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On enveloping type-definable structures

Published online by Cambridge University Press:  12 March 2014

Cédric Milliet
Cédric Milliet*
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan, UMR 5208 CNRS, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
*
Université Galatasaray, Faculté de Sciences et de Lettres, Çirağan Caddesi n°36, 34357 Ortaköy, Istamboul, Turkey, E-mail: milliet@math.univ-lyonl.fr
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Abstract

We observe simple links between equivalence relations, groups, fields and groupoids (and between preorders, semi-groups, rings and categories), which are type-definable in an arbitrary structure, and apply these observations to the particular context of small and simple structures. Recall that a structure is small if it has countably many n-types with no parameters for each natural number n. We show that a ∅-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, rings, categories, groupoids and preorders which are ∅-type-definable in a small structure. For an A-type-definable group GA(where the set A may be infinite) in a small and simple structure, we deduce that

(1) if GA is included in some definable set X such that boundedly many translates of GA cover X, then GA is the conjunction of definable groups.

(2) for any finite tuple ḡ in GA, there is a definable group containing ḡ.

Keywords

Small theorysimple theorytype-definable groupsemi-groupfieldringpreorderequivalence relationcategorygroupoidenvelope
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 3 , September 2011 , pp. 1023 - 1034
Copyright
Copyright © Association for Symbolic Logic 2011

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