Download COORDINATIZING A PROJECTIVE PLANE II 1. Addition in γ We

COORDINATIZING A PROJECTIVE PLANE II
TIMOTHY VIS
1. Addition in γ
We now define an addition on γ that mimics the addition in a field plane and
preserves one of its features.
y
(a, a + b)
(0, b)
x
(a, 0)
Figure 1
In Figure 1 showing an affine field plane, we see that the vertical line through
the point (a, 0) and the line of slope one through the point (0, b) meet in the point
(a, a + b), as we would expect. In our coordinatization of a projective plane, this
vertical line would necessarily pass through the point (∞) = Y , and would thus be
the line AY . Similarly, all lines of slope one would pass through the point (1) = I, so
that this second line becomes BI. The point of intersection then becomes AY ∩BI,
having second coordinate a + b. We therefore define a + b in this manner.
Definition 1. Given a, b in γ, define a + b as follows. Let A = (a, 0), B = (0, b),
I = (1), and Y = (∞). Define the point P = AY ∩ BI. Then P = (a, c). Define
a + b = c.
We next determine the algebraic structure of (γ, +).
Proposition 2. 0 + a = a + 0 = a.
Proof. The sum 0+a is defined by considering points A = (0, 0), B = (0, a), I = (1),
and Y = (∞). Then the point P = AY ∩ BI = B = (0, a), so that 0 + a = a. See
Figure 2.
The sum a + 0 is defined by considering points A = (a, 0), B = (0, 0), I = (1),
and Y = (∞). Then the point P = AY ∩ BI = AY ∩ OI and, lying on the line OI
has the same first and second coordinates. So P = (a, a) and a + 0 = a. See Figure
3.
Date: April 21, 2009.
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Y
I
(0, a)
(0, 0 + a)
X
(0, 0)
Figure 2
Y
I
(a, a)
X
(0, 0)
(a, 0)
Figure 3
So our addition gives an identity element. We wish also to show that we may
uniquely solve equations with this addition.
Proposition 3. Given a and c, there exist unique x and y such that x + a = c and
a + y = c.
Proof. Define points A = (a, 0), P = (a, c), I = (1), Y = (∞). Then OY ∩ P I
has coordinates (0, y), and by the construction of these points, a + y = c. This
construction forces y to be unique as any other y would give a point B ′ with the
line B ′ I intersecting the line BI in two distinct points, a contradiction. See Figure
4.
Similarly, define points X = (0), B = (0, a), C = (0, c), I = (1). The point
XC ∩ BI has coordinates (x, c), and if we construct A = (x, 0), x + a = c. As
before, the construction forces x to be unique, and any other x would give a point
A′ with A′ Y intersecting AY in two distinct points, a contradiction. See Figure 5
Theorem 4. (γ, +) is a loop with identity 0.
Proof. Propositions 2 and 3 show that (γ, +) satisfies all axioms of a loop.
COORDINATIZING A PROJECTIVE PLANE II
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Y
I
(a, c)
(0, y)
X
(a, 0)
O
Figure 4
Y
I
(x, c)
(0, a)
X
(x, 0)
O
Figure 5
2. Multiplication in γ
We can also define a multiplication in γ that preserves features of the multiplication in a field.
y
y = xm
(a, am)
x
(a, 0)
Figure 6
In Figure 6 showing again an affine field plane, we see that the vertical line
through the point (a, 0) and the line y = xm of slope m through the origin meet,
as expected, in the point (a, am). In our coordinatization of a projective plane, the
vertical lines pass through the point Y = (∞) and the line of slope m is the line
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OM , where M = (m). So this point of intersection is AY ∩ OM and should have
second coordinate a · m. We therefore define a · m in this manner.
Definition 5. Given a, m in γ, define a · m as follows. Let A = (a, 0), O = (0, 0),
M = (m), and Y = (∞). Define the point P = AY ∩ OM . Then P = (a, c). Define
a · m = c.
We now determine the algebraic structure of (γ, ·).
Proposition 6. 0 · m = a · 0 = 0.
Proof. The product 0 · m is defined by considering A = (0, 0), O = (0, 0), M = (m),
and Y = (∞). Clearly A = O so that AY ∩ OM = (0, 0) and 0 · m = 0. See Figure
7
Y
(m)
X
(0, 0)
(0, 0 · m)
Figure 7
The product a · 0 is defined by considering A = (a, 0), O = (0, 0), X = (0), and
Y = (∞). Now A ∈ OX, so that AY ∩ OX = (a, 0) and a · 0 = 0. See Figure 8
Y
X
(a, 0)
(a, a · 0)
O
Figure 8
So zero behaves as we might expect.
COORDINATIZING A PROJECTIVE PLANE II
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Proposition 7. 1 · a = a · 1 = a.
Proof. The product 1 · a is defined by considering A = (1, 0), O = (0, 0), M = (a),
and Y = (∞). Then P = AY ∩ OM has coordinates (1, 1 · a). But the coordinates
of M were defined as the second coordinate of this point, so that certainly a = 1 · a.
See Figure 9
Y
(a)
(1, 1 · a)
O
(1, 0)
X
Figure 9
On the other hand, a · 1 is defined by considering A = (a, 0), O = (0, 0), I = (1),
and Y = (∞). Then P = AY ∩ OM has coordinates (a, a · 1) and lies on the line
OI. But every point on OI has the same first and second coordinates, so that
a · 1 = a. See Figure 10
Y
I
(a, a · 1)
O
(a, 0)
X
Figure 10
So our multiplication also has an identity element. We finally have to show that
equations can be uniquely solved under multiplication.
Proposition 8. Given a and c in γ \ {0}, there exist unique x and y such that
x · a = c and a · y = c.
Proof. Define points M = (a), C = (0, c), Y = (∞), X = (0), O = (0, 0). Let
P = OM ∩ CX. Then P has coordinates (x, c). Define A = P Y ∩ OX, which then
has coordinates (x, 0). By this construction, x·a = c, and x is uniquely determined,
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as any other solution would yield a point A′ , with A′ Y and AY intersecting in both
P and Y . See Figure 11
Y
(0, c)
(a)
(x, x · a)
X
(x, 0)
O
Figure 11
On the other hand, define points A = (a, 0), C = (0, c), Y = (∞), X = (0),
O = (), 0). Let P = AY ∩ CX, having coordinates (a, c). Now let M = OP ∩ XY
with coordinates (y). Then a · y = c, and y is uniquely determined, as the existence
of any other value would give a point M ′ where M O and M ′ O intersect in two
distinct points. See Figure 12
Y
(0, c)
(y)
(a, a · y)
X
O
(a, 0)
Figure 12
Once again, we have all of the structure necessary for a loop, so long as we
exclude zero.
Theorem 9. (γ \ {0} , ·) is a loop.
Proof. Proposition 7 establishes the existence of an identity element. Proposition
8 establishes unique solutions to equations over γ, and Proposition 6 ensures that
the solutions are never zero. Thus, all axioms of a loop are satisfied.
COORDINATIZING A PROJECTIVE PLANE II
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3. The Planar Ternary Ring
In a field plane, all of the lines may be described as the sets of points satisfying
a linear equation y = x · m + c. Unfortunately, the addition and multiplication
we have defined may not allow us to describe lines in this way. Thus, we define a
ternary relation to describe the sets of points on a line.
Definition 10. Given x, m, and c in γ, we define the ternary operation x ◦ m ⋆ c
as follows. Let A = (x, 0), Y = (∞), M = (m), and C = (0, c). Then AY ∩ CM
has coordinates (x, d). Define x ◦ m ⋆ c = d. See Figure 13.
Y
(m)
(x, x ◦ m ⋆ c)
(0, c)
O
(x, 0)
X
Figure 13
It follows immediately from the definition that a point (x, y) is on the line through
(0, c) and (m) precisely when y = x ◦ m ⋆ c. We can also derive our addition and
multiplication as special cases of this ternary operation.
Notice that if c = 0, we have C = O, and our definition for x ◦ m ⋆ c coincides
exactly with the definition of x · m. Similarly if m = 1, M = I and our definition
for x ◦ m ⋆ c coincides exactly with the definition of x + c. Notice also that if x = 0,
A = O and AY ∩ CM = C, so that 0 ◦ m ⋆ c = c. Similarly, if m = 0, M = X and
every point of XC has second coordinate c so that x ◦ 0 ⋆ c = c.
We can also derive certain algebraic properties from geometric properties of the
plane. For instance, given x, m, and y, there is a unique point of intersection of
the lines OY and h(x, y) , (m)i, and thus a unique c such that x ◦ m ⋆ c = y. Given
m1 6= m2 , c1 6= c2 we find two lines determined by the equations y = x ◦ m1 ⋆ c1
and y = x ◦ m2 ⋆ c2 , which meet in a unique point (x, y) giving the unique x such
that x ◦ m1 ⋆ c1 = x ◦ m2 ⋆ c2 (two lines meet in a unique point). Finally, given two
points (x1 , y1 ) and (x2 , y2 ), there is a unique line joining these two points. This
line intersects OY in a unique point (0, c), and intersects XY in a unique point
(m), giving the unique (m, c) such that both x1 ◦ m ⋆ c = y1 and x2 ◦ m ⋆ c = y2 .
We summarize these results in the following proposition.
Proposition 11. In (γ, +, ·, (◦, ⋆)),
(1) x ◦ 1 ⋆ c = x + c.
(2) x ◦ m ⋆ 0 = x · m.
(3) 0 ◦ m ⋆ c = x ◦ 0 ⋆ c = c.
(4) Given x, m, y, there exists a unique c such that x ◦ m ⋆ c = y.
(5) Given m1 6= m2 , c1 6= c2 , there exists a unique x such that x ◦ m1 ⋆ c1 =
x ◦ m 2 ⋆ c2 .
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(6) Given x1 , x2 , y1 , y2 , there exists a unique pair (m, c) such that x1 ◦m⋆c = y1
and x2 ◦ m ⋆ c = y2 .
Definition 12. The system (γ, +, ·, (◦, ⋆)) is called the planar ternary ring. If
s ◦ m ⋆ c = x · m + c, the planar ternary ring is said to be linear.