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--- sav08:mapping_fixpoints_under_lattice_morphisms [2009/03/26 13:51]
vkuncak
+++ sav08:mapping_fixpoints_under_lattice_morphisms [2009/03/26 13:54]
vkuncak
@@ Line 1: / Line 1: @@
 ====== Mapping Fixpoints Under Lattice Morphisms ======
-**Definition:** Let $(X,\le)$ and $(Y,\sqsubseteq)$ be complete [[lattices]]. We call and $F : X \to Y$ a function. We ca
+**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
+\[
+   F(\sqcup X_1) = \sqcup \{ F(a).\ a \in X_1 \}
+\]
-**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 morphisms (they distribute through arbitrary least upper bound) such that
+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
 \[
     F(\Gamma(y)) \le \Gamma(F^\#(y))