Properties of Monoids That Are Presented By Finite Convergent String-Rewriting Systems - a Survey

In recent years a number of conditions has been established that a monoid must necessarily satisfy if it is to have a presentation through some finite convergent stringrewriting system. Here we give a survey on this development, explaining these necessary conditions in detail and describing the relationships between them. 1 Introduction String-rewriting systems, also known as semi-Thue systems, have played a major role in the development of theoretical computer science. On the one hand, they give a calculus that is equivalent to that of the Turing machine (see, e.g., [Dav58]), and in this way they capture the notion of `effective computability' that is central to computer science. On the other hand, in the phrase-structure grammars introduced by N. Chomsky they are used as sets of productions, which form the essential part of these grammars [HoUl79]. In this way string-rewriting systems are at the very heart of formal language theory. Finally, they are also used in combinatorial semig..