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On sets with rank one in simple homogeneous structures
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
2015 (Engelska)Ingår i: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, s. 223-250Artikel i tidskrift (Refereegranskat) Published
Abstract [en]

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.

Ort, förlag, år, upplaga, sidor
2015. Vol. 228, s. 223-250
Nyckelord [en]
model theory, homogeneous structure, simple theory, pregeometry, rank, reduct, random structure
Nationell ämneskategori
Algebra och logik
Forskningsämne
Matematik
Identifikatorer
URN: urn:nbn:se:uu:diva-243006DOI: 10.4064/fm228-3-2ISI: 000352858400002OAI: oai:DiVA.org:uu-243006DiVA, id: diva2:785724
Tillgänglig från: 2015-02-03 Skapad: 2015-02-03 Senast uppdaterad: 2017-12-05Bibliografiskt granskad
Ingår i avhandling
1. Limit Laws, Homogenizable Structures and Their Connections
Öppna denna publikation i ny flik eller fönster >>Limit Laws, Homogenizable Structures and Their Connections
2018 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Alternativ titel[sv]
Gränsvärdeslagar, Homogeniserbara Strukturer och Deras Samband
Abstract [en]

This thesis is in the field of mathematical logic and especially model theory. The thesis contain six papers where the common theme is the Rado graph R. Some of the interesting abstract properties of R are that it is simple, homogeneous (and thus countably categorical), has SU-rank 1 and trivial dependence. The Rado graph is possible to generate in a probabilistic way. If we let K be the set of all finite graphs then we obtain R as the structure which satisfy all properties which hold with assymptotic probability 1 in K. On the other hand, since the Rado graph is homogeneous, it is also possible to generate it as a Fraïssé-limit of its age.

Paper I studies the binary structures which are simple, countably categorical, with SU-rank 1 and trivial algebraic closure. The main theorem shows that these structures are all possible to generate using a similar probabilistic method which is used to generate the Rado graph. Paper II looks at the simple homogeneous structures in general and give certain technical results on the subsets of SU-rank 1.

Paper III considers the set K consisting of all colourable structures with a definable pregeometry and shows that there is a 0-1 law and almost surely a unique definable colouring. When generating the Rado graph we almost surely have only rigid structures in K. Paper IV studies what happens if the structures in K are only the non-rigid finite structures. We deduce that the limit structures essentially try to stay as rigid as possible, given the restriction, and that we in general get a limit law but not a 0-1 law.

Paper V looks at the Rado graph's close cousin the random t-partite graph and notices that this structure is not homogeneous but almost homogeneous. Rather we may just add a definable binary predicate, which hold for any two elemenets which are in the same part, in order to make it homogeneous. This property is called being homogenizable and in Paper V we do a general study of homogenizable structures. Paper VI conducts a special case study of the homogenizable graphs which are the closest to being homogeneous, providing an explicit classification of these graphs.

Ort, förlag, år, upplaga, sidor
Uppsala: Department of Mathematics, 2018. s. 43
Serie
Uppsala Dissertations in Mathematics, ISSN 1401-2049 ; 104
Nyckelord
Model theory, random structure, finite model theory, simple theory, homogeneous structure, countably categorical, 0-1 law
Nationell ämneskategori
Algebra och logik
Forskningsämne
Matematisk logik; Matematik
Identifikatorer
urn:nbn:se:uu:diva-330142 (URN)978-91-506-2672-8 (ISBN)
Disputation
2018-02-16, Polhemssalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, 13:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2018-01-17 Skapad: 2017-11-28 Senast uppdaterad: 2018-02-09

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