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**The theory of semirings with applications in mathematics and theoretical computer science.**
*(English)*
Zbl 0780.16036

Pitman Monographs and Surveys in Pure and Applied Mathematics. 54. Harlow: Longman Scientific and Technical. New York: Wiley. xiv, 318 p. (1992).

Semirings and also semimodules over them have become an important tool in such diverse areas as combinatorics, functional analysis, topology, graph theory, Euclidean geometry, ring theory including partially ordered rings, optimization theory, automata theory, formal language theory, coding theory, and the mathematical modeling of quantum physics and parallel computing systems. However, results on semirings are scattered through the literature of the last five or six decades and an increasing number of papers dealing with their applications in the last two decades. So it is very important to have now this first monography on semirings, which contains also an immense collection of their various applications in all the fields count up above, together with detailed references to the nearly 600 items in its excellent bibliography.

In this book a semiring is defined as a nonempty set \(R\) with two binary operations such that (1) \((R,+)\) is a commutative monoid with identity 0; (2) \((R,\cdot)\) is a commutative monoid with identity \(1_ R\); (3) Multiplication distributes over addition from either side; (4) \(0r = r0 = 0\) for all \(r\in R\) and (5) \(1\neq 0\) (whereas rings are mostly denoted by \(S\)). The first chapters, dealing with the structure of those semirings, present among other topics the building of new semirings from old ones, the concepts of complement elements, of ideals, of factor semirings and of morphisms of semirings and their kernels. In the following chapters, factor semirings, Euclidean semirings and additively regular semirings are investigated. The Chapters 13-17 are devoted to semimodules over semirings, including free, projective and injective semimodules, the localization of semimodules and to linear algebra over semirings. Then partially-ordered and lattice-ordered semirings are considered. Another important tool, in particular for applications, are infinite sums in semirings, defined in an abstract way and subjected to a set of axioms. If each family of elements of a semiring is summable, the semiring is called complete. The last both Chapters 21 and 22 deal with complete lattice-ordered semirings and with fixed points of affine maps \(\lambda: H \to M\), where \(M\) is a semimodule over a semiring.

In the literature, the terminology about semirings is far away from being unique. Moreover, many authors define concepts and properties and their names (caused by the special situation they are dealing with) in a way which excludes e.g. the case of rings or which does not fit with generally adopted concepts and terms in a similar or even more general situation. Some of those less adequate definitions occur also in this book. For instance, a semiring \((R,+,\cdot)\) is called “simple” if \(r+1 = 1\) holds for all \(r\in R\), whereas “simple” in theory of universal algebra means that \((R,+,\cdot)\) has only the trivial congruences. There are also a few minor errors in the mathematics.

In this book a semiring is defined as a nonempty set \(R\) with two binary operations such that (1) \((R,+)\) is a commutative monoid with identity 0; (2) \((R,\cdot)\) is a commutative monoid with identity \(1_ R\); (3) Multiplication distributes over addition from either side; (4) \(0r = r0 = 0\) for all \(r\in R\) and (5) \(1\neq 0\) (whereas rings are mostly denoted by \(S\)). The first chapters, dealing with the structure of those semirings, present among other topics the building of new semirings from old ones, the concepts of complement elements, of ideals, of factor semirings and of morphisms of semirings and their kernels. In the following chapters, factor semirings, Euclidean semirings and additively regular semirings are investigated. The Chapters 13-17 are devoted to semimodules over semirings, including free, projective and injective semimodules, the localization of semimodules and to linear algebra over semirings. Then partially-ordered and lattice-ordered semirings are considered. Another important tool, in particular for applications, are infinite sums in semirings, defined in an abstract way and subjected to a set of axioms. If each family of elements of a semiring is summable, the semiring is called complete. The last both Chapters 21 and 22 deal with complete lattice-ordered semirings and with fixed points of affine maps \(\lambda: H \to M\), where \(M\) is a semimodule over a semiring.

In the literature, the terminology about semirings is far away from being unique. Moreover, many authors define concepts and properties and their names (caused by the special situation they are dealing with) in a way which excludes e.g. the case of rings or which does not fit with generally adopted concepts and terms in a similar or even more general situation. Some of those less adequate definitions occur also in this book. For instance, a semiring \((R,+,\cdot)\) is called “simple” if \(r+1 = 1\) holds for all \(r\in R\), whereas “simple” in theory of universal algebra means that \((R,+,\cdot)\) has only the trivial congruences. There are also a few minor errors in the mathematics.

Reviewer: H.J.Weinert (Clausthal-Zellerfeld)

### MSC:

16Y60 | Semirings |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

68Q70 | Algebraic theory of languages and automata |

06F25 | Ordered rings, algebras, modules |

06F05 | Ordered semigroups and monoids |

16W80 | Topological and ordered rings and modules |

68Q45 | Formal languages and automata |

40J05 | Summability in abstract structures |