September 2011 On enveloping type-definable structures
Cédric Milliet
J. Symbolic Logic 76(3): 1023-1034 (September 2011). DOI: 10.2178/jsl/1309952532


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 \bar{g} in GA, there is a definable group containing \bar{g}.


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Cédric Milliet. "On enveloping type-definable structures." J. Symbolic Logic 76 (3) 1023 - 1034, September 2011.


Published: September 2011
First available in Project Euclid: 6 July 2011

zbMATH: 1244.03112
MathSciNet: MR2849257
Digital Object Identifier: 10.2178/jsl/1309952532

Primary: 03C45 , 03C60 , 20L05 , 20M99

Keywords: category , envelope , equivalence relation , field , groupoid , preorder , Ring , semi-group , simple theory , Small theory , type-definable group

Rights: Copyright © 2011 Association for Symbolic Logic


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Vol.76 • No. 3 • September 2011
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