2 ℵ No pairwise non-isomorphic maximal-closed subgroups of Sym(N) via the classification of the reducts of the Henson digraphs
Abstrakt
Given two structures M and N on the same domain, we say that N is a reduct of M if all EMPTY SET -definable relations of N are EMPTY SET -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 ℵ 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 that there are 2 ℵ 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 ℵ 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.