Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let   M     M        be an o-minimal expansion of a densely ordered group and   H     H        be a pairwise disjoint collection of dense subsets of  M    such that   ⋃H     ⋃  H        is definably independent in   M     M       .  study the structure   (M,(H) H∈H )     (  M  ,    (  H  )    H  ∈  H    )       . Positive results include that every open set definable in   (M,(H) H∈H )     (  M  ,    (  H  )    H  ∈  H    )        is definable in   M     M       , the structure induced in   (M,(H) H∈H )     (  M  ,    (  H  )    H  ∈  H    )        on any   H 0 ∈H       H    0    ∈  H        is as simple as possible (in a sense that is made precise), and the theory of   (M,(H) H∈H )     (  M  ,    (  H  )    H  ∈  H    )        eliminates imaginaries and is strongly dependent and axiomatized over the theory of   M     M        in the most obvious way. Negative results include that   (M,(H) H∈H )     (  M  ,    (  H  )    H  ∈  H    )        does not have definable Skolem functions and is neither atomic nor satisfies the exchange property.  also characterize (model-theoretic) algebraic closure and thorn forking in such structures. Throughout,  compare and contrast our results with the theory of dense pairs of o-minimal structures.
['Alfred Dolich', 'Christopher J. Miller', 'Charles Steinhorn']
I am writing a research paper on a idea Expansions of o-minimal structures by dense independent sets
[{"from": "human", "value": "I am writing a research paper on a idea Expansions of o-minimal structures by dense independent sets"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let   M     M        be an o-minimal expansion of a densely ordered group and   H     H        be a pairwise disjoint collection of dense subsets of  M \u00a0  such that   \u22c3H     \u22c3  H        is definably independent in   M     M       .  study the structure   (M,(H) H\u2208H )     (  M  ,    (  H  )    H  \u2208  H    )       . Positive results include that every open set definable in   (M,(H) H\u2208H )     (  M  ,    (  H  )    H  \u2208  H    )        is definable in   M     M       , the structure induced in   (M,(H) H\u2208H )     (  M  ,    (  H  )    H  \u2208  H    )        on any   H 0 \u2208H       H    0    \u2208  H        is as simple as possible (in a sense that is made precise), and the theory of   (M,(H) H\u2208H )     (  M  ,    (  H  )    H  \u2208  H    )        eliminates imaginaries and is strongly dependent and axiomatized over the theory of   M     M        in the most obvious way. Negative results include that   (M,(H) H\u2208H )     (  M  ,    (  H  )    H  \u2208  H    )        does not have definable Skolem functions and is neither atomic nor satisfies the exchange property.  also characterize (model-theoretic) algebraic closure and thorn forking in such structures. Throughout,  compare and contrast our results with the theory of dense pairs of o-minimal structures."}]