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This is perhaps an easy question, but... Let $M$ be a matroid on a ground set $E$, and let $A$ and $B$ be non-disjoint subsets of $E$ such that $M|A$ and $M|B$ are both connected. Is $M|(A\cup B)$ then necessarily connected? Clearly this is true for graphic matroids, but I can't find any results in the literature regarding the general case. |
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Yes. Let $E$ be the ground set of a matroid. Define an equivalence relation $\sim$ on $E$ by imposing that $i \sim j$ if $i$ and $j$ are in the same circuit of $E$, and taking the transitive closure of this. Then the equivalence classes of $\sim$ are the connected components of the matroid. Any circuit of $M|_A$ is also a circuit of $M$, so if two elements of $A$ are in the same connected component of $M|_A$ then they are in the same connected component of $M$. (The converse is not true.) Let $x$ be in $A \cap B$. For any $a \in A$, since $A$ is connected, we have that $a$ is in the same connected component of $M|_A$ as $x$ is. By the observation of the previous paragraph, this means that $a$ and $x$ are in the same connected component of $M$. Similarly, every $b \in B$ is in the same connected component of $M$ as $x$ is. So all of $M$ is one connected component. |
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