An order topologyfor finitely generated free monoids

In this article a method is given for embedding a finitely generated free monoid as a dense subset of the unit interval. This gives an order topology for the monoid such that the submonoids generated by an important class of maximal codes occur as “thick” subsets. As an ordered topological space, the notion of thickness in a frec monoid can be interpreted in a number of ways. One such notion is that of density. In particular, subsets of a free monoid that fail to meet all two sided ideals (the thin sets, of which recognizable codes are an example) are shown (corollary 4.2) to be nowhere dense. Furthermore, it is shown (corollary 5.1) that a thin code is maximal if and only if the submonoid that it generates is dense on some interval. Thus thin codes that are maximal are precisely those that generate thick submonoids. Another notion of thickness is that of category. The embedding allows the free monoid to be viewed as a subspace of the unit interval. In theorem 5.6 it is shown that a thin code is maximal just in case the closure of the submonoid that it generates is second category in the unit interval. A mild connection with Lebesque measure is then made. In what follows, all free monoids are assumed to be generated by a finite set of at least two elements. IfA is such a set, thenA ^* denotes the free monoid generated byA. The setA is called an alphabet, the elements ofA ^* are called words, ande denotes the empty word inA ^*. Topological terminology and notation follows that of Kelley 2].