Disjoint n-Amalgamation and Pseudofinite Countably Categorical Theories

Kruckman, Alex

doi:10.1215/00294527-2018-0025

Cited by 7 publications

(9 citation statements)

References 26 publications

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“…In general, the variables ξ s encode the |s|-ary information about the structure M. When M factors through substructures, the only dependence f i has on the information of arity strictly less than | rng x| is already realized by the lower arity part of the structure M. This means that the functions f i have no "hidden" information: all the information in ξ s , ≺ s is represented in M| s . The construction in the following proof is essentially the frame-wise uniform measure introduced in [10]; see also [20].…”

Section: Distributions With Enough Amalgamationmentioning

confidence: 99%

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Relatively Exchangeable Structures

Crane¹,

Towsner²

2018

J. symb. log.

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We study random relational structures that are relatively exchangeable-that is, whose distributions are invariant under the automorphisms of a reference structure M. When M is ultrahomogeneous and has trivial definable closure, all random structures relatively exchangeable with respect to M satisfy a general Aldous-Hoover-type representation. If M also satisfies the n-disjoint amalgamation property (n-DAP) for all n ≥ 1, then relatively exchangeable structures have a more precise description whereby each component depends locally on M.Date: September 18, 2018. 1991 Mathematics Subject Classification. 03C07 (Basic properties of first-order languages and structures); 03C98 (Applications of model theory); 60G09 (exchangeability). 1 2 HARRY CRANE AND HENRY TOWSNER L-structure on M . We call a pair X and Y of random L-structures on M equal in distribution, written X = D Y, if P(X| S = S) = P(Y| S = S) for every S ∈ L S , for all finite S ⊆ M .1.2. Relative exchangeability. Special cases of L-structures include binary relations, set partitions, undirected graphs, triangle-free graphs, as well as composite objects, e.g., a set together with a binary relation, a pair of graphs, etc. We are particularly interested in random L-structures that satisfy natural invariance properties with respect to the symmetries of another structure, of which exchangeability is a special case.We also call a probability measure µ exchangeable whenever X ∼ µ is an exchangeable L-structure.Given a large structure U = (Ω, R 1 , . . . , R r ) and a probability measure µ on Ω, we can obtain an exchangeable random L-structure X = (N, X 1 , . . . , X r ) by sampling elements φ(1), φ(2), . . . independently and identically distributed (i.i.d.) from µ and then defining X = U φ . Explicit representations of exchangeable structures are detailed in the work of de Finetti [15], Aldous [5], Hoover [17], and Kallenberg [18]. As a special case, the Aldous-Hoover theorem [5,17] characterizes the exchangeable random k-ary hypergraphs X = (N, X )-that is, the exchangeable random structures with a single symmetric k-ary relation-through the decomposition (1)x ∈ X ⇐⇒ f ((ξ s ) s⊆rng x ) = 1, where f is Borel measurable, the random variables ξ s are i.i.d. Uniform[0, 1], and rng x is the set of distinct elements in x. For instance, an exchangeable random graph can be generated by specifying a function f : [0, 1] 4 → {0, 1} with f (·, b, c, ·) = f (·, c, b, ·), selecting independent Uniform[0, 1] parameters ξ ∅ , ξ {i} for each i ∈ N, and ξ {i,j} for each pair i < j, and including the edge {i, j} exactly when f (ξ ∅ , ξ {i} , ξ {j} , ξ {i,j} ) = 1.Exchangeable structures not only play a fundamental role in probability theory [6,18], Bayesian inference [15], and applications in population genetics [19] but also have a natural place in the study of homogeneous structures in combinatorics [23] and mathematical logic [2][3][4]. In many applications, e.g., spin-glass models in statistical physics [8] and combinatorial stochastic processes [9,24], a random structure X is...

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Section: Distributions With Enough Amalgamationmentioning

confidence: 99%

“…The construction in the following proof is essentially the frame-wise uniform measure introduced in [10]; see also [20]. Let A ⊆ age n (M) be the set of amalgams and let A be a choice of representatives from each isomorphism class of A. n-DAP ensures that A, and therefore A , is nonempty.…”

Section: F R Is Isomorphic To M With Probability 1 and Factors Thmentioning

confidence: 99%

Relatively Exchangeable Structures

Crane¹,

Towsner²

2018

J. symb. log.

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“…The question of whether T m,n has a prime model seems to be a hard combinatorial problem; in the case m = n = 2, we show that it is equivalent to a longstanding open problem in the theory of projective planes (Theorem 3.8).Finally, from the point of view of classification theory, the generic theory recipe has been a fruitful source of examples of simple theories (see [6], for example). Recently, many examples of generic theories which are not simple have been shown to be NSOP 1 (see [8], [19], [20]). The theories T m,n are further examples of this phenomenon, and they provide good combinatorial examples of properly NSOP 1 theories.…”

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confidence: 99%

Independence in Generic Incidence Structures

Conant¹,

Kruckman²

2019

J. symb. log.

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We study the theory T m,n of existentially closed incidence structures omitting the complete incidence structure K m,n , which can also be viewed as existentially closed K m,n -free bipartite graphs. In the case m = n = 2, this is the theory of existentially closed projective planes. We give an ∀∃-axiomatization of T m,n , show that T m,n does not have a countable saturated model when m, n ≥ 2, and show that the existence of a prime model for T 2,2 is equivalent to a longstanding open question about finite projective planes. Finally, we analyze model theoretic notions of complexity for T m,n . We show that T m,n is NSOP 1 , but not simple when m, n ≥ 2, and we show that T m,n has weak elimination of imaginaries but not full elimination of imaginaries. These results rely on combinatorial characterizations of various notions of independence, including algebraic independence, Kim independence, and forking independence. 1The theory T m,n is also interesting from the perspective of the model theory of graphs. The class of graphs has a model companion, the unique countable model of which is known as the "random graph". This structure has a bipartite counterpart, the "random bipartite graph". When it is viewed as a structure in a language with unary predicates for the bipartition, the theory of the random bipartite graph is the model companion of the theory of arbitrary incidence structures. Both the random graph and the random bipartite graph have well-understood theories: they are ℵ 0 -categorical, simple, unstable, and have quantifier elimination and trivial algebraic closure.In the same way, T m,n can be viewed as the theory of existentially closed bipartite graphs omitting the complete bipartite graph K m,n . Thus the theories T m,n are bipartite counterparts to the theories T n of generic K n -free graphs (existentially closed graphs omitting a complete subgraph of size n). The theories T n , introduced by Henson [13], are important examples of countably categorical theories exhibiting the properties TP 2 , SOP 3 , and NSOP 4 (see [24], and also [9] for discussion of the model theoretic properties of Henson graphs). We show that for m, n ≥ 2, T m,n also has TP 2 , but it is NSOP 1 , hence tamer, in a sense, than the Henson graphs.In another contrast to the Henson graphs, T m,n is not countably categorical when m, n ≥ 2. In fact, we show that in this case, T m,n has continuum-many types over the empty set, and hence has no countable saturated model (Theorem 3.3). The question of whether T m,n has a prime model seems to be a hard combinatorial problem; in the case m = n = 2, we show that it is equivalent to a longstanding open problem in the theory of projective planes (Theorem 3.8).Finally, from the point of view of classification theory, the generic theory recipe has been a fruitful source of examples of simple theories (see [6], for example). Recently, many examples of generic theories which are not simple have been shown to be NSOP 1 (see [8], [19], [20]). The theories T m,n are further examples of this phenomen...

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“…This exists since P has a -minimum element. We can generalize the strong amalgamation property to higher dimensions, following Section 3 of [15], suitably modified for classes of structures. Definition 2.18.…”

Section: Bymentioning

confidence: 99%

Products of Classes of Finite Structures

Guingona¹,

Parnes²,

Scow³

2021

Preprint

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We study the preservation of certain properties under products of classes of finite structures. In particular, we examine age indivisibility, indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interact with the wreath product, direct product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products.

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Disjoint n-Amalgamation and Pseudofinite Countably Categorical Theories

Cited by 7 publications

References 26 publications

Relatively Exchangeable Structures

Relatively Exchangeable Structures

Independence in Generic Incidence Structures

Products of Classes of Finite Structures

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