Problem Set - Ordered sets

Problem Set - Ordered sets

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

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)