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Suppose I have a matroid $M = (E, \mathcal{I})$. It is a known fact that given any two bases $X_0$ and $X_n$, we can transform $X_0$ into $X_n$ by repeatedly applying the basis exchange axiom. So there is a sequence of bases $X_0, \ldots, X_n$ such that $X_{i+1}$ is obtained by applying basis exchange to $X_i$ and $X_n$.

I am considering an extension of this. Suppose I have two bases $A_0$ and $B_0$ which partition the ground set. That is, $A_0$ and $B_0$ are disjoint bases whose union is the ground set $E$.

Now we are given two other bases $A_n$ and $B_n$. The goal is to transform the partition $A_0, B_0$ into $A_n, B_n$ by repeatedly applying symmetric basis exchange between $A_i$ and $B_i$ to obtain $A_{i+1}$ and $B_{i+1}$. In other words, we are looking for a sequence of partitions $A_0 \cup B_0, \ldots, A_n \cup B_n$ such that $A_i$ and $B_i$ are disjoint bases for each $i$ and $A_i \cup B_i = E$.

Is this always possible? Do we need to make some assumption about the matroid, such as base orderability?

Thank you

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