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∈E
such that if
y∈E
then
x≤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)