Finite Derivation Type Implies The Homological Finiteness Condition FP3

A monoidM that admits a finite convergent presentation satisfies the homological finiteness condition FP∞ and Squierʹs combinatorial property of having finite derivation type. Although Squier has given an example of a finitely presented monoidS1 that satisfies the condition FP∞, but that does not have finite derivation type, the exact relationship between these two conditions is unsolved. Here we establish a partial result by showing that for finitely presented monoids the property of having finite derivation type implies the homological finiteness conditions FP3 . Hence, the property FP3 is strictly weaker than the property of having finite derivation type.