Last updates: 7 December 2009
Define the following and give an example for each:
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An ordered set
$S$ has the least upper bound property if it satisfies:
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An ordered set
$S$ is well ordered if it satisfies:
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An ordered set
$S$ is totally ordered if it satisfies:
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An ordered set
$S$ is a lattice if it satisfies:
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Show that
$\mathbb{Q}$ does not have the least upper bound property. | |
Show that
$\mathbb{R}$ has the least upper bound property. | |
Which of
${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},\u2102$ have the least upper bound property? | |
Which of
${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},\mathbb{Q},\mathbb{R},\u2102$ are well ordered? | |
Which of
${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},\mathbb{Q},\mathbb{R},\u2102$ are totally ordered? | |
Which of
${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},\mathbb{Q},\mathbb{R},\u2102$ are lattices? | |
Let
$S$ be a set. Show that the set of subsets of
$S$ is partially ordered by inclusion. | |
Define the following and give examples of each:
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Let
$S$ be an ordered field.
Prove the following:
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Let $S$ be an ordered group and let $x\in G$. Define the absolute value of $x$. |
[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)