9 found
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  1.  16
    Expansions of o-Minimal Structures by Dense Independent Sets.Alfred Dolich, Chris Miller & Charles Steinhorn - 2016 - Annals of Pure and Applied Logic 167 (8):684-706.
  2.  24
    Dp-Minimality: Basic Facts and Examples.Alfred Dolich, John Goodrick & David Lippel - 2011 - Notre Dame Journal of Formal Logic 52 (3):267-288.
    We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian (...)
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  3. Forking and Independence in o-Minimal Theories.Alfred Dolich - 2004 - Journal of Symbolic Logic 69 (1):215-240.
  4.  76
    Model Completeness for Trivial, Uncountably Categorical Theories of Morley Rank 1.Alfred Dolich, Michael C. Laskowski & Alexander Raichev - 2006 - Archive for Mathematical Logic 45 (8):931-945.
    We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.
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  5.  12
    Extensions of Ordered Theories by Generic Predicates.Alfred Dolich, Chris Miller & Charles Steinhorn - 2013 - Journal of Symbolic Logic 78 (2):369-387.
    Given a theoryTextending that of dense linear orders without endpoints, in a language ℒ ⊇ {<}, we are interested in extensionsT′ ofTin languages extending ℒ by unary relation symbols that are each interpreted in models ofT′ as sets that are both dense and codense in the underlying sets of the models.There is a canonically “wild” example, namelyT= Th andT′ = Th. Recall thatTis o-minimal, and so every open set definable in any model ofThas only finitely many definably connected components. But (...)
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  6.  5
    Connectedness in Structures on the Real Numbers: O-Minimality and Undecidability.Alfred Dolich, Chris Miller, Alex Savatovsky & Athipat Thamrongthanyalak - 2022 - Journal of Symbolic Logic 87 (3):1243-1259.
    We initiate an investigation of structures on the set of real numbers having the property that path components of definable sets are definable. All o-minimal structures on $$ have the property, as do all expansions of $$. Our main analytic-geometric result is that any such expansion of $$ by Boolean combinations of open sets either is o-minimal or defines an isomorph of $$. We also show that any given expansion of $$ by subsets of $\mathbb {N}^n$ has the property if (...)
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  7. A Note on Definable Completeness and Weak o-Minimality.Alfred Dolich - 2007 - Notre Dame Journal of Formal Logic 48:281-292.
  8.  47
    A Note on Weakly O-Minimal Structures and Definable Completeness.Alfred Dolich - 2007 - Notre Dame Journal of Formal Logic 48 (2):281-292.
    We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures.
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  9.  7
    Weak Dividing, Chain Conditions, and Simplicity.Alfred Dolich - 2004 - Archive for Mathematical Logic 43 (2):265-283.
    We study the properties of the independence relation given by weak dividing in simple theories. We also analyze abstract independence notions satisfying various axioms and relate these to the simple case.
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