An Axiomatic Approach to Free Amalgamation

Conant, Gabriel

doi:10.1017/jsl.2016.42

Cited by 12 publications

(32 citation statements)

References 26 publications

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“…All mentioned properties of the generic tetrahedron-free 3-hypergraph, except for the 1-basedness, have been known for a long time. Results which imply that it is 1-based were recently proved by Conant [6] and by the present author [21].…”

Section: Simple Homogeneous Structuressupporting

confidence: 62%

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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Section: Simple Homogeneous Structuressupporting

confidence: 62%

Binary simple homogeneous structures

Koponen

2018

Annals of Pure and Applied Logic

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“…Moreover, we should remove the requirement that a should enumerate a closed set and instead deal with the general case. Since our assumptions are slightly different from those in Conant's original result ( [12], Lemma 5.5), we repeat the proof and adapt it to our setting. Proposition 8.13.…”

Section: Proofmentioning

confidence: 99%

“…These results are used in Section 7 to classify T * Sq in terms of the dividing lines of first order theories: T * Sq is a new example of a theory with TP 2 and NSOP 1 . In Section 8 we use the approach developed in [12] to show that T * Sq has elimination of hyperimaginaries and weak elimination of imaginaries.…”

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confidence: 99%

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Model theory of Steiner triple systems

Barbina

Casanovas

2019

J. Math. Log.

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A Steiner triple system is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite Steiner triple systems has a Fraïssé limit M F . Here we show that the theory T * Sq of M F is the model completion of the theory of Steiner triple systems. We also prove that T * Sq is not small and it has quantifier elimination, TP 2 , NSOP 1 , elimination of hyperimaginaries and weak elimination of imaginaries.

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“…Forbidding 3-irreducible structures. The notion of 'indecomposable structure' used by Henson in [11] has been generalized by Conant to the notion of 'k-irreducible structure' in [8]. We say that a structure A (in any relational language) is k-irreducible if for any choice of k elements a 1 , .…”

Section: Examplesmentioning

confidence: 99%

On Constraints and Dividing in Ternary Homogeneous Structures

Koponen¹

2018

J. symb. log.

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Let M be ternary, homogeneous and simple. We prove that if M is finitely constrained, then it is supersimple with finite SU-rank and dependence is k-trivial for some k < ω and for finite sets of real elements. Now suppose that, in addition, M is supersimple with SU-rank 1. If M is finitely constrained then algebraic closure in M is trivial. We also find connections between the nature of the constraints of M, the nature of the amalgamations allowed by the age of M, and the nature of definable equivalence relations. A key method of proof is to "extract" constraints (of M) from instances of dividing and from definable equivalence relations. Finally, we give new examples, including an uncountable family, of ternary homogeneous supersimple structures of SU-rank 1.

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An Axiomatic Approach to Free Amalgamation

Cited by 12 publications

References 26 publications

Binary simple homogeneous structures

Binary simple homogeneous structures

Model theory of Steiner triple systems

On Constraints and Dividing in Ternary Homogeneous Structures

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