Abstract
An algebraA satisfiesTC (the term condition) if\(p(\alpha ,\tilde x) = (\alpha ,\tilde y)iffp(b,\tilde x) = (b,\tilde y)\) for any\(\alpha ,b \in A,\tilde x,\tilde y \in A^n \) and anyn + 1-ary termp.TC algebras have been extensively studied. We previously determined the structure of allTC semigroups. We use this result to show that ifS is aTC semigroup thenSE = {a e S | ax is an idempotent for somex e S} is an inflation ofSReg (the set of regular elements ofS) andSReg ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup. As a corollary of this result, we show thatS is a semisimpleTC semigroup iffS ≅H × A × B whereH is an abelian group,A is a left zero semigroup, andB is a right zero semigroup.
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