WEAK CANONICAL BASES IN NSOP THEORIES

Abstract

AbstractWe study the notion of weak canonical bases in an NSOP $_{1}$ theory T with existence. Given $p(x)=\operatorname {tp}(c/B)$ where $B=\operatorname {acl}(B)$ in ${\mathcal M}^{\operatorname {eq}}\models T^{\operatorname {eq}}$ , 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 $\approx $ of collinearity of an indiscernible sequence is type-definable. Secondly, we prove that given a total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequence I in p, the weak canonical base of $\operatorname {tp}(I/B)$ is $\operatorname {acl}(e)$ , if the hyperimaginary $I/\approx $ 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 intersection of the weak canonical bases of all total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p over B. However, while we investigate some examples, we point out that given two weak canonical bases of total $\mathop {\smile \hskip -0.9em ^| \ }^K$ -Morley sequences in p need not be interalgebraic, contrary to the case of simple theories. Lastly we suggest an independence relation relying on weak canonical bases, when T has those. The relation, satisfying transitivity and base monotonicity, might be useful in further studies on NSOP $_1$ theories .