Endomorphisms of Fraïssé limits and automorphism groups of algebraically closed relational structures

Semigroups of endomorphisms;Semigroups;Maximal subgroups

McPhee, Jillian Dawn ; Quick, M. R. (Martyn R.) ; Quick, M. R. (Martyn R.)

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Semigroups of endomorphisms;    Semigroups;    Maximal subgroups;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/3358/JillianMcPheePhDThesis.pdf?sequence=6&isAllowed=y

来源: DR-NTU

【 摘 要 】

Let Ω be the Fraïssé limit of a class of relational structures. We seek toanswer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graphR, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to3) 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). It has been 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 todetermine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph andΩ = R,D,B,then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsetsof 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 is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist2[superscript aleph-naught] J-classes of End(Ω). Additionally we showthat if Ω = R,D then all regular D-classes contain2[superscript aleph-naught] group H-classes. On the other hand, we show that whenΩ = B,Q there exist regular D-classeswhich contain countably many group H-classes.