10 found
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  1.  25
    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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  2.  23
    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.
  3. Forking and independence in o-minimal theories.Alfred Dolich - 2004 - Journal of Symbolic Logic 69 (1):215-240.
  4.  13
    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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  5.  83
    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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  6.  49
    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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  7. A note on definable completeness and weak o-minimality.Alfred Dolich - 2007 - Notre Dame Journal of Formal Logic 48:281-292.
  8.  11
    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 $(\mathbb {R},<)$ have the property, as do all expansions of $(\mathbb {R},+,\cdot,\mathbb {N})$. Our main analytic-geometric result is that any such expansion of $(\mathbb {R},<,+)$ by Boolean combinations of open sets (of any arities) either is o-minimal or defines an isomorph of $(\mathbb N,+,\cdot )$. We also show that any given expansion of $(\mathbb (...)
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  9.  4
    Topological properties of definable sets in ordered Abelian groups of burden 2.Alfred Dolich & John Goodrick - 2023 - Mathematical Logic Quarterly 69 (2):147-164.
    We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp‐rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle‐third set (Theorem 2.9). If it has burden 2 and both an infinite (...)
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  10.  8
    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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