  We prove that a monoid is sofic, in the sense recently introduced by
Ceccherini-Silberstein and Coornaert, whenever the J-class of the identity is a
sofic group, and the quotients of this group by orbit stabilisers in the rest
of the monoid are amenable. In particular, this shows that the following are
all sofic: cancellative monoids with amenable group of units; monoids with
sofic group of units and finitely many non-units; and monoids with amenable
Schutzenberger groups and finitely many L-classes in each D-class. This
provides a very wide range of sofic monoids, subsuming most known examples
(with the notable exception of locally residually finite monoids). We conclude
by discussing some aspects of the definition, and posing some questions for
future research.
