On translational hulls of type-A semigroups

In this paper, the translational hull of a type-A semigroup is considered. Our main result is to prove that the translational hull of a semilattice of cancellative monoids (respectively a proper type-A semigroup, an E-reflexive type-A semigroup) is still the same type of semigroups. This answers an open problem posted by Petrich on translational hulls of semigroups in 1984.