Homogeneous 1‐based structures and interpretability in random structures

Koponen, Vera

doi:10.1002/malq.201400096

Cited by 5 publications

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“…Remark 2.2. The definition of binary random structure above coincides with the one given in [13] and is equivalent to the definition given in [2]. Clearly, the Rado graph is a binary random structure according to the definition given here.…”

Section: Preliminariesmentioning

confidence: 91%

“…The context of this article is the same as that of [2,13] and therefore we refer to those articles (any one of them will do) for basics and relevant facts about homogeneous structures, simple structures and the extension M eq of M by imaginaries. However we repeat the following definitions here: we say that N is canonically embedded in M eq if N is a ∅-definable subset of M eq and for every 0 < n < ω and every relation R ⊆ N n that is ∅-definable in M eq there is a relation symbol in the vocabulary of N which is interpreted as R, and the vocabulary of R contains no other symbols.…”

Section: Preliminariesmentioning

confidence: 99%

“…Section 2 gives a few definitions and remarks of relevance for this article. Since the work here takes place within the same context as [2,13] we refer to the preliminary section of any one of these articles for more detailed explanations of notions and known results that will be used (concerning homogeneous and ω-categorical structures with simple theories and about imaginary elements).…”

Section: Introductionmentioning

confidence: 99%

“…This is the only part which may not be immediate from[13, Section 3]. However, if C has all properties (i)-(iv) but not (v), then we can let C ′ = {c ∈ C : c ∈ crd(a) for some a ∈ M } and it is straightforward to verify (using that M is ω-categorical) that C ′ satisfies (i)-(v), because for every a ∈ M , crd(a) is the same whether computed with respect to C or with respect to C ′ 7.…”

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confidence: 99%

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Binary Primitive Homogeneous Simple Structures

Koponen¹

2017

J. symb. log.

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Abstract. Suppose that M is countable, binary, primitive, homogeneous, and simple, and hence 1-based. We prove that the SU-rank of the complete theory of M is 1. It follows that M is a random structure. The conclusion that M is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is defined in terms of imaginary elements. This is partly why we also characterize equivalence relations which are definable without parameters in the context of ω-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.

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Section: Preliminariesmentioning

confidence: 91%

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Binary Primitive Homogeneous Simple Structures

Koponen¹

2017

J. symb. log.

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“…If for every homogeneous simple structure M its complete theory has trivial dependence and finite SU-rank, then the behavior of dependence in simple homogeneous structures parallels that of stable homogeneous structures (see [22] for a survey of stable homogeneous structures). The reader is referred to [2,1,19] for more results about simple homogeneous structures.…”

Section: Introductionmentioning

confidence: 99%

Binary simple homogeneous structures are supersimple with finite rank

Koponen

2015

Proc. Amer. Math. Soc.

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Binary simple homogeneous structures

Koponen

2018

Annals of Pure and Applied Logic

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We describe all binary simple homogeneous structures M in terms of ∅definable equivalence relations on M , which "coordinatize" M and control dividing, and extension properties that respect these equivalence relations. The expressions finitely homogeneous and ultrahomogeneous are also used for the same notion. 1 c R c, b ⌣ | c Rdand tp(a/acl(c R )) = tp(b/acl(c R )), where 'acl' is taken in M eq , then there is e ∈ M such that tp(e, c) = tp(a, c) and tp(e,d) = tp (b,d). Otherwise such e may not exist (in any elementary extension of M), not even whend is a single element.and tp(a/acl(c R )) = tp(b/acl(c R )), are just the premisses (in the present context) of the independence theorem for simple theories. So the interesting part, with respect to (d)(ii), is that if (for every R as in (i)) these premisses are not satisfied, then a "common extension" may not exist. Thus we do not, in general, get anything "for free" beyond what the independence theorem guarantees. From this, one may get the impression that common extensions of types like in (d) are unusual. But note that, by part (i) of (d), we can always find a ∅-definable equivalence relation R such that c ⌣ | c Rd . Therefore I would say that (by part (ii) of (d)), in a binary simple structure, common extensions of two types do exist as long as we respect all ∅-definable equivalence relations and some other "reasonable" conditions related to them. The examples in sections 7.1 -7.3 show that these conditions are, in fact, necessary. The reason that (d) only considers an extension of two 1-types (one of which has only one parameter c) is that, since M is binary with elimination of quantifiers, the problem of extending more than two k-types (with finite parameter sets) can be reduced to a finite sequence of "extension problems", each of which involves only two 1-types and one of the types has only one parameter. More about this is said in the beginning of Section 6.From the proofs of the main results, one can extract information about ω-categorical (not necessarily binary or homogeneous) supersimple structures with finite SU-rank and trivial dependence. This information is presented in Corollaries 5.3 and 5.4, and may be useful in future studies of nonbinary simple homogeneous structures. Now we turn to problems about simple homogeneous structures. If M is stable and homogeneous, then M has the finite submodel property 3 and T h(M) is decidable. (For the first result, see [23, Proposition 5.1] or [17, Lemma 7.1]; for the second, see the proof of Theorem 5.2 in [23].) It is still not settled whether every binary simple homogeneous structure has the finite submodel property, nor whether its theory must be decidable. But my guess is that the answer is 'yes' to both questions. 4 Regarding nonbinary simple homogeneous structures, I would say that all core problems are unsolved. The answer is unknown to each of these questions, where we assume that M is (nonbinary) simple and homogeneous: Must T h(M) be supersimple? If T h(M) is supersimple, must it have finite SU-rank?. M...

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Homogeneous 1‐based structures and interpretability in random structures

Cited by 5 publications

References 29 publications

Binary Primitive Homogeneous Simple Structures

Binary Primitive Homogeneous Simple Structures

Binary simple homogeneous structures are supersimple with finite rank

Binary simple homogeneous structures

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