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${2^{{\aleph _0}}}$ PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS

Published online by Cambridge University Press:  01 August 2018

LOVKUSH AGARWAL  and
MICHAEL KOMPATSCHER
LOVKUSH AGARWAL
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF LEEDS LEEDS, LS2 9JT UKE-mail:[email protected]
MICHAEL KOMPATSCHER
Affiliation:
DEPARTMENT OF ALGEBRA MFF UK SOKOLOVSKA 83 186 00 PRAHA 8 CZECH REPUBLICE-mail:[email protected]
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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.

Keywords

03C0703C1503C4020B2720B35maximal-closed subgroups of Sym(ℕ)reductsHenson digraphsfirst-order definabilityhomogeneous structurestypesorbitscanonical functions
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 2 , June 2018 , pp. 395 - 415
Copyright
Copyright © The Association for Symbolic Logic 2018 

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