Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures
Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures
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Fraïssé极限的自同态和代数封闭关系结构的自同构群
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发表时间：
2012
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通讯作者：
J. McPhee
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作者：
J. McPhee
设Ω是一类关系结构的Fraisse极限。我们试图回答以下关于Ω的半群理论问题。End（Ω）的群H -类，即极大子群是什么？我们回答这个问题的Fraisse极限包括随机图R、随机有向图D、随机竞赛图T、随机二部图B、Henson图Gn（n ≥ 3）和全序Q。End（Ω）的极大子群与由End（Ω）的幂等象诱导的关系结构的自同构群密切相关。在[BD 00]和[Dol 12]中，证明了由来自End（Ω）的幂等元的像诱导的关系结构是代数封闭的。因此，我们研究哪些群可以实现为代数闭关系结构的自同构群，以确定在每种情况下End（Ω）的最大子群。特别地，我们证明了：如果Γ是可数图且Ω = R，D，B，则End（Ω）存在2个与Aut（Γ）同构的<$0极大子群.此外，我们提供了一个完整的描述的子集Q是一个幂等元的图像从结束（Q）。我们称这些子集为Q的收缩集，并证明了：如果Ω是全序，f：Ω → Q是一个嵌入，使得im f是Q的收缩集，则End（Q）存在2个<$0极大子群同构于Aut（Ω）.我们还证明了对于某些n ∈ N，End（Q）的任何可数极大子群都必须与Z同构。作为所发展的方法的结果，我们还能够证明当Ω = R，D，B，Q时存在End（Ω）的2 <$0正则D-类，当Ω = R，D，B时存在End（Ω）的2 <$0正则J -类。另外，我们证明了若Ω = R，D，则所有正则D-类都包含2个<$0群H -类.另一方面，我们证明了当Ω = B，Q时，存在包含可数个群H -类的正则D-类.
Let Ω be the Fraisse limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group H -classes, i.e. the maximal subgroups, of End(Ω)? Fraisse limits for which we answer this question include the random graph R, the random directed graph D, the random tournament T , the random bipartite graph B, Henson’s graphs Gn (n ≥ 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). In [BD00] and [Dol12] it was shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B, then there exist 2א0 maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f : Ω → Q is an embedding such that im f is a retract of Q, then there exist 2א0 maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Z for some n ∈ N. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2א0 regular D-classes of End(Ω) and when Ω = R,D,B there exist 2א0 J -classes of End(Ω). Additionally we show that if Ω = R,D then all regular D-classes contain 2א0 group H -classes. On the other hand, we show that when Ω = B,Q there exist regular D-classes which contain countably many group H -classes.