Introduction

It is a well-known fact that the boolean calculus is one of the mathematical foundations of electronic computers. This explains the important role of the boolean semiring in computer science. The aim of this paper is to present other semirings that occur in theoretical computer science. These semirings were named tropical semirings by Dominique Perrin in honour of the pioneering work of our Brazilian colleague and friend Imre Simon, but are also commonly known as (min, +)-semirings.

The aim of this paper is to present tropical semirings and to survey a few problems relevant to them. We shall try to give an updated status of the different questions, but detailed solutions of most problems would be too long and technical for this survey. They can be found in the other papers of this volume or in the relevant literature. We have tried to keep the paper selfcontained as much as possible. Thus, in principle, there are no prerequisites for reading this survey, apart from a standard mathematical background. However, it was clearly not possible to give a full exposition of the theory of automata within 20 pages. Therefore, suitable references will be given for readers who would like to pursue the subject further and join the tropical community.

The paper is organized as follows. The main definitions are introduced in Section 2. Two apparently disconnected applications of tropical semirings are presented: the Burnside type problems in group and semigroup theory in Section 3, and decidability problems in formal language theory in Section 4. The connection between the two problems is explained in Section 5. A conclusion section ends the paper.

Introduction

This chapter is devoted to a study of van der Waerden's theorem, which is, according to Khinchin, one of the “pearls of number theory.” This theorem illustrates a principle of unavoidable regularity: It is impossible to produce long sequences of elements taken from a finite set that do not contain subsequences possessing some regularity, in this instance arithmetic progressions of identical elements.

During the last fifty years, van der Waerden's theorem has stimulated a good deal of research on various aspects of the result. Efforts have been made to simplify the proof while at the same time generalizing the theorem, as well as to determine certain numerical constants that occur in the statement of the theorem. This work is of an essentially combinatorial nature. More recently, results from ergodic theory have led to the discovery of new extensions of van der Waerden's theorem, and, as a result, to a topological proof.

The plan of the chapter illustrates this diversity of viewpoints. The first section, after a brief historical note, presents several different formulations of van der Waerden's theorem. The second section gives a combinatorial proof of an elegant generalization due to Grünwald. The third section, which concerns “cadences,” gives an interpretation of the theorem in terms of the free monoid. In the fourth section is presented a topological proof of van der Waerden's theorem, due to Fürstenberg and Weiss.

Abstract

Algebraic semigroup theory has made great strides in the recent years. It is a remarquable fact that these new results did require the introduction of some auxiliary structures. The aim of this article is to emphasize the role of three of these tools: topology, partial order and categories. We survey in particular Reiterman's theorem on identities for pseudovarieties, an extension of Eilenberg's variety theorem to ordered semigroups and some decomposition theorems on semigroups that require the notion of a derived category.

I met Professor A. H. Clifford just once, in 1984. It was my first semigroup conference in the USA and I felt very honored when he told me very kindly how he enjoyed the way automata theory and semigroups mixed together. Ten years later, it was again a great honor for me to participate to the Tulane conference in his memory.

Introduction

Algebraic semigroup theory has made great strides in the recent years. It is a remarkable fact that these new results did require the introduction of some auxiliary structures. In this article, I would like to emphasize the role of three of these tools: topology, partial orders and categories. The three parts are relatively independent.

Identities and Topology

The results presented in this section are a good illustration of the following quotation of Marshall Stone [34]: ‘A cardinal principle of modern mathematical research may be stated as a maxim: “One must always topologize”.’ Varieties of finite semigroups are a good example where Stone's principle was applied successfully.