Example of a geometric lattice whose dual is not geometric.

Question

I'm looking for an example of a geometric lattice (a finite semimodular lattice in which every element is a join of atoms) whose dual (the lattice obtained by reversing the order relation) is not geometric.

Thank you!

Answer 1

Let me see if I get it right, this time...
Consider the partition lattice $\Pi(4)$ of a four element set, say $\{a,b,c,d\}$.
Partition lattices are geometric.
But $\Pi(4)$ is not dually semimodular, that is, it doesn't satisfy $$x \preceq y \Longrightarrow x \wedge z \preceq y \wedge z.$$ For let $x=\{\{a,c\},\{b,d\}\}$, $y=\{\{a,b,c,d\}\}$, and $z=\{\{a,d\},\{b,c\}\}$.
Then $y \wedge z = z$, but $x \wedge z=\{\{a\},\{b\},\{c\},\{d\}\}$, and there exist two partitions in the middle: