Let $\mathfrak{A}$ be a poset. For $a, b \in \mathfrak{A}$ we will denote $a \not\asymp b$ if only if there are a non-least element $c$ such that $c \leqslant a \wedge c \leqslant b$.
Let $\mathfrak{A}$, $\mathfrak{B}$ are posets. I call a pointfree funcoid a pair $\left( \alpha ; \beta \right)$ of functions $\alpha : \mathfrak{A} \rightarrow \mathfrak{B}$, $\beta : \mathfrak{B} \rightarrow \mathfrak{A}$ such that $ \forall x \in \mathfrak{A}, y \in \mathfrak{B}: \left( y \not\asymp^{\mathfrak{B}} \alpha \left( x \right) \Leftrightarrow x \not\asymp^{\mathfrak{A}} \beta \left( y \right) \right) . $
Conjecture If $\left( \alpha ; \beta \right)$ is a pointfree funcoid and $\alpha$ is a bijection $\mathfrak{A} \rightarrow \mathfrak{B}$, then $\alpha$ is an order isomorphism $\mathfrak{A} \rightarrow \mathfrak{B}$.
A weaker conjecture:
Conjecture If $\left( \alpha ; \beta \right)$ is a pointfree funcoid and $\alpha$ is a bijection $\mathfrak{A} \rightarrow \mathfrak{B}$ and $\beta$ is a bijection $\mathfrak{B} \rightarrow \mathfrak{A}$, then $\alpha$ is an order isomorphism $\mathfrak{A} \rightarrow \mathfrak{B}$.
If these conjectures are false, what additional conditions we may add to make them true? (Maybe, these are true for lattices? distributive lattices?)