  We prove new results on inheritance of Green's relations by subsemigroups in
the presence of stability of elements. We provide counterexamples in other
cases to show in particular that not all right-stable semigroups are embeddable
in left-stable semigroups. This is carried out in the context of a survey of
the various closely related notions of stability and minimality of Green's
classes that have appeared in the literature over the last sixty years, and
which have sometimes been presented in different forms.
