Pairwise Nonisomorphic Maximal-Closed Subgroups of Sym(ℕ) via the Classification of the Reducts of the Henson Digraphs

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.

Permutation groups on countable vector spaces over prime fields

“…Since any such structure is ω-categorical, this is equivalent to there being only a finite number of closed permutation groups containing the automorphism group of A. This conjecture has been verified for numerous structures including the order of the rationals [14] and the random graph [20], but is still open [21,4,16,9,18,17,10,8,1,19,2,7,6,11]. For countable homogeneous structures in a finite non-relational signature, it turns out to be false: the countable vector space over the two-element field, with an additionally distinguished non-zero vector, has an infinite number of first-order reducts [12], although it has only finitely many (four) first-order reducts if it is not equipped with any additional structure beyond the vector space structure [13].…”

Classification of -Categorical Monadically Stable Structures

“…By Theorem 3.41 we know that G is isomorphic to some group G(H ; N 0 , ... , N k ) where N 0 , N 1 , ... , N k , H ∈ H n-1 . Let Y i := Dom(N i ), H i := H (Y i ) , 2 . By the induction hypothesis this implies that we have finitely many choices for the group H * up to isomorphism.…”

“…Let Δ := (Dom(H ) × )2 and∇ := {((a, n), (b, n)) : a, b ∈ H, n ∈ }. Then P := (∅, Δ, ∇) is an -partition of Gwith all components isomorphic to H. Corollary 3.18.…”

Logic Blog 2020

“…Kaplan and Simon [24] showed that it is a maximal closed subgroup (that is also countable). Agarwal and Kompatscher [1] have provided continuum many maximal-closed groups that are not even algebraically isomorphic, using "Henson digraphs" that were introduced in a paper of Henson.…”