9 found
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  1.  12
    Small Skew Fields.Cédric Milliet - 2007 - Mathematical Logic Quarterly 53 (1):86-90.
    Wedderburn showed in 1905 that finite fields are commutative. As for infinite fields, we know that superstable (Cherlin, Shelah) and supersimple (Pillay, Scanlon, Wagner) ones are commutative. In their proof, Cherlin and Shelah use the fact that a superstable field is algebraically closed. Wagner showed that a small field is algebraically closed , and asked whether a small field should be commutative. We shall answer this question positively in non-zero characteristic.
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  2.  3
    On the Definability of Radicals in Supersimple Groups.Cédric Milliet - 2013 - Journal of Symbolic Logic 78 (2):649-656.
  3.  47
    On Enveloping Type-Definable Structures.Cédric Milliet - 2011 - Journal of Symbolic Logic 76 (3):1023 - 1034.
    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 θ-type-definable group in a small structure is the conjunction of definable groups, and extend the result to semi-groups, fields, (...)
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  4.  40
    Stable Division Rings.Cédric Milliet - 2011 - Journal of Symbolic Logic 76 (1):348 - 352.
    It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings.
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  5.  4
    Groupes Fins.Cédric Milliet - 2014 - Journal of Symbolic Logic 79 (4):1120-1132.
    We investigate some common points between stable structures and weakly small structures and define a structureMto befineif the Cantor-Bendixson rank of the topological space${S_\varphi }\left} \right)$is an ordinal for every finite subsetAofMand every formula$\varphi \left$wherexis of arity 1. By definition, a theory isfineif all its models are so. Stable theories and small theories are fine, and weakly minimal structures are fine. For any finite subsetAof a fine groupG, the traces on the algebraic closure$acl\left$ofAof definable subgroups ofGover$acl\left$which are boolean combinations of (...)
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  6.  15
    Fields with Few Types.Cédric Milliet - 2013 - Journal of Symbolic Logic 78 (1):72-84.
    According to Belegradek, a first order structure is weakly small if there are countably many $1$-types over any of its finite subset. We show the following results. A field extension of finite degree of an infinite weakly small field has no Artin-Schreier extension. A weakly small field of characteristic $2$ is finite or algebraically closed. A weakly small division ring of positive characteristic is locally finite dimensional over its centre. A weakly small division ring of characteristic $2$ is a field.
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  7.  6
    Corps Stables.Cédric Milliet - 2011 - Journal of Symbolic Logic 76 (1):348-352.
    On y montre qu'un corps stable de caractéristique positive est de dimension finie sur son centre, puis on généralise la chose aux corps simples.
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  8.  3
    Variations Sur Un Thème de Aldama Et Shelah.Cédric Milliet - 2016 - Journal of Symbolic Logic 81 (1):96-126.
  9.  5
    On Properties of (Weakly) Small Groups.Cédric Milliet - 2012 - Journal of Symbolic Logic 77 (1):94-110.
    A group is small if it has only countably many complete n-types over the empty set for each natural number n. More generally, a group G is weakly small if it has only countably many complete 1-types over every finite subset of G. We show here that in a weakly small group, subgroups which are definable with parameters lying in a finitely generated algebraic closure satisfy the descending chain conditions for their traces in any finitely generated algebraic closure. An infinite (...)
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