1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 17

In that case we shall say that f introduces a two-order in S (the one-

order being the linear order).

An order isomorphism between two two-ordered sets (S 1 , Ji) and (S 2 , f 2 )

is a bijective function g : S1 --+ S 2 which either preserves or reverses the

order, i.e., such that

f1(p,q,r) = μfz(g(p),g(q),g(r))

whereμ is either +1 or -1. Ifμ = +1, it is said that g is direct, and if

μ = -1, that g is opposite.

Next, let F be a field that is also a two-ordered set. Then F is called

a two-ordered field if the mappings

ga : x ----+ x +a and ha : x ----+ax, a -:f. 0

are order isomorphisms. If both are direct, F will be called a direct two-

ordered field.

In the complex field C a two-order can be introduced by defining

1 1 1

f(zi, z2, za) = sgn z1 z2 z2

Z-1 Z-2 za

where sgnr = r/lrl for r # 0 real, and sgnO = 0. Furthermore, it is easy
to check that

and
f(z1,z2,za) = f(az1,az2,aza), a# 0
so that both ga and ha are dired order automorphisms of C. Hence it
follows that C is a direct two-ordered field.

It can be shown that any direct two-ordered field that has the so-called

supremum property is isomorphic to C. For the details, see L. Gutierrez

Novoa [12].

1.5 The Complex System as a Linear System and as an Algebra

The complex number system with the definitions of equality and addition

as given in Section Li, together with property (1.2-2), namely,

r(a,b) = (ra,rb)

taken for the definition of the product of the real number r by the complex

number (a, b ), constitutes a linear or vector system of dimension two over