lattices made by $a\wedge b = ab$ and $a\vee b=a+b-ab$

Question

We define the binary operations $\vee$ and $\wedge$ on $\Bbb R$ by $a\wedge b = ab$ and $a\vee b=a+b-ab$. Then for $A=\{0\}$ and $A=\{1\}$ and $A=\{0,1\}$, the set $(A,\wedge,\vee)$ is a lattice.

Is there any other $A\subset \Bbb R$ such that $(A,\wedge,\vee)$ is a lattice?

Answer 1

In a lattice, $\vee$ and $\wedge$ are idempotent. So here $a = a \wedge a = a^2$, which implies...