Pre-Semirings, Semirings and Dioids

As an introduction to this first chapter, we show, by discussing four characteristic examples, that even with internal operations with limited properties – in particular those are not invertible – there exist nonetheless algebraic structures in which it is possible to solve fixed-point type equations and obtain eigenvalues and eigenvectors of matrices. It will be seen throughout this book that it is possible to reconstruct, in such structures, a major part of classical algebra.

This first chapter is composed of two parts. The first is devoted to some basic properties and to a typology of algebraic structures formed by a set endowed with a single internal operation: semigroups and monoids in Sect. 2, ordered monoids in Sect. 3.

The second part is devoted to the basic properties and typology of algebraic structures formed by a set endowed with two internal operations: pre-semirings in Sect. 4, semirings in Sect. 5 and dioids in Sect. 6.

For each of these structures, the most important subclasses are pointed out and the basic terminology to be used in the following chapters is introduced.