The following is a well known fact:
A frame morphism $\Phi : B \to A$, considered as a map of posets has a right adjoint $\Psi : A \to B$.
My question is, under which hypotesis $\Phi \Psi \Phi = \Phi $ or $\Psi \Phi \Psi = \Psi$? I am asking because I do believe this might happen very often.
I believe the correct answer is: it's always true.
Observe that since you are dealing with adjunctions between posets you have the units $$\eta_X \colon X {\longrightarrow} \Psi\Phi(X)$$ and counits $$\epsilon_Y \colon \Phi\Psi(Y){\longrightarrow} Y$$ applying to them the monotone mappings (functors) $\Phi$ and $\Psi$ respectively you get $$\Phi(\eta_X) \colon \Phi(X) \to \Phi\Psi\Phi(X)$$ and $$\Psi(\epsilon_Y) \colon\Psi\Phi\Psi(Y) \to \Psi(Y)$$ which testify that $\Phi(X) \leq \Phi\Psi\Phi(X)$ and $\Psi\Phi\Psi(Y)\leq\Psi(Y)$ for each $X \in B$ and $Y \in A$.
On the other hand we have also the morphisms $$\eta_{\Psi(Y)}\colon \Psi(Y) \to \Psi\Phi\Psi(Y)$$ and $$\epsilon_{\Phi(X)} \Phi\Psi\Phi(X) \to \Phi(X)$$ which testify that $\Psi(Y) \leq \Psi\Phi\Psi(Y)$ and $\Phi\Psi\Phi(X)\leq \Phi(X)$.
By anti-symmetric property of the orders you get that $\Phi(X)=\Phi\Psi\Phi(X)$ and $\Psi(Y)=\Psi\Phi\Psi(Y)$ for every $X \in B$ and $Y \in A$.
