WEAK CANONICAL BASES in NSOP1THEORIES

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Abstract

We study the notion of weak canonical bases in an NSOP1theory T with existence. Given p(x) = tp(c=B) where B = acl(B) in Meqj= Teq, the weak canonical base of p is the smallest algebraically closed subset of B over which p does not Kim-fork. With this aim we firstly show that the transitive closure ≈ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total K-Morley sequence I in p, the weak canonical base of tp(I=B) is acl(e), if the hyperimaginary I/≈ is eliminable to e, a sequence of imaginaries. We also supply a couple of criteria for when the weak canonical base of p exists. In particular the weak canonical base of p is (if exists) the intersec-tion of the weak canonical bases of all total K-Morley sequences in p over B. However, while we investigate some examples, we point out that given two weak canonical bases of total K-Morley sequences in p need not be interalgebraic, contrary to the case of simple theories. Lastly we suggest an independence relation re-lying on weak canonical bases, when T has those. The relation, satisfying transitivity and base monotonicity, might be useful in further studies on NSOP1theories.

Original languageEnglish
JournalJournal of Symbolic Logic
DOIs
Publication statusAccepted/In press - 2021

Bibliographical note

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All Science Journal Classification (ASJC) codes

  • Philosophy
  • Logic

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