Mapping Fixpoints Under Lattice Morphisms

Definition: Let $(X,\le)$ and $(Y,\sqsubseteq)$ be complete lattices. We call $F : X \to Y$ a complete join-morphism iff for each set $X_1 \subseteq X$ we have

$\begin{equation*} F(\sqcup X_1) = \sqcup \{ F(a).\ a \in X_1 \} \end{equation*}$

For example, $F(a_1 \sqcup a_2 \sqcup a_3) = F(a_1) \sqcup F(a_2) \sqcup F(a_3)$ .

Lemma: Let $(X,\le)$ and $(Y,\sqsubseteq)$ be complete lattices, and $F : X \to X$ , $\Gamma : X \to Y$ , $F^\# : Y \to Y$ be complete join-morphisms such that

$\begin{equation*} F(\Gamma(y)) \le \Gamma(F^\#(y)) \end{equation*}$

for all $y \in Y$ . If $lfp$ denotes least fixpoint of a function, then

$\begin{equation*} lfp(F) \le \Gamma(lfp(F^\#)) \end{equation*}$

In other words, we can approximate $lfp(F)$ by computing $lfp(F^\#)$ .