The Finiteness of Finitely Presented Monoids

Introduction. It is undecidable whether a monoid given by a nite presentation is nite (see, e.g., [1], pp. 157{160). On the other hand, with the mere knowledge that the monoid is nite, one can e ectively construct the multiplication table of the monoid, and thus obtain a complete understanding of its structure. This paper will present this construction method in detail (Section 2) and o er some remarks about its computational complexity (Section 3). The notation here will be based on [1], whose nal chapter furnishes much of the background and will be referenced frequently.