Section 7.6: Partial Orderings

1. Section 7.6: Partial Orderings Def: A relation R on a set S is called a partial ordering (or partial order) if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set (or poset). Ex: The set of integers with the relation  = {(a, b) | a  b} is a poset. Ex: The set of integers with the relation  = {(a, b) | a  b} is a poset. Ex: The set of positive integers with the relation {(a, b) | a divides b} is a poset. Ex: Let A be a set. Then the relation  is a partial ordering on P(A). Remark: The posets in the first two examples are very different from the posets in the next two examples. In the first example, if we take any two distinct elements of the set, either (a, b) or (b, a)  .

2. Def: The elements a and b of a poset (S, R) are called comparable if either aRb or bRa. When a and b are elements of S such that neither aRb nor bRa then a and b are called incomparable. Ex: In the poset (Z+, |), the elements 3 and 9 are comparable since 3 | 9, but the elements 5 and 7 are incomparable since neither 5 | 7 nor 7 | 5. Remark: A partial ordering is called “partial” because it is possible for two elements to be incomparable. So in a sense, only part of the poset is ordered. If we have a partial ordering in which every pair of elements in the poset are comparable then we call it a total ordering. Def: If (S, R) is a poset such that every pair of elements of S are comparable, then R is called a total ordering (or total order, linear ordering, or linear order). (S, R) is called a totally ordered set (or linearly ordered set). We also call such a poset a chain. Ex: The poset (Z, ) is a totally ordered set or chain.

3. Def: If (S, R) is a totally ordered set such that every nonempty subset of S has a least element then R is called a well ordering. Ex: We already know that Z+ with the usual  ordering is a well ordered set. However, we have seen that Z with  is not well ordered. Remark: Even though (Z, ) is not a well ordered set, this doesn’t mean that we can’t find a well ordering on Z. We can well order Z with the ordering 0, 1, -1, 2, -2, 3, -3, 4, -4, … where an element a to the left of another element b means that aRb. So 0R1, 0R-1, 0R2, … This is not the only well ordering on Z. In fact there are infinitely many well orderings that can be put on Z. We could simply interchange two elements in the above sequence for example. Theorem: Any countable set has a well ordering. In fact, there are infinitely many well orderings that can be put on a countable set. It is even possible to put a well ordering on some uncountable sets. However, this is a deep result that is beyond the scope of this course.

4. Recall the digraph representation of a relation using nodes and edges. This representation can be used for a poset since it is just a relation. However, we can simplify the digraph representation significantly when we are dealing with a poset. The result is a Hasse diagram. Ex: Consider the poset ({1, 2, 3, 4}, ). [Draw the digraph] Since a partial order must be reflexive, we can dispense with the loops that we know must be present at every vertex in order to unclutter the diagram. [Remove loops] We can also take advantage of antisymmetry. Since we know that we can never have a pair of edges in opposite directions, we can arrange the nodes so that all edges in the digraph will point upwards. After we have done this, we can remove the arrows on our edges since we already know that all of them must point upwards. [Rearrange nodes] Finally, we can take advantage of transitivity by removing all edges that we know must be present due to transitivity. [Remove edges]

5. Ex: Draw the Hasse diagram for the poset ({1, 2, 3, 4, 6, 8, 12}, |). Ex: Draw the Hasse diagram for the poset (P({a, b, c}), ). Ex: Draw the Hasse diagram for the poset (Z, ). Ex: Draw the Hasse diagram for the poset (Z+, ). Def: An element of a poset is called maximal if it is not ‘less than’ any other element. An element of a poset is called minimal if there are no other elements ‘less than’ it. Ex: 8 and 12 are maximal elements of the poset ({1, 2, 3, 4, 6, 8, 12}, |). 1 is the only minimal element of this poset. Ex: {a, b, c} is the only maximal element of the poset (P({a, b, c}), ). is the only minimal element of this poset. Ex: (Z, ) has no maximal elements nor any minimal elements. Ex: (Z+, ) has no minimal elements but 1 is a maximal element.

6. Def: If there is an element g in a poset (S, R) such that every other element in S is ‘less than’ g, then g is called the greatest element. If there is an element s in a poset (S, R) such that s is ‘less than’ every other element in S, then s is called the least element. A greatest and/or least element of a poset may or may not exist, but is unique if it does. Ex: The poset ({1, 2, 3, 4, 6, 8, 12}, |) has no greatest element. 1 is the least element of this poset. Ex: {a, b, c} is the greatest element of the poset (P({a, b, c}), ). is the least element of this poset. Ex: (Z, ) has no greatest element nor any least element. Ex: (Z+, ) has no least element but 1 is the greatest element. Remark: Note that the greatest element (if it exists) is necessarily a maximal element. Also the least element (if it exists) is necessarily a minimal element.

7. Def: Let (S, R) be a poset and let A be a subset of S. If there is an element u in S that is ‘greater than or equal to’ all of the elements in A, then u is called an upper bound of A. If there is an element d in S that is ‘less than or equal to’ all of the elements in A, then d is called a lower bound of A. Remark: Note that an upper or lower bound for A does not have to be a member of A, but it can be. Also note that if we take A = S, then an upper bound for A is the same as a greatest element of S. Ex: The set {2, 4, 8} has lower bounds of 1 and 2 and upper bound of 8 for the poset ({1, 2, 3, 4, 6, 8, 12}, |). Ex: The set {{c}, {a, b}} has  as a lower bound and {a, b, c} as an upper bound for the poset (P({a, b, c}), ). Ex: The set E has no lower bounds or upper bounds for the poset (Z, ). Ex: The set {1, 9} has upper bound 1 for the poset (Z+, ). The lower bounds of {1, 9} are 9, 10, 11, 12, 13, …

8. Def: Let (S, R) be a poset and let A  S. If there is an element u in S that is both an upper bound for A and also less than all other upper bounds for A then u is called the least upper bound of A. If there is an element d in S that is both a lower bound for A and also greater than all other lower bounds for A then d is called a greatest lower bound of A. Ex: The set {2, 4, 8} has 2 as its greatest lower bound for the poset ({1, 2, 3, 4, 6, 8, 12}, |). It has 8 as the least upper bound. Ex: The set {{c}, {a, b}} has  as its greatest lower bound and has {a, b, c} as its least upper bound for the poset (P({a, b, c}), ). Ex: The set E has no greatest lower bound or least upper bound for the poset (Z, ). Ex: The set {1, 9} has 1 as its least upper bound for the poset (Z+, ). The greatest lower bound of {1, 9} is 9.

9. Def: A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice. Ex: The poset ({1, 2, 3, 4, 6, 8, 12}, |) is not a lattice. For example, the set {8, 12} has no least upper bound (it has no upper bounds). Ex: The poset (P({a, b, c}), ) is a lattice. Think about how you might find the glb or lub of some pair of elements of this set. Ex: The poset (Z, ) is a lattice. For any pair of integers a and b that you take, the glb of {a, b} is min(a, b) and the lub of {a, b} is max(a, b). Ex: The poset (Z+, ) is a lattice. For any pair of positive integers a and b that you take, the glb of {a, b} is max(a, b) and the lub of {a, b} is min(a, b). [where max and min are with respect to the usual order] Ex: The poset (Z+, |) is a lattice. For any pair of positive integers a and b that you take, the glb of {a, b} is gcd(a, b) while the lub of {a, b} is lcm(a, b).

10. Homework 7 • Average: 70 Median: 75 • http://www.cs.virginia.edu/~cmt5n/cs202/hw7/ • Homework 8 • Average: 74 Median: 79 • http://www.cs.virginia.edu/~cmt5n/cs202/hw8/ • #36 was commonly missed • The solution key presents a rather tedious, straightforward application of the counting techniques we discussed in class. • There is a more clever way to solve this problem but it takes a bit of creative insight beyond what we discussed in class. I did not present this more clever solution in the key because I want you to see how it can be solved without this insight. • If you found the clever solution, that is much better.