JOURPZAL OF ALGEBRA 137, 297-337 (1991) Topologies for the Free Monoid* J. E. PIN C.N.R.S., L.I.T.P. Universitd Paris VI, Tour 55-45, 4 place Jussieu, 75252 Paris Cedex 05, France Communicated by T. E. Hall Received March 11, 1988 The finite group (or protinite) topology was first introduced for the free group by M. Hall, Jr. and by Reutenauer for free monoids. This is the initial topology defined by all the monoid morphisms from the free monoid into a discrete finite group. The p-adic topology is defined in the same way by replacing “group” by “p-group” in the definition. In this paper we study the properties of these topologies and their connexions with the theory of formal languages. (0 1991 Academic Press, Inc. The finite group (or prolinite) topology was first introduced for the free group by M. Hall, Jr. [S] and extended by Reutenauer [ 151 to the case of free monoids. This is the initial topology defined by all the monoid morphisms from the free monoid into a discrete finite group. The p-adic topology is defined in the same way by replacing “group” by “p-group” in the definition. (Similarly, the “prolinite topology” introduced by D. Scott [17] is obtained by replacing “group” by “monoid” in the definition, but we are not concerned with this topology in this paper.) The aim of this paper is to found a “descriptive theory” of these topologies for the free monoid. That is, one restricts one’s attention to “simple” subsets of the free monoid and one tries to decide whether these sets are open, closed, com- pact, or if one can describe their topological closure, etc. The “simple” sets we have in mind in this paper are the recognizable (or regular) sets of automata theory. It is fair to say immediately that we are not completely successful, and therefore part of our paper is devoted to the discussion of various conjectures. The main tools for this study are the free group (especially the topologies for the free group) and the theory of finite automata. It is well known that every subset L of the free monoid A* defines in a natural way a congruence on A*, called the syntactic congruence. The sub- sets of A* for which this congruence is of finite index are called l Supported by the PRC Mathdmatiques et Informatique. 297 0021~8693/91 $3.00 Copyright Q 1991 by Academic Press, Inc. All rights 01 reproduction in any form reserved. 298 J. E. PIN recognizable. If L is a recognizable subset of A*, the quotient of A* by the syntactic congruence of L is a finite monoid M, called the syntactic monoid of L. The image P of L in M is the syntactic image of L. This syntactic monoid turns out to be a very powerful concept in language theory, and a number of fascinating results relate the combinatorial properties of recognizable languages with algebraic properties of their syntactic monoid (see [4, 7, 121 for more details). A central idea of this paper is that topological properties of L are also reflected by algebraic properties of it4 and P. For instance, it is easy to show that if L is closed, then the property holds that for every s, t E M, and for every idempotent e E M (set E P 3 st E P). (*) We conjecture that the converse is true in general, and we prove this con- jecture in two significant particular cases: if P is a submonoid of M, or if the idempotents of M commute. This conjecture is not only interesting by itself, but also has some surprising consequences. For instance, a positive answer to the conjecture would solve an open problem on finite semigroups for the solution of which J. Rhodes has recently offered $100. Next, this conjecture, if true, gives an algorithm to decide whether a given recognizable set is open or closed. In fact, we show that it also gives an algorithm to compute the closure of a given recognizable set. Two such algorithms are presented in this paper. One of the main results of the paper shows that open sets are preserved under concatenation products of the form (L,, L,, . Lk) -+ L,a,L,az ... akL,. This result is based on a detailed analysis of a counting operation and furnishes many explicit examples of open sets. Another interesting result is that every open free submonoid of the free monoid is also closed. This is reminiscent of the well-known fact that every open subgroup of a topological group is closed, but here the condition “free” is required. The paper requires a basic knowledge of finite automata, finite semi- groups, and elementary topology, However, we have tried to keep the paper self-contained as far as possible. The paper is organized as follows. Section 1 contains the basic definitions and some useful results (including some new results) used in the paper. Section 2 introduces the definition and the basic properties of the topologies for the free monoid. Section 3 is devoted to the proof of a language theoretic result on concatenation product with counter. Properties of open sets are analysed in Section 4. Here we show that open sets are preserved under some natural operation of language theory: quotients, concatenation product, plus operation, inverse (non-empty) substitutions, etc. and we discuss the connections between the topologies for the free monoid and the free group. In Section 5, we study the topological properties of recognizable sets. In particular, we characterize the clopen recognizable sets and the compact recognizable sets. FREE MONOID 299 Even more is known on the submonoids of the free monoid, the properties of which are presented in Section 6. Section 7 is devoted to a detailed discussion of our major conjectures and their consequences. Part of the results of this paper have been announced in [ 111. 1. PRELIMINARIES In this section, we briefly recall the main definitions or properties that are used throughout this paper. For more details, the reader is referred to [l, 4, 7, 123 for languages and to [3] for topology. (a) Semigroups and Monoids A semigroup S is a set equipped with an associative operation, usually denoted multiplicatively. A monoid is a semigroup with identity. Given two monoids M and N, a map cp:M+ N is a monoid morphism if 1~ = 1 and (mn)q = (mcp)(ncp)for every m, n E M. A relational morphism z: M + N is a map from M into the subsets of N such that (1) for every m E M, mr is nonempty, (2) 1 ElZ, (3) for every m, n E M, (mz)(nz) c (mn)z. Let r: M -+ N be a relational morphism. If M’ is a submonoid of M, then M’z is a submonoid of N. Similarly, if N’ is a submonoid of N, N’z-I= {meMImznN’#IZ(} is a submonoid of M. Further properties of relational morphisms can be found in [ 12, 211. We just mention here a new result, inspired by an unpublished paper of Schtitzenberger. PROPOSITION 1.1. Let G be a finite group, H an arbitrary group, and let z: G + H be a surjective relational morphism. Then G’ = lz-’ is a normal subgroup of G, H’ = lr is a normal subgroup of H, and G/G’ = H/H’. Furthermore, for every gE G, gz is a coset of H’. ProoJ: We first claim that for every g E G, h E gz implies h - ’ E g- ‘2. Indeed, since r is surjective, there exists x E G such that h - ’ E xz. Let n be the order of G. Then hk’ = (h-l)” h”-‘E (xz)” (gz)“-’ c (xngn-‘)z= g-lz, since x”= g” = 1. In particular, H’ = 17 is a subgroup of H. Furthermore, if h E gz and XE 17, then by the claim, hxh-’ E (gt)(lr)(g-‘t)c (gg-‘)r = 17. Thus H’ is normal in H. Furthermore, the following property holds: for every h E gz, gz = H’h. (1) 481/137/2-4 300 J. E. PIN Indeed, H’h c (lz)(gz) c gr and, on the other hand, (gz) h ’ c (gz)(g-‘z) c 1~. Thus z induces a group morphism from G onto H/H’, the kernel of which is the group N= {gEGJ gt= lr}. We claim that N = G’ = {g E G 1 1 E gz}. Indeed, since 1 E lz, N is contained in G’. Conversely, if g E G’, then 1 E gr, whence gr = (17). 1 = It by (1). Thus N = G’, G’ is normal in G and G/G’ = H/H’. 1 (b) Languages Let A be a finite set, called the alphabet. We denote by A* the free monoid over A and by F(A) the free group over A. A letter is an element of A, a word is an element of A * and a language is a subset of A*. The empty word is denoted by 1. The product of two words u and u is simply denoted by uv. Thus every word u can be written in an unique way as u=a,a,~.~a,, where a,, . a, are letters. In this case, n is the length of u, denoted by (u(. In particular, the empty word 1 is the only word of length 0. If u is a word, we set u” = 1, and, for every n 2 0, u”+ i = uu”. If L is a language, L+ (resp. L*) denotes the subsemigroup (resp.