# Differences

This shows you the differences between two versions of the page.

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==== |

Trace: