Certain Dense Embeddings of Regular Semigroups

In a previous paper, the author has introduced a number of homomorphisms of an arbitrary semigroup into the translational hull of certain Rees matrix semigroups or orthogonal sums thereof. For regular semigroups, it is proved here that all of these homomorphisms have the property that the image is a densely embedded subsemigroup, i.e., is a densely embedded ideal of its idealizer, and that the corresponding Rees matrix semigroups are regular. Several of these homomorphisms are 1-1, in each case they furnish a different dense embedding of an arbitrary regular semigroup into the translational hull of a regular Rees matrix semigroup or orthogonal sums thereof. A new representation for regular semigroups is introduced.