Problem Set - Ordered sets

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

Ordered sets

	Show that if a greatest lower bound exists, then it is unique.
	Show that if $S$ is a lattice then the intersection of two intervals is an interval.
	A poset $S$ is left filtered if every subset $E$ of $S$ has an upper bound. A poset $S$ is right filtered if every subset $E$ of $S$ has an lower bound. Let $S$ be a poset and let $E$ be a subset of $S$ . A minimal element of $E$ is an element $x \in E$ such that if $y \in E$ then $x \leq y$ . A poset $S$ is well ordered if every subset $E$ of $S$ has a minimal element. Show that every well ordered set is totally ordered.
	Show that there exist totally ordered sets that are not well ordered.

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)