Partial Orders: Understanding Reflexive, Antisymmetric, and Transitive Relations - Prof. M, Quizzes of Discrete Structures and Graph Theory

Download Partial Orders: Understanding Reflexive, Antisymmetric, and Transitive Relations - Prof. M and more Quizzes Discrete Structures and Graph Theory in PDF only on Docsity! Partial Orders Margaret M. Fleck 17 November 2008 Finishes the dicussion of partial orders (section 8.6 of Rosen). 1 Announcements Last honors homework will be released shortly, probably due last Monday of classes. Quiz 3 will be in class, Wednesday after break. 2 Recap Recall from last Friday that a partial order is a relation that is reflexive, antisymmetric, and transitive. Unlike a total (linear) order, it lacks a guar- antee that two elements will be “comparable”, i.e. related in one or the other direction. So ≤ is a total order on the integers because for every p, q, either p ≤ q or q ≤ p. The divides relation is a partial order, because some pairs of numbers (e.g. 3 and 5) don’t divide one another in either order. The relation < isn’t a partial order, because it’s not reflexive, it’s irreflex- ive. Relations that are irreflexive, antisymmetric, and transitive are strict partial orders. We will be concentrating on the normal (non-strict) partial orders. The strict ones are closely related and fairly similar. 1 A set with a partial order relation on it is called a partially ordered set, frequently abbreviated as poset. We also saw a new type of stripped down graph representation, used only for partial orders, called a Hasse diagram. In this representation, you leave out the reflexive loop and arcs you can infer from transitivity. You also arrange the nodes on the page so all the arrows point up, then delete the arrowheads. 3 More examples of partial orders Suppose we have a set A and build a collection S of subsets of A. Then set inclusion ⊆ is a partial order on S. For example, suppose A = {a, b, c} and S contains all subsets of A. Then we have {a, b, c} {a, b} {b, c} {a, c} {a} {b} {c} ∅ Notice that {a, b} and {b, c} aren’t comparable, because neither is a subset of the other. Partial orders can be used to represent type relations, e.g. for variables types or classes in programming languages or for modelling our knowledge about the real world for artificial intelligence. These relationships often gen- erate graphs that are a bit like trees, but have crossing branches. 2 shoes coat pants socks shirt shower breakfast Notice that some pairs of tasks can be done in either order, e.g. you can put on your socks before or after your shirt. Sometimes a set with a partial order has a greatest element, i.e. an element that is ≥ all the other elements in the set. This was true for our subset and type examples. However, in this case, our graph has two elements at with nothing larger: “shoes” and “coat”. In such cases, both top elements are called maximal. Officially, if (A,) is a partially ordered set, then x ∈ A is maximal if and only if there is no y ∈ A such that x 6= y and x  y. Similarly, a partial order sometimes has a least element, i.e. an element that is smaller than all other elements. In this example (and in the types example above), we instead have two minimal elements. 6 Topological sorting If we have a partial order showing the constraints, we can create one or more total orders that obey the constraints. These are called total orders compatible with the partial order or a topological sort or linearization of the partial order. Typically, there are many total orders compatible with a given partial order. For example, here’s two topological sorts of our scheduling example, which represent two perfectly reasonable ways to get ready in the morning. • shower pants socks shirt breakfast shoes coat 5 • breakfast shower shirt pants socks shoes coat Examples like this occur frequently in Artificial Intelligence, and in more serious-sounding scheduling applications (e.g. business, military, transporta- tion). You can compute a topological sort by repeatedly removing a minimal element from the input set and adding it to the end of the output totally ordered set. 7 Upper and lower bounds Suppose we have a partially ordered set (A,). Suppose we pick a subset S of A. Then x is an upper bound for S if x ≤ y for all y ∈ S. For example, rememeber the divides relation on the set A = {1, 15, 12, 3, 5, 2, 4, 6}. 15 12 46 5 3 2 1 The elements 6 and 12 are both upper bounds for the subset S = {1, 2, 3, 6}. In this case, 6 is smaller than all the other upper bounds for S, so it is called the least upper bound. There isn’t always a least upper bound. Suppose we looked at the divide relation on the set {1, 3, 5, 30, 45}. Then 30 and 45 are both upper bounds for the subset T = {1, 3, 5} but 30 and 45 aren’t comparable to one another So T doesn’t have a least upper bound. Lower bounds and greatest lower bounds are defined similarly. 6