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Samuel Braunfeld

Publications and source records attributed to Samuel Braunfeld.

29 records · Page 2Linked to original sources
arXiv2021

Mutual algebraicity and cellularity

We prove two results intended to streamline proofs about cellularity that pass through mutual algebraicity. First, we show that a countable structure $M$ is cellular if and only if $M$ is $ω$-categorical and mutually algebraic. Second, if a countable structure $M$ in a finite relational language is mutually algebraic non-cellular, we show it admits an elementary extension adding infinitely many infinite MA-connected components. Towards these results, we introduce MA-presentations of a mutually algebraic structure, in which every atomic formula is mutually algebraic. This allows for an improved quantifier elimination and a decomposition of the structure into independent pieces. We also show this decomposition is largely independent of the MA-presentation chosen.

math.LO↗
arXiv2021

The undecidability of joint embedding for 3-dimensional permutation classes

As a step towards resolving a question of Ruškuc on the decidability of joint embedding for hereditary classes of permutations, which may be viewed as structures in a language of 2 linear orders, we show the corresponding problem is undecidable for hereditary classes of structures in a language of 3 linear orders.

math.LO↗
arXiv2021

Monadic stability and growth rates of $ω$-categorical structures

For $M$ $ω$-categorical and stable, we investigate the growth rate of $M$, i.e. the number of orbits of $Aut(M)$ on $n$-sets, or equivalently the number of $n$-substructures of $M$ after performing quantifier elimination. We show that monadic stability corresponds to a gap in the spectrum of growth rates, from slower than exponential to faster than exponential. This allows us to give a nearly complete description of the spectrum of slower than exponential growth rates (without the assumption of stability), confirming some longstanding conjectures of Cameron and Macpherson and proving the existence of gaps not previously recognized.

math.LO↗
arXiv2021

Counting siblings in universal theories

We show that if a countable structure $M$ in a finite relational language is not cellular, then there is an age-preserving $N \supseteq M$ such that $2^{\aleph_0}$ many structures are bi-embeddable with $N$. The proof proceeds by a case division based on mutual algebraicity.

math.LO↗
arXiv2020

$Λ$-ultrametric spaces and lattices of equivalence relations

For a finite lattice $Λ$, $Λ$-ultrametric spaces have, among other reasons, appeared as a means of constructing structures with lattices of equivalence relations embedding $Λ$. This makes use of an isomorphism of categories between $Λ$-ultrametric spaces and structures equipped with certain families of equivalence relations. We extend this isomorphism to the case of infinite lattices. We also pose questions about representing a given finite lattice as the lattice of $\emptyset$-definable equivalence relations of structures with model-theoretic symmetry properties.

math.RA↗
arXiv2020

Homogeneous 3-dimensional permutation structures

We provide a classification of the homogeneous 3-dimensional permutation structures, i.e. homogeneous structures in a language of 3 linear orders, partially answering a question of Cameron. We also arrive at a natural description of all known homogeneous finite-dimensional permutation structures by modifying the language used in the construction from a previous paper, completing the "census" begun there.

math.LO↗
arXiv2020

The classification of homogeneous finite-dimensional permutation structures

We classify the homogeneous finite-dimensional permutation structures, i.e., homogeneous structures in a language of finitely many linear orders, giving a nearly complete answer to a question of Cameron, and confirming the classification conjectured by the first author. The primitive case was proven by the second author using model-theoretic methods, and those methods continue to appear here.

math.LO↗
arXiv2018

Infinite Limits of Finite-Dimensional Permutation Structures, and their Automorphism Groups: Between Model Theory and Combinatorics

In the course of classifying the homogeneous permutations, Cameron introduced the viewpoint of permutations as structures in a language of two linear orders, and this structural viewpoint is taken up here. The majority of this thesis is concerned with Cameron's problem of classifying the homogeneous structures in a language of finitely many linear orders, which we call finite-dimensional permutation structures. Towards this problem, we present a construction that we conjecture produces all such structures. Some evidence for this conjecture is given, including the classification of the homogeneous 3-dimensional permutation structures. We next consider the topological dynamics, in the style of Kechris, Pestov, and Todorčević, of the automorphism groups of the homogeneous finite-dimensional permutation structures we have constructed, which requires proving a structural Ramsey theorem for all the associated amalgamation classes. Because the $\emptyset$-definable equivalence relations in these homogeneous finite-dimensional permutation structures may form arbitrary finite distributive lattices, the model-theoretic algebraic closure operation may become quite complex, and so we require the framework recently introduced by Hubička and Nešetril. Finally, we consider the decision problem for whether a finitely-constrained permutation avoidance class is atomic, or equivalently, has the joint embedding property. As a first approximation to this problem, we prove the undecidability of the corresponding decision problem in the category of graphs. Modifying this proof also gives the undecidability, in the category of graphs, of the corresponding decision problem for the joint homomorphism property, which is of interest in infinite-domain constraint satisfaction problems.

math.LO↗
arXiv2017

Ramsey expansions of $Λ$-ultrametric spaces

For a finite lattice $Λ$, $Λ$-ultrametric spaces are a convenient language for describing structures equipped with a family of equivalence relations. When $Λ$ is finite and distributive, there exists a generic $Λ$-ultrametric space, and we here identify a family of Ramsey expansions for that space. This then allows a description the universal minimal flow of its automorphism group, and also implies the Ramsey property for all known homogeneous finite-dimensional permutation structures, i.e. structures in a language of finitely many linear orders. A point of technical interest is that our proof involves classes with non-unary algebraic closure operations. As a byproduct of some of the concepts developed, we also arrive at a natural description of the known homogeneous finite-dimensional permutation structures, completing our previously begun "census".

math.LO↗