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**Power algebras over semirings. With applications in mathematics and computer science.**
*(English)*
Zbl 0947.16035

Mathematics and its Applications (Dordrecht). 488. Dordrecht: Kluwer Academic Publishers. x, 203 p. (1999).

Let \((R,+,\cdot)\) be an additively commutative semiring with absorbing zero \(0\) and identity \(1\not= 0\). By a power algebra over \(R\) the author means the set \(R^A\) of all \(R\)-valued mappings on some non-empty set \(A\) together with one or more operations which can be defined depending on special properties of \((R,+,\cdot)\) or on additional operations or order relations on \(A\). The first chapters 0 to 4 of the book present a lot of examples and general constructions of this kind including, e.g., matrix semirings or semirings of power series. Then, in the remaining four chapters, the cases that \((A,*)\) is a semigroup, a monoid or a group or that \((A,\oplus)\) is an \(S\)-module for some ring \(S\) or that \((A,\oplus,\odot)\) is itself a semiring are discussed in detail. Every chapter contains some accurately written theory with complete proofs and illustrating examples. A carefully written subject index helps the reader to find the numerous concepts defined throughout the book. Also an extensive bibliography (428 titles) is included which shows the wide range of applications of the developed concepts in many areas of mathematics and computer science. (Also submitted to MR).

Reviewer: U.Hebisch (Freiberg)

### MSC:

16Y60 | Semirings |

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

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

68Q45 | Formal languages and automata |