Problem Set - Orders

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 7 December 2009

Orders

	Define the following and give an example for each: (a)   partial order, (b)   total order, (c)   order, (d)   ordered set, (e)   maximum, (f)   minimum, (g)   upper bound, (h)   lower bound, (i)   bounded above, (j)   bounded below, (k)   least upper bound, (l)   greatest lower bound, (m)   supremum, (n)   infimum, (o)   intervals.
	An ordered set $S$ has the least upper bound property if it satisfies: If $E\subseteq S$, $E\ne \varnothing$, and $E$ is bounded above then $\sup \left(E\right)$ exists in $S$.
	An ordered set $S$ is well ordered if it satisfies: If $E\subseteq S$ then $E$ has a minimal element.
	An ordered set $S$ is totally ordered if it satisfies: If $x,y\in S$ then $x<y$ or $x<y$.
	An ordered set $S$ is a lattice if it satisfies: If $x,y\in S$ then $\sup \{x,y\}$ and $\inf \{x,y\}$ exist.
	Show that $\mathbb{Q}$ does not have the least upper bound property.
	Show that $ℝ$ has the least upper bound property.
	Which of ${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},ℂ$ have the least upper bound property?
	Which of ${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},\mathbb{Q},ℝ,ℂ$ are well ordered?
	Which of ${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},\mathbb{Q},ℝ,ℂ$ are totally ordered?
	Which of ${\mathbb{Z}}_{>0},{\mathbb{Z}}_{\ge 0},\mathbb{Z},\mathbb{Q},ℝ,ℂ$ 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: (a)   ordered monoid, (a)   ordered group, (a)   ordered ring, (a)   ordered field,
	Let $S$ be an ordered field. Prove the following: (a)   If $a\in S$ and $a>0$ then $-a<0$. (b)   If $a\in S$ and $a>0$ then $a^{-1}>0$. (c)   If $a,b\in S$, $a>0$ and $b>0$ then $ab>0$.
	Let $S$ be an ordered group and let $x\in G$. Define the absolute value of $x$.

References [PLACEHOLDER]

[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)