1. Partial Orders

2. Definition
A relation R on a set S is a partial ordering
if it is reflexive, antisymmetric, and transitive
A set S with a partial ordering R is called
a partially ordered set or a poset and
is denoted (S,R)
It is a partial ordering because pairs of elements
may be incomparable!

3. Assume a poset (S,R) does have a cycle (a,b),(b,c),(c,a)
A poset has no cycles
Proof
Assume a poset (S,R) does have a cycle (a,b),(b,c),(c,a) a b c
A poset is reflexive, antisymmetric, and transitive
(a,b) is in R and (b,c) is in R
consequently (a,c) is in R, due to transitivity
(c,a) is in R, by our assumption above
(c,a) is in R and (a,c) is in R
this is symmetric, and contradicts our assumption
consequently the poset (S,R) cannot have a cycle
What kind of proof was this?

4. Example
show it is
reflexive
antisymmetric
transitive

6. The equations editor has let me down

7. Definition

8. Example
3 and 9 are comparable
3 divides 9
5 and 7 are incomparable
5 does not divide 7
7 does not divide 5

9. Example

10. Definition

11. Example
reflexive
antisymmetric
transitive
totally ordered
all pairs are comparable
every subset has a least element
note: Z+ rather than Z

12. Read
… about lexicographic ordering
pages 417 and 418

13. Hasse Diagrams
A poset can be drawn as a digraph
it has loops at nodes (reflexive)
it has directed asymmetric edges
it has transitive edges
Draw this removing all redundant information
a Hasse diagram
remove all loops
(x,x)
remove all transitive edges
if (x,y) and (y,z) remove (x,z)
remove all direction
draw pointing upwards

14. Example of a Hasse Diagram
The digraph of the above poset (divides) has
loops and an edge (x,y) if x divides y

15. Example of a Hasse Diagram1 7 4 9 10 11 12 3 2 6 8 5

16. Exercise of a Hasse Diagram
Draw the Hasse diagram for the above poset
consider its digraph
remove loops
remove transitive edges
remove direction
point upwards

17. Exercise of a Hasse Diagram
{(5,5),(5,4),(5,3),(5,2),(5,1),(5,1),
(4,4),(4,3),(4,2),(4,1),(4,0),
(3,3),(3,2),(3,1),(3,0),
(2,2),(2,1),(2,0),
(1,1),(1,0),
(0,0)} 5 4 3 2 1 5 4 3 1 2

18. Maximal and Minimal Elements
Maximal elements are at the top of the Hasse diagram
Minimal elements are at the bottom of the Hasse diagram

19. Example of Maximal and Minimal Elements1 7 4 9 10 11 12 3 2 6 8 5
Maximal set is {8,12,9,10,7,11}
Minimal set is {1}

20. Greatest and Least Elements1 7 4 9 10 11 12 3 2 6 8 5
There is no greatest Element
The least element is 1
Note difference between maximal/minimal and greatest/least

21. Lattices
Read pages

22. Topological Sorting
Read pages

23. fin