Simple automorphism groups of cycle-free partial orders
Simple automorphism groups of cycle-free partial orders
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无环偏序的简单自同构群
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
R. Warren
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文献类型：
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作者：
M. Droste;J. Truss;R. Warren
摘要本文的目的是表明，自同构群的许多'无圈'偏序研究[沃伦，R.：K-C S-传递无圈偏序的结构。Memoirs of the American Mathematical Society（1997），to appear] and [Creed，P.，Truss，J. K. Warren，R.：具有无限链的k-CS-传递无圈偏序的结构是简单的。这与树的情况形成了强烈的对比，它们形成了一种自然的概括。它显示在[Droste，M.，荷兰，W.C. Macpherson，H. D.：无限半线性阶的自同构群（I）和（II）. Proc.伦敦Math.Soc.58（1989），454-478和479-494]证明了任何充分传递树的自同构群至少有正规子群。所有的无限链无环偏序研究[Creed，P.，Truss，J. K. Warren，R.：具有无限链的k-CS-传递无圈偏序的结构，出现]有单自同构群。有限链的情况更加复杂;其中Dedekind-MacNeille完备化的链上的序可以表示为非平凡离散（传递）序的字典积（由群尊重），自同构群不是简单的。对于有限链和无限链的情形，简单自同构群分为两类：一类是需要共轭个数有界（≤ 2）来表示一个非单位元，另一类是没有这样的界。
Abstract The purpose of this paper is to show that the automorphism groups of many of the ‘cycle-free’ partial orders studied in [Warren, R.: The structure of k-C S-transitive cycle-free partial orders. Memoirs of the American Mathematical Society (1997), to appear] and [Creed, P., Truss, J. K. and Warren, R.: The structure of k-C S-transitive cycle-free partial orders having infinite chains, to appear] are simple. This contrasts strongly with the situation for trees, of which they form a natural generalization. It was shown in [Droste, M., Holland, W.C. and Macpherson, H.D.: Automorphism groups of infinite semilinear orders (I) and (II). Proc. London Math. Soc. 58 (1989), 454–478 and 479–494] that the automorphism group of any sufficiently transitive tree has at least normal subgroups. All the infinite chain cycle-free partial orders studied in [Creed, P., Truss, J. K. and Warren, R.: The structure of k-C S-transitive cycle-free partial orders having infinite chains, to appear] have simple automorphism groups. The finite chain case is more involved; where the ordering on chains of the Dedekind-MacNeille completion can be expressed as a lexicographic product by a non-trivial discrete (transitive) ordering (respected by the group), the automorphism group is not simple. For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (≤ 2) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.