${2^{{\aleph _0}}}$ PAIRWISE NONISOMORPHIC MAXIMAL-CLOSED SUBGROUPS OF SYM(ℕ) VIA THE CLASSIFICATION OF THE REDUCTS OF THE HENSON DIGRAPHS. (1st August 2018)

• Record Type: Journal Article
• Authors: AGARWAL, LOVKUSH; KOMPATSCHER, MICHAEL
• Is Part Of: Journal of symbolic logic. Volume 83:Number 2(2018)
• Page Start: 395
• Page End: 415
• Publication Date: 2018-08-01
• Subjects: 03C07, -- 03C15, -- 03C40, -- 20B27, -- 20B35; maximal-closed subgroups of Sym(ℕ), -- reducts, -- Henson digraphs, -- first-order definability, -- homogeneous structures, -- types, -- orbits, -- canonical functions; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1017/jsl.2017.74
• Languages: English
• ISSNs: 0022-4812

Abstract

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.