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

On properties not inherited by monoids from their Schutzenberger groups