1. 8.6 Partial Orderings

2. Definition Partial ordering– a relation R on a set S that is Reflexive, Antisymmetric, and Transitive Examples? R={(a,b)| a is a subset of b } R={(a,b)| a divides b } on {1,2,3,4} – R={(1,1),(1,2),(1,3),(1,4),(2,2),…} R={(a,b)| a≤ b } R={(a,b)| a=b+1 }

3. Partially ordered set (poset) (S,R) -- a set S and a relation R on S, that is R, A, and T. Often we use (S, ≼ ) Note: ≼ is a generic symbol for R It includes the usual ≤, but it is more general. It also covers other poset relations: divides, subset,… We say a ≼ b iff aRb Also a ≺ b iff a ≺ b and a≠b

4. Examples and non-examples of posets (S, ≼) 1. (Z, ≤) proof 2. (Z, ≥)

5. More examples 3. (Z, |) where | is “divides” 4. ( Z+, |)

6. …examples 5. (P(S),  ) where S={1,2,3} and P(S) is the power set 6. (P(S),  ) where S is a set and P(S) is the power set

7. Comparable Def: The elements a and b of a poset (S, ≼) are said to be “comparable” if either a ≼b or b ≼a. Otherwise, they are “incomparable.”

8. Comparable, incomparable elements For each set, find comparable elementsincomparable (if any): 1.(Z, ≤ ) using the usual ≤ 2. (Z+, |) 3. (P(S), ) where S={1,2,3}

9. totally (linearly) ordered set Def: A poset (S, ≼ ) is a totally (linearly) ordered set if every two elements of S are comparable. ≼ is then a total order, and S is a chain.

10. Are these examples total orders or not? (Z, ≤ ) (Z+, |)

11. Lexicographic Order (dictionary) Things to consider: Longer lengths or different lengths in words Ex: Discreet<discrete Discreet<discreetness Discrete<discretion

12. Lexicographic order Suppose (A1, ≼ 1 ) and (A2, ≼ 2 ) are two posets. Let (a 1, a 2 ), (b 1, b 2 )  A 1 xA 2 Let (a 1, a 2 ) ≺ (b 1, b 2 ) in case either a 1 ≺ 1 b 1 or (a 1 =b 1 and a 2 ≺ 2 b 2 ) Letter or number examples

13. (A1xA2, ≼ ) is a poset Proof Method? Proof – see book

14. Hasse diagram Hasse diagram—a diagram that contains sufficient information to find a partial ordering Algorithm: – create a digraph with directed edges pointing up – remove all loops (reflexive is assumed) – remove any (a,c) where (a,b) and (b,c) are present (transitivity assumed) – remove arrows (direction up is assumed)

15. Ex. 1. S={1,2,3,4}; poset (S, ≤) Original digraphreduced diagram 4 | 3 | 2 | 1

16. Ex. 2: (S, ≼) where S={1,2,3,4,6,8,12} and ≼ ={(a,b)|a divides b} Shorthand: ({1,2,3,4,6,8,12}, | ) 8 12| 46| 23 | 1

17. Ex 3: Hasse diagram of (P({a,b,c}), )

18. Ex. 4: Hasse of ({2,4,5,10,12,20,25,}, | )

19. Maximal, minimal… Def: Let (S, ≼ ) be a poset and a S. – a is maximal in (S, ≼ ) if there does not exist b  S such that a ≺ b. – a is minimal in (S, ≼ ) if there does not exist b  S such that b ≺ a. – a is the greatest element of (S, ≼ ) if b ≼ a for all b  S. – a is the least element of (S, ≼ ) if a ≼ b for all b  S. Find examples of maximal, greatest elements,… in above examples.

20. greatest element Claim: The greatest element, when it exists, is unique. Proof: – Method? Similarly, the least element, when it exists, is unique.

21. Upper bound,… Def: Let (S, ≼ ) be a poset and A  S. – If u  S and a ≼ u for all a  A,u is an upper bound of A. – If l  S and l ≼ a for all a  A, l is an lower bound of A. – x is a least upper bound of A, lub(A), if x is an upper bound and x ≼ z for every upper bound z of A. – y is a greatest lower bound of A, glb(A), if y is a lower bound and z ≼ y for every lower bound z of A. – Remark: lub and glb are unique when they exist.

22. Ex. 5 (S, ≼ ) A={b,d,g}, B=(d,e} hiupper bounds of A: |lub(A)= gflower bounds of A: ||glb(A)= de ||upper bounds of B bc lower bounds of B a find lub and glb

23. Ex. 6: A={4,6,8} with “divides” relation lub(A)= glb(A)= Note: lub=? glb=?

24. Well-ordered set Def: (S, ≼ ) is well-ordered set if it is a poset such that ≼ is a total ordering and every nonempty subset of S has a least element. Find Ex and non-ex.: (Z+, ≤) (Z, ≤) (Z+ x Z+, lexicographic order) (R+, ≤)

25. Topological sorting Use: for project ordering Def: A total ordering ≼ is compatible with the partial order R if a ≼ b whenever aRb. The construction of such a total order is called a topological sorting. Lemma: Every finite non-empty poset (S, ≼ ) has a minimal element.

26. ({2,4,5,10,12,20,25}, | ) Recall Hasse diagram for ({2,4,5,10,12,20,25}, | ) Create several topological sorts.

27. House Ex- book

28. Advising example