22 found
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  1.  24
    Pseudofinite Structures and Simplicity.Darío García, Dugald Macpherson & Charles Steinhorn - 2015 - Journal of Mathematical Logic 15 (1):1550002.
    We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are (...)
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  2.  50
    A Version of o-Minimality for the P-Adics.Deirdre Haskell & Dugald Macpherson - 1997 - Journal of Symbolic Logic 62 (4):1075-1092.
  3.  35
    Vapnik–Chervonenkis Density in Some Theories Without the Independence Property, II.Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson & Sergei Starchenko - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):311-363.
    We study the Vapnik–Chervonenkis density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm {U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.
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  4.  8
    On Variants of o-Minimality.Dugald Macpherson & Charles Steinhorn - 1996 - Annals of Pure and Applied Logic 79 (2):165-209.
  5.  13
    Cell Decompositions of C-Minimal Structures.Deirdre Haskell & Dugald Macpherson - 1994 - Annals of Pure and Applied Logic 66 (2):113-162.
    C-minimality is a variant of o-minimality in which structures carry, instead of a linear ordering, a ternary relation interpretable in a natural way on set of maximal chains of a tree. This notion is discussed, a cell-decomposition theorem for C-minimal structures is proved, and a notion of dimension is introduced. It is shown that C-minimal fields are precisely valued algebraically closed fields. It is also shown that, if certain specific ‘bad’ functions are not definable, then algebraic closure has the exchange (...)
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  6.  6
    Model Theory of Finite and Pseudofinite Groups.Dugald Macpherson - 2018 - Archive for Mathematical Logic 57 (1-2):159-184.
    This is a survey, intended both for group theorists and model theorists, concerning the structure of pseudofinite groups, that is, infinite models of the first-order theory of finite groups. The focus is on concepts from stability theory and generalisations in the context of pseudofinite groups, and on the information this might provide for finite group theory.
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  7.  5
    Omega-Categoricity, Relative Categoricity and Coordinatisation.Wilfrid Hodges, I. M. Hodkinson & Dugald Macpherson - 1990 - Annals of Pure and Applied Logic 46 (2):169-199.
  8.  10
    Strongly Determined Types.Alexandre A. Ivanov & Dugald Macpherson - 1999 - Annals of Pure and Applied Logic 99 (1-3):197-230.
    The notion of a strongly determined type over A extending p is introduced, where p .S. A strongly determined extension of p over A assigns, for any model M )- A, a type q S extending p such that, if realises q, then any elementary partial map M → M which fixes acleq pointwise is elementary over . This gives a crude notion of independence which arises very frequently. Examples are provided of many different kinds of theories with strongly determined (...)
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  9.  8
    Finite Axiomatizability and Theories with Trivial Algebraic Closure.Dugald Macpherson - 1991 - Notre Dame Journal of Formal Logic 32 (2):188-192.
  10.  15
    Interpreting Groups in Ω-Categorical Structures.Dugald Macpherson - 1991 - Journal of Symbolic Logic 56 (4):1317-1324.
    It is shown that no infinite group is interpretable in any structure which is homogeneous in a finite relational language. Related questions are discussed for other ω-categorical structures.
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  11.  6
    Unexpected Imaginaries in Valued Fields with Analytic Structure.Deirdre Haskell, Ehud Hrushovski & Dugald Macpherson - 2013 - Journal of Symbolic Logic 78 (2):523-542.
    We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.
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  12.  20
    Reconstruction of Homogeneous Relational Structures.Silvia Barbina & Dugald Macpherson - 2007 - Journal of Symbolic Logic 72 (3):792 - 802.
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  13.  35
    Extending Partial Orders on o‐Minimal Structures to Definable Total Orders.Dugald Macpherson & Charles Steinhorn - 1997 - Mathematical Logic Quarterly 43 (4):456-464.
    It is shown that if is an o-minimal structure such that is a dense total order and ≾ is a parameter-definable partial order on M, then ≾ has an extension to a definable total order.
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  14.  3
    Binary Relational Structures Having Only Countably Many Nonisomorphic Substructures.Dugald Macpherson & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):876-884.
  15.  14
    Uncountable Homogeneous Partial Orders.Manfred Droste, Dugald Macpherson & Alan Mekler - 2002 - Mathematical Logic Quarterly 48 (4):525-532.
    A partially ordered set is called k-homogeneous if any isomorphism between k-element subsets extends to an automorphism of . Assuming the set-theoretic assumption ⋄, it is shown that for each k, there exist partially ordered sets of size ϰ1 which embed each countable partial order and are k-homogeneous, but not -homogeneous. This is impossible in the countable case for k ≥ 4.
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  16.  16
    A Note on Valuation Definable Expansions of Fields.Deirdre Haskell & Dugald Macpherson - 1998 - Journal of Symbolic Logic 63 (2):739-743.
  17.  13
    Binary Relational Structures Having Only Countably Many Nonisomorphic Substructures.Dugald Macpherson & James H. Schmerl - 1991 - Journal of Symbolic Logic 56 (3):876-884.
  18.  3
    A. H. Lachlan. On Countable Stable Structures Which Are Homogeneous for a Finite Relational Language. Israel Journal of Mathematics, Vol. 49 , Pp. 69–153. - G. Cherlin and A. H. Lachlan. Stable Finitely Homogeneous Structures. Transactions of the American Mathematical Society, Vol. 296 , Pp. 815–850. [REVIEW]Dugald Macpherson - 1993 - Journal of Symbolic Logic 58 (1):350-352.
  19.  5
    Review: A. H. Lachlan, On Countable Stable Structures Which Are Homogeneous for a Finite Relational Language; G. Cherlin, A. H. Lachlan, Stable Finitely Homogeneous Structures. [REVIEW]Dugald Macpherson - 1993 - Journal of Symbolic Logic 58 (1):350-352.
  20.  22
    Recent Developments in Model Theory, Notre Dame Journal of Formal Logic, Vol.54, Nos. 3-4, 2013.Dugald Macpherson - 2014 - Bulletin of Symbolic Logic 20 (3):357-359.
  21.  12
    Review: Ya'acov Peterzil, Sergei Starchenko, A Trichotomy Theorem for O-Minimal Structures. [REVIEW]Dugald Macpherson - 1999 - Journal of Symbolic Logic 64 (2):908-910.
  22.  12
    Ya'acov Peterzil and Sergei Starchenko, A Trichotomy Theorem for o-Minimal Structures, Proceedings of the London Mathematical Society, Ser. 3 Vol. 77 , Pp. 481–523. [REVIEW]Dugald Macpherson - 1999 - Journal of Symbolic Logic 64 (2):908-910.