Semirings and their applications.

*(English)*Zbl 0947.16034
Dordrecht: Kluwer Academic Publishers. xi, 381 p. (1999).

This book is a carefully revised and enhanced version of “The theory of semirings, with applications in mathematics and theoretical computer science” [Longman Sci. Tech., Harlow (1992; Zbl 0780.16036)] by the same author which was the first monograph on semirings. There are three new chapters namely on sets and relations with values in a semiring, on factor semimodules and on complete semimodules. (The first topic is also the theme of “Power algebras over semirings” [Kluwer Academic Publishers, Dordrecht (1999; see the following review Zbl 0947.16035)] by the same author.) Also new is an index of applications where a reader can find those parts where semirings occurring in a specific area of mathematics or computer science are described. Due to the large amount of included material the book is very concise and requires some effort by the reader. Especially in the case of many examples it will be helpful to see the cited literature (about 700 titles) for more details.

Chapter headings: Hemirings and semirings: definitions and examples; Sets and relations with values in a semiring; Building new semirings from old; Some conditions on semirings; Complemented elements in semirings; Ideals in semirings; Prime and semiprime ideals in semirings; Factor semirings; Morphisms of semirings; Kernels of morphisms; Semirings of fractions; Euclidean semirings; Additively-regular semirings; Semimodules over semirings; Factor semimodules; Some construction for semimodules; Localization of semimodules; Linear algebra over a semiring; Partially-ordered semirings; Lattice-ordered semirings; Complete semirings; Complete semimodules; Complete-lattice-ordered semirings; Fixed points of affine maps. (Also submitted to MR).

Chapter headings: Hemirings and semirings: definitions and examples; Sets and relations with values in a semiring; Building new semirings from old; Some conditions on semirings; Complemented elements in semirings; Ideals in semirings; Prime and semiprime ideals in semirings; Factor semirings; Morphisms of semirings; Kernels of morphisms; Semirings of fractions; Euclidean semirings; Additively-regular semirings; Semimodules over semirings; Factor semimodules; Some construction for semimodules; Localization of semimodules; Linear algebra over a semiring; Partially-ordered semirings; Lattice-ordered semirings; Complete semirings; Complete semimodules; Complete-lattice-ordered semirings; Fixed points of affine maps. (Also submitted to MR).

Reviewer: U.Hebisch (Freiberg)