Infinitely many reducts of homogeneous structures

Bodor, Bertalan; Cameron, Peter J‎.; Szabó, Csaba

doi:10.1007/s00012-018-0526-8

Cited by 6 publications

(3 citation statements)

References 20 publications

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“…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].…”

Section: Introductionmentioning

confidence: 99%

Permutation groups on countable vector spaces over prime fields

Bodor¹,

Pinsker²,

Schiffer³

et al. 2021

Preprint

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Section: Introductionmentioning

confidence: 99%

Permutation groups on countable vector spaces over prime fields

Bodor¹,

Pinsker²,

Schiffer³

et al. 2021

Preprint

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“…The list includes the rationals with the usual ordering [25], the countably infinite random graph [37], the homogeneous universal -free graphs [38], the expansion of by a constant [31], the universal homogeneous partial order [35], the random ordered graph [19], and many more [1, 2, 12, 13]. Note that if we drop the assumption that the signature of the homogeneous structure is relational, then Thomas’ conjecture is false even if we keep the assumption that is -categorical: already the countable atomless Boolean algebra has infinitely many first-order reducts [21].…”

Section: Introductionmentioning

confidence: 99%

Classification of -Categorical Monadically Stable Structures

BODOR

2023

J. symb. log.

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A first-order structure $\mathfrak {A}$ is called monadically stable iff every expansion of $\mathfrak {A}$ by unary predicates is stable. In this paper we give a classification of the class $\mathcal {M}$ of $\omega $ -categorical monadically stable structure in terms of their automorphism groups. We prove in turn that $\mathcal {M}$ is the smallest class of structures which contains the one-element pure set, is closed under isomorphisms, and is closed under taking finite disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal {M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal {M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas’ conjecture for the class $\mathcal {M}$ .

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“…On the other hand, recent work gives evidence that the conjecture is false. In [8], it is shown that the countable homogeneous Boolean-algebra has infinitely many reducts. (This is not a counter-example to Thomas' Conjecture because the structure is homogeneous in a functional language, not a relational one.)…”

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confidence: 99%

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

Agarwal

Kompatscher²

2018

J. symb. log.

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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.

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Infinitely many reducts of homogeneous structures

Cited by 6 publications

References 20 publications

Permutation groups on countable vector spaces over prime fields

Permutation groups on countable vector spaces over prime fields

Classification of -Categorical Monadically Stable Structures

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

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