1 results
${2^{{\aleph _0}}}$ PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS
-
- Journal:
- The Journal of Symbolic Logic / Volume 83 / Issue 2 / June 2018
- Published online by Cambridge University Press:
- 01 August 2018, pp. 395-415
- Print publication:
- June 2018
-
- Article
- Export citation
-
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.
