On Residual Finiteness of Monoids, their Schutzenberger Groups and Associated Actions
arXiv:1003.3176 · doi:10.1016/j.jalgebra.2014.02.025
Abstract
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.
27 pages, 3 figures, revised according to referee's comments, to appear in Journal of Algebra