Theory of Statistics - George Mason University

628 0 Statistical Mathematics
0.0.3.6 Ordered Fields
We have defined ordered sets in Section 0.0.1.4 in terms of the existence of a
binary relation. We now define ordered fields in terms of the field operations.
Definition 0.0.4 (ordered field)
A field S is said to be ordered if there is a subset P of S such that
• P ∩ (−P) = ∅;
• P ∪ {0} ∪ (−P) = S;
• x, y ∈ P ⇒ x + y, x ◦ y ∈ P.
Notice that this is only a partial ordering; it is neither a linear ordering nor
a well-ordering. Applying this definition to the real numbers, we can think of
P as the positive reals, and the elements of −P as the negative reals. Notice
that 1 must be in P (because a ∈ P ⇒ a 2 ∈ P and b ∈ −P ⇒ b 2 ∈ P) and so
an ordered field must have characteristic 0; in particular, a finite field cannot
be ordered.
We define the binary relations “≤”, “” in the ordered field
S in a way that is consistent with our previous use of those symbols. For
example, for x, y ∈ S, x < y or y > x implies y − x ∈ P.
We now define a stronger ordering.
Definition 0.0.5 (Archimedean ordered field)
An ordered field S defined by the subset P is said to be Archimedean ordered
if for all x ∈ S and all y ∈ P, there exists a positive integer n such that
ny > x.
An Archimedean ordered field must be dense, in the sense that we can find
an element between any two given elements.
Theorem 0.0.3
Let S be an Archimedean ordered field, and let x, y ∈ S such that x < y. Then
there exists integers m and n such that m/n ∈ S and
x < m
n
< y.
Proof.
Exercise.
The practical application of Theorem 0.0.3 derives from its implication
that there exists a rational number between any two real numbers.
0.0.4 Linear Spaces
An interesting class of spaces are those that have a closed commutative and
associative addition operation for all elements, an additive identity, and each
Theory of Statistics c○2000–2013 James E. Gentle