Abstract
By the following simple formula $$\forall x \exists y (x = xyy, y = xyx)$$ (1) We characterize semigroups from the title. Considering a local property of their ℋ-classes we get bands and Boolean groups as extreme cases of semigroups with that property. We also provide a construction showing that ℋ-classes can be sufficiently complicated (at least as Abelian groups are). Then we permute right-hand sides of identities in (1) getting Boolean semigroups (x3=x) and so-called anti-inverse semigroups. Finally we show that Boolean semigroups are a proper subclass of the intersection of anti-inverse semigroups and unions of dihedral groups.
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