On some problems of Petrich concerning Bruck and Reilly semigroups

For an inverse semigroupS with its idempotents dually well-ordered, we prove thatS is isomorphic to the semigroup of all one-to-one partial right translations ofS. Also, we prove for a Bruck semigroupS=B(T, α) thatS isE-unitary if and only ifT isE-unitary and α is an idempotent pure homomorphism. Moreover, we characterize allE-unitary covers ofB(T, α), whereT is a finite chain of groups.