G14FUN: Functional Analysis, Introductory material on totally - PDF document

G14FUN: Functional Analysis, Introductory material on totally ordered sets and partially ordered sets Dr J. F. Feinstein January 25, 2010 1

1 Total orders Recall that we say that we have a relation ρ on a set X , if we have a rule which allows us to determine, for each pair of elements x and y in X , whether or not x ρ y . You will probably be familiar with equivalence relations and their properties. (If not, it would be a good idea to read up about them.) However here we will discuss various kinds of order relations. Usually, rather than ρ , we will < or simply ≤ for these. use the notation ∼ You will already be familiar with the usual relation of ≤ for sets of numbers such as R , N , Z , Q . We will be looking at this relation and other relations which share some, but not necessarily all of the same properties. 2

Definition 1.1 A total order on a set X is a < on X satisfying the following four relation ∼ conditions, for all x, y, z in X : < x ; (a) x ∼ < y and y ∼ < z then x ∼ < z ; (b) if x ∼ < y and y ∼ < x then y = x ; (c) if x ∼ < y and y ∼ < x holds. (d) at least one of x ∼ < ) In this case we say that the ordered pair ( X, ∼ is a totally ordered set (or that X is a totally < ) ordered set with order relation ∼ 3

The most obvious examples of total orders are to take the usual order on any of the sets of numbers N , Z , Q , R . Another common example is to take the reverse order on these sets. For < by example, on Z , we could define the relation ∼ < y if (and only if) y ≤ x . This is saying that x ∼ obviously a special case of reversing a given order. Exercise. Show that the reverse of any total order on a set X is always a total order on X . < on Exercise. Find an example of a total order ∼ N × N . Proposition 1.2 Every subset of a totally ordered set is also totally ordered, using the same order relation (restricted to the subset). 4

Total orders are also sometimes called linear orders . Also, totally ordered sets are sometimes called simply ordered sets . In the next section we will see what happens if you weaken the conditions on your order relations slightly, and work instead with partial orders . 5

2 Partial orders 2.1 Definitions and examples In this section we look at relations that satisfy the first three conditions for a total order but not necessarily the fourth condition. Definition 2.1 A partial order on a set X is a < on X satisfying the following three relation ∼ conditions, for all x, y, z in X : < x ; (a) x ∼ < y and y ∼ < z then x ∼ < z ; (b) if x ∼ < y and y ∼ < x then y = x ; (c) if x ∼ 6

< ) In this case we say that the ordered pair ( X, ∼ is a partially ordered set (or that X is a < ). partially ordered set with order relation ∼ We may also abbreviate partially ordered set by poset . NOTE: every total order is a partial order, but not every partial order is a total order! Exercise: is the equality relation a partial order < b if a = b )? on R ( a ∼ All our earlier examples of total orders are also partial orders. Partial orders which are not total orders include the following examples, whose properties you should check: < n if n is divisible by m ; • X = N , m ∼ < ( x 2 , y 2 ) if both • X = R × R , ( x 1 , y 1 ) ∼ x 1 ≤ x 2 and y 1 ≤ y 2 . 7

• Let X be the set of all polynomial functions with real coefficients. For such polynomials p , q < q if the graph of p we may define a relation p ∼ never rises above the graph of q , i.e. (more formally) for all x in R we have p ( x ) ≤ q ( x ) . • (More abstract, but very useful!) Let X be the set of all possible subsets of N , which may be denoted by P ( N ) or 2 N . We can then use set inclusion as our relation: given A , B in X (in other words A , B are subsets of N ), we say < B if A ⊆ B . A ∼ • Partially ordered sets which have only finitely many elements can be drawn in the form of Hasse diagrams , which will be discussed briefly in lectures. 8

Remember! < must satisfy • To be a total order, a relation ∼ all conditions (a) to (d). < must satisfy • To be a partial order, ∼ conditions (a) to (c). • If any of the conditions (a) to (c) fail for a < then ∼ < is not a partial order, and relation ∼ hence is not a total order either. You need not check the remaining conditions! < then it is • If condition (d) fails for a relation ∼ certainly not a total order (you need not check the other conditions to deduce this). However, < might still be a partial order: to determine ∼ this you need to look at conditions (a) to (c). 9

As in the previous section, we can reverse a given order relation and see what happens. Exercise. Show that the reverse of any partial order on a set X is always a partial order on X . We also have the following result corresponding to Proposition 1.2. Proposition 2.2 Every subset of a partially ordered set is also also partially ordered, using the same order relation (restricted to the subset). 10