Unions of dihedral groups (Q1070357)

By the following simple formula (1) \(\forall x\exists y\) \((x=xyy\), \(y=xyx)\) we characterize semigroups from the title. Considering a local property of their \({\mathcal H}\)-classes we get bands and Boolean groups as extreme cases of semigroups with that property. We also provide a construction showing that \({\mathcal H}\)-classes can be sufficiently complicated (at least as Abelian groups are). Then we permute right-hand sides of identities in (1) getting Boolean semigroups \((x^ 3=x)\) 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.