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sav08:galois_connection_on_lattices [2008/05/07 10:03] vkuncak |
sav08:galois_connection_on_lattices [2008/05/07 10:14] vkuncak |
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==== Galois Connection ==== | ==== Galois Connection ==== | ||
- | Galois connection is defined by two monotonic functions $\alpha : C \to A$ and $\gamma : A \to C$ between [[partial order]]s $\leq$ on $C$ and $\sqsubseteq$ on $A$, such that | + | Galois connection is defined by two monotonic functions $\alpha : C \to A$ and $\gamma : A \to C$ between [[:partial order]]s $\leq$ on $C$ and $\sqsubseteq$ on $A$, such that |
\begin{equation*} | \begin{equation*} | ||
\alpha(c) \sqsubseteq a\ \iff\ c \leq \gamma(a) \qquad (*) | \alpha(c) \sqsubseteq a\ \iff\ c \leq \gamma(a) \qquad (*) | ||
Line 19: | Line 19: | ||
* $\gamma$ is an [[wk>injective function]] | * $\gamma$ is an [[wk>injective function]] | ||
+ | ===== Using Galois Connection ===== | ||
+ | |||
+ | Define $sp^\#$ using $\alpha$ and $\gamma$: | ||
+ | ++++| | ||
+ | \[ | ||
+ | sp^\#(a,r) = \alpha(sp(\gamma(a),r) | ||
+ | \] | ||
+ | ++++ | ||
+ | |||
+ | ===== Best Abstract Transformer ===== |