Lecture 13 – Introduction to Posets

You may want to download the lecture slides that were used for these videos (PDF).

1. Posets

This video defines partially ordered sets, called posets. Then, we discuss two examples of partial orders. (10:57)

2. Linear Orders

A linear (or total) order is a partial order where any two numbers can always be compared. (1:38)

3. Covers in a Poset

When we have a poset P, and we have two distinct points x and y, we say that x is covered by y when x < y and there is no point z in P with x < z < y. (4:16)

4. Cover Graphs and Order Diagrams

This video introduces cover graphs and order diagrams. When P is a poset, we associate with P a graph G called the cover graph of P. The vertices of G are the points of P. When x and y are distinct points in P, they are adjacent in G if x covers y or y covers x in P.  An order diagram (or Hasse diagram) is like the cover graph, except that whenever x covers y, x must appear above y in the diagram.  (6:21)

5. Order Diagrams and Binary Relations

Using the same poset we drew before, let’s see if we can write down the binary relation. (7:57)

6. Order Diagrams and Cover Graphs

In this video we explore relationships between cover graphs and order diagrams. (7:30)

7. Determining Relations in Order Diagrams?

When looking at an order diagram, we can say that x < y exactly when there is a path starting at x that only moves upwards and eventually hits y. (2:59)

8. Comparability

Here, we define when two points are comparable or incomparable.  Then, we talk about the comparability graph and incomparability graph. (3:31)

9. An Alternate Definition

This video gives an alternate definition of a poset, just in terms of the <, rather than ≤. (3:10)

10. Maximal and Minimal Points

In this video we define maximal and minimal points, and discuss a few examples. (4:36)

11. Chains, Maximal Chains, and Antichains

In this video, we introduce antichains and chains.  Then, we discuss maximal chains and the height of a poset.  Paralleling this, we define maximal antichains and the width of a poset.  We review these concepts again in the next lecture. (8:28)