For Groups the Property of Having Finite Derivation Type is Equivalent to the Homological Finiteness Condition FP_3
For Groups the Property of Having Finite Derivation Type is Equivalent to the Homological Finiteness Condition FP_3
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对于群，具有有限导数类型的性质等价于同调有限性条件 FP_3
DOI：
10.1006/jsco.1996.0046
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发表时间：
1996
期刊：
影响因子：
--
通讯作者：
F. Otto
中科院分区：
文献类型：
--
作者：
Robert Cremanns;F. Otto
同调有限性FP 3和具有有限导子型的组合性质是双表示么半群允许有限收敛表示的必要条件。对于一般的幺半群，具有有限导子型的性质意味着性质FP 3，甚至存在非FP 3的幺半群，但不具有有限导子型（Cremanns and Otto，1994）。在这里，对比这个结果，我们表明，这两个属性是等价的。证明是基于这样一个结果：通过有限表示给出的群G具有有限导子型当且仅当与群G相关联的关系之间的恒等式的ZG-模是有限生成的。这个结果在（Cremanns and Otto，1994）中被宣布，以概念上简单的方式被证明，大大改进了（Cremanns and Otto，1994）中仅概述的原始证明。然后，使用基本的代数参数，我们得出我们的主要结果，而不使用太多的同源性理论，从而使证明容易获得的计算机科学家和数学家在代数和重写理论的一些背景。
The homological finiteness propertyFP3and the combinatorial property of having finite derivation type are both necessary conditions for finitely presented monoids to admit finite convergent presentations. For monoids in general, the property of having finite derivation type implies the propertyFP3, and there even exist finitely presented monoids that areFP3, but that do not have finite derivation type (Cremanns and Otto, 1994). Here, contrasting this result, we show that for groups these two properties are equivalent. The proof is based on the result that a groupG, which is given through a finite presentation 〈X; R〉 has finite derivation type if and only if the ZG-module of identities among relations that is associated with 〈X; R〉 is finitely generated. This result, which was announced in (Cremanns and Otto, 1994), is proved in a conceptually simple manner, greatly improving upon the original proof that was only outlined in (Cremanns and Otto, 1994). Then, using elementary algebraic arguments we derive our main result without using much of homology theory, thus making the proof easily accessible to computer scientists and mathematicians with some background in algebra and rewriting theory.