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Paper   IPM / M / 16205
School of Mathematics
  Title:   Pseudofiniteness in Hrushovski Constructions
  Author(s):  Massoud Pourmahdian (Joint with A. Valizadeh)
  Status:   Published
  Journal: Notre Dame J. Formal Logic
  No.:  1
  Vol.:  61
  Year:  2020
  Pages:   1-10
  Supported by:  IPM
In a relational language consisting of a single relation R, we investigate pseudofiniteness of certain Hrushovski constructions obtained via predimension functions. It is notable that the arity of the relation R plays a crucial role in this context. When R is ternary, by extending the methods recently developed by Brody and Laskowski, we interpret ⟨Q+,<⟩ in the ⟨K+,≤∗⟩-generic and prove that this structure is not pseudofinite. This provides a negative answer to the question posed in an earlier work by Evans and Wong. This result, in fact, unfolds another aspect of complexity of this structure, along with undecidability and the strict order property proved in the mentioned earlier works. On the other hand, when R is binary, it can be shown that the ⟨K+,≤∗⟩-generic is decidable and pseudofinite.

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