Partial Orders on the integers. In this case ( a , b ) R if a b . a a - PowerPoint PPT Presentation

� � � � � � � � Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide01.html Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide02.html Partial Orders prev | slides | next prev | slides | next A relation R on a set S is a partial ordering or partial order if it is reflexive, antisymmetric and transitive. A set S together with a partial order R is called a partially ordered set or poset and is denoted ( S , R ). For example, the relation "less than or equal to" is a partial ordering Partial Orders on the integers. In this case ( a , b ) R if a b . a a so R is reflexive. a b implies that b a only when a = b so R is antisymmetric. if a b and b c then a c so R is transitive. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 So ( Z , ) is a poset. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:52 PM 1 of 1 09/11/2003 03:52 PM Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide03.html Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide04.html Partial Orders Partial Orders prev | slides | next prev | slides | next Show that the divisibility relation is a partial ordering on Z + : It is important to note that while this partial ordering is on Z + , there are pairs of elements a and b from Z + such that neither ( a , b ) nor ( a , b ) R if a | b ( b , a ) are in R , the divisibility relation. Need to show that R is 3 does not divide 5, nor does 5 divide 3. reflexive, The elements a and b of a poset ( S , R ) are called comparable if either ( a , b ) R or ( b , a ) R . When a and b are elements of S such that antisymmetric, neither ordered pair of them is in R then a and b are called incomparable . transitive. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:52 PM 1 of 1 09/11/2003 03:52 PM

Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide05.html Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide06.html Partial Orders Partial Orders prev | slides | next prev | slides | next Lexicographic Order If ( S , R ) is a poset and every two elements of S are comparable, S is called a totally ordered set and R is called a total order . This is sometimes called the "dictionary ordering" because of it’s similarity to how words are arranged in a dictionary. Consider the The "less than or equal to" relation on Z is totally ordered. following order: The "divides" relation on Z + is not totally ordered. lash < lashing < lass < lasso < last < lasting ( S , R ) is a well-ordered set if it is a poset such that R is a total order The length of the word has little to do with the order. What system and such that every nonempty subset of S has a least element. is actually used to determine the order? 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 What system is used to determine the order of these ordered pairs? (2,3) < (3,2) < (3,3) < (3,4) < (4,1) < (4,9) < (5,2) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:52 PM 1 of 1 09/11/2003 03:53 PM Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide07.html Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide08.html Partial Orders Partial Orders prev | slides | next prev | slides | next Lexicographic Order Hasse Diagrams The lexicographic ordering on A × B for two posets ( A , R 1 ) and Consider the partial ordering on {1, 2, 3} ( B , R 2 ) is defined R = {(1,1), (1,2), (1,3), (2,2), (2,3), (3,3)} ( a 1 , b 1 ) < ( a 2 , b 2 ) and it’s diagraph if a 1 < a 2 or ( a 1 = a 2 and b 1 < b 2 ) where a 1 , a 2 A and b 1 , b 2 B . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:53 PM 1 of 1 09/11/2003 03:53 PM

Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide09.html Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide10.html Partial Orders Partial Orders prev | slides | next prev | slides | next Hasse Diagrams Hasse Diagrams Since to be a partial ordering the relation R must be reflexive we Since partial ordering relations are transitive, we can omit the know that there will be loops at each vertex. Thus we can omit "short cut" edges and, as before, understand them to be present. them and understand them to be present. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:53 PM 1 of 1 09/11/2003 03:53 PM Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide11.html Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide12.html Partial Orders Partial Orders prev | slides | next prev | slides | next Hasse Diagrams Hasse Diagrams Finally, because partial order relations are antisymmetric, we can Exercise: Construct the Hasse Diagram for ({2, 5, 8, 10, 20}, |). The always draw the digraph so that all edges (except the "loop" edges relation here is the "divides" relation. which we’ve already omitted) point up. If we agree to always do this then we can change the arrows to lines; we are coding the (Solution is on the next slide...) direction in the orientation of the graph rather than in the arrows. Our graph now looks like 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 This diagram is an example of a Hasse Diagram . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:53 PM 1 of 1 09/11/2003 03:53 PM

Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide13.html Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide14.html Partial Orders Partial Orders prev | slides | next prev | slides | next Hasse Diagrams Hasse Diagrams The Hasse diagram for ({2, 5, 8, 10, 20}, |): If there is only one maximal element then it is the greatest element . If there is more than one maximal element then there is no greatest element. Similarily, if there is a unique minimal element then it is the least element . If there is more than one minimal element then there is no least element. Consider a subset A of a poset ( S , R ). The upper bounds of A are the elements that are "above" every element in A . This means that The maximal elements of this poset are the "tops"; in this case {8, that every upper bound can be obtained by tracing up from each 20}. element in A . The minimal elements of this poset are the "bottoms", in this case The lower bounds of A are defined in a similar way; they are {2, 5}. "below" all elements in A . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:53 PM 1 of 1 09/11/2003 03:53 PM Partial Orders http://localhost/~senning/courses/ma229/slides/partial-orders/slide15.html Partial Orders prev | slides | next Hasse Diagrams If A ={ a , b , c } then the upper bounds are { e , f , h , j } and the least upper bound is e . If A ={ d , e , f } the upper bounds are { f , h , j }, the least upper bound is f . The lower bounds are { a , b } and the greatest lower bound is b . The least element is a and there is no greatest element. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 of 1 09/11/2003 03:53 PM