Free lattice-ordered groups represented as $o$-$2$ transitive $l$-permutation groups
Free lattice-ordered groups represented as $o$-$2$ transitive $l$-permutation groups
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自由格序群表示为 $o$-$2$ 传递 $l$-排列群
DOI：
10.1090/s0002-9947-1985-0787955-7
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发表时间：
1985
影响因子：
1.3
通讯作者：
S. McCleary
中科院分区：
文献类型：
--
作者：
S. McCleary
对于秩为tj > 1的自由格序群Fv，很容易提出问题，其答案s2是“明显的”，但很难验证。例如：1；F的中心是什么？2. Fv是直接不可分解的吗？3. Fv有基本元素吗？4. Fv是完全分配律吗？问题1最近由Medvedev回答，问题1和问题2由Arora和McCleary回答，他们使用Conrad通过自由群Gv的右序来表示Fv。在这里，我们用一个完全不同的工具来回答这四个问题：Fv的（忠实）表示为一个o-2传递/置换群，它是病态的（没有有界支持的非恒等元素）。这个表达式是Glass为大多数无穷大的tj建立的，并在这里推广到所有的q >1。奇怪的是，Fv的传递表示的存在性（通过Kopytov的结果）意味着在Conrad表示中存在Gv的某种正确排序，它本身就足以给出Fv的忠实表示。对于有限tj，我们发现Fv的每一个传递表示都可以由一个病态的0 -2传递表示构成，通过将点扩展到0块；并且Fv的每一个病态o-2传递表示都可以推广到F的病态o-2传递表示。当tj有限时，F是否具有病态的o-2传递表示被Glass描述为一个“基本未解决问题”[7，第138页]。适当地，确立这样一种表象的存在将使这四个介绍性问题非常容易回答。有关背景，请参见[7或8]。本文几乎完全独立于[l]。1. 背景。设x是/-群F的一个子集，如果从x到任意/-群H的每个函数都能唯一地扩展到从F到H的/-同态，则F在x上是自由的。如果F在某个子集x上是自由的，则F是自由的。F的秩表示x的基数，它由[1，命题1]定义。1984年2月21日收到。1980数学学科分类。主要06 f15。
It is easy to pose questions about the free lattice-ordered group Fv of rank tj > 1 whose answers2 are "obvious", but difficult to verify. For example: 1. What is the center of F,? 2. Is Fv directly indecomposable? 3. Does Fv have a basic element? 4. Is Fv completely distributive? Question 1 was answered recently by Medvedev, and both 1 and 2 by Arora and McCleary, using Conrad's representation of Fv via right orderings of the free group Gv. Here we answer all four questions by using a completely different tool: The (faithful) representation of Fv as an o-2-transitive /-permutation group which is pathological (has no nonidentity element of bounded support). This representation was established by Glass for most infinite tj, and is here extended to all q > 1. Curiously, the existence of a transitive representation for Fv implies (by a result of Kopytov) that in the Conrad representation there is some right ordering of Gv which suffices all by itself to give a faithful representation of Fv. For finite tj, we find that every transitive representation of Fv can be made from a pathologically o-2-transitive representation by blowing up the points to o-blocks; and every pathologically o-2-transitive representation of Fv can be extended to a pathologically o-2-transitive representation of F . Whether F has a pathologically o-2-transitive representation when tj is finite was described by Glass as a "basic unsolved problem" [7, p. 138]. Appropriately, establishing the existence of such a representation will make the four introductory questions exceedingly easy to answer. For background, see [7 or 8]. The present paper is almost completely independent of[l]. 1. Background. Let x be a subset of an /-group F. F is free on x if every function from x into an arbitrary /-group H can be extended uniquely to an /-homomorphism from F into H. Any two /-groups free on sets of the same cardinality r\ are /-isomorphic. F is free if it is free on some subset x. The rank of F means the cardinality of x, which is well defined by [1, Proposition 1]. Received by the editors February 21, 1984. 1980 Mathematics Subject Classification. Primary 06F15.