A new finiteness condition for monoids presented by complete rewriting systems ( after Craig

Recently, Craig Squier introduced the notion of finite derivation type to show that some finitely presentable monoid has no presentation by means of a finite complete rewriting system. A similar result was already obtained by the same author using homology, but the new method is more direct and more powerful. Here, we present Squier’s argument with a bit of categorical machinery, making proofs shorter and easier. In addition we prove that, if a monoid has finite derivation type, then its third homology group is of finite type. An invariant for a structure is something which can be calculated in many ways, but only depends on the structure itself. Typical examples are the dimension of a vector space or the genus of a surface. Squier’s finiteness condition for monoids is of this kind: It can be defined in terms of a finite presentation, but does not depend on the choice of this presentation. To begin with a simpler case, consider the following theorem, which is not hard to prove: If M is a finitely presentable monoid, Σ a finite alphabet and φ a surjective morphism from the free monoid Σ∗ to M , then the congruence on Σ∗ induced by φ is generated by some finite set R ⊂ Σ∗×Σ∗. In other words, the existence of a finite presentation for M does not depend on the choice of the set of generators, provided it is finite. The invariance of Squier’s finiteness condition is of the same nature, but it is a 2-dimensional word problem in the sense of [Bu]. Therefore, we have to introduce a little more algebraic material (sections 1–2) before we get to the heart of the matter (sections 3–6). With this geometrical viewpoint, the connection with homology becomes quite natural (section 7). I am grateful to Volker Diekert for pointing out the preprint of Squier, and to Friedrich Otto for the bibliographical references. Theorem 3 has been proved independently by Robert Cremanns and Friedrich Otto [CrOt]. ∗Address: 163 avenue de Luminy, case 930, 13288 MARSEILLE CEDEX 9, FRANCE.