Example of a function between boolean lattices that preserves $(\top,\bot,\wedge,\vee)$ but not complements.

Question

Its easy to find boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ such that $f$ preserves both top and bottom elements, as well as binary meets, but not complements.

Example. Let $A$ and $B$ denote the four and two element boolean lattices respectively, and let $f$ map every element of $A$ to $\bot_B,$ except for $\top_A$ which is mapped to $\top_B$.

Note however that the function $f : A \rightarrow B$ defined above does not preserve joins; in particular, letting $a$ and $a'$ denote the two "middle" elements of $A$, we have the following.

• $f(a \vee a') = f(\top_A) = \top_B$
• $f(a) \vee f(a') = \bot_B \vee \bot_B = \bot_B$

Question. What's an example of boolean lattices $A$ and $B$ together with a function $f : A \rightarrow B$ subject to the following constraints?

1. $f$ preserves top and bottom elements
2. $f$ preserves binary meets and joins
3. $f$ does not preserve all complements

Answer 1

There is no such function. Given a boolean algebra, observe that $\lnot a$ is the unique element satisfying $a \land \lnot a = \bot$ and $a \lor \lnot a = \top$; thus, any function that preserves $\bot$, $\top$, $\lor$, and $\land$ must also preserve $\lnot$.