A finiteness condition for rewriting systems

We present a purely combinatorial approach to the question of whether or not a finitely presented monoid has a finite canonical presentation. Our approach is based on the notion of “finite derivation type” which is a combinatorial condition satisfied by certain rewriting systems. Our main result states that if a monoid M has a finite canonical presentation, then all finite presentations of M have finite derivation type. By proving that a certain monoid S1 does not have finite derivation type we show in addition that the homological finiteness condition FP ∞ is not sufficient to guarantee that a finitely presented monoid with a decidable word problem has a finite canonical presentation.