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sav08:mapping_fixpoints_under_lattice_morphisms [2009/03/26 13:53] vkuncak |
sav08:mapping_fixpoints_under_lattice_morphisms [2009/03/26 13:53] vkuncak |
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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 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) |
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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)) | F(\Gamma(y)) \le \Gamma(F^\#(y)) | ||