PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS

Abstract

AbstractGiven 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 supergroupsG≤ 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.