On some problems of Petrich concerning Bruck and Reilly semigroups (Q1204135)

A one-to-one partial right translation of a semigroup \(S\) is a one-to-one partial transformation \(\rho\), the domain of which is a left ideal \(I\) of \(S\), such that \((xy)\rho = x\cdot (y)\rho\) for each \(x\in S\), \(y\in I\). It is proved that if \(S\) is a regular semigroup with its set of idempotents dually well-ordered, then \(S\) is isomorphic to the set of all one-to-one partial right translations of \(S\). It is also proved that a Bruck semigroup \(B(T,\alpha)\) is \(E\)-unitary if and only if \(T\) is \(E\)- unitary and \(\alpha\) is an idempotent pure homomorphism. The construction of \(E\)-unitary covers of Bruck semigroups \(B(T,\alpha)\), where \(T\) is a finite chain of groups, is given.