Non-Unique Factorizations
Non-Unique Factorizations
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DOI：
10.1007/s00233-009-9194-5
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发表时间：
2010-04
期刊：
影响因子：
0.7
通讯作者：
P. A. García-Sánchez
中科院分区：
文献类型：
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作者：
P. A. García-Sánchez
这本书是在格拉茨的卡尔-弗兰泽斯大学写的，这并不令人惊讶，因为人们预计作者有一天会完成如此巨大的任务。近几十年来，格拉茨学派关于非唯一因式分解的研究一直很有成效，在某种程度上，这项研究应该在一份手稿中实现，就像正在审查的手稿一样。这本专著收集了几乎所有关于非唯一因式分解理论的已知结果，或者通过证明它们，或者通过给出精确的参考文献。有许多说明性的例子，每当一个蕴涵不成立时，在大多数情况下，都会给出一个反例来解释为什么它不能被证明。最后，读者发现了一系列的附录，强调了卷的自包含性质。熟悉半群理论的读者会发现一些记号很奇怪。例如，半群是具有单位元的交换半群，也就是通常所说的交换么半群；而么半群总是意味着可消的。这一约定源于这样一种观点，即经典非唯一因式分解问题中感兴趣的么半群是整环上的乘性么半群，其中消去律成立。因为这在正文的开头已经解释过了，所以这不是问题。
It is not a surprise that this book has been written at Karl-Franzens Universität in Graz, as it was expected that the authors would some day accomplish such a gigantic task. The Graz school on non-unique factorizations has been very productive in recent decades, and in some way this research should be materialized in a manuscript like the one under review. This monograph collects almost all known results on non-unique factorization theory, either by proving them or by giving precise references. There are many illustrative examples, and whenever an implication does not hold, in most cases, a counterexample is given to explain why it cannot be proved. At the end, the reader finds a series of appendices that emphasize the self-contained character of the volume.Readers familiar with semigroup theory would find some notation strange. For instance, a semigroup is a commutative semigroup with a unit element, that is, what is usually known as a commutative monoid; and a monoid is always meant to be cancellative. This convention comes from the idea that the monoids of interest in classical non-unique factorization problems are multiplicative monoids of integral domains, where the cancellation law holds. Since this is explained at the beginning of the text, it is not a problem.