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Koponen, Vera

Open this publication in new window or tab >>Binary simple homogeneous structures### Koponen, Vera

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##### Abstract [en]

##### Keywords

model theory, homogeneous structure, simple theory, stability theory, classification theory
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-364743 (URN)10.1016/j.apal.2018.08.006 (DOI)000448496300007 ()
#####

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Available from: 2018-11-01 Created: 2018-11-01 Last updated: 2019-06-27Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

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.

Open this publication in new window or tab >>On Constraints And Dividing In Ternary Homogeneous Structures### Koponen, Vera

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_some",{id:"formSmash:j_idt184:1:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_1_j_idt188_otherAuthors",{id:"formSmash:j_idt184:1:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_1_j_idt188_otherAuthors",multiple:true}); 2018 (English)In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 83, no 4, p. 1691-1721Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

model theory, homogeneous structure, simple theory, constraint, dividing, amalgamation
##### National Category

Algebra and Logic
##### Identifiers

urn:nbn:se:uu:diva-373241 (URN)10.1017/jsl.2018.61 (DOI)000454236900023 ()
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Available from: 2019-01-14 Created: 2019-01-14 Last updated: 2019-01-14Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.

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 < omega 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.

Open this publication in new window or tab >>Binary primitive homogeneous simple structures### Koponen, Vera

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##### Abstract [en]

##### Keywords

model theory, homogeneous structure, simple theory, random structure
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-320741 (URN)10.1017/jsl.2016.51 (DOI)000397110700011 ()
#####

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Available from: 2017-04-24 Created: 2017-04-24 Last updated: 2017-04-26Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Suppose that M is countable, binary, primitive, homogeneous, and simple. We prove that the SU-rank of the complete theory of M is 1 and hence 1-based. 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 omega-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.

Open this publication in new window or tab >>Homogeneous 1-based structures and interpretability in random structures### Koponen, Vera

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_some",{id:"formSmash:j_idt184:3:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_3_j_idt188_otherAuthors",{id:"formSmash:j_idt184:3:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_3_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 6-18Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Wiley-VCH Verlagsgesellschaft, 2017
##### Keywords

model theory, homogeneous structure, simple theory, random structure
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-322293 (URN)10.1002/malq.201400096 (DOI)000400361900001 ()
#####

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Available from: 2017-05-18 Created: 2017-05-18 Last updated: 2017-06-26Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.

Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let math formula be countable V-structure which is homogeneous, simple and 1-based. The first main result says that if math formula is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if math formula is “coordinatized” by a set with SU-rank 1 and there is no definable (without parameters) nontrivial equivalence relation on M with only finite classes, then math formula is strongly interpretable in a random structure.

Open this publication in new window or tab >>Random l-colourable structures with a pregeometry### Ahlman, Ove

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.### Koponen, Vera

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_some",{id:"formSmash:j_idt184:4:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_4_j_idt188_otherAuthors",{id:"formSmash:j_idt184:4:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_4_j_idt188_otherAuthors",multiple:true}); 2017 (English)In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 32-58Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Wiley-VCH Verlagsgesellschaft, 2017
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-321515 (URN)10.1002/malq.201500006 (DOI)000400361900003 ()
#####

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Available from: 2017-05-06 Created: 2017-05-06 Last updated: 2017-11-28Bibliographically approved

We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

Open this publication in new window or tab >>Binary simple homogeneous structures are supersimple with finite rank### Koponen, Vera

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_some",{id:"formSmash:j_idt184:5:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_5_j_idt188_otherAuthors",{id:"formSmash:j_idt184:5:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_5_j_idt188_otherAuthors",multiple:true}); 2016 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, no 4, p. 1745-1759Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Model theory; homogeneous structure; simple theory; stable theory; rank
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-275007 (URN)10.1090/proc/12828 (DOI)000369298400034 ()
#####

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Available from: 2016-01-27 Created: 2016-01-27 Last updated: 2017-11-30Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.

Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous, then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set.

Open this publication in new window or tab >>Limit laws and automorphism groups of random nonrigid structures### Ahlman, Ove

### Koponen, Vera

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_some",{id:"formSmash:j_idt184:6:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_6_j_idt188_otherAuthors",{id:"formSmash:j_idt184:6:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_6_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Journal of Logic and Analysis, ISSN 1759-9008, E-ISSN 1759-9008, Vol. 7, no 2, p. 1-53, article id 1Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

finite model theory, limit law, zero-one law, random structure, automorphism group
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-248078 (URN)10.4115/jla.2015.7.2 (DOI)000359802400001 ()
#####

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Available from: 2015-03-26 Created: 2015-03-26 Last updated: 2017-12-04Bibliographically approved

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.

A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

Open this publication in new window or tab >>On compactness of logics that can express properties of symmetry or connectivity### Koponen, Vera

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.### Hyttinen, Tapani

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_some",{id:"formSmash:j_idt184:7:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_7_j_idt188_otherAuthors",{id:"formSmash:j_idt184:7:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_7_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Studia Logica: An International Journal for Symbolic Logic, ISSN 0039-3215, E-ISSN 1572-8730, Vol. 103, no 1, p. 1-20Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Springer, 2015
##### Keywords

Abstract logic, Model theoretic logic, Compactness, Completeness, Automorphism, Connectivity, Random graph theory
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-244545 (URN)10.1007/s11225-013-9522-3 (DOI)000349360100001 ()
#####

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Available from: 2015-02-17 Created: 2015-02-17 Last updated: 2017-12-04Bibliographically approved

Helsingfors Universitet.

A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The basic idea underlying the results and examples presented here is that it is possible to construct a countable first-order theory T such that every model of T has a very rich automorphism group, but every finite subset T' of T has a model which is rigid.

Open this publication in new window or tab >>On sets with rank one in simple homogeneous structures### Ahlman, Ove

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.### Koponen, Vera

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_some",{id:"formSmash:j_idt184:8:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_8_j_idt188_otherAuthors",{id:"formSmash:j_idt184:8:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_8_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, p. 223-250Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

model theory, homogeneous structure, simple theory, pregeometry, rank, reduct, random structure
##### National Category

Algebra and Logic
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:uu:diva-243006 (URN)10.4064/fm228-3-2 (DOI)000352858400002 ()
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Available from: 2015-02-03 Created: 2015-02-03 Last updated: 2017-12-05Bibliographically approved

We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

Open this publication in new window or tab >>Typical automorphism groups of finite nonrigid structures### Koponen, Vera

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_some",{id:"formSmash:j_idt184:9:j_idt188:some",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt184_9_j_idt188_otherAuthors",{id:"formSmash:j_idt184:9:j_idt188:otherAuthors",widgetVar:"widget_formSmash_j_idt184_9_j_idt188_otherAuthors",multiple:true}); 2015 (English)In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 54, no 5-6, p. 571-586Article in journal (Refereed) Published
##### National Category

Algebra and Logic
##### Research subject

Mathematical Logic
##### Identifiers

urn:nbn:se:uu:diva-260072 (URN)10.1007/s00153-015-0428-9 (DOI)000358581600006 ()
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Available from: 2015-08-14 Created: 2015-08-14 Last updated: 2017-12-04Bibliographically approved