Abstract
We study semigroups that behave nicely with respect to a distinguished subset of idempotents E, both in terms of the extended Green’s relations $$\widetilde{\mathcal {K}}_E$$ and as unary semigroups. New structure theorems are given, notably in the case of central idempotents. Finally, the decomposition theorems are applied to the study of regular semigroups with particular generalized inverses.
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