Word problems and a homological niteness condition for monoids

Our purpose is to prove that a monoid which has a 'nice' solution to its word problem satisfies a certain homological finiteness condition. More precisely, we prove: if a monoid S has a finite terminating Church-Rosser presentation, then S is (FP)3; this is Theorem 4.1 below. (See Section 2 for the definition of "terminating" and "Church-Rosser".) Examples of groups that are not (FP)3 are known; see Section 4 for a brief description of several of these. For completeness, we provide an example of a monoid that is not (FP)3' In each case, the monoid (or group) is finitely-presented and has a solvable word problem. These examples answer (in the negative) the following question of Jantzen [15]: does a finitelypresented monoid with a solvable word problem have a finite terminating Church-Rosser presentation? The Church-Rosser property was discovered by Church and Rosser [9] during the course of research on the A-calculus. Properties of terminating relations were investigated by Newman [16]. For a systematic treatment of both topics together with further references, see [14]. Monoids with terminating Church-Rosser presentations have been studied by Nivat [17] and others. See [5] for a recent survey. We conclude this introduction with a brief outline of what follows and some further discussion. Section 1 contains basic results on Noetherian relations. In particular, we develop some tools for dealing with free abelian groups which have a basis ordered by a Noetherian relation. Section 2 introduces terminating and Church-Rosser presentations. (Because of difficulties in verifying that the relation -..,) defined in Section 2 is Noetherian, it is common to assume that the rewriting rules R are length-reducing: if (r, s) E R, then Irl > lsi. We specifically do not make this assumption, so that our terminology differs, for example, from that of [5].) Variations of Theorem 2.1, which gives