non-prime maximal ideal in a complemented lattice

Question

Is there a complemented lattice, which has a non-prime maximal ideal?

Answer 1

Consider the lattice $\mathbf{M}_3=\{0,a,b,c,1\}$ where $0$ is the bottom element, $1$ is the top, and $a,b,c$ are the atoms. $\mathbf{M}_3$ is complemented since 1 has complement 0, 0 has 1, $a$ has $b$, $b$ has $a$, and $c$ has $a$.

Consider the ideal $I=\{0,a\}$. $I$ is maximal since if we add either $b$ or $c$, we then get $1=a\vee b=a\vee c\in I$. However, $I$ is not prime, since $b\wedge c=0\in I$ but neither $b$ nor $c$ is in $I$.