If $M=(E,S)$ and $N=(E,F)$ are 2 partition matroids, and $I=S \cap F $ . Is there a matroid with $I$ being its set of independent sets?

Question

If $M=(E,S)$ and $N=(E,F)$ are 2 partition matroids, and $I=S \cap F $ . Is there a matroid with $I$ being its set of independent sets?

My intuition says it's correct because $M,N$ are partition matroids, but I dunno how to prove this.

Answer 1

Let $E=\{0,1,2\}$. Let $\mathscr{P}_S=\big\{\{0,1\},\{2\}\big\}$ be the partition corresponding to $S$, and let

$$S=\left\{A\subseteq E:|A\cap P|\le 1\text{ for each }P\in\mathscr{P}_S\right\}\;.$$

Let $\mathscr{P}_F=\big\{\{0\},\{1,2\}\big\}$ be the partition corresponding to $F$, and let

$$F=\left\{A\subseteq E:|A\cap P|\le 1\text{ for each }P\in\mathscr{P}_F\right\}\;.$$

Verify that

$$I=\big\{\varnothing,\{0\},\{1\},\{2\},\{0,2\}\big\}\;,$$

and show that the augmentation (or exchange) property fails for $\{1\}$ and $\{0,2\}$.