Counterexamples Concerning Inverse Monoids and Graded Algebras

Pride raised two questions. the first one is: for any n, is there a graded algebra of type fp n but not of type fp n+1? the second one is: the following implication bi-fp n ⇒ left-fp n + right-fp n is reversible for groups, is it reversible for inverse monoids? in this paper, we give two examples of graded algebras of type fp0 but not of type fp1. also, we show that the above implication is not reversible for inverse monoids when n =1 by giving an example of an inverse monoid of type fp1 but not of type bi-fp1.