*(English)*Zbl 1072.16040

The book deals with semirings $(S,+,\xb7)$ in the most inclusive sense, i.e. $(S,+)$ and $(S,\xb7)$ are arbitrary semigroups such that multiplication distributes over addition from both sides. After a short introduction with historical remarks on the origin of semirings and related algebraic structures follows an explanation of the five main streams of semiring theory: 1. pure theoretical investigations, 2. relations to arithmetic and number theory, 3. connection with logic, including non-classical and multiple-valued logics, 4. geometrical and topological investigations and idempotent analysis, 5. relations to the theories of automata, formal languages, optimization and other parts of applied mathematics.

Accordingly, part one of the book presents on nearly 90 pages a short guide through the literature of different branches of semiring theory: algebraic theory of semirings, linear algebra over semirings (including semimodules), ordered structures and semirings, and selected applications of semirings. This last section is subdivided in the following topics: 1. geometrical and topological investigations and connections to categories, 2. semirings in modern analysis, measure theory, idempotent analysis, differential equations, optimization, 3. more about optimization, discrete event systems, and semirings with idempotent addition, 4. semirings and seminearrings in process algebras and related topics, 5. semirings in automata theory and the theory of formal languages.

In part two follows a (nearly) complete bibliography of semirings on about 290 pages. Finally, an excellent index is included which helps the reader to find very quickly the topic of interest and the relevant literature on it.