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

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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

• rder relations. Usually, rather than ρ, we will

use the notation ∼ < or simply ≤ for these. 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

• ther relations which share some, but not

necessarily all of the same properties.

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

• rdered set with order relation ∼

<)

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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 example, on Z, we could define the relation ∼ < by saying that x ∼ < y if (and only if) y ≤ x. This is

• bviously a special case of reversing a given
• rder.
• Exercise. Show that the reverse of any total
• rder on a set X is always a total order on X.
• Exercise. Find an example of a total order ∼

< on N × N. Proposition 1.2 Every subset of a totally

• rdered set is also totally ordered, using the

same order relation (restricted to the subset).

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Total orders are also sometimes called linear

• rders. 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.

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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 relation ∼ < on X satisfying the following three conditions, for all x, y, z in X: (a) x ∼ < x; (b) if x ∼ < y and y ∼ < z then x ∼ < z; (c) if x ∼ < y and y ∼ < x then y = x;

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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

• n R (a ∼

< b if a = b)? All our earlier examples of total orders are also partial orders. Partial orders which are not total

• rders include the following examples, whose

properties you should check:

• X = N, m ∼

< n if n is divisible by m;

• X = R × R, (x1, y1) ∼

< (x2, y2) if both x1 ≤ x2 and y1 ≤ y2.

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• Let X be the set of all polynomial functions

with real coefficients. For such polynomials p, q we may define a relation p ∼ < q if the graph of 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 2N. We can then use set inclusion as our relation: given A, B in X (in

• ther words A, B are subsets of N), we say

A ∼ < B if A ⊆ B.

• 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.

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Remember!

• To be a total order, a relation ∼

< must satisfy all conditions (a) to (d).

• To be a partial order, ∼

< must satisfy conditions (a) to (c).

• If any of the conditions (a) to (c) fail for a

relation ∼ < then ∼ < is not a partial order, and hence is not a total order either. You need not check the remaining conditions!

• If condition (d) fails for a relation ∼

< then it is 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).

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As in the previous section, we can reverse a given order relation and see what happens.

• Exercise. Show that the reverse of any partial
• rder 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

• rdered set is also also partially ordered, using

the same order relation (restricted to the subset).

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