# Finiteness and Regularity in Semigroups and Formal Languages

• Aldo de Luca
• Stefano Varricchio
Book

1. Front Matter
Pages I-X
2. Aldo de Luca, Stefano Varricchio
Pages 1-30
3. Aldo de Luca, Stefano Varricchio
Pages 31-76
4. Aldo de Luca, Stefano Varricchio
Pages 77-152
5. Aldo de Luca, Stefano Varricchio
Pages 153-177
6. Aldo de Luca, Stefano Varricchio
Pages 179-194
7. Aldo de Luca, Stefano Varricchio
Pages 195-227
8. Back Matter
Pages 229-242

### Introduction

The aim of this monograph is to present some recent research work on the combinatorial aspects of the theory of semigroups which are of great inter­ est for both algebra and theoretical computer science. This research mainly concerns that part of combinatorics of finite and infinite words over a finite alphabet which is usually called the theory of "unavoidable" regularities. The unavoidable regularities ofsufficiently large words over a finite alpha­ bet are very important in the study of finiteness conditions for semigroups. This problem consists in considering conditions which are satisfied by a fi­ nite semigroup and are such as to assure that a semigroup satisfying them is finite. The most natural requirement is that the semigroup is finitely gener­ ated. Ifone supposes that the semigroup is also periodic the study offiniteness conditions for these semigroups (or groups) is called the Burnside problem for semigroups (or groups). There exists an important relationship with the theory of finite automata because, as is well known, a language L over a fi­ nite alphabet is regular (that is, recognizable by a finite automaton) if and only if its syntactic monoid S(L) is finite. Hence, in principle, any finite­ ness condition for semigroups can be translated into a regularity condition for languages. The study of finiteness conditions for periodic languages (Le. , such that the syntactic semigroup is periodic) has been called the Burnside problem for languages.

### Keywords

Monoid algebra combinatorics combinatorics on word combinatorics on words computer science finiteness conditions formal language formal languages language mathematics regularity conditions semigroups theoretical computer science unavoidable regularities

#### Authors and affiliations

• Aldo de Luca
• 1
• Stefano Varricchio
• 2
1. 1.Dipartimento di MatematicaUniversità di Roma „La Sapienza€RomaItaly
2. 2.Dipartimento di MatematicaUniversità di Roma „Torvergata€RomaItaly

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-59849-4
• Copyright Information Springer-Verlag Berlin Heidelberg 1999
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-63771-4
• Online ISBN 978-3-642-59849-4
• Series Print ISSN 1431-2654
• Buy this book on publisher's site