  We give an example of a monoid with finitely many left and right ideals, all
of whose Schutzenberger groups are presentable by finite complete rewriting
systems, and so each have finite derivation type, but such that the monoid
itself does not have finite derivation type, and therefore does not admit a
presentation by a finite complete rewriting system. The example also serves as
a counterexample to several other natural questions regarding complete
rewriting systems and finite derivation type. Specifically it allows us to
construct two finitely generated monoids M and N with isometric Cayley graphs,
where N has finite derivation type (respectively, admits a presentation by a
finite complete rewriting system) but M does not. This contrasts with the case
of finitely generated groups for which finite derivation type is known to be a
quasi-isometry invariant. The same example is also used to show that neither of
these two properties is preserved under finite Green index extensions.
