A relation for general and inverse semigroups

The relation in the title is S defined by on an arbitrary semigroup. We investigate antisymmetry of S by means of a (minimal) family whose members can not appear as subsemigroups. Transitivity of S is characterized similarly by means of the family and homomorphic images of a certain semigroup. We study the transfer of certain properties of a monoid T and the Bruck semigroup B(T,) over T. The paper concludes with a consideration of certain properties of the relation S on inverse semigroups.