FINITE DERIVATION TYPE IMPLIES THE HOMOLOGICAL FINITENESS CONDITION FP3

A monoid M that admits a finite convergent presentation satisfies the homological finiteness condition FPinfinity and Squier's combinatorial property of having finite derivation type. Although Squier has given an example of a finitely presented monoid S-1 that satisfies the condi tion FPinfinity, but that does not have finite derivation type, the ex act relationship between these two conditions is unsolved. Here we est ablish 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.