Lecture Videos Lecture 13 – Introduction to Posets

# 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)