Binary Primitive Homogeneous Simple Structures

Koponen, Vera

doi:10.1017/jsl.2016.51

Cited by 5 publications

(20 citation statements)

References 17 publications

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“…Before recalling what is known from before about binary simple homogeneous structures, we give the definition of trivial dependence (also called 'trivial forking', or 'totally trivial' in [14]). [21] and in Remark 6.6 of the same article.…”

Section: Simple Homogeneous Structuresmentioning

confidence: 71%

“…part (a) of the 'main results' in the introduction), it follows that part (i) of Fact 2.5 still holds if the assumption about 'primitivity' is replaced with the condition that there is no ∅-definable equivalence relation on M which has infinitely many infinite equivalence classes. Fact 2.5 (i) fails without the binarity condition as shown by Example 2.7 in [21], which is primitive, homogeneous, and superstable with SU-rank 2 (but nonbinary). It is also not a random structure.…”

Section: Simple Homogeneous Structuresmentioning

confidence: 99%

“…We already mentioned that T h(M), the complete theory of M, is supersimple with finite SU-rank. It is also known that T h(M) is 1-based and has trivial dependence/forking [21,Fact 2.6 and Remark 6.6]. If M is, in addition, primitive, then M has SU-rank 1 and is a random structure [21].…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Simple Homogeneous Structuresmentioning

confidence: 71%

Section: Simple Homogeneous Structuresmentioning

confidence: 99%

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Binary simple homogeneous structures

Koponen

2018

Annals of Pure and Applied Logic

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“…it has a nontrivial ∅-definable equivalence relation on its universe. A different kind of example of SU-rank 2 which is primitive is Example 3.3.2 in [23], which is also discussed in [16,Example 2.7]. This example is not 2-transitive.…”

Section: 5mentioning

confidence: 99%

“…one with relational vocabulary) will be called k-ary if no relation symbol of its vocabulary has arity higher than k. We say binary and ternary instead of 2-ary and 3-ary, respectively. The fine structure of all binary simple homogeneous structures is well understod in terms of supersimplicity, SU-rank, pregeometries, the behaviour of dividing and the nature of definable sets of SU-rank 1 [2,15,16,17]. However, the arguments for binary structures do not carry over to ternary structures.…”

Section: Introductionmentioning

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

Conant¹

2017

J. symb. log.

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We use axioms of abstract ternary relations to define the notion of a free amalgamation theory. These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP 2 and, assuming modularity, with NSOP 3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraïssé limits with free amalgamation, which provides new context for the fact that the generic Kn-free graphs are SOP 3 , while the higher arity generic K r n -free r-hypergraphs are simple.

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Binary Primitive Homogeneous Simple Structures

Cited by 5 publications

References 17 publications

Binary simple homogeneous structures

Binary simple homogeneous structures

On Constraints and Dividing in Ternary Homogeneous Structures

An Axiomatic Approach to Free Amalgamation

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