Partial Orders

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!

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?

Example show it is reflexive antisymmetric transitive

The equations editor has let me down

Definition

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

Example

Definition

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

Read … about lexicographic ordering pages 417 and 418

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

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

Example of a Hasse Diagram 1 7 4 9 10 11 12 3 2 6 8 5

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

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

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

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

Greatest and Least Elements 1 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

Lattices Read pages 423-425

Topological Sorting Read pages 425-427

fin