  In this paper we discuss connections between the following properties: (RFM)
residual finiteness of a monoid M; (RFSG) residual finiteness of Schutzenberger
groups of M; and (RFRL) residual finiteness of the natural actions of M on its
Green's R- and L-classes. The general question is whether (RFM) implies (RFSG)
and/or (RFRL), and vice versa. We consider these questions in all the possible
combinations of the following situations: M is an arbitrary monoid; M is an
arbitrary regular monoid; every J-class of M has finitely many R- and
L-classes; M has finitely many left and right ideals. In each case we obtain
complete answers.
