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sav08:mapping_fixpoints_under_lattice_morphisms [2009/03/26 13:53] vkuncak |
sav08:mapping_fixpoints_under_lattice_morphisms [2015/04/21 17:30] |
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| - | ====== Mapping Fixpoints Under Lattice Morphisms ====== | ||
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| - | **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 | ||
| - | \[ | ||
| - | F(\sqcup X_1) = \sqcup \{ F(a).\ a \in X_1 \} | ||
| - | \] | ||
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| - | For example, $F(a_1 \sqcup a_2 \sqcup a_3) = F(a_1) \sqcup F(a_2) \sqcup F(a_3) | ||
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| - | **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 | ||
| - | \[ | ||
| - | F(\Gamma(y)) \le \Gamma(F^\#(y)) | ||
| - | \] | ||
| - | for all $y \in Y$. If $lfp$ denotes least fixpoint of a function, then | ||
| - | \[ | ||
| - | lfp(F) \le \Gamma(lfp(F^\#)) | ||
| - | \] | ||
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| - | In other words, we can approximate $lfp(F)$ by computing $lfp(F^\#)$. | ||
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