an inflation of S-Reg (the set of regular elements of S) and S-Reg congruent to H x A x B where H is an abelian group, A is a left zero semigroup, and B is a right zero semigroup. As a corollary of this result, we show that S is a semisimple TC semigroup iff S congruent to H x A x B where H is an abelian group, A is a left zero semigroup, and B is a right zero semigroup.