For finitely presented monoids the homological finiteness conditions and coincide

is a homological finiteness condition of monoids that was introduced by Pride and Wang. For finitely presented groups this property is equivalent to the homological finiteness condition , but for finitely presented monoids in general it is strictly stronger than the conditions left- and right- together. Here we define the notion of for monoids, using bimodules instead of left- or right-modules, and by establishing an exact sequence involving the relation bimodule of a monoid, we prove that for finitely presented monoids the property is equivalent to the finiteness condition .