2^_0 pairwise non-isomorphic maximal-closed subgroups of Sym(N) via the classification of the reducts of the Henson digraphs [article]

Lovkush Agarwal, Michael Kompatscher
2015 arXiv   pre-print
Given two structures M and N on the same domain, we say that N is a reduct of M if all ∅-definable relations of N are ∅-definable in M. In this article the reducts of the Henson digraphs are classified. Henson digraphs are homogeneous countable digraphs that omit some set of finite tournaments. As the Henson digraphs are _0-categorical, determining their reducts is equivalent to determining all closed supergroups G< Sym(N) of their automorphism groups. A consequence of the classification is
more » ... there are 2^_0 pairwise non-isomorphic Henson digraphs which have no proper non-trivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are 2^_0 pairwise non-conjugate maximal-closed subgroups of Sym(N). By the reconstruction results of Rubin, these groups are also non-isomorphic as abstract groups.
arXiv:1509.07674v1 fatcat:4ebw7uzecfc6dagqntd62ttovu