$2^{{\aleph }_0}$ pairwise non-isomorphic maximal-closed subgroups of Sym$(\mathbb{N})$ via the classification of the reducts of the Henson digraphs

Abstract

Given two structures $\mathcal{M}$ and $\mathcal{N}$ on the same domain, we say that $\mathcal{N}$ is a reduct of $\mathcal{M}$ if all $\varnothing$-definable relations of $\mathcal{N}$ are $\varnothing$-definable in $\mathcal{M}$. 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 ${\aleph }_0$-categorical, determining their reducts is equivalent to determining all closed supergroups $G<$ Sym$(\mathbb{N})$ of their automorphism groups. A consequence of the classification is that there are $2^{{\aleph }_0}$ 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 $2^{{\aleph }_0}$ pairwise non-conjugate maximal-closed subgroups of Sym$(\mathbb{N})$. By the reconstruction results of Rubin, these groups are also non-isomorphic as abstract groups.

References in corpus (1)

• Reducts of the Generic Digraph

Cited by in corpus (2)

• Canonical Functions: a proof via topological dynamics
• Structures with Small Orbit Growth