On residual finiteness of monoids, their Schützenberger groups and associated actions

In this paper we discuss connections between the following properties: (RFM) residual finiteness of a monoid M; (RFSG) residual finiteness of Schützenberger 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, which are summarised in a table.