  In the present paper, a series of results and examples that explore the
structural features of H-commutative semigroups are provided. We also
generalise a result of Isbell from commutative semigroups to H-commutative
semigroups by showing that the dominion of an H-commutative semigroup is
H-commutative. We then use this to generalise Howie and Isbell's result that
any H-commutative semigroup satisfying the minimum condition on principal
ideals is saturated.
