closedness of a sublattice under complements

Question

Let $A$ be a bounded sublattice of the bounded lattice $(X,\le)$ with

$$\max A=\max X, ~~\min A=\min X$$

Let $a,b\in X$ be complements and $a\in A$.

Is $b\in A$?

Answer 1

Counterexample: let $X=\{\emptyset,\{a\},\{b\},\{a,b\}\}$ (where $a\ne b$), and let $A=\{\emptyset,\{a\},\{a,b\}\}$. Then $X$ is a bounded lattice; $A$ is a bounded sublattice with the same $0$ and $1$; $\{a\},\{b\}$ are complements in $X$; $\{a\}\in A$, but $\{b\}\notin A$.