# Differences

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complete_lattices [2010/07/29 15:46]
127.0.0.1 external edit
complete_lattices [2012/06/16 00:02] (current)
jipsen
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subsets of $L$ to elements of $L$ and subsets of $L$ to elements of $L$ and
+$\langle L,\vee,\wedge\rangle$ is a [[lattices|lattice]] where $x\vee y=\bigvee\{x,y\}$, $x\wedge y=\bigwedge\{x,y\}$ and
-$\langle L,\vee,\wedge\rangle$ is a [[Lattices]]+$\bigvee S$ is the least upper bound of $S$,
+$\bigwedge S$ is the greatest lower bound of $S$.
-$\bigvee S$ is the least upper bound of $S$
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-$\bigwedge S$ is the greatest lower bound of $S$
==Morphisms== ==Morphisms==
Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices. Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices.
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$h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$ $h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$
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====Examples==== ====Examples====
Example 1: $\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets. Example 1: $\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets.
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====Basic results==== ====Basic results====