ABSTRACT
In Chapter 9, we proved that a language is recognisable precisely when its syntactic monoid is finite. When the language is recognisable, the syntactic monoid can easily by calculated as the transition monoid of the minimal au tomaton of the language. The goal of algebraic language theory is to answer questions about recognisable languages by studying their syntactic monoids. To do this, we first have to learn more about finite semigroups. This is the subject of Section 10.1. To demonstrate that the algebraic approach is viable, we show in Section 10.2 that many of the results about recognisable languages proved in Chapters 3 and 4 using automata can easily be proved using finite monoids. The real vindication of this approach, however, will come in Chap ter 11, when we prove a new result about recognisable languages using the syntactic monoid. The results of this chapter lead us to look for a correspon dence between recognisable languages and finite monoids. In Section 10.3, we prove that this correspondence cannot be interpreted in a naive way. The correct correspondence is described in Chapter 12.
