Here is the question I am trying to solve:
Show that the lattice of flats of $M(K_n)$ is isomorphic to the partition lattice $\pi_n$. Definition: $\pi_n$ is the set of partitions of $[n]$, partially ordered by $\tau \leq \pi$ if every block of $\tau$ is contained in a block of $\pi$.
I was thinking about trying to guess the map from the following figure (in Oxley book, second edition):
But still I am not quite sure how to guess it, I know that for the edge 6, we picked the block $\{c,d\}$ because this edge attached these two vertices and then we will take all the other vertices in the top diagram as singletons to form a map at the edge $6$ between $\pi_4$ and the lattice $\mathcal{L}(M(K_4)).$ but how can I conclude a pattern for the map incase of $\pi_n$?
And how to show the lattice homomorphism of this map?
Could anyone clarify this for me please?
In the graphic matroid of the complete graph $K_n$, the closure of an edge set $S$ is, effectively, the transitive closure of $S$ as a relation: for any vertices $x,y$, if there is a $x-y$ path whose edges lie in $S$, then we add the edge $xy$ to the closure. When we look at $S$ and its closure as subgraphs of $K_n$, the closure of $S$ turns every component of $S$ into a clique on the same set of vertices.
As a result, the flats (or closed sets) correspond to the subgraphs of $K_n$ where every component is a clique.
The correspondence between flats of $M(K_n)$ and partitions of $[n]$ is as follows: we think of the vertex set of $K_n$ as $[n]$, and a partition $\pi$ corresponds to the flat of $M(K_n)$ whose connected components are exactly the blocks of $\pi$. Let's write $F(\pi)$ for the flat corresponding to $\pi$ in this way.
To check that this is a lattice isomorphism, we check that $\tau \le \pi$ if and only if $F(\tau) \subseteq F(\pi)$.
What does $\tau \le \pi$ mean? It means that whenever two points $x,y\in [n]$ are in the same block of $\tau$, they are also in the same block of $\pi$. Being in the same block of $\tau$ is equivalent to being in the same connected component of $F(\tau)$, and because all the connected components are cliques, it is equivalent to $xy$ being an edge of $F(\tau)$. So an equivalent way of writing $\tau \le \pi$ is that whenever $xy$ is an edge of $F(\tau)$, it is an edge of $F(\pi)$ - and this is exactly the statement $F(\tau) \subseteq F(\pi)$ we wanted.