##
**Non-unique factorizations. Algebraic, combinatorial and analytic theory.**
*(English)*
Zbl 1113.11002

Pure and Applied Mathematics (Boca Raton) 278. Boca Raton, FL: Chapman & Hall/CRC (ISBN 1-58488-576-9/hbk). xxi, 700 p. (2006).

This huge work presents the current state of the theory of factorizations in integral domains, and, more generally, in certain classes of semigroups. It has been realized in recent years that factorization properties in domains are related to similar properties in simpler algebraic structures, and the authors apply this idea to describe a wealth of results concerning several aspects of factorization theory.

They deal, in particular, with the structure of sets of lengths of factorizations, half-factorial sets, various combinatorial constants in finite Abelian groups, related to factorizations, and appropriate analytical theory. Many asymptotic results concerning numbers with several factorization properties in global fields are proved in a uniform way, via the Tauberian theorem of Ikehara-Delange. One finds in the book all known results related to factorizations, either with complete proofs, or with a reference (the bibliography consists of 400 items). Several open problems are mentioned with comments about their status. This is a very useful book.

They deal, in particular, with the structure of sets of lengths of factorizations, half-factorial sets, various combinatorial constants in finite Abelian groups, related to factorizations, and appropriate analytical theory. Many asymptotic results concerning numbers with several factorization properties in global fields are proved in a uniform way, via the Tauberian theorem of Ikehara-Delange. One finds in the book all known results related to factorizations, either with complete proofs, or with a reference (the bibliography consists of 400 items). Several open problems are mentioned with comments about their status. This is a very useful book.

Reviewer: Władysław Narkiewicz (Wrocław)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11R27 | Units and factorization |

11R47 | Other analytic theory |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13G05 | Integral domains |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20D60 | Arithmetic and combinatorial problems involving abstract finite groups |

20K01 | Finite abelian groups |

20M14 | Commutative semigroups |