Groups whose Geodesics are Locally Testable
Groups whose Geodesics are Locally Testable
复制标题
测地线可局部测试的群
DOI：
10.1142/s0218196708004676
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Sarah Rees
中科院分区：
文献类型：
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作者：
S. Hermiller;D. Holt;Sarah Rees
如果一个单词在集合中的成员资格是由其某个有界长度 k 的子单词的性质决定的，则该单词的常规集合是 (k-) 局部可测试的。在本文中，我们研究了所有测地线词的集合相对于某个生成集是（k-）局部可测试的群，我们称这样的群是（k-）局部可测试的。我们证明群是 1-局部可测的当且仅当它是自由阿贝尔群。我们证明了（k-）局部可测试群的类在采用有限直接乘积的情况下是封闭的。我们还表明，局部可测试群具有有限多个扭转单元的共轭类。我们的工作涉及对特定群体的计算机研究，为此，我们在 GAP 中实现了一种算法，从短词自动结构开始，计算有限状态自动机，其语言等于群体的所有测地线的集合（假设该语言是规则的）。我们提供该算法的简要描述。
A regular set of words is (k-)locally testable if membership of a word in the set is determined by the nature of its subwords of some bounded length k. In this article we study groups for which the set of all geodesic words with respect to some generating set is (k-)locally testable, and we call such groups (k-)locally testable. We show that a group is 1-locally testable if and only if it is free abelian. We show that the class of (k-)locally testable groups is closed under taking finite direct products. We show also that a locally testable group has finitely many conjugacy classes of torsion elements. Our work involved computer investigations of specific groups, for which purpose we implemented an algorithm in GAP to compute a finite state automaton with language equal to the set of all geodesics of a group (assuming that this language is regular), starting from a shortlex automatic structure. We provide a brief description of that algorithm.