${2^{{\aleph _0}}}$ PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS. (1st August 2018) - AGARWAL, LOVKUSH
Abstract: Given two structures ${\cal M}$ and ${\cal N}$ on the same domain, we say that ${\cal N}$ is a reduct of ${\cal M}$ if all $\emptyset$ -definable relations of ${\cal N}$ are $\emptyset$ -definable in ${\cal M}$ . 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 the closed supergroups G ≤ Sym(ℕ) of their automorphism groups. A consequence of the classification is that there are ${2^{{\aleph _0}}}$ pairwise noninterdefinable Henson digraphs which have no proper nontrivial reducts. Taking their automorphisms groups gives a positive answer to a question of Macpherson that asked if there are ${2^{{\aleph _0}}}$ pairwise nonconjugate maximal-closed subgroups of Sym(ℕ). By the reconstruction results of Rubin, these groups are also nonisomorphic as abstract groups.