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TC semigroups and inflations

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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 thenS E = {a ε S | ax is an idempotent for somex ε S} is an inflation ofS Reg (the set of regular elements ofS) andS Reg ≅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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Authors and Affiliations

  1. Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, 31261, Dhahran, Saudi Arabia

    R. J. Warne

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  1. R. J. Warne
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Communicated by László Márki

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Warne, R.J. TC semigroups and inflations. Semigroup Forum 54, 271–277 (1997). https://doi.org/10.1007/BF02676610

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  • DOI: https://doi.org/10.1007/BF02676610

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