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Partial Orders

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

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Partial Orders

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  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. A poset has no cycles Proof a b c Assume a poset (S,R) does have a cycle (a,b),(b,c),(c,a) • 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

  5. The equations editor has let me down

  6. Definition

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

  8. Example

  9. Definition

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

  11. Read • … about lexicographic ordering • pages 417 and 418

  12. 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

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

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

  15. 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

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

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

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

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

  20. Lattices Read pages 423-425

  21. Topological Sorting Read pages 425-427

  22. fin

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