1. 1 Partial Orderings Aaron Bloomfield CS 202 Rosen, section 7.6

2. 2 Introduction An equivalence relation is a relation that is reflexive, symmetric, and transitive A partial ordering (or partial order) is a relation that is reflexive, antisymmetric, and transitive Recall that antisymmetric means that if (a,b)  R, then (b,a)  R unless b = a Recall that antisymmetric means that if (a,b)  R, then (b,a)  R unless b = a Thus, (a,a) is allowed to be in R Thus, (a,a) is allowed to be in R But since it’s reflexive, all possible (a,a) must be in R But since it’s reflexive, all possible (a,a) must be in R A set S with a partial ordering R is called a partially ordered set, or poset Denoted by (S,R) Denoted by (S,R)

3. 3 Partial ordering examples Show that ≥ is a partial order on the set of integers It is reflexive: a ≥ a for all a  Z It is reflexive: a ≥ a for all a  Z It is antisymmetric: if a ≥ b then the only way that b ≥ a is when b = a It is antisymmetric: if a ≥ b then the only way that b ≥ a is when b = a It is transitive: if a ≥ b and b ≥ c, then a ≥ c It is transitive: if a ≥ b and b ≥ c, then a ≥ c Note that ≥ is the partial ordering on the set of integers (Z, ≥) is the partially ordered set, or poset

4. 4 Symbol usage The symbol   is used to represent any relation when discussing partial orders Not just the less than or equals to relation Not just the less than or equals to relation Can represent ≤, ≥, , etc Can represent ≤, ≥, , etc Thus, a  b denotes that (a,b)  R Thus, a  b denotes that (a,b)  R The poset is (S,  ) The poset is (S,  ) The symbol  is used to denote a  b but a ≠ b If  represents ≥, then  represents > If  represents ≥, then  represents > Fonts for this lecture set (specifically for the  and  symbols) is available on the course website

5. 5 Comparability The elements a and b of a poset (S,  ) are called comparable if either a  b or b  a. Meaning if (a,b)  R or (b,a)  R Meaning if (a,b)  R or (b,a)  R It can’t be both because  is antisymmetric It can’t be both because  is antisymmetric Unless a = b, of course If neither a  b nor b  a, then a and b are incomparable If neither a  b nor b  a, then a and b are incomparable Meaning they are not related to each other This is definition 2 in the text This is definition 2 in the text If all elements in S are comparable, the relation is a total ordering

6. 6 Comparability examples Let  be the “divides” operator | In the poset (Z +,|), are the integers 3 and 9 comparable? Yes, as 3 | 9 Yes, as 3 | 9 Are 7 and 5 comparable? No, as 7 | 5 and 5 | 7 No, as 7 | 5 and 5 | 7 Thus, as there are pairs of elements in Z + that are not comparable, the poset (Z +,|) is a partial order

7. 7 Comparability examples Let  be the less than or equals operator ≤ In the poset (Z +, ≤ ), are the integers 3 and 9 comparable? Yes, as 3 ≤ 9 Yes, as 3 ≤ 9 Are 7 and 5 comparable? Yes, as 5 ≤ 7 Yes, as 5 ≤ 7 As all pairs of elements in Z + are comparable, the poset (Z +, ≤ ) is a total order a.k.a. totally ordered poset, linear order, chain, etc. a.k.a. totally ordered poset, linear order, chain, etc.

8. 8 A bit of Star Wars humor…

9. 9 End of lecture on 28 April 2005

10. 10 Well-ordered sets (S,  ) is a well-ordered set if: (S,  ) is a totally ordered poset (S,  ) is a totally ordered poset Every non-empty subset of S has at least element Every non-empty subset of S has at least element Example: (Z,≤) Is a total ordered poset (every element is comparable to every other element) Is a total ordered poset (every element is comparable to every other element) It has no least element It has no least element Thus, it is not a well-ordered set Thus, it is not a well-ordered set Example: (S,≤) where S = { 1, 2, 3, 4, 5 } Is a total ordered poset (every element is comparable to every other element) Is a total ordered poset (every element is comparable to every other element) Has a least element (1) Has a least element (1) Thus, it is a well-ordered set Thus, it is a well-ordered set

11. 11 Lexicographic ordering Consider two posets: (S,  1 ) and (T,  2 ) We can order Cartesian products of these two posets via lexicographic ordering Let s 1  S and s 2  S Let s 1  S and s 2  S Let t 1  T and t 2  T Let t 1  T and t 2  T (s 1,t 1 )  (s 2,t 2 ) if either: (s 1,t 1 )  (s 2,t 2 ) if either: s1 1 s2s1 1 s2s1 1 s2s1 1 s2 s 1 = s 2 and t 1  2 t 2 Lexicographic ordering is used to order dictionaries

12. 12 Lexicographic ordering Let S be the set of word strings (i.e. no spaces) Let T bet the set of strings with spaces Both the relations are alphabetic sorting We will formalize alphabetic sorting later We will formalize alphabetic sorting later Thus, our posets are: (S,  ) and (T,  ) Order (“run”, “noun: to…”) and (“set”, “verb: to…”) As “run”  “set”, the “run” Cartesian product comes before the “set” one As “run”  “set”, the “run” Cartesian product comes before the “set” one Order (“run”, “noun: to…”) and (“run”, “verb: to…”) Both the first part of the Cartesian products are equal Both the first part of the Cartesian products are equal “noun” is first (alphabetically) than “verb”, so it is ordered first “noun” is first (alphabetically) than “verb”, so it is ordered first

13. 13 Lexicographic ordering We can do this on more than 2-tuples (1,2,3,5)  (1,2,4,3) When  is ≤ When  is ≤

14. 14 Lexicographic ordering Consider the two strings a 1 a 2 a 3 …a m, and b 1 b 2 b 3 …b n Here follows the formal definition for lexicographic ordering of strings If m = n (i.e. the strings are equal in length) (a 1, a 2, a 3, …, a m )  (b 1, b 2, b 3, …, b n ) using the comparisons just discussed (a 1, a 2, a 3, …, a m )  (b 1, b 2, b 3, …, b n ) using the comparisons just discussed Example: “run”  “set” Example: “run”  “set” If m ≠ n, then let t be the minimum of m and n Then a 1 a 2 a 3 …a m, is less than b 1 b 2 b 3 …b n if and only if either of the following are true: Then a 1 a 2 a 3 …a m, is less than b 1 b 2 b 3 …b n if and only if either of the following are true: (a 1, a 2, a 3, …, a t )  (b 1, b 2, b 3, …, b t ) (a 1, a 2, a 3, …, a t )  (b 1, b 2, b 3, …, b t ) Example: “run”  “sets” (t = 3) (a 1, a 2, a 3, …, a t ) = (b 1, b 2, b 3, …, b t ) and m < n (a 1, a 2, a 3, …, a t ) = (b 1, b 2, b 3, …, b t ) and m < n Example: “run”  “running”

15. 15 Hasse Diagrams Consider the graph for a finite poset ({1,2,3,4},≤) When we KNOW it’s a poset, we can simplify the graph 43214321 43214321 43214321 43214321 Called the Hasse diagram

16. 16 Hasse Diagram For the poset ({1,2,3,4,6,8,12}, |)

17. 17 Not being covered The remainder of 7.6 is not being covered due to lack of time Maximal and minimal elements Maximal and minimal elements Lattices Lattices Topological sorting Topological sorting

18. 18 Quick survey I felt I understood the material in this slide set… I felt I understood the material in this slide set… a) Very well b) With some review, I’ll be good c) Not really d) Not at all

19. 19 Quick survey The pace of the lecture for this slide set was… The pace of the lecture for this slide set was… a) Fast b) About right c) A little slow d) Too slow

20. 20 Quick survey How interesting was the material in this slide set? Be honest! How interesting was the material in this slide set? Be honest! a) Wow! That was SOOOOOO cool! b) Somewhat interesting c) Rather borting d) Zzzzzzzzzzz

21. 21 A bit of humor…